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Solar radiation induced perturbations and control of satellite trajectories Van Der Ha, Jozef Cyrillus 1977

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SOLAR RADIATION INDUCED PERTURBATIONS AND CONTROL OF SATELLITE TRAJECTORIES by JOZEF CYRILLUS^VAN DER HA M.Sc.s TECHNOLOGICAL UNIVERSITY EINDHOVENs THE NETHERLANDS, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1977 © Jozef Cyrillus Van der Ha, 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that publication, in part or in whole, or the copying of this thesis for financial gain shall not be allowed without my written permission. JOZEF C. VAN DER HA Department of Mechanical Engineering The University of British Columbia, Vancouver, Canada, V6T 1W5 i i ABSTRACT The long-term orbital perturbations due to solar radiation forces as well as ways to u t i l i z e these effects for corrections in the orbit are investigated. In order to obtain familiarity with relative merits of the formulations and methods relevant to the present objective, the special case of an orbit in the ecliptic plane and a force along the radiation is considered f i r s t . The long-term valid analysis is based upon the two-variable expansion method and incorporates the apparent motion of the sun by treating the sun's position as a quasi-orbital element. Analytical representations for orbital elements are derived and the perturbations are conveniently summarized in the form of polar plots showing the long-term evolution of the eccentricity vector. While the eccentricity is periodic with period close to one year, the argument of the perigee contains secular terms. The total energy and thus major axis remain conserved in the long run. However, in the course of one year, the effect of the earth's shadow may lead to small secular changes in the major axis thereby modifying the satellite's period. Next, the analysis is extended to orbits of an arbitrary inclination with closed-form analytical solutions established in some special cases. An interesting relation between the long-term behavior of the orbital in-clination and the in-plane perturbations is discovered. Also, more general sat e l l i t e configurations are studied: e.g., spacecrafts modelled as a plate in an arbitrary fixed orientation with respect to the earth or solar radia-tion as well as platforms kept fixed to the inertia! space. In a l l appli-i i i cations a r e a l i s t i c solar radiation force allowing for diffuse and/or specular reflection as well as for re-emission of absorbed radiation is considered. In a few cases, the analysis is extended to include arbi-t r a r i l y shaped sa t e l l i t e bodies modelled by a number of surface components of homogeneous material characteristics. After establishing a comprehensive spectrum of the qualitative and quantitative aspects of solar radiation induced orbital perturbations, the attention is focused on the development of control strategies involving the rotation of solar panels attached to the s a t e l l i t e to manipulate both the direction and magnitude of the resulting force. A few on-off switching strategies are explored and the most effective switching locations for several specific objectives, e.g. maximization of the major axis, are de-termined. The switching strategies explored here constitute an attractive possibility for orbital corrections. The concept is particularly of inte-rest to modern communications sa t e l l i t e technology since i t allows their normal operation to remain unaffected over approximately half the time. Although on-off switching may lead to substantial changes in the major axis, i t is not necessarily the best policy when time-varying orientations are also taken into consideration. The optimal control strategy for maxi-mization of the major axis over one revolution is determined by means of the numerical steepest-ascent iteration procedure, and its effectiveness is com-pared with that of the switching programs. The solution should prove to be of interest in several future missions including the launching of a solar sail from a geocentric orbit into a heliocentric or escape trajectory. Subsequently, solar radiation effects upon a s a t e l l i t e (usually a solar sail) in a heliocentric orbit are explored. F i r s t , the sail is i v taken in a fixed but arbitrary orientation to the local frame. Using specific i n i t i a l conditions, exact solutions in the form of conic sections and three-dimensional logarithmic spirals are established. For an arbi-trary i n i t i a l orbit, long-term approximate representations of the orbital elements are derived. An effective out-of-plane spiral transfer trajectory is obtained by reversing the force component normal to the orbit at speci-fied positions. By choosing the appropriate control angles, any point in space can eventually be reached. Finally, time-varying optimal control strategies are explored for increasing the total energy (and angular momentum) during one revolution. While analytical approximate results can be established for near-circular orbits, in the general case a numerical steepest-ascent technique is em-ployed. The results are compared with those from the constant sail setting indicating that the latter is a near-optimal strategy for low eccentricity starting orbits. V TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 1.1 Preliminary Remarks i 1.2 Review of the Literature 4 1.2.1 Solar radiation induced orbital perturbations . . 4 1.2.2 Orbital control using solar radiation forces . . . 9 1.2.3 Small-thrust trajectories 11 1.2.4 Optimal trajectories 13 1.3 Scope and Objective of the Study . . . . . . . 15 2. SOLAR RADIATION EFFECTS UPON AN ORBIT IN THE ECLIPTIC PLANE . 21 2.1 Preliminary Remarks 21 2.2 General Formulation of the Solar Radiation Force . . . . 23 2.3 Plate Normal to Radiation 25 2.3.1 Short-term valid approximations 29 2.3.2 Rectification/iteration procedure 34 2.4 Two-Variable Expansion Procedure 35 2.4.1 Long-term valid results, case e = 0(6) 41 2.4.2 Long-term valid results, case e = 0(6 ) 47 2.5 Discussion of Results 48 2.5.1 Case £ = 0(6) 49 2.5.2 Case e = 0 ( 6 2 ) 67 2.6 Evaluation of the Shadow Effects 69 2.6.1 Short-term shadow effects 70 2.6.2 Long-term shadow effects . . . 73 2.6.3 Discussion of results 77 vi Chapter Page 2.7 Concluding Remarks 81 3. SOLAR RADIATION INDUCED PERTURBATIONS OF AN ARBITRARY GEOCENTRIC ORBIT 84 3.1 Derivation of the Perturbation Equations 84 3.2 Determination of the Solar Radiation Force 92 3.3 Plate Normal to Radiation 94 3.3.1 Short-term analysis 96 3.3.2 Long-term approximations 99 3.3.3 Discussion of results . 105 3.4 Satellite in Arbitrary Fixed Orientation to Radiation . . 114 3.5 Plate in Arbitrary Fixed Orientation to Local Frame . . . 119 3.6 Concluding Remarks 126 4. GEOCENTRIC ORBITAL CONTROL USING SOLAR RADIATION FORCES . . . 127 4.1 Preliminary Remarks 127 4.2 Switching at Perigee and Apogee 128 4.3 Systematic Increase in Angular Momentum 130 4.4 Systematic Increase in Total Energy 133 4.5 Optimal Orbit Raising 141 4.6 Orientation Control of the Orbital Plane 148 4.6.1 Control of the inclination 150 4.6.2 Control of the line of nodes 151 4.7 Half-Yearly Switching 154 4.8 Concluding Remarks 157 5. HELIOCENTRIC SOLAR SAILING WITH ARBITRARY FIXED SAIL SETTING . 159 5.1 Preliminary Remarks 159 5.2 Formulation of the Problem 160 vi i Chapter Page 5.3 Three Dimensional Spiral Trajectories 165 5.4 Out-of-Plane Spiral Transfer 172 5.5 Arbitrary Initial Conditions . . . . 177 5.5.1 Short-term approximate solution . . . . 177 5.5.2 Long-term behavior of the elements 178 5.5.3 Higher-order contributions 185 5.5.4 Discussion of results 187 5.6 Concluding Remarks 190 6. DETERMINATION OF OPTIMAL CONTROL STRATEGIES IN HELIOCENTRIC ORBITS 1 9 2 6.1 Preliminary Remarks 192 6.2 Formulation of the Problem 193 6.3 Maximization of Total Energy 195 6.4 Maximization of Angular Momentum . . . . 201 6.5 Discussion of Results 203 6.6 Concluding Remarks 212 7. CLOSING COMMENTS 214 7.1 Summary of Conclusions 214 7.2 Recommendations for Future Work 215 BIBLIOGRAPHY 217 APPENDIX I - EVALUATION OF THE INTEGRALS A . AND B 227 nk nk APPENDIX II - EVALUATION OF THE FOURIER COEFFICIENTS a^ R, d ^ . . 233 APPENDIX III - DERIVATION OF HIGHER-ORDER EQUATIONS 236 v i i i LIST OF TABLES Table Page 2.1 Material Parameters for a Few Typical Spacecraft Components 24 4.1 Comparison of Control Strategies (e = 0.0002) . . 147 6.1 Response a(2ir) for Optimal Control Strategy and for a = arcsin(3" 1 / 2) 207 ix LIST OF FIGURES Figure Page 1-1 A schematic diagram showing the concept of solar sail 3 1- 2 Schematic overview of the plan of study: (a) geocentric orbits; , 19 (b) heliocentric orbits 20 2- 1 Geometry of sun, earth and s a t e l l i t e including shadow region 26 2-2 Loci in a^, e plane having the same period of long-term perturbations 50 2-3 Long-term variations in eccentricity as predicted by the three methods 52 2-4 Long-term behavior of semi-latus rectum and argument of the perigee 54 2-5 Secular variation of the argument of the perigee for c <_ X <_ b 55 2-6 Long-term variations in the semi-major axis and orbital period 56 2-7 Polar plots showing long-term behavior of eccentricity vector e_: (a) case X = c, = 0; 59 (b) case 3aQQ/2 < X < c, e 0 Q •= 0.5; . . . . . . ^ 59 (c) case c < X < b, e 0 Q = 0.109; 60 (d) case X. = b, e Q 0 = 0.05 60 2-8 Long-term orbital behavior showing traversing of spatial region 63 2-9 Actual path of s a t e l l i t e during f i r s t revolution (for exagge-rated solar parameter, e = 0.02) 65 2-10 Loci of i n i t i a l conditions, X = constant, leading to the same extrema of eccentricity 66 2-11 Representative polar plots for the Communications Technology Satellite: (a) e Q 0 = 0; 68 (b) e 0 Q = 0.1 68 Figure Page 2- 12 Comparison of the analytical long-term approximate solutions for the shadow effects upon: (a) semi-major axis; 79 (b) eccentricity 79 3- 1 General three-dimensional configuration of the earth, s a t e l l i t e and the sun 87 3-2 Idealized s a t e l l i t e configurations: (a) sphere; 93 (b) f l a t plate or surface component in arbitrary orientation to orbital plane . 93 3-3 Polar plots, illustrating long-term behavior of the eccentri-city vector e: (a) e Q 0 = 0; 107 (b) e Q 0 = 0.5 107 3-4 Typical long-term behavior of the longitude of nodes as affected by the i n i t i a l solar aspect angle 109 3-5 Long-term behavior of the inclination for i n i t i a l l y circular equatorial orbit I l l 3-6 Variations in orbital inclination for i n i t i a l l y equatorial orbit of eccentricity 0.1 112 3-7 Behavior of orbital inclination for i n i t i a l l y equatorial orbit of eccentricity 0.5 113 3-8 Typical polar plots for plate maintaining fixed orientation to radiation, a = i> + arctan{cos(i) tan fi} and: (a) 3 - 23°; 118 (b) 6 = 0 118 3- 9 Long-term in-plane perturbations for plate fixed to local frame: (a) semi-major axis; 125 (b) polar plot for eccentricity vector e_ 125 4- 1 Configuration of switching points for controlled orbital change 129 4-2 Results of perigee-apogee switching strategy 131 4-3 Behavior of semi-latus rectum in (v^, v^) switching program . 134 xi Figure Page 4-4 Controlled variation of the semi-major axis for ( v ^ , v^) switching program and optimal control strategy 139 •4-5 Long-term variations in eccentricity during ( v ^ , v^) switching program 140 4-6 (a) Optimal control strategy for maximization of Aa 146 (b) Corresponding optimal orientation of solar sail . . . . . 146 4-7 Controlled change in inclination for various i n i t i a l conditions 152 4-8 Behavior of eccentricity for switching program of Equations (4.23) 153 4-9 (a) Behavior of eccentricity for switching program of Equations (4.25) " . . 155 (b) Controlled change in position of the line of nodes . . . . 155 4- 10 Effect of half-yearly switching upon eccentricity and inclination 156 5- 1 (a) Configuration of the sun and solar sail in a heliocentric trajectory 161 (b) Successive rotations a, (3 (and y) for defining arbitrary orientation of solar sail 161 5-2 (a) Optimal sail orientation and corresponding spiral angle for various values of r e f l e c t i v i t y 169 (b) Actual planar spiral trajectory for es = 0.15 and p = 1 . 169 5-3 Potential for near-circular interplanetary transfer by solar sail for a few values of e 170 s 5-4 (a) Orientation of the osculating plane as affected by a constant force normal to i t 173 (b) Switching strategy leading to a systematic increase in orbital inclination . 173 5-5 (a) Combinations of inclination and radial distance attainable after a given time 176 (b) Levelcurves for constant |S| and |T| 176 (c) Growth of inclination for pure out-of-plane transfer . . . 176 5-6 Comparison of the analytical results for the long-term behavior of the semi-latus rectum 188 XI 1 Figure Page 5- 7 Long-term behavior of semi-major axis and eccentricity as predicted by the zeroth-order solution 189 6- 1 Comparison of analytical and numerical optimal controls for e $ = 0.015: (a) e Q Q = 0.2; 204 (b) e 0 Q = 0.4 204 6-2 Optimal sail setting for e = 0.09, to™ = 0 and a few values of e Q 0 206 6-3 Optimal control programs for e = 0.15, e n n = 0.2 and a few values of O J q o 208 6-4 Actual trajectory under optimal sail setting showing inter-ception with Mars' and Venus' orbits 210 6-5 Control strategies for maximization of angular momentum . . 211 ACKNOWLEDGEMENT The author wishes to express his sincere gratitude to Dr. V.J. Modi for his patient guidance and stimulating encouragement throughout the preparation of this thesis. The investigation reported here was partially supported by the National Research Council of Canada, Grant No. A2181. xiv LIST OF SYMBOLS semi-major axis total energy, -1/a vector of orbital elements (a, I, p, q, i,.Q, n) vector of orbital elements a_ excluding n semi-major axis of reference (24-hour period) orbit, 42,241 km 6 semi-major axis of earth's orbit,lA.U.= 1.496 x 10 km slowly varying Fourier coefficients, Appendix II constant, c(l + d 2 ) 1 / 2 constant relating e and 6, (6/e) a^g 2 constant relating e and 6 2, (62/e) a"^ 2 constant in spiral trajectory, -u'(v)/u(v) constant, Equation (5.9) constant, e/6 constant, 3 e a j ^ 2 / (26) eccentricity eccentricity vector, pointing from origin to perigee with length e, Figure 3-1 emissivities of front , and back side of surface element modified eccentricity, [ e ^ + 2e sp 0 QR + e 2 R 2 ] ^ 2 / (l-e sR). vector function denoting the right-hand-side of the perturbation equation vector function f_ excluding equation for n vector function denoting the right-hand-sides of Eouations (4.15) angular momentum (per unit mass) vector, _r * y_ components of along x n and y n axes, respectively inclination of orbital plane with respect to ecliptic unit vectors along the x, y, z axes, respectively (r = 0, 1, 2) unit vectors along the x , y r , axes unit vectors along the x , y n > z n axes auxiliary element, e COSOJ = p cos^ + q sin^ auxiliary element, e sinw = a costy - p sin<j; semi-latus rectum auxiliary variable, ln(ji) modified semi-latus rectum, &oo/C-esR) mass of sate!1ite number of illuminated surface components auxiliary vector, (0, 0, sinB^) auxiliary vector, -(cosg^ cosc^, cosg^ sina^, 0) auxiliary vector, -(cosB^ sinc^, - C O S B L. cosa^.'O) auxiliary element, e cos to auxiliary element, e sinoj radius vector, pointing from origin to s a t e l l i t e , Figure 3-1 ratio R /I e sign of U = (u_n. u_ ) auxiliary vector, (0, 0, s i n i Q 0 s i n n Q 0 ) auxiliary vector, -(K-|Q> K^Q, 0) auxiliary vector, -(K 9 N, - K L N, 0) time (nondimensional) inverse radius, 1/r (u£x, u£ , uj^ z), unit vector normal to surface component A^ with components along local coordinate axes s s s (u , u , u ), unit vector along direction of radiation x y z with components along local coordinate axes component of u_£ along local vertical , -cosa^ COSB^ component of u_£ along local horizontal, -sina^ cosB^ component of u_£ normal to orbit, sinp^ component of u_s along local vertical, 0 0 /\ -cos (i/2) cos(v-if)-fi) - sin (i/2) sin(v-ijM-n) component of u_s along local • horizontal , 2 2 /\ cos (i/2) sin(v-i^-n) + sin (i/2) sin(v-^+n) component of u_s normal to orbit, sin(i) sinfj velocity vector, r_ auxiliary element, 1 - (1 - e 2 ) 1 / / 2 rotation vector along instantaneous radius vector, (rF z/h)i_ auxiliary element, e s'i.n(n-to) = p sinn - q cosn auxiliary element, e cos (n-to) = p cosn + q sinri moving frame of reference fixed to osculating plane (geocentric case): x along the radial, y along the circumferential and z along the orbit-normal direction, Figure 3-1 intermediate frame of reference after rotation of surface element by a, Figure 3-2b reference frame fixed to the plate after Eulerian rotations a and 3 , Figure 3-2b XVI 1 A 0(v), B 0(v) Aj(v), B^v) A nk' nk AJ' (a_Q), B J ' ( a J ^0' reference frame with x n and y^ axes in the osculating plane (x along line of nodes) and z normal to orbital plane total effective illuminated surface area of the s a t e l l i t e , cross-sectional area for spherical s a t e l l i t e auxiliary functions in expression for Mg, Equation (5.21) (j = 2,3,4,5) auxiliary functions in second-order results, Equations (5.41) k-th surface component (nondimensional) (n = 1,2,...; k = 0,1,2,...) integrals, defined and evaluated in Appendix I vectors of Fourier coefficients in expansion of f_(a^,v) B_^ ' (a_Q) 0 f (a^x) \ COS(JT) ) sin(jr) d i / T f C0(v) k^ auxiliary constant, c sT/(2S) = esT/C constant in spiral trajectory, u(v) a{v) 2 2 1/2 auxiliary function of v, (AQ + BQ) (j = 0,1,2) auxiliary vector, |U k|{a l ksJ + [ a 2 k + p kU k] r^} A k 2 1/2 abbreviation for 3 arcsin(s) - s(l - s ) -2 1/2 integral constant, sin(igg) (1 - jgg) integral constant, [k 00 - d (1 - e2QQ) sinn 0 03 s i n ( i Q Q ) e o o ) V 2 + d k00 s i n i^00 ] c o s ^ i 0 0 ) integral constant, [(1 + d j00 c o s^00 (j = 0,1,...,4) auxiliary vector, Equations (3.26) constant, D 3 / [ ( l + d 2) (1 - D 2 ) ] 1 / 2 — = ^x' ^ v' ^2) s o ^ a r r a d i a t i o n force (Equation 3.7) with components ^ along local coordinate axes (Equations 3.8) xvi n f_(a_,v) vector function, f i f v outside and 0_ i f v inside I F-](e,x)j F,,(e,x) auxiliary functions, Equations (4.3) G(v) auxiliary function of v, Equation (3.21) H, Hamiltonian, Equation (6.4) and ( 6 .19 ) , respectively H(e,x) auxiliary function, Equation (4.8) I g shadow interval, (n+fT-3-j > r]+i\+^2^ I ^ auxiliary integral, Appendix I I interval where force is switched on I ^ interval where force is switched off K, KR component of K_ along orbit-normal and K(VL.), respectively K_ = (M, L, K) unit vector along Z-axis with components along local coordinate axes K-j(v), K^(v) auxil iary functions, Equations (3.15) K 1 Q constant, cos ( i 0 0 / 2 ) cos(n 0 0+^ 0 0) + sin ( i ' 0 0 / 2 ) cos(n 0 0-^ 0 0) K 2 Q constant, cos ( i 0 0 / 2 ) sin(n 0 0+^ 0 0) - sin ( i Q 0 / 2 ) s i n ( n 0 0 - * 0 0 ) L component of _K along local horizontal M, component of K along local vertical and M(v^), respectively P, Q auxiliary functions, Equations (6.10) P^ , auxiliary functions, Equations (6.22) R component of R_ along local vertical R_ = (R, S, T) functions of rotation angles a and 3, and material properties, Equation (5.4), with components along local reference frame Rg nondimensional earth's equatorial radius (6378 km), 0.151 S component of-R_ along local horizontal S c solar constant, 1-35 kW/m S1 solar radiation pressure, S /(velocity'of l i g h t ) , 4.51 x i o " 6 N/m2 xi x T component of R_ along orbit-normal Tp, temperature of front , and back side of surface element U, dot-products (u_n «ijs) and (u_£ • u_s) , respectively UQ constant, sin3 sinlipg) s 1 n^Q0 U-j constant, cosg [K^g cosa + K^Q sina] U2 constant, cosB [K 2Q cosa - sina] r • W_ rotation vector, w + v X, Y, Z inertia! reference axes, Figures 3-1, 5-1 a a modified control angle, a + v a = (a,3,Y) Eulerian control angles defining orientation of surface element with respect to orbital plane, Figures 3-2, 5-1b a*(v) optimal control vector OQ(V) starting value for control vector a-| angle between switching locations, Vg, and sun-earth line a^, 6^  Eulerian control angles for surface component a n phase angle in periodic long-term variations of orbital P elements, arctan {bx 0 0/[cy Q 0 - 3ag 0(l - e Q Q ) ] / 2 / 2 ] } a g spiral angle, arctan(c s) 6-|, $2 shadow angles, Figure 2-1 6 slow angular rate of motion of the sun with respect to earth, 1/365.2422 6-j, &2 auxil iary angl es , Equations (3.27) 63, 64 auxiliary angles, Equations (4.6) 6a, 6a variations in a and a , respectively e ratio of solar radiation and gravity forces for geocentric orbits, 2 S'(A/m) a 2 / u = 4.0 x 10"5 (A/m) XX e s ratio of solar radiation and gravity forces for heliocentric orbits, 2S1 (A/m) a 2 /y = 1.52 x 10"3 (A/m) n solar aspect angle measured from inertia! X.axis A. n modified solar aspect angle measured from line of nodes, r\-Q 6 true anomaly, measured from the instantaneous perigee axis, cj> - CO = v - to K material parameter, (e fT^ - e bT^)/(e.T^ + A constant characterizing the i n i t i a l conditions, c(l - e 2 ) 1 / 2 + 3 y Q 0 a 2 0/2 A_(v) vector of adjoint variables 14 3 2 y earth's gravitational parameter, 3.986 x 10 m /s 20 3 2 y s sun's gravitational parameter, 1.326 x 10 m /s v quasi-angle in osculating plane, v = <f> + cos(i), v(0) = 0: employed as independent variable v slow independent variable, ev v slow independent variable, 6v v-|, X)^ points of entry into and exit out of shadow cylinder length of interval before rectification v . (j = 1,2,...,6) switching points, Figure 4-1 vJ v k (k = 0,1,2,...) abbreviation for v R = kir/ (1 + B 2 ) 1 / 2 von' v o f f o n _ a n c* °ff~switching points, respectively £p, n0» CQ reference axes, fixed to osculating plane in heliocentric orbits, £;Q along the local vertical, n Q along the local hori-zontal and CQ along the orbit-normal, Figure 5-1 a , n-i» intermediate frame of reference after rotation of solar sail by a , Figure 5-1b r), c reference frame fixed to solar sail after rotations by a and 3, Figure 5-1b p material parameter characterizing specular r e f l e c t i v i t y of surface component, p - j P ^ p-j portion of incident photons which are reflected P 2 portion of reflected photons which are reflected specularly p., p f specular r e f l e c t i v i t y for back and front side of surface element, respectively p ratio R 11 Ke e specular r e f l e c t i v i t y for surface component o material parameter for homogeneous f l a t plate, + + p , or homogeneous sphere, (1 - T)/2 + 2a2/3 a-| , a 2 material parameters, o-. = (1 - p - x)/2 and o2 = [p 1(1 - p 2 ) + K(1 - p 1 - T)]/3 a-| k, o-| and a 2 for surface component A^ T material parameter denoting transmissivity of surface element (J) argument of latitude, i.e. position angle of s a t e l l i t e as measured from the line of nodes, 9 + co, (for ecliptic orbits: <$> = v), Figure 3-1 X angle between projection of sun-earth line and perigee axis (for ecliptic orbits: x = n - co), Figure 4-1 angle characterizing shift of orbital plane, v - § to argument of the perigee with respect to the line of nodes co position of the perigee measured in osculating plane from axis v = 0 (for ecliptic orbits: oo = oo), co + \p oip modified position of the perigee, arctan [ P Q O ^ P O O + e s R ^ Aa_ first-order changes in orbital elements after one revolution, ea^ (2TT) A vector of influence functions, Equation (4.17) $ , ^ auxiliary variables, p cosv + q sinv and p sinv - q cosv, respectively XXI 1 $Q , H'Q auxiliary variables, p R 0 cosv + q Q 0 sinv and PQQ sinv - cosv , respectively ft longitude of ascending node, measured from the autumnal equinox, Figure 3-1 Single subscripts refer to the order of the perturbation terms; 00 indicates i n i t i a l conditions; dots and primes refer to differentiation with respect to time and v, respectively. The norm | |a(v) | | stands for the integral over (0,2TT) of the dot-product of a(v) with i t s e l f . It should be mentioned that the branches of the inverse trigonometric functions (e.g.. arctan) are not explicitly specified but are readily determined by the i n i t i a l conditions involved and by continuity. 1. INTRODUCTION 1.1 Preliminary Remarks Since 1965, when the f i r s t Intelsat spacecraft, appropriately named 'Early Bird', provided 240 transatlantic telephone ci r c u i t s , on-board power requirements for communications satellites have been growing steadily. This in turn has led to the use of larger solar panels, the most widely used source of the photovoltaic power. For example, the experimental Canada/ U.S.A. Communications Technology Satellite (CTS), launched on January 17, 1976, is provided with two solar panels 7.32 m * 1.14 m each, generating up to 1.3 kW. The trend suggests future communication systems using more sophisticated satellites with increased capabilities and accommodating a larger number of smaller receiving ground terminals. It is likely that in the near future, motive power for interplanetary explorations and geocentric transfer missions will be provided by the Solar Electric propulsion Stage (SEP or SEPS). The electrical power needed for its ion engines i s , typically, of the order of 25 kW and, in the present state of the art, will be generated by two, 120 square meter, arrays of solar ce l l s . Advances in space science and technology are overtaking our wildest imagination. Launching of the space shuttle is about to open up avenues for assembling and servicing of space vehicles in orbit. It is likely to bring into the realm of reality by the turn of the century the concept of Solar Satellite Power Stations (SSPS) equipped with lightweight arrays of solar c e l l s , a few kilometers in area, generating around 5 GW and relaying 2 this energy by means of microwave transmission to receiving stations on earth. A promising possibility for large-scale exploration of the planetary system is provided by the concept of solar sailing where the spaceship is propelled by solar radiation forces arising from the impingement of photons upon large sails made of aluminized Mylar or Kapton. A technology assess-ment of a solar sail mission to Halley's comet in the beginning of the next decade is undertaken by NASA's Jet Propulsion Laboratory. It appears feasible to transfer a scientific package of approximately 850 kg into a trajectory for a rendezvous with the comet using an 850 m * 850 m aluminum-coated 0.1 mil Mylar s a i l . Figure 1-1 shows the concept of a solar s a i l . A characteristic common to a l l these space programs is the presence of large, lightweight appendages exposed to the solar radiation. Due to the high area/mass ratios involved, substantial perturbative accelerations of the spacecraft may be produced by the solar radiation forces. In fact, this is precisely the intention in the case of a spacecraft equipped with a solar s a i l . In other situations however, these perturbations may become detrimental to the spacecraft's performance, e.g., a communications satel-l i t e may d r i f t away from its preferred location. In any case, a detailed knowledge of the nature of the long-term orbital effects of solar radiation forces would f a c i l i t a t e the process of eliminating undesired influences in certain applications and of enhancing desired capabilities in other situa-tions. With this as a background, the thesis aims at providing better under-standing of the long-term orbital implications of the solar radiation forces as well as at assessing the f e a s i b i l i t y of u t i l i z i n g them for effecting pre-scribed orbital changes. Figure 1-1 A schematic diagram showing the concept of solar sail 4 1.2 Review of the Literature 1.2.1 Solar radiation induced orbital perturbations The fact that light carries momentum and exerts pressure when i t ' is incident upon a surface, was known long before the advent of the space 2 age and is inherent in Einstein's famous E = mc law. Nevertheless, the f i r s t exhibition of solar radiation effects upon an earth's s a t e l l i t e (the Vanguard I, launched on March 17, 1958) caught the observers by sur-prise: only the classical perturbations due to the higher harmonics of the earth's potential f i e l d and luni-solar gravitational.influences were taken into consideration. Subsequently, Musen et a l J included the solar radiation effect in an attempt to account for the observed discrepancy (of amplitude 2 km and period of 850 days) in its perigee height and found i t to be f u l l y responsible. The f i r s t theoretical analysis of the effect (Musen ), deriving the equations governing the evolution of the orbital elements by means of the vectorial method, appeared soon after. A few of the basic features of the solar radiation pressure effects were discovered, e.g., a significant perturbation in the perigee height and only small short-term periodic variations in the semi-major axis. Furthermore, he established that for certain combinations of altitude and inclination, the solar radiation force interacts with the perturbations due to the earth's oblateness: the most interesting of these so-called 'resonance' cases is the one where the perigee closely follows the motion of the sun producing a long-period, large ampli-tude variation in the perigee height which could seriously affect the l i f e -3 time of the s a t e l l i t e . A study by Parkinson et a l . with reference to the lifetime of the Beacon sa t e l l i t e further emphasizes this point. 5 It was the passive communications balloon-satellite Echo I, launched on August 12, 1960, which provided a dramatic indication as to the possible severity of solar radiation induced orbital change: in about five months the perigee altitude decreased from more than 1500 km to 930 km and subsequently increased again to almost its original value. The property which made Echo I extremely sensitive to solar radiation effects was its high area/mass ratio: the s a t e l l i t e consisted of an aluminum-coated half-mil thick mylar sphere, 30 meter in diameter, weighing some 70 kg. Many papers are devoted 4-8 to the perturbations of the Echo I ; the latter two of these provide com-prehensive analyses of its orbital behavior. A very readable account on 9 sunlight pressure induced perturbations is given by Shapiro et a l . , who describe the effect upon orbiting dipoles in the West Ford experiment. A l l a n ^ extended Musen's results by including the effect of the shadow in the pertubation equations and provided some numerical results. By inte-gration of the classical Lagrange's perturbation equations in terms of the eccentric anomaly, Kozai^ was probably the f i r s t one to establish general 12 analytical results, valid for a short duration. Bryant has indicated how the method of averaging may be employed in deriving the equations under-lying the long-term orbital perturbations, but does not provide any results. A very comprehensive account on the effect of solar radiation, including 13 the shadow effect, on the orbital period is given by Wyatt , who derives short-term analytical results for several special cases. An admirable attempt to obtain first-order analytical results for the combined effects of solar radiation and the earth's second zonal harmonic was undertaken 14 by Koskela , but the validity of the application of the approach beyond the f i r s t few revolutions must be questioned. Under certain simplifying 6 15 assumptions, Cook et a l . have derived an elegant approximate solution to this problem for near-circular, non-resonant orbits in the context of 16 the West Ford experiment. An interesting, subsequent paper by Cook finds a good agreement between this solution and the observed motion of the Echo I and Explorer 9 satellites. After these pioneering contributions, the attention was directed either to refinements in the basic understanding of solar radiation effects or to applications where the force can be employed in bringing about de-sired changes. The primary emphasis was focused on the effects of reflected radiation from the earth, e.g. the excellent work of .Wyatt^, which was 18 later extended by Baker . Under no circumstances, however, can this in-fluence rival the dominance of direct solar radiation effects. A very extensive summary of a l l aspects of solar radiation effects as well as the 19 more traditional sources of orbital perturbations is given by Shapiro Especially of interest is his exposition on stable near-earth orbits, having the characteristic of constant eccentricity under the combined in-fluences of solar radiation and the second harmonic of the earth's poten-tial f i e l d . Another enlightening review of the main solar radiation 20 features, stemming from the Russian literature, is by Polyachova : although the t i t l e s of a few figures are interchanged, this paper provides the most detailed information on resonance conditions. Using a formulation 21 in terms of the Hamiltonian expressed in Delaunay variables, Brouwer in-vestigated resonance in the case of polar orbits and finds general agree-22 ment with numerical results. Later, Hori extended the analysis to general orbits. It can be concluded that resonance does not occur for orbits with a semi-major axis exceeding three times the earth's radius, except when the eccentricity is very large. A numerical study of the solar radiation pressure effects on satellites with several different configurations is 23 24 presented by Lubowe . Levin provides a fresh insight into the nature of the solar radiation effects by analysing the behavior of the radial dis-25 26 tance for i n i t i a l l y circular orbits. Zee ' has presented an approximate analytical study of the combined influences of gravitational and solar ra-diation forces for near-circular equatorial orbits. On the other hand, 27 Lidov employs double averaging, both in the motion of the s a t e l l i t e and that of the sun to obtain approximate results valid for extremely long 28 29 duration. The results obtained by Isayev et a l . ' . are valid for a short interval only since the position of the sun is kept constant with respect to the earth providing a uniform force f i e l d . A high-precision short and long-term numerical integration scheme based on Kozai's equations^ in-30 31 32 eluding the shadow effect was recently presented by Aksnes . Sehnal ' has summarized various aspects of solar radiation influences. A.satellite whose orbital behavior attracted almost as much attention as Echo I is Pageos, launched in 1966. Pageos consists of a balloon quite similar in size, structure and mass to Echo I, but its shape approximates a prolate spheroid. Many studies are devoted to explaining the anomalies in its 33 34 35 36 orbital behavior: Sehnal , Prior , Fea , and Gambis analyse the in-fluence of earth-reflected radiation upon this spacecraft. At present, its orbital anomalies are believed to be caused by a unique interplay of 37 38 attitude and orbital perturbations ' : the orientation of the satellite's spin-axis as well as its spin-rate are changing continually due to solar radiation torques thereby producing a time-dependent orbital perturbation force. 8 It should be emphasized that almost a l l studies employ a simplified solar radiation force model taking a constant-magnitude force along the direction of radiation. A more r e a l i s t i c formulation was provided by 39 Georgevic who includes the effects of diffuse reflection and re-emission of absorbed energy. This model proved capable of predicting the actual magnitude of the solar radiation force upon the Mariner 9 Mars orbiter within 0.1 %. A number of papers are devoted specifically to the effects of the 40 earth's shadow. Escobal presents a detailed analysis of the points of entry and exit of the shadow region. The fraction of the orbit spent in 41 darkness, expressed in true anomaly, is determined by Karymov , and by 42 Zhurin in terms of time. An interesting approach for incorporating the shadow effect in the analytical treatment of solar radiation perturbations 43 is proposed by Ferraz-Mello , who multiplies the perturbation potential by a shadow function, being unity outside and zero inside the shadow inter-val. After development of this function in terms of Fourier series, a first-order solution in the form of in f i n i t e trigonometric series in the mean anomaly is obtained for the Delaunay variables. Since the computation of the coefficients is extremely laborious, the practicability of the ap-proach must be considered limited. Other shortcomings are pointed out by 44 the author himself in a subsequent paper undertaking a new attack using Von Zeipel's method and a Hamiltonian in extended Delaunay variables. The main outcome of the analysis is the absence of secular perturbations 45 in semi-major axis, eccentricity and inclination. Vilhena de Moraes has found a close correspondence between the outcome of Ferraz-Mello's model applied to the Vanguard II sa t e l l i t e and results by Kozai. Short-term 9 46-48 semi-analytical results were obtained by Lala et a l . , developing their 49 own shadow function but keeping the sun in a fixed position. Meeus studied the observed orbital behavior of a few satellites and found that, in general, the effect of the shadow makes the semi-major axis increase (decrease) when the eccentricity is diminishing (growing). 1.2.2 Orbital control using solar radiation forces Whereas all of the previous references deal with natural perturbations in the sense that the librational motion of the s a t e l l i t e body is not delibe-rately manipulated, the following category of papers studies the effects of controlled changes in the orientation of the reflecting surface and thus the resulting solar radiation force. The f e a s i b i l i t y of u t i l i z i n g solar radiation forces for controlled orbital change was assessed quite early in the space age. 50 In 1958, Garwin envisioned an exploration of the solar system by means of large solar sails made of aluminized•Mylar. Considering heliocentric solar 51 sail trajectories Tsu derived an approximate solution in the form of a pla-nar logarithmic spiral neglecting the small radial velocity component. 52 London remedied this shortcoming and determined, graphically, the best sail setting and corresponding spiral angle for minimum-time transfer. The spiral 53 solution, naturally, allows only specific i n i t i a l conditions. Pozzi et a l . suggested an iteration scheme to accomodate more general i n i t i a l conditions. A f a i r l y complete survey of solar sail trajectories and possible missions is 54 55 given by Kiefer . Modi et a l . proposed and on-off strategy with the sail normal to or aligned with the radiation leading to a significant elongation of the orbit as the perigee moves towards and the apogee dr i f t s away from the sun. 10 Other studies foresaw opportunities for using solar sails in geocen-56 t r i e orbits. Sands proposed to rotate the sail about an axis perpendicular to the orbital plane at half the rate of the satellite's motion around the earth. This strategy enables the s a t e l l i t e to reach an escape trajectory 57 eventually. For an orbit in a plane normal to the ecliptic Fimple deter-mined the control strategy which maximizes the component of the solar radia-tion force along the instantaneous velocity, thereby continuously increasing the total energy and semi-major axis. Cohen et a l . ^ achieved substantial changes in the orbital elements of an orbit in the e c l i p t i c plane by means of an on-off switching strategy: during the on-phase,.when the s a t e l l i t e moves away from the sun, the plate is aligned with the radius vector and kept normal to the orbital plane, while during the off-phase the plate is along the radiation. The effects upon a large earth-orbiting mirror in the 59 ecliptic plane reflecting sunlight to the earth were determined by Bosch under certain simplifying assumptions. Ahmad et a l . ^ considered this pro-blem as well as that of a perfect absorber facing the sun in a more r e a l i s t i c equatorial orbit and obtained the orbital perturbations using a numerical technique. Furthermore, the forces and torques required to maintain the de-c i sired orientations were calculated. Shrivastava et a l . determined the panel orientation for obtaining maximum changes in various orbital elements. The f e a s i b i l i t y of east-west station-keeping of communications satellites by means 6 2 of controlled solar radiation forces was demonstrated by Shrivastava et a l . 63 and further substantiated by Modi et a l . A different concept for u t i l i z i n g solar radiation forces in orbital 64 control is presented by Buckingham , who studied a balloon with different reflective and absorptive characteristics on either side permitting control 11 of the force through rotation of the body. The same concept applied to 65 plates is investigated by Black . 1.2.3 Small-thrust trajectories The problem of controlled orbital change by means of solar radiation forces may be studied within the general framework of small-thrust trajec-tories which normally consider perturbing forces due to micro-thruster units. Although obvious differences exist in the nature of these two sources of orbital change (because of the constraints imposed by the instantaneous po-sition of the sun and thus the direction of the force), a knowledge of the methods and results of the more classical f i e l d of small-thrust trajectories would certainly be valuable. The smallness of the thrust is capitalized upon by modelling the problems in terms of perturbation theory using expan-sions in terms of the ratio of thrust/gravity forces. The problem of either tangential or radial constant small thrust for circular orbits was studied f i r s t ^ - ^ . A comprehensive analysis including intermittent thrust by means 69 of the Krylov-Bogoliubov method has been presented by Lass et a l . , provi-ding the following results: a constant radial force causes the axis of the orbit to precess, while a tangential thrust changes an i n i t i a l l y circular orbit into a spiral. Rider^ proposed a control strategy for changing the inclination and longitude of nodes of an orbit while Lass et a l . ^ study 72 73 the effects of a thrust normal to the orbital plane. Zee ' refined the analysis for a small tangential thrust and discovered small oscillations in the spiral trajectory. The Russian literature, naturally, abounds with studies related to small-thrust problems as a consequence of the epoch-making work of Krylov, Bogoliubov and M i t r o p o l s k i i ^ in the f i e l d of nonlinear o s c i l -12 7 R 7f< lations. Laricheva et a l . ' illustrated some of the p i t f a l l s of the method of averaging by a few il l u s t r a t i v e examples: for orbits with i n i t i a l eccentricity smaller or of the same order as the small perturbation para-meter, indiscriminate application of averaging may lead to qualitatively incorrect results. Taking a constant tangential acceleration, Okhotsimskii^ analysed the resulting motion in detail using asymptotic representations 78 near e = 0 and e = 1, while Cohen has presented an approximate solution accounting for the variation of the mass of the.satellite due to the burning of fuel. The more general problem of a constant small thrust under an arbitrary but fixed angle to the local vertical gained the attention of the investiga-79 tors next. Johnson et a l . derived a solution valid for short duration only. Introducing an independent slow variable in the radial distance and separating the oscillatory and non-oscillatory terms in an ad-hoc manner, 80 Ting et a l . offered a prelude to the application of the two-variable ex-81 pansion procedure to this problem. A later paper by Brofman also treats the case of tangential thrust with variable mass and orbital decay due to drag in a similar manner. Nayfeh found essentially the same results using his more systematic derivative-expansion method. While a l l these studies consider an i n i t i a l l y circular orbit, the problem in its most gene-ral form, including a starting orbit of an arbitrary eccentricity, was solved 83 by Shi et a l . using the two-variable expansion method. Due to the fact that the ratio of thrust/gravity does not remain small for ascending tra-jectories, their results do not predict the radial distance correctly near 84 escape. This deficiency is redressed by the same authors through a careful analysis of the rate of change of radial distance in three different regions : 13 gravity dominant, gravity and thrust of the same order, and thrust dominant. Incorporating the change in the mass of the s a t e l l i t e , Moss^ studied c i r -86 cumferential thrust by the same (two-variable expansion) method. Flandro obtained approximate long-term solutions for the orbital elements under a low thrust normal to the orbital plane. For an il l u s t r a t i v e description of the two-variable expansion method 87 one is referred to the original presentation by Cole and Kevorkian and 88 89 the more comprehensive treatment by Kevorkian . Morrison points out the consistency between the results obtained by this method and those derived by the modified method of averaging. A more fundamental. treatment of these 90 91 92 methods can be found in Perko and Klimas . Kevorkian established the equivalence of the Von Zeipel and the two-variable expansion methods up to 93 first-order in the small parameter. Nayfeh has described the various per-turbation methods and their relative advantages in detail. 1.2.4 Optimal trajectories Finally, a few papers using optimal control theory in determining the best steering and/or thrust program to accomplish a given, objective in a prescribed manner should be mentioned. The f i e l d of optimal control theory. fostered by the calculus of variations, has become a full-grown science i n ' 94 i t s e l f . A theoretical foundation is given by Lee et a l . and a practical 95 summary is provided by Bryson et a l . . The application of optimal control theory in rocket and sat e l l i t e trajectories is manifested in numerous papers. A problem which has attracted continuous attention over the last two decades concerns the optimal transfer, i.e. determination of the thrust direction for reaching a prescribed final orbit from a given i n i t i a l orbit with 14 minimum fuel consumption. Early contributions dealing with various + 4-u- un k , A 9 6 > 9 7 c IA 98,99 M 1 k 100,101 aspects of this problem are by Lawden , Faulders , Melbourne 102 and Hinz . Of particular interest is the conclusion by Lawden that the optimal thrust orientation approximately bisects the tangential and circum-ferential directions. A comprehensive analytical solution for transfer be-103 104 tween two close ellipses is presented by Edelbaum . Breakwell et a l . studied the problem of reaching a specified energy level with minimum fuel expenditure. An higher-order analytical treatment of the linearized equa-105 tions for near-circular transfer is presented by Mclntyre et a l . . A review of the early papers on optimal trajectories is given by B e l l ^ . j n the Russian literature, the development of the maximum principle by Pontrya-gin and B o l t y a n s k i i ^ has stimulated many researchers in the space sciences. Of particular interest is the work by Lebedev and others^^"^ who consider the minimum-time transfer between coplanar circular orbits by means of a so-lar s a i l : a numerical iteration method is employed to solve a system of differential equations with partly i n i t i a l , partly final boundary conditions. An interesting attempt to find an approximate solution to the problem of transfer between two coplanar orbits in minimum time using the method of averaging is presented by Avramchuk et a l J ^ ; unfortunately, only the ad-joint equations are amenable to closed-form solutions. The book by Grod-112 zovskii et a l . provides a somewhat outdated, but exhaustive treatment on various aspects of small-thrust and optimal trajectories. More recently, 113 Brusch has presented a comprehensive treatment of the minimum-fuel trans-fer from an i n i t i a l circular to a prescribed coplanar, e l l i p t i c orbit. An analytical solution to the optimal (in the sense of least fuel) escape from a circular orbit in terms of a straightforward perturbation solution was 15 114 given by Anthony et a l . . An essentially similar problem is treated more 115 accurately by Jacobson et a l . by means of the two-variable expansion pro-cedure. They discovered small (order e = thrust/gravity) oscillatory terms in the near-tangential optimal control strategy. These results were sub-stantiated by Reidelhuber et al using a different formulation. An ex-haustive review (up to 1965) of papers using optimal control theory with emphasis on f l i g h t mechanics is given by Paiewonsky^^. Unfortunately, analytical (approximate) solutions to optimal control problems may be derived in very limited situations only. Therefore, many numerical methods have been developed, specifically for this purpose. A very attractive procedure is the steepest-ascent (or gradient) method in-volving a generalization of a problem in the ordinary calculus, viz. the maximization of a function subject to constraints. An heuristic description 118 of the method is given by Kelley , while a general treatment is presented 119 120 by Bryson et a l . " and Campbell et a l . 1.3 Scope and Objective of the Study The literature survey indicates that many aspects of solar radiation induced orbital perturbations have been investigated. Resonance conditions leading to large amplitude variations of the orbital elements are well esta-2 20 22 1 1 1 3 14 bl ished ' ' . Short-term valid analytical results are available ' ' and approximate representations for the long-term behavior are explored for 15 24 26 43 27 certain special cases: near-circular ' ' ' or low-inclination orbits The available solutions are based upon a model where the force is taken along the radiation, which is justified only when the s a t e l l i t e can be modelled as a sphere with homogeneous surface characteristics or as a plate 16 kept normal to the radiation. In the present investigation an attempt is made to obtain long-term valid analytical solutions for the orbital elements with no restrictions imposed on the i n i t i a l orbit and the apparent motion of the sun accounted for. Because of the successful application of the two-variable expansion procedure 83 \ in small-thrust trajectories , i t is f e l t that an approach along these'lines should deserve attention in the present situation. In addition to providing valuable information as to the long-term evolution of orbits in general, a comprehensive understanding of the qualitative aspects of solar radiation effects would be a valuable guide in exploring control strategies for desired orbital changes. Furthermore, the analysis is based upon a r e a l i s t i c force model allowing for diffuse and/or specular reflection as well as for re-emis-sion of absorbed radiation. In some cases, the investigation is extended to include arbitrarily shaped sa t e l l i t e structures modelled by a number of f l a t surface elements of homogeneous material characteristics. Other applications such as space platforms modelled as a f l a t plate in an arbitrary fixed orien-tation with respect to the earth as well as those kept fixed to inertial space are also studied. It should be mentioned that the effects of other perturbation forces are ignored in the present investigation. For an equatorial geosynchronous orbit, the magnitude of the major perturbing forces as compared with the local gravity force are of the following order: -5 i) solar radiation force . : 4 (A/m) 10 ; i i ) out-of-plane oblateness force: 10 i i i ) in-plane oblateness force : 4 x 10 ; iv) lunar attraction force : 1.5 x 10 ; v) solar attraction force : 7 x 10"^ ; 17 Hence, for satellites with a large A/m ratio (e.g., the SSPS and particularly the solar s a i l ) , radiation forces would be the predominant source of pertur-bations. However, for spacecrafts with a relatively small A/m ratio (e.g., communications satellites) the traditional perturbations, especially those due to the earth's oblateness, need to be incorporated. Except in the reso-nance cases, the wellknown secular effects caused by the classical perturba-tions could simply be added to the results obtained for the solar radiation induced orbital changes in the first-order approximation. Another part of the thesis is concerned with the development of control strategies, involving the rotation of solar panels attached to the main body, thereby producing variations in both the magnitude and direction of the resul-ting solar radiation force. Considerable attention is given to on-off swit-ching programs, where the plate is aligned with the radiation during the off-phase and normal to the radiation, generating the largest possible force, in the on-phase. The optimal locations for switching are determined for a few specific objectives such as maximum increase in total energy. While on-off switching may lead to substantial changes in the major axis, i t is not neces-sarily the optimal strategy when time-varying orientations are also taken into consideration. Therefore, the determination of the optimal control strategy for maximization of the major axis after one revolution is underta-ken and the effectiveness of this control program is compared with that of the switching strategies. This investigation is of relevance for raising a solar sail from a geocentric to a heliocentric or escape trajectory. Subsequently, the orbital behavior of satellites in an heliocentric orbit is studied in detail. The resulting orbital behavior of a spacecraft in a fixed orientation to the local frame is explored in terms of exact so-18 1uti ons (specific i n i t i a l conditions) or approximate long-term valid repre-sentations (general case). The potential of out-of-plane spiral transfer trajectories is assessed. The results are mainly of interest for inter-planetary solar sail missions. While some aspects of interplanetary trans-fer have been explored^^ no studies on optimal escape are reported. Therefore, time-varying optimal control strategies are investigated with the objective to maximize the increase in total energy and angular momentum per revolution. In addition, these results may be used for assessing the rela-tive effectiveness of constant sail settings. A schematic overview of the plan of study is presented in Figures 1-2 a and b. SOLAR RADIATION PRESSURE INDUCED PERTURBATIONS AND CONTROL OF SATELLITE ORBITS o r b i t a l perturbations plate o r i e n t a t i o n normal to r a d i a t i o n e c l i p t i c shadow eff e c t numerical int egratio n • short-term a n a l y t i c a l • r e c t i f i c a t i o n / i t e r a t i o n • two-variable expansion Chapter 2 1 a r b i t r a r y fixed angle to l o c a l frame a r b i t r a r y o r b i t a l plane geocentric o r b i t s helio . . -.-^"n next centric o r b i t s | p a g e -a r b i t r a r y e c c e n t r i c i t y a r b i t r a r y s a t e l l i t e body a r b i t r a r y fixed angle to r a d i a t i o n or i n e r t i a l space a r b i t r a r y o r b i t a l plane •short-term a n a l y t i c a l • r e c t i f i c a t i o n / i t e r a t i o n Chapter 3 o r b i t a l control normal to rad i a t i o n 1 time dependent plate o r i e n t a t i o n a r b i t r a r y o r b i t a l plane switching at switching to control plane of o r b i t l h a l f - y e a r l y switching veloc i t y normal to radia t i o n e c l i p t i c plane X sun-earth l i n e apogee/ perigee l i n e of nodes i n c l i -nation control of ec c e n t r i -c i t y control of major axis c o n t r o l of l a t u s rectum l_ i i i 1 1 maximi-zation of major axis • short-term a n a l y t i c a l • r e c t i f i c a t i o n / i t e r a t i o n and/or • two-variable expansion • numerical steepest-ascent i t e r a t i o n Chapter 4 Figure 1 - 2 Schematic overview of the plan of study: (a) geocentric orbits a r b i t r a r y fixed to l o c a l frame prescribed i n i t i a l v e l o c i t y vector a r b i t r a r y o r b i t a l plane h e l i o c e n t r i c o r b i t s I plate o r i e n t a t i o n - a r b i t r a r y arbit orbit plane :rary :al 3 time dependent optimal control of s a i l o r i e n t a t i o n maximization of t o t a l energy per revolution maximization of angular momentum per revolution out-of-plane tr a n s f e r by switching in-plane transfer o r b i t evolution exact a n a l y t i c a l s o l u t i o n s : conic sections, logarithmic s p i r a l s •short-term a n a l y t i c a l • two-variable expansion • numerical Chapter 5 approximate a n a l y t i c a l numerical steepest-ascent i t e r a t i o n Chapter 6 approximate a n a l y t i c a l Figure 1-2 Schematic overview of the plan of study: (b) heliocentric orbits 2. SOLAR RADIATION EFFECTS UPON AN ORBIT IN THE ECLIPTIC PLANE 2.1 Preliminary Remarks In this chapter, the perturbations of a sat e l l i t e orbit in the ecliptic plane subjected to solar radiation forces are studied. Long-term valid approximate solutions for the orbital elements are derived, by means of the two-variable expansion procedure while accounting for the apparent motion of the sun around the earth. The results are compared with those obtained by repeated rectification of the short-term valid solutions obtained by a straightforward perturbation method and their relative accuracies assessed using a double precision numerical integration routine. In the analysis,the solar radiation force is taken along the direction of the sun-earth line which is considered to be coincident with the sun-satellite line, since for a geocentric orbit the relative distance of a sat e l l i t e to the earth in comparison with that to the sun can be ignored. Taking the resulting radiation force along the sun-earth line is justified in the case of a sat e l l i t e with large solar panels kept normal to the incident radiation for maximum on-board power production, e.g., communications satellites or the proposed SEPS mentioned before. A spherical s a t e l l i t e with homogeneous surface characteristics would also experience a solar radiation force along the sun-earth line. Two cases of practical importance are studied separately: f i r s t , the nondimensional solar radiation force parameter referred to as 'solar parameter' (e) is taken to be of the same order of magnitude as the 'frequency parameter' (6) designating relative motion of the sun in the ec l i p t i c plane. This assumption is valid for satellites with a relatively large area/mass ratio, e.g., Echo I (A/m = 10 m /kg) 1?1 1?? 1?3 or the proposed SSPS ' ' . In the other case, the solar 2 parameter e is taken to be of the order 6 representing a class of satellites with relatively small solar radiation perturbations like the CTS 1 2 4. By expressing the perturbation equations in terms of p = e cos oo and q=e since, the singularity in co for e = 0 is avoided making the analysis uniformly valid for both circular and e l l i p t i c a l osculating orbits. A comprehensive picture of the long-term orbital perturba-tions is provided by polar plots (p, q-diagrams) for the eccentricity and argument of the perigee. The effect of the earth's shadow is investigated separately. Note that this influence is likely to be strongest for orbits in the ec l i p t i c plane since the sa t e l l i t e is now eclipsed in every revolution. Both short and long-term analytical representations have been established. The qualitative and quantitative understanding of long-term perturbations of orbits in the ecliptic plane may serve as a guide in predicting the behavior of near-ecliptic, including equatorial, orbits. Furthermore, the analysis yields considerable insight into the nature and range of validity of the approximate methods, thus providing a basis for establishing a rational approach for the following chapters. 2.2 General Formulation of the Solar Radiation Force A r e a l i s t i c model for the solar radiation force acting upon 39 a sa t e l l i t e has been provided by Georgevic in his detailed analysis of the radiation force upon the Mariner 9 spacecraft. In case of a sat e l l i t e in a geocentric orbit up to the geosynchronous altitude, fluctuations in the local value of the solar constant are almost entirely due to the seasonal variations in the solar constant i t s e l f , caused by the eccentricity of the earth's orbit. These variations amount to about 3.4% from the mean value and are ignored. The solar radiation force upon an arbitrarily shaped s a t e l l i t e in a geocentric orbit can be represented in the following general form: 2S 1 I i n-u. S | |cr-,+ la2 + p(un-u.S)] u n | dA , • (2. 1) where u 0 is the unit-normal to the surface element dA and A denotes the total effective surface area of the sa t e l l i t e illuminated by the sun. The absolute sign around u_n • u_s is necessary to ensure that the force has a non-negative component along the direction of the radiation, _u . The material parameters o-., and p may vary over the surface area and are determined by the re f l e c t i v i t y and emissivity of the surface element dA : 24 p = p ^ p 2 ; a-| = (1 - p - T)/2 ; a 2 = [p-j (1 - p 2 ) + K(1 - p-j - T) ]/3 ..... (2.2) where p-j denotes the total fraction of the incident photons which are reflected, p 2 the portion of these photons which are reflected specularly, and T the portion of photons transmitted through the surface. The constant K depends upon the temperatures and emissiv-it i e s of the front and back sides of the surface element: K = <ef T f - eb Tb> / <ef T f + eb Tb> • • Variations in the material parameters with time due to deterioration of the surface or due to changes in temperature f a l l outside the scope of the present investigation. The following table gives an idea of the values of the 39 material constants for a few typical spacecraft components including 125 aluminum-coated mylar solar sails : Table 2.1 Material Parameters for a Few Typical Spacecraft Components Components p l p2 T e f eb K P a l a 2 a Solar panel 0 21 1 .00 0 0 81 0 81 0 0 21 0 39 0 0 60 Hi gh-gai n Antenna 0 30 0 67 0 0 84 0 06 0 .87 0 20 0 40 0.23 0 83 Solar Sail 0 88 0 94 0 0 05 0 60 -0 .85 0 83 0 09 -0.02 0 90 25 In most practical cases the total surface area can be divided into components representing different parts of the s a t e l l i t e , each with its own homogeneous material parameters, so that the integral of Equation (2.1) can be written as a summation over the various compon-ents. In many applications, most notably solar sail and SSPS, the magnitude of the force upon one component, namely the sail and solar panels, is so predominant over the sum of the forces upon a l l other components that, effectively, the s a t e l l i t e can be modelled as a plate with homogeneous material characteristics. 2.3 Plate Normal to Radiation For satellites which can be effectively modelled as a homogen-eous plate normal to the incident radiation, the solar radiation force of Equation (2.1) can be simplified as F_= 2a S1A _us , since u_n and u5 coincide for that case. It is interesting that the force upon a spherical s a t e l l i t e with homogeneous surface characteristics, takes on the same form with a equal to (1 - x)/2 + 2[p-) (1 - p 2) + K( 1 - p-j T ) ] / 9 as obtained by integration over the spherical surface. In this case A represents the cross-sectional area of the sphere. In an inertial reference frame fixed to the earth, the equations of motion in polar coordinates r and v become: r - r v 2 = -p/r 2 - 2a S'(A/m)cos[v - n(v)] ; rv+2fv = 2a S'(A/m)sin [ v-n ( v ) ] . (2.3) 26 sun shadow i Figure 2-1 Geometry of sun, earth and sa t e l l i t e including shadow region The solar aspect angle n(v) denotes the sun's position, Figure 2-1. For the analysis to be valid over a long term, the relative motion of the sun needs to be taken into account: since the sun completes one revolution per year, i.e. 1/6 = 365.2422 days, i t follows that 3 1/2 n(v) = 6t(v)(y/ap + n Q 0 . (2.4) It is convenient to nondimensionalize the equations by introducing 3 1/2 the reference length and time units a r = 42,241 km and (a /p) =1/(2TT) day. Forces are nondimensionalized through a / (u rn ) and become, mathematically, indistinguishable from accelerations. The form of Equations (2.3) is not convenient for finding analytical solutions, therefore, a transformation u = l / r as in the derivation of the classical Keplerian equations is performed, and the angle v is taken as independent variable (v = h/r ) leading to the (nondimensional) equations: u" (v) + u(v) = l/X,(y)+e |cos[v - n(v) ] - u' (v)sin[v - n(v) ]/ u 2(v) }/£(v) ; V (v) = 2 e sin [v - n(v)] / u 3(v) ; t'(v) = l/(uV/2) ; n(v) = 6t(v) + n (2.5) The solar parameter e is defined as e = 2S 1 (A/m)(a*/u) = 4.0 x 10"5 (A/m) . It should be noted that the parameter a is taken equal to unity (i.e., p = l , T = 0) in the present analysis. A different value of a can readily be accommodated by modifying the parameter e accordingly. Since the solar parameter is very small, i t may be justified to postulate solutions for the radial distance in the form of conic sections with slowly changing orbital elements, i.e. u(v) is written in the form: u(v) = [ 1 + p(v)cos v + q(v) sin v ] / £(v) , (2. where p (= e cos to), q (= e sin to) and I are slowly varying orbital elements. At any instant v = v-j, the 'ellipse' with elements p-j =p(v^), q-j and £^ is referred to as the osculating ellipse. This orbit may be interpreted as the e l l i p t i c trajectory that would be followed by the sa t e l l i t e i f the perturbation force were to vanish at v = instantaneously. This can be seen by taking e = 0 for v >_ v-j in Equations (2.5). It can also be understood that both the radius and velocity vectors at any point in the actual (perturbed) trajectory are identical to those of the osculating ellipse corresponding to that point. This is referred to as the condition of osculation 29 and can be stated mathematically as u'(v) = (-p sin v + q cos v)/£ . The second-order equation for u(v) can now be replaced by an equivalent system of two first-order equations for p(v) and q(v). Thus, the complete system of equations to be studied becomes: 2 p'(v) = el -|-sinri+ (p + cos v) si n (v - n)/ (1 + P cos v + q si n v \ / 2 ( 1 + p cos v + q sin v) ; q' (v) = e £ 2 | cos n+ (q + sin v) sin(v-n)/(l+pcosv + q s i n v ) j / ( 1 + p cos v + q sin v) ; 3 "3 V (v) = 2 e Z sin (v - n)/(l + p cos v + q sin v) ; n ' ( v ) = 6 £ 3 / 2 / ( 1 +pcosv + q s i n v ) 2 . (2.7) It should be noted that the solar aspect angle n ( v ) is treated here as a quasi-orbital element. The system of equations (2.7) will be written symbolically as a/ (v) = e f (a_,v) and arbitrary i n i t i a l conditions a^ (O) = a^g, with the vector a^  containing the pertinent orbital elements. Note that f is periodic in the variable v . 2.3.1 Short-term valid approximations A short-term valid approximation for the orbital elements can readily be obtained by means of an expansion of the elements in 30 terms of a simple perturbation series. In case e is of the same order of magnitude as 6 , the expansion may be taken in the form N-1 a(v) = I e J a.(v) + 0 (e N ) j=0 J ( 2 . 8 ) On substitution of this series into Equations ( 2 . 7 ) , i t follows that a^(v) = a^g and integration of the first-order equations leads to or explicitly: 'PT(V) = ^ o O i c o s n n n [ p n n B ^ (v) + B„(v) / 2 ] - sin r, n n[A 9 n(v) '00 L P 0 0 " 3 1 v v / " 32 0 0 L " 2 0 * + P 0 0 A 3 1(v) + A 3 0(v) / 2 + A 3 2(v) / 2 ] q 1 (v) = £ Qg | cos n 0 0 [A 2 Q(v) + q Qg B 3 1 (v) + A30(.v) / 2 - A 3 2 (v ) / 2 ] sin n 0 0 [ q 0 Q A 3 ] (v) + B„(v) / 2 ] 32 v ^ (V) 2 £ 0 0 B 3 1(v) cosrigg - A 3 1(v) sinr,g 0 n i ( v ) = c(l - e 2 0 ) 3 / 2 A 2 Q(v) ( 2 . 9 ) 31 where the integrals A n k(v) and B n k ( v ) , which depend on p Q 0 and q Q 0 , are defined and evaluated in Appendix I. With these and after con-siderable amount of algebraic manipulation, the orbital elements can be expressed expli c i t l y in terms of i n i t i a l conditions as follows: a, (v) ^a00 00 cos(v - TIQQ) 1 + p Q 0 cos v + q Q 0 sinv) Pi (v ) a00 ( 1 " e n n ) / 2 00; (1 - e Q 0 ) s i n v sin(v - n n n) 00; (1 +p Q 0 cos v + q Q 0 sinv)' + 3 ( p Q 0 sin v - q n ncos v ) sin n 00" '00 1 + P 0 0cos V + q n n sin v 3 A 1 Q ( v ) s i n n 0 0 00 2 2 x / 0 f ( 1 - e 0 0 ) cosv s in (v-n 0 0) q^v) •= -a^ n ( l -eJ n)/2 i 0 0 0 0 1 ( l + p 0 0 c o s v + q n n s i nv )^ 100" ( p 0 Q sin v - q Q 0 cosv) cos n Q Q + 3 - 3A-|Q(V)COSTI 1 + p 0 Q c o s v + q Q 0 sinv 00 ( p n n sin v - q n n cos V) t \ n 2 xl/2 J i\ f \ V K00 M00 n^v) = c(l - e Q 0) \ A i n(v) - — l + p 0 0 cosv + q o o S 1 n v ...(2.10) It is seen that after one orbit (V = 2TT), only the terms containing A-JQ(V) do not vanish. While these results provide a reasonable approximation to the orbital elements of the osculating ellipse at any point during the f i r s t few revolutions, i t is of particular interest to consider the orbital elements at v = 2TT. The terms which vanish at v = 2TT can then be identified as short-term periodic contributions and are of secondary importance in the long-term behavior of the orbital elements. Writing Aa_ = ea^ (2TT), one obtains by substituting v = 2TT into the integrals of Equations (2.10): Ap = - S T r e a ^ d - e 2 Q ) 1 / 2 sin r , 0 0 ; 2 2 1/2 Aq = 37rea 0 0(l - e 0 Q ) cos n Q 0 ; A£ = 6 ^ £ a o o ( 1 " e 0 0 ) 1 / 2 [ p 0 0 s i n n 0 0 - q 0 0 C O S T 1 0 0 ] ' An = 2T\ 6 a ^ 2 At = S T r e a ^ 2 {(4 + P 0 0 ) c o s n 0 0 + 6 q 0 0 s i n n 0 0 } + O ( e 2 0 ) (2. Here the expression for At is obtained by expanding the elements in 2 1/2' the integrand r /£ for small e o Q . The change in semi-major axis can be expressed in terms of the results of Equations (2.11) yielding 33 Aa=0, so that the major axis and hence the total energy return to their original values after one revolution in the first-order theory: the energy added while moving away from the sun is balanced by that removed during the motion towards the sun. In case e^^O, the changes in eccentricity and argument of the perigee can also be expressed in terms of Equations (2.11): 2 2 1/2 Ae = (p 0 0A p + q 0 0Aq)/e 0 0 = - 3 u e a Q 0 ( l - e Q 0) s i n ( n 0 0 - u ) Q 0 ) ; Aa) = (p 0 0Aq-q 0 0Ap)/e 2 0 = 3^ e a 2 Q ( l - e 2 0 ) 1 / 2 c o s ( n 0 0 - ^ ^ / e ^ . (2.12) It is evident that these first-order solutions represent a valid approximation only for a limited duration as the elements tend to move away from their reference values with the passage of time. Eventually, the solution becomes unreliable since i t is unable to distinguish long-periodic from truly secular trends. In the, follow-ing sections, a few approaches for obtaining long-term approximate solutions are studied. 2.3.2 Rectification/iteration procedure The short-term solutions obtained in the previous subsection can be employed in a scheme to extend the interval of validity of these solutions. Thereto, a certain interval over which the first-order straightforward perturbation solutions provide sufficiently accurate approximations is selected, say (0, v^). For convenience, but not out of necessity, is usually taken as 2TT . At v = , the f i r s t -order changes in the elements are added to the i n i t i a l values, i.e., rectification of the i n i t i a l conditions: ^ V r e c t . = ^ 0 + e i l ( v f > • Subsequently, the adjusted value a . ( v f ) r e c t is treated as the i n i t i a l condition for the next interval, say (v^, 2v^), and again the f i r s t -order changes in a^ (v) at v = 2v^ are calculated and the elements are updated. All elements as well as the solar aspect angle are treated in this manner and the procedure can be repeated as often as needed. Eventually, however, neglected second-order influences will affect the desired accuracy adversely. Mathematically, the procedure is described as follows: the system of differential equations a_' (v) = ef(a_,v) and ajO) = a^ Q is written in integral form: a(v) = ^ + e T f [a(x),T] dx . (2. J0 35 Application of the first-order straightforward perturbation expansion proposed in Equations (2.9) over the interval [kv f,(k+1)v f], k = 0,1,2,---, leads to the result: a-,[(k + l ) v f ] (k+l)v f f_[ajkvp) ,T] dx (2.14) kv. Thus, the rectification/iteration procedure can be interpreted as replac-ing the integral in Equation (2.13) at v = Nvf by the modified Riemann sum: N-1 (k+l)v f T a(Nvf) = a ^ + e I k=0 kv f f[a(kv f),x] dx j=l (2.15) Successive calculation of a^(jVf) , j = l,2,---,N by means of Equation (2.14) leads to a piecewise constant approximation for the slowly varying elements. The accuracy of the approximation depends on the choice of the number of intervals N or the length of the interval . By taking N sufficiently large or sufficiently small, any desired accuracy can be attained. In fact, in the limit N + °° (or v^^-0) , the approximation becomes the exact solution. Consequently, accuracies exceeding those obtained by second and higher-order expansions without rectification can be attained by simply choosing a sufficiently small 126 interval before rectification of the first-order results (Lubowe ). Apart from providing physical insight through interpretation of the first-order results, the rectification/iteration procedure is perfectly suited for execution by a digital computer at a considerable saving in cost and effort as compared to a numerical integration of the original system of equations. 2.4 Two-Variable Expansion Procedure A relatively recent, but extremely popular method for establish ing long-term valid asymptotic representations for the solutions of a 87-93 set of differential equations is the two-variable expansion method It involves the introduction of a so-called slow variable which is to be treated as distinctly independent of the regular independent variable, transforming ordinary into partial differential equations in the two independent variables. The solution of this transformed problem will contain certain indeterminate functions of the slow variable to be ascertained by postulating the mathematical constraint that the problem possesses a consistent asymptotic expansion uniformly valid for times of the order of the reciprocal of the small parameter. Physically, the imposed constraint may be interpreted as the elimination of secular terms. Formally, the orbital elements (including the solar aspect angle) are expanded in asymptotic series: 37 N-l a(v) = I ej z. (v,v) +0(eN) , (2.16) j=0 with the slow variable v defined by v=ev. Substituting these series into the perturbation equations a_'(v) = ef(a,v) and collecting terms of like order in e yields 3CIQ/3V = 0 for the zeroth-order elements so that aQ = aQ(v) with a^(0) = a^ Q . The unknown slowly varying functions a^ (v) will be determined by requiring that the first-order contributions a_-|(v,v) remain bounded as a function of v (elimination of secular terms). This condition is equivalent (at least in the problems considered here) to the mathematical constraint mentioned before. The first-order equations are of the following general form: 3a-, da~ — 1 = ~ — + L [io(v),v] , a,(0) = 0 ; (2.17) 3v dv ^ 1 with the functions f_ periodic in the variable v . A convenient way of separating the terms leading to unbounded contributions from those producing bounded results is by expanding the right-hand-side of Equation (2.17) in terms of Fourier series with slowly varying coefficients, 3i] da-0 r 2 T r 3v dv + 1 f [ ^ ( V J . T ] dx/ (2rr) + I |AJ ( 3 ^ ) 0 0 5 jv 0 j = l + BJ (a n)sin jv \ . (2.18) 38 It should be realized that the slow variable v is treated as independent of v during the integrations (cf. the method of averaging where the slowly changing mean variables are considered constants during integra-tion). The vector functions A J [ a i Q ( v ) ] and B_ J[a^ ( v ) ] can be evaluated explicitly in an obvious and straightforward manner in terms of the Fourier coefficients a ^ , bj^, c ^ and d ^ , Appendix II. From Equati on (2.18) i t is apparent that a_^(v,v) will be bounded (in fact, periodic) as a function of v i f the following relation for a ^ ( v ) is satisfied: dv f [ a ^ v ) ^ ] dx/(2Tr) , (2.19) meaning that the slow rate of change of a^v) must equal the averaged (over one revolution) value of the right-hand-side of the perturbation equations. The similarity of the zeroth-order two-variable results, Equation (2.19) with those from first-order averaging is quite apparent: in fact, the equation obtained from first-order averaging is identical to Equation (2.19). It is interesting to compare the zeroth-order terms obtained by the two-variable method with the results from rectification/iteration, written as (2rr) - ^ ( 0 ) ] / (2ir) 2TT f [ a , N , T ] dx/(27r) (2.20) 0 Comparing the expressions in Equations (2.19) and (2.20), one can interpret a^(v) in terms of the rectification procedure as portraying a continual rectification (i.e., interval before rectification is infinitesimal) of the first-order results while the periodic dependence of f upon v has been eliminated by averaging. (Note that the left-hand-side of Equations (2.19) and (2.20) may be inter-preted as a differential and difference quotient, respectively.) Conse quently, the function JLQ(V) will generally be a better approximation to the exact solution than the results obtained by repeated rectification of the first-order straightforward perturbations when the interval of rectification is = 2TT . However, in order to improve upon a certain accuracy, one needs to solve for the higher-order equations in _a-|(v,v) etc., in case of the two-variable expan-sion procedure, while the accuracy of the rectification/iteration method can be enhanced by simply choosing a smaller interval before rectification of the first-order straightforward perturbation results. The first-order solutions a_^(v,v) may be obtained immediately by integration of the remainder of Equation (2.18), yielding: oo a-|(v,v) = I (1/j) | A ^ a ^ s i n jv - B^a^cos jv j + a^ (v) , j=l (2 where the as yet unknown functions jL-|(v) must be determined from a con-straint (similar as the one upon a,) upon the behavior of a~(v,v). 40 The second-order equations can be obtained from a/ = ef_(a_,v) by means of a Taylor expansion of f_(£,v) around a. ^ JLQ , leading to 9a^ 2 3a_-| 3\T = " 9\T + 3f. • a^v.v) ; a 2(0) = 0 - = iO (2.22) Again, a Fourier series expansion of the right-hand-side is used for the separation of the bounded and unbounded contributions and differential equations for a_-|(v) are obtained when requiring that a_2 be bounded as a function of v . This process can be continued for higher orders, i f desired, but usually the contributions beyond the first-order can not be expressed in analytical form. Therefore, a sensible policy would consist of attempting to solve for the lower-order two-variable results and, i f unsuccessful, or in case a better accuracy is needed, employing the rectification/iteration procedure with a sufficiently small interval v f . Note that v f must be smaller than 2TT i f the accuracy of the zeroth-order two-variable terms is to be exceeded. A fortunate consequence of the similarities of the expressions in Equations (2.19) and (2.20) is that i t allows us to write down, automatically, the zeroth-order two-variable equations, once the first-order straightforward solutions at v = 2TT are known (and vice versa). 41 2.4.1 Long-term valid results, case e =0(6) In this section the two-variable expansion method will be applied to obtain long-term valid approximations for the orbital elements. First, the case where the solar parameter is of the same order of magni-tude as the frequency parameter of the sun in the ecl i p t i c plane is considered. Applying the resulting expression of Equation (2.19), yields the following zeroth-order equations: D P0 _ „2 d v d v £Q | c o s n 0 [p QB 3 1(2^) + B32(2TT)/2] - s i n n Q [A2Q(2TT) + P 0A 3 1(2^) + A30(2TT)/2 + A32(2TT)/2] }/(2TT) ; D Q0 _ n2 £J j cosn 0 [A20(2TT) + q QB 3 1(2^) + A3Q(2TT)/2 A 3 2(2^)/2] - s i n n 0 [ q 0 A 3 1 ( 2 T r ) + B 3 2 ( 2 T T ) / 2 ] /(2IT) ; dv £ Q j B 3 1 (2TT)COS PQ - A 3 1 (2fT)sin nQ \ I TT d n . d v c (1 - PQ " %)3/2 A 2 0 C2ir)/(2ir) ; (2.23) 42 with i n i t i a l conditions a^O) = a ^ . The similarity in the structure of Equations (2.23) and the short-term results of Equations (2.10) with v = 2TT is evident indeed, as explained in the previous subsection. The integrals A^UTT) and B^{2T\) now contain the slowly varying zeroth-order elements PQ(V), qQ(v), etc. Upon calculation and sub-stitution of the integrals in Equations (2.23), a coupled nonlinear system of differential equations is obtained: 3 2 2 1/2 p Q(v) = - ^  a Q 0 (1 - e Q) sin nQ 3 2/-, 2x1/2 q 0(v) = 2 a00 ^ " V C 0 S n0 3 2 1/2 £Q(V) = 3 a Q 0 (1 - e Q) [p Q sin nQ - q Q cos nQ] nQ(v) = c (2.24) 2 2 2 where equals p^ + q^ . It is seen that ng(v) = HQQ+cv , denoting that the long-term behavior of the solar aspect angle is a Tinear function of the slow variable v in the zeroth-order approximation. Also i t follows readily from Equations (2.24) that ag(v) = (write a0 = &Q/O - en,))' s o t n a t t n e J° r a x i s and total energy remain con-served in the long run in this approximation. Another integral can be derived quite readily from the system of Equations (2.24): [1 - 6 Q ( V ) ] 1 / 2 + 3aQ 0y Q (v) / 2 = X = constant ( 2 . 2 5 ) 43 Introducing auxiliary orbital elements x and y defined by r \ X sinri -cos n cos n sinn V = e sin(n - to)" cos(r) - O J ) ^ (2.26) so that x 2 + y 2 = p 2 + q 2 = e 2 , i t follows from Equations (2.24) that x (v) and y Q(v) satisfy the following set of equations xQ' (v) + bL x Q(v) = 0 , y 0 ^ + c x 0 ^ = 0 ' (2.27) with i n i t i a l conditions x Q = x Q 0 , x'(0) = c y 0 0 - 3 a 2 0 ( l - e 2 Q ) 1 / 2 / 2 and YQ(0) = y Q 0 . The constants x Q 0 and y Q 0 can be expressed in terms of the usual orbital elements according to Equation (2.26). The solutions XQ(V) and y Q(v) can readily be determined from Equations (2.27) x Q(v).= (b 2 - A 2 ) 1 / 2 sin (bv + a ) / b , y 0(v) '= [ c ( b 2 - A 2 ) 1 / 2 cos (bv + a ) + 3a 2 0A/2] / b 2 , (2.28) The elements P Q(v) and q Q(v) become P 0(v) = (b 2 - X 2 ) 1 / 2 [ s i n ( b v + a p ) s i n n Q + c cos(bv+ a p)cos ri 0/b ]/b + 3 A a 2 Q cosn Q/(2b 2) 44 q n(v) = (b 2 - X 2) 1 /' 2[ ccos(bv + a n) sin nn/b - sin(bv+a ,) * *cosn Q]/b - i 3 A d 0 0 bin n0/(2b") . (2.29) The conventional elements e^, £g and ojg can be determined from the results of Equations (2.28) and (2.29) . e Q(v) = { l - [cA - | a 2 0 ( b 2 - A 2 ) 1 / 2 c o s ( b v + a p ) ] 2 / b 4 ^ 1 / 2 £g(v) = a Q 0 [ c A - | a 2 Q ( b 2 - A 2) 1 / 2cos(bv + a p ) ] 2 / b 4 , w0 (v) = n 0(v) -arcsin { (b2 - A 2 ) 1 / 2 s i n ( b v + a p)/[be Q(v)] .....(2.30) The result for ojg(v) is meaningful only i f eg(v) does not vanish. If egg is small i t is recommended to calculate the argument of the perigee from the relation w0(v) = arctan [q Q(v) / p 0(v)] If A < c , the argument of the arcsin function can be shown to pass through one and the arcsin function to increase continuously. In case A > c , the argument remains less than one and the arcsin function 45 keeps on oscillating between slowly changing upper and lower bounds. Physically, the two cases correspond to the major axis oscillating around its i n i t i a l orientation or following the motion of the sun, respectively. After the determination of the zeroth-order results, the f i r s t -order terms can be obtained by explicit calculation of the Fourier co-efficients A J(a^) and B^a^) of Equation (2.21) for the present case. It follows that: P^v.v) = £ Q co sn 0 I (1/j) | [ P 0 b 3 i + b32 / 2^ s i n j v + [ p0 d31 + d^ 2/2](l - cos jv) j - £ 2 s i n n 0 . H V J ) J E a ^ + P 0 a31 + a30 / 2 + a32 7 2 ] S i n j V + [ C20 + P0C31 + C 3 0 / 2 + c^/2 ] (1 - cos jv) | + p^v) ; q^v.v) = £ Q co sn 0 I (1/j) j [a^ 0 + q Q b^ +a^Q/2 - a^2/2 ] sin jv j=l + [c^ 0 + q Q d^ +c33Q/2 - c^/2 ] (1 - cos jv) 46 j=l £0s i n n 0 E C/J) -j [q na^ n + b^9/2] sin jv (T31 u32' + [ q 0 c ^ + d^2/2](l - cos jv) \ + q ] (v) • a i ( v ^ } = 2 a o o / ( 1 " e o } I O/j) cos n Q | [q Q a^ Q + ] sin jv + [ q 0 c 2 0 + d21 ] ( 1 " c o s j v ) } " s i n n 0 { C p 0 a 2 0 + a ^ ] sin jv+ [p Q cjjQ + c ^ ] ( l - cos jv) j + a^v) ; n-| (v,v) c(l - e 2 ) 3 / 2 I (1/j) j=l a ^ sin jv + c^ Q ( l - cos jv) + n-, (v) (2.31) The Fourier coefficients a J. , bJ, , etc. are defined and calculated in nk ' nk ' Appendix II and are functions of p Q, q Q, etc. The unknown 'slow' functions p-j(v), q-|(v), etc. are to be determined from the boundedness cons t ra in t imposed upon the second-order terms and vanish for v = 0. For small e^, the Four ier c o e f f i c i e n t s are proport ional to ( _ e g ) J a n c l converge very rap id l y so that usual ly only the f i r s t few terms need to be c a r r i e d . The per iod ic terms in Equations (2.31) stay with in a band of a width of order e around the long-term zeroth-order so lut ions a^(v) . The secular contr ibut ions of a_-|(v) in Equations (2.31) are of order e f o r v up to order 1/e, i . e . , up to about 800 days fo r e = 0.0002. 2.4.2 Long-term va l i d r e s u l t s , case e =0 (6 ) In the case that the so lar parameter e i s of comparable magni-2 tude with 6 , a s im i l a r ana lys i s as in the previous subsection can be fol lowed when the slow var iab le i s taken as v = 6v and the elements are expanded in ser ies of powers of 6 rather than e . The zeroth-order 3/2 ~ re su l t s become P 0 = P 0 0> ^ o ^ O O ' £0 = £00 a n d n0^ v^ = a00 v + n 0 0 ' w h i l e the f i r s t - o r d e r re su l t s are ~ 3 2 2 1/2 P-|(v) =' 2 a 0 0 ^ " e 0 0 ^ [cos nQ - cos n0Q] / c ] , q i ( v ) = | a 2 Q ( l - e ^ ) 1 ' 2 [ s i n n o - s i n n n o ] / ^ , ^ ( v ) =-3a 0 3 0 ( l , - e 2 Q Q ) y 2 [ p Q 0 ( c o s n 0 - c o s n 0 0 ) q 0 0 ( s i n nQ - s in n 0 0 ) ] / c i 48 n^v) = l^2 I (l/j) j a ^ s i n jv + c ^ O -cos jv) J , ( 2 . 3 2 ) j=l with a^Q and C^Q dependent on p ^ , q , etc. In the usual manner, expressions for e-j(v) and O J ^ ( V ) can be written down, while for small e^Q, to becomes indeterminate and the argument of the perigee needs to be found from p-j and q-j : to-|(v) = arctan q Q 0 + 6q 1(v) + 0(6 2) P 0 0 + 6p-,(v) + 0(6 2) ( 2 . 3 3 ) From this relation and Equations ( 2 . 3 2 ) , i t follows that for small v and S Q Q ^ O : C O ^ ( V ) = PQQ + T T / 2 , reaffirming the well-known fact that for an i n i t i a l l y circular Orbit, the perigee will appear 9 0 ° ahead of the sun-earth line. 2 . 5 Discussion of Results To assess the validity of the approximate approaches de-veloped in the previous sections, the results are compared with those from a numerical integration of the perturbation equations. The parameters involved were taken corresponding to situations of practical interest: an Echo-type sat e l l i t e and the SSPS representing 2 the case e =0(6) and the CTS illustrating the situation e=o(S ). The i n i t i a l orbital geometry and solar aspect angle were varied system-at i c a l l y . 2.5.1 Case e = 0(6) From the results derived in Section 2.4.1, i t is apparent that e n(v) and &n(v) change periodically with period In any case, this period is less than one year and smaller the e , the closer i t approaches one year, which is indeed the period in the case e = 0(6 ) as found in Section 2.4.2. Also, the period increases with decreasing a ^ (Figure 2-2). For example, taking aQ  = 1 and 2 e = 0.0002, i.e., A/m = 10m /kg and a = 0.5,'which is the case for an SSPS with p = K = T = 0 o r a spherical sat e l l i t e with p = 1. and x = 0, the resulting period of the long-term perturbations is approximately 363 days. Equations (2.30). Limiting A to the physically meaningful domain 3 2 -p a n n < X < b , the eccentricity e n(v) lies between a 3 7 2 / (be) = 1 / ( 6 2 + 9 e 2a n n/4) days (2. The extrema of e n(v) and £n(v) can be determined from 0,max = [3Xa2 II + c ( b 2 - A 2 ) 1 / 2 ] / b 2 at v 51 eO,min 3A a20 / 2 - c ( b 2 - A 2 ) 1 / 2 /b 2 , at v = [(2n + l) i r - a ] / (be) , (2.35) with n = 0 ,1 ,2,'" and v > 0 . It is seen that for A = b', the minimum and maximum values are the same. Hence, eg and £g remain constant: eg(v) = egg and £g (v ) = £gg . If A = c, the trajectory will become circular at some point as e n . =0 . r 0,min Figure 2-3 shows the accuracy of the zeroth-order two-variable solution and the rectification/iteration (with interval = 2TT) results in comparison with a double precision Runge-Kutta integration routine. The approximate results proved to be quite effective and their com-parison is purposely limited to one case: egg = 0.5 and rigg = TT . The two-variable expansion procedure predicts the eccentricity correctly to three decimal places, while the rectification/iteration method yields results correct to two places. The comparisons were made at v = 2Trn , n= 1,2,«••,1200 . It should be noted that the first-order changes in time are incorporated in the rectification/iteration procedure, whereas time is taken proportional to v in the zeroth-order two-variable expansion results. As can be expected, the value of the i n i t i a l solar aspect angle has no influence on the resulting behavior of the eccentricity when egg = 0, curve (d). The fluctuations in eg (v ) can be as large as 0.2, curves (b) and (d). However, a suit-able combination of i n i t i a l parameters may also result in very small perturbations as indicated by curve (c). In fact, in the 52 e Days Figure 2-3 Long-term variations in eccentricity as predicted by the three methods 53 limiting case of DQQ = 0 and = 0.109 the variations in eg(v) disappeared completely (case A = b). Figure 2-4 shows the predictions of the approximate methods as to the behavior of the semi-latus rectum and argument of the perigee for A<c . In case A>c , the precession of the major axis is described by a large linear secular variation with a small amplitude periodic motion superimposed on i t as shown in Figure 2-5. For A = c , the argument of perigee shows periodic discontinuities with a jump through 180°. Note also that in the case A = b (i.e., e Qg = 0.109 and ngg=0 here), the periodic component disappears completely leaving only the linear variation: the major axis keeps on pointing towards the sun, while the eccentricity remains constant. In the case A<c , the axis oscillates between slowly moving upper and lower bounds. Figure 2-6 shows the very small long-periodic variations in the semi-major axis and the osculating periods for a few values of the i n i t i a l solar aspect angle obtained from the numerical integration routine. Note that the analytical methods predict that a(v) remains con-stant in the first-order, so that the variations depicted here are second-order effects. The orbital elements affected most severely by solar radiation forces are eccentricity, semi-latus rectum and argument of the perigee. Since the semi-major axis is not affected in the f i r s t -order, the changes in semi-latus rectum can be expressed in terms of those of the osculating eccentricity. Complete visualization of the first-order changes in orbital geometry is thus provided by the two 54 pert. + rect. - t w o var.exp. £=0.0002 , a o o=1 , n 0 o=n .^oo = 0. Days Figure 2-4 Long-term behavior of semi-latus rectum and argument of the perigee Days Figure 2-5 Secular variation of the argument of the perigee for c < A < b : P p 1 • co0 0 =0 , E = . 0 0 0 2 , a o o = i , e 0 0 = 0 . 1 Days Figure 2-6 Long-term variations in the semi-major axis and orbital period CTl 57 elements e and co (or p and q). Complete comprehension of the nature of the orbital perturbations could be obtained from plots showing the long-term behavior of e and co for various i n i t i a l conditions and solar aspect angles. One attractive possibility is depicting e and co as a polar plot in the p,q-plane with e the length and co the argument of the eccentricity vector e_. It can be observed from Equations (2.24) that the slope of the polar plot as a function of v is determined from ^ (v) = tan (6 a 3 / 2 v + n Q 0 + TT/2) . (2.36) d Po Considering, for ill u s t r a t i o n , an orbit with i n i t i a l l y OJQQ = 0 (so that the polar plot starts out from the q = 0 axis) i t follows that i n i t i a l l y the angle at which the tangent to the curve q Q = q0(P()) i s inclined to the p axis equals n Qg + TT/2. A S V advances the tangent rotates slowly in an anti-clockwise manner. At v = 2TT/(CE;) , i.e. after slightly less than one year, the tangent returns to its original value indicating that the polar plot describes anti-clockwise loops in the p,q-plane, Figures 2-7a-d. This type of plots allows an easy visualization of the orientation of the major axis as well as the eccentricity of the orbit over a long duration. The i n i t i a l configuration is best characterized by the parameter X as defined by the i n i t i a l orbital elements and solar 2 aspect angle. It can be shown that c > 3a n n/2 provided that 26/(3agQ ), which covers a l l practical cases. The physically use-ful range of A is limited to 3agg/2 <_ A _< b . It is informative to study a few special cases: (a) A = c : This case defines the locus of i n i t i a l conditions for which the ensuing trajectory w i l l have a circular osculating orbit at some time within one year: the corresponding polar plots pass through the origin p = q = 0 . Figure 2-7a presents a few examples belonging 2 2 to this class. For any 6QQ Ji 3C8QQ / b (=0.220 in the example), an appropriate value HQQ can be found so that the resulting curve goes through the origin. It can be seen that the argument of the perigee jumps through 180° at the origin. (b) 33^^/2 < A < c : Here, the polar plots do not pass through nor encircle the origin and the eccentricity oscillates between the values e0 min a n d e0 max determined in Equations (2.35). The curves in Figures 2-7b qualitatively indicate the behavior for = 0.5. While going around in the anti-clockwise manner, a slow precession in the clockwise sense is superimposed on the motion and the result is a trajectory describing loops between the two concentric circles of radii e n . and e n . As pointed out before, the period of os c i l -0,rmn 0,max r r lation in eccentricity is close to but less than one year. Interest-ing is the behavior of argument of the perigee, co : after one complete cycle of eg(v), co has decreased by -2TT(1 -c/b), amounting to a precession of -2.14° per year in the example. As the factor 1 - c/b 0.2 q 0.1 o -0.1 -0.2 -0.3 eoo=0.109 -0.1 (c) 0 0.1 - n0=o • \o=% € = 0 . 0 0 0 ? 0.2 -0.1 eoo=0.05 ^oo=0 c < A < b (d) 0 Figure 2-7 Polar plots, showing long-term behavior of eccentricity vector e: (c) case c < A < b; '00 0.109; (d) case A = b; '00 0.05 ve =0.167 0.2 q 0.1 0 -0.1 -0.2 -0.3 0.1 0.2 cn O 61 increases with increasing e , the precession will be faster for larger e . Note also that the periodic variations in co become smaller for increasing e^: physically, the major axis is more 'rigid' for larger e00' (c) A - 3aQQ / 2 : This case represents the locus of i n i t i a l conditions which eventually yield a parabolic trajectory (e = l ) . However, since the semi-major remains constant in the first-order, e =1 would imply that the perigee coincides with the center of attraction. Obviously, the l i f e of the sat e l l i t e would end long before e= 1 is reached. (In fact, for 9QQ = 1» e = 0.84 will be the maximum physically meaningful eccentricity). The minimum eccentricity to reach an escape trajectory is e0,min = ( c 2 - 9 a 0 0 / 4 ) / b 2 ' f ° r n 0 0 = T r . In the example, e ^ ^ = 0.976, 2 hence the locus A = 38QQ/2 is not attainable. (d) c < X < b : For these values of X the variation of co is pre-dominantly linear in character, increasing continuously while the curves in the polar plot ci r c l e around the origin, Figure 2-7d. From the definition of X , the criterion for encirclement of the origin (c < A) can be expressed in terms of the i n i t i a l conditions: 2 2 2 2 e Q 0 < 3a Q Q c cos(n 0 0 - coQO)/[c + 9a Q 0 cos (n Q 0 - w 0 0)/4] (2.37) Note that for 90° < PQQ - wQ 0 _< 270° the loci can not enclose the origin. 62 (e) X = b : This interesting case represents only one possible i n i t i a l configuration, namely, e^Q = 3aQQ/(2b), i.e. 0.109 in the example, and NQQ=WQQ . The resulting eccentricity does not change at a l l , i.e. eg (v ) = throughout while OJQ(V)= OQ(V) , so that the major axis keeps on pointing towards the sun and the shape of the orbit re-mains unchanged. The corresponding polar plot consists of a circ l e of radius 3aQQ/(2b) around the origin, Figure 2-7c. The case is interesting since a large region in space can be traversed by a sat e l l i t e satisfying X = b without altering the shape of the orbit. The annular region is contained by the two concentric circles whose radii are the perigee and apogee heights 9QQ(1 -GQQ) and SQQ(1 + eoo)> respectively. In the example, the distance between the circles amounts to more than 9,200 km. The actual orbit based on the result, r Q ( v , v ) = lQ{\>) I [1 + p Q ( v)cos v + q Q ( v ) sin v ] , is depicted, for a typical case, in Figure 2-8 i11ustrating "the differ-ences in the osculating ellipses at 90 day intervals: the wide band of spatial region reached by the sat e l l i t e is quite apparent. This is significant in designing a mission aimed at scie n t i f i c measurements over a vast area in space. Since the short-term solutions are also known (Section 2.3.1): r(v) = r n ( v ) + ery(v) , 63 3 o o = 1 e 0o = 0.1 € =0.0002 •oo / ' '2 day 0 - . d a y go day 180 day 270 1.2 0.8 r sinv 0.4 0 -0 .4 - 0 . 8 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 r c o s v Figure 2-8 Long-term orbital behavior showing traversing of spatial region -1.2 with r-j(v) readily expressed in terms of £-|, p-j, and q-j by expansion, i t is interesting to show the actual path of the sate l l i t e during its f i r s t revolution. This has been done in Figure 2-9 for an exaggerated 2 A/m ratio of 1,000 m /kg : note that the point of minimum distance to earth occurs about 90° ahead of the sun-earth line. Finally, loci of A = constant are plotted in the e^, TIQQ plane (Figure 2-10). They can be used to advantage in assessing the bounds of eccentricity as e n and e n . depend on A only for given J 0,max 0,mm r J 3 e and a^ (Equations 2.35). Obviously, the lowest and highest values of e^g on each curve correspond to the limiting values m f l x and eg m i- n belonging to that locus. Thus, for any combination of i n i t i a l eccentricity and solar aspect angle, the corresponding extremes of eccentricity of the ensuing trajectory can be assessed immediately. This should prove useful during the preliminary planning of a mission as i t provides a convenient way of determining whether a given sate-l l i t e will dip into the free molecular environment or not. As can be expected, the point A = b moves to the right for larger e with corresponding increase in fluctuations of the eccentricity. The area designated by A > c corresponds with polar plots encircling the origin and the locus A = c denotes i n i t i a l conditions with polar plots passing through the origin. 65 r cos v Figure 2-9 Actual path of sate!1ite during f i r s t revolution (for exaggerated solar parameter, e =0.02) Figure 2-10 Loci of i n i t i a l conditions, A = constant, leading to the same extrema of eccentricity 67 2.5.2 Case e = 0(5 2) It is apparent here that e-j ( v ) is periodic with a period of exactly one year and the range of eccentricity is given by el,max " el,min 3 e ^ 2 / 6 regardless of i n i t i a l solar aspect angle. For a satellite with the 6 parameters of CTS (i.e. e = 1.4 x 10" , c-j = 5.4) and agg = 1, the _3 variation in eccentricity is 1.50x10 . In terms of perigee distance, this result translates to a maximum fluctuation of some 63km in six months. In Figure 2-11, the polar plots for egg - 0 and 0.1 are shown for a few.values of n.QQ. The slope of the polar plots turns out to be exactly the same as in the previous section so that the influence of i n i t i a l solar aspect angle presented qualitatively in Figures 2-7 remains valid for this case. The periodic variation in co is over _2 180° in Figure 2-1 la but is reduced to the order 10 in Figure 2-lib where egg = 0.1 . A polar plot in the form of a c i r c l e around the origin as in the previous section, can also be found here, namely for ? 7 4 2 1 / 2 ? e Qg = 3Sagg/(4c^.+ 9agQ6 ) " , i.e. 0.75x10 in the CTS example, and Hgg = 0. In these circumstances, the major axis follows the motion of the sun. Interestingly, the formula for egg found here is identical to the expression of the previous section. If e QQ<6Sag 0/(4c 2+gajgfi 2)^ 2, i.e. 1.50xlO - 3 here, the plots enclose the origin or pass through i t provided appropriate values for the solar aspect angle are taken. For larger than this value, the orbit will always remain e l l i p t i c . In conclusion, the results here are qualitatively consistent with those of the previous section with two differences: (i) absence of the slow clockwise precession of the polar plot; ( i i ) amplitude of variation in eccentricity does not depend on i n i t i a l solar aspect angle. 2.6 Evaluation of the Shadow Effects The existence of a shadow region, making the solar radiation forces vanish whenever the sun as seen from the sa t e l l i t e is eclipsed by the earth, presents a major obstacle for obtaining rea l i s t i c long-term solutions for the orbital elements. It is generally assumed that the umbra and penumbra regions may well be replaced by an equivalent simple circular cylinder of radius Rg and axis along the sun-earth line. The space within this cylinder is taken to be completely dark with an abrupt transition to f u l l illumination out-side the shadow region. The points of entry and exit of the shadow cylinder satisfy a quartic equation in terms of the cosine of the true anomaly. In general, its solution is too unwieldy for practical use and numerical methods (e.g., successive substitution) is to be preferred. A few special cases exist, however, where the points of entry and exit appear in a more tractable form, e.g. when the 70 instantaneous orbit is circular or when the orbit lies in the ecl i p t i c plane. In the present section, the influence of the shadow upon both the short- and long-term results for the orbital elements derived in the previous sections for ecl i p t i c orbits, are determined. An interesting approximate relationship linking the small long-term variations in the semi-major axis to the behavior of the auxiliary element y = ecos (n- w) is established. While these analytical results match quite well with those found by repeated short-interval rectification and iteration over the f i r s t 100 revolutions, relatively large discrepancies arise for longer spans of time. These must be attributed to second-order effects. As to the behavior of the eccentricity, i t is found that the no-shadow results are correct to at least two decimals over the f i r s t year. 2.6.1 Short-term shadow effects In case of a prograde* orbit in the ecl i p t i c plane, the points of entry (v-|) and exit (v 2) of the shadow cylinder are determined by the equations, r(v-j) sin (v-j - p) = Rg , n + TT/2 < v-j < n + TT , r (v 2 ) sin (v 2 - n) = -R , n + TT < v^ < n + 3TT/2 . (2.38) * Prograde means that the motion of the satellite is in the same direction as that of the sun with respect to the earth. 71 Introduction of the shadow angles 3-j and 3 2 (Figure 2-1) through B.| = n + TT - v.j. and 3 2 = v2 ~ (f| + T T ) ' a n c' substitution of these angles into Equation (2.38) leads to two quadratic equations from which 3-j and 3 2 can be expressed in terms of x = n - to and s = R g/£ : 3-| 2 = arcsin { 2 2 2 1/2 s ( l + e s s i n x ) - e s c o s x ( l - s ± 2es sin x + e s ) , . - . . 2 2 ? ? 1 ± 2es sin x + e s (2.39) Note that for a circular orbit, this result simplifies to 3-j = 3 2 = arcsin s and v-j 2 = n + IT + arcsin s . A first-order approximation for the changes in the orbital elements can be obtained by integration of the perturbation equations as in Section 2.3.1, while excluding the contributions over the shadow interval I . Only the resulting change in semi-major axis can be expressed in a tractable form in case of an orbit of arbitrary eccentricity: Aa = 2ea2£ | (1 - s 2 + 2es sinx + e 2 s 2 ) 1 / 2 - (1 - s2-2es sin- x + e 2 s 2 ) 1 / 2 + e 2s sin 2 X j / (1 - e 2cos 2 x) , (2.40) where the integral over the fu l l cycle (0,2TT) vanishes. The result indicates that the major axis remains unaffected when the sun lie s on the major axis or the orbit is circular. Note that the change in the major axis is maximal for x = ± Tr/2 > i.e. when the radiation 72 is normal to the axis. For instance, taking e = 0.1, a = l and e = 0.0002, _5 i t follows that ^ a m x equals 1.2x10 amounting to 0.5 km and a change in the period of about 1.5 sec per revolution. In general, the major axis increases (decreases) i f the point of entry is farther (closer) to the sun than the point of exit, which is evident from physical conside-rations. For small eccentricity, the results for the orbital elements p and q can be written explicitly: Ap = -ea 2 sin n j 3TT - C g - 2es[(l - s 2) cos x + s 2 sin x/tan n] + o(e2) j , 2 f 2 2 2 1 Aq = ea cos r\ < 3u - C g - 2es[(l - s ) cos x - s sin x tan n] + 0(e ) > , Aa = 4ea3es sinx | 1/(1 - s 2 ) 1 / 2 + e c o s x + 0(e 2) | , (2.41) where the expansion of Equation (2.40) is also added. Note that the contribution represented by the factor 3TT originates from the integration over the f u l l cycle and the abbreviation C s stands for 2 1/2 o arcsin (s) - s(l - s T • In case e / 0, the changes in eccentric-ity and argument of the perigee are obtained quite readily from Equations (2.41): Ae = -ea 2 sin x j 3TT - C g - 2es cos x + 0(e 2) j , 2 f 2 2 eAw = ea cos x j 3TT - C - 2es [(1-s ) cos x + s sin x tan x ] + 0(e 2) 1 . (2.42) 73 It is interesting that for a circular orbit, the shadow effect upon the elements p, q, e and co can be accounted for by simply multiplying the 'no-shadow' results by a factor 1 -C s/(3TT) . For a geosynchronous orbit, this factor is approximately 0.97 so that the shadow effect reduces the 'no-shadow' perturbations in e and co by about three percent. 2.6.2 Long-term shadow effects The long-term implications of the shadow effects upon the orbital elements will be assessed both analytically (for near-circular orbits) and semi-analytically by numerical rectification and iteration of the short-term results. The interval before rectification i s , usually, taken as TT/2 and, for assessing the accuracy of the results, a few runs with an interval of TT/3 are performed. Since the short-term (i.e., within one revolution) perturbations in the semi-major axis could be larger than the net long-term changes, care must be taken for proper separation of the latter effects from the former ones. The elements at = 2 TT k, k = l,2,3,-*' are taken as representative for the long-term trend. The upcoming points of entry and exit of the shadow region are reassessed after each interval by substituting the most recent orbital elements into Equations (2.39). It is e s t i -mated that a rectification interval of TT/2 predicts the semi-major axis accurately to four decimal places and the elements e and co to at least two decimals uniformly over a 400 day time-span. These accura-cies were established by means of a comparison with results obtained by rectification after TT/3 radians. For near-circular orbits, i t is possible to describe the long-term evolution of the orbital elements analytically by means of the two-variable expansion procedure. Provided the i n i t i a l posi-tion of the sun is close to the perigee axis, the eccentricity w i l l not become much greater than i t s i n i t i a l value egg and a general 1/2 2 upper limit for e Q(v) may be taken eoo + 3 c e a n o ^ + 9 a 0 0 c e / / 4 ^ regardless of rigg . These results have been established in the previous sections disregarding the shadow effects. Presuming that this influence does not affect the order of magnitude of the perturbations in eccentricity, the aforementioned value may be used for assessing whether the eccentricity w i l l remain sufficiently small throughout or not. (This is mainly determined by egg and the parameter c £ = e/S). As in Section 2.4, the orbital elements (including n) are expanded in asymptotic series in v and v . While the zeroth-order results readily lead to a^ = a^(v), the first-order equations can be written symbolically as 9a_^  da^ j dv + F ( i n . v) 3n-| drig 9v dv + a 3 / 2 / [ c £ ( l +p 0 cos v + q Q s i n v ) 2 ] , (2 where F_ equals _£ except in the interval I s(v) where £ vanishes. In the present order of approximation, the shadow interval lies between v-j (SQ) = TT + n 0 - 31 {&Q) , and V ^ C I Q ) = TT + n Q + &2(^Q) • The slowly varying shadow angles 3-|(a^ ) and &2^^ a r e "identical in structure as in Equation (2.39) except for the fact that e, s and x are now al l dependent upon v . The vector-function £(a^,v) , though discontinuous, is 2iT-periodic in the fast variable v and can, in principle at least, be expanded in Fourier series with coefficients depending on the slow variable v . In practice, however, these series converge much slower than those for the corresponding continuous vector-function f_(.a_Q,v) discussed in Section 2.4. Nevertheless, represen-tative trends are illustrated by the zeroth-order solutions. The requirement that first-order terms a^ remain bounded in the variable v , leads to the following constraints (Equations 2.43), io(v) = j F [ a ^ v ) , T ] dx/ (2TT) , ( 2 . • 2TT-I s n Q ( v ) = a 3 / 2 (v) / c £ Performing the integrations, a set of coupled differential equations in terms of ag, PQ and qg is obtained. This system can readily be reduced to the following set of equations in ag, Xg = egSinxg and y 0 = e 0 c o s x 0 : 76 a Q(v) = 2R ea 2x 0/Tr [ a 0 + y 0 ( a 2 - R e 2 ) 1 / 2 ] / ( a 2 - R e 2 ) 1 / 2 + O(e 3) , 3/2 ? X'Q(V) = YQBQ' / c £ - 33Q [TT-arcsin(R e/a 0)]/(2Tr) -Re(ao " Re) 1 / 2/( 2^) + R e ( a 0 " ^ V ^ C ^ + 0 ( eO } ' y Q(v) = - x 0 a Q / 2 / c £ + R 3x 0 / ( T r a Q ) + 0(e 2) , (2.45) for uniformly small eccentricity. As mentioned before, the maximum eccentricity will be of the order c £ for a near-circular i n i t i a l orbit. 2 2 For consistency, terms of the order c^eg and c £ must be treated as eg . The following expression for a n(v) is obtained from Equations (2.45) when terms of the orders e^ , (R e/agg) 4 and higher are ignored: A ( ) W = a 0 0 - R e c £ a00 2 ^ ^ l o ^ O ^ - ^ (2-46) Utilizing this result, the equations for XQ and yg can be written as V o ^ ^ ^ ^ e = 3 a 0 0 2 / 2 " Re a00 2 ^ ( v ) ] ^ > x Q(v) = - y 0 ( v ) / c , .....(2.47) 2 3 where terms of order eg and (RQ/SQQ) have been neglected. These 77 equations are solved readily , y Q(v) = c £ a j / 2 [ 3 / 2 - R e / ( 7 r a 0 0 ) ] [ l - c o s l f i ^ ) ] + y Q 0 cos [fi-jv) - x Q 0 sin ( i^v) x 0 ( ^ = { y 0 0 " c e a00 2 [ 3 / 2 - R e / ^ a 0 0 ) ] } s i n ( V } + x Q 0 cos (fi-jv) , (2.48) showing that y Q and x Q are periodic with a slightly modified frequency as compared to the no-shadow case. The parameter fi-j stands for: fil = a00 2 ^ + R e C e / ^ a O 0 1 / 2 ^ 1 / 2 / C e " <2-49> It has been checked that the solutions of Equations (2.46) and (2.48) after substitution of R g=0 are identical to the expansions for small e Q of the long-term no-shadow solutions of Section 2.4.1. 2.6.3 Discussion of results The validity of the approximate long-term analytical solution has been assessed by comparing the results with those from repeated 78 rectification and iteration of the first-order short-term solutions. Figure 2-12 shows the comparison for a satellite with e=0.0002 in a geosynchronous orbit with i n i t i a l eccentricity of 0.1 and solar aspect angle HQQ - TT/2 over a 400 day time-span. The solid line represents the most accurate result obtained by rectification after not more than TT/3 radians where a l l orbital elements, the solar aspect angle as well as the next point of entry of the shadow region are reassessed. By taking larger intervals before rectification, the maximum discrepancy -4 in semi-major axis compared to the solid line is found to be 2x10 for an interval of TT and 5x10 for an interval of TT/2 (not shown) over a 400 day period. Also shown in Figure 2-12a is the result obtained by rectification after a f u l l revolution (2TT). The differ-ence between this and the aforementioned approximations is quite notable and must be attributed to the fact that second-order contri-butions are not picked up in this case. The importance of higher-order terms may be evaluated by considering the no-shadow situation, where in the first-order theory, the semi-major axis returns to i t s original value.after one complete cycle. Precise numerical inte-gration, however, reveals variations in the semi-major axis up to -3 an amplitude of almost 10 in the long run due to higher order influences (Figure 2-6). When the effect of the shadow is in-corporated in the analysis, the first-order changes in semi-major axis are caused by a difference in the distance of the points of entry and exit with respect to the sun. The change in semi-major 1.002 a .998 .996 two var iable expansion approx imat ions rect if icat ion , 6x per revolution , interval < TJ/3 2 1 < TT < 2 n 0 100 200 300 400 100 0 200 300 400 Days Figure 2-12 Comparison of the analytical long-term approximate solutions for the shadow effects upon: (a) semi-major axis; (b) eccentricity 80-axis over one revolution amounts to approximately A e e P ^ a ^ s i n x (Equations 2-41). Since Rg and e are small and e as well as sinxare often oscillatory in the long run, i t is not surprising that the total of the higher-order effects (enhanced by the addition of 'interrupted' periodic terms) can build up to and even exceed the magnitude of the first-order shadow effect. The dotted curve in Figure 2-12a represents the long-term approximate analytical solution a^iv) of Equation (2.46). Since only the first-order shadow effect is incorporated in this solution, i t is evident that i t is closer to the 2iT-rectification approximation than to the actual solution. Nevertheless, the analytical solution provides a reasonable prediction for the behavior of the semi-major axis over the f i r s t half year. The main objective in determining the perturbations of the semi-major axis is to evaluate changes in the orbital period which is of interest for assessing the d r i f t in the overhead position of the sa t e l l i t e . A change in the semi-major axis of 0.002 (after about 200 days) translates to a change in the orbital period of more than four minutes and a d r i f t in overhead position of 120 km per revolution at geo-synchronous altitude. Figure 2-12b shows the comparison for the eccentricity in the same circumstances. In contrast to the behavior of the semi-major axis, the eccentricity exhibits f a i r l y large perturbations in the first-order 'no-shadow theory' so that in comparison the shadow affects the resulting perturbations only in a minor way (due to the factor 1 - Cs/(3TT) in the short-term Equations 2.41). Comparing the results 81 with those obtained by neglecting the shadow effect, i t is found that, in the case of eccentricity, the influence of the shadow does not show up in the f i r s t two decimal places over a 400 day time-span. Nevertheless, its effect is more dominant than that of the higher-order terms in eccentricity which are not f e l t up to three decimal places over 1200 days. When studying the observed perigee distances and orbital 49 periods of the Echo I, Pageos and 1963-30D satellites, Meeus con-jectured the following rule: "The orbital period (and thus also the semi-major axis) diminishes when the orbit becomes more eccentric and increases when the eccentricity is decreasing." The results depicted in Figure 2-12 seem to obey this rule quite well. However, from Equations (2.46) a slightly modified rule can be formulated: "the changes in the major axis due to the shadow effect are proportional to the behavior of the slowly varying function -ecos(n- O J ) . " In the case where the major axis follows the sun's motion, which happens i f egg is sufficiently small and the i n i t i a l solar aspect angle is close to the perigee axis, n-w will be nearly constant and the two rules are consistent. 2.7 Concluding Remarks The important conclusions of the present chapter may be summarized as follows: (i) Considering a satellite in the ecli p t i c plane and taking the solar radiation force along the direction of radiation, both short and long-teirm valid approximations for the orbital elements are derived using a straightforward and a two-variable perturbation method, respectively. ( i i ) The two-variable expansion procedure is found convenient for deriving closed-form analytical results for the long-term orbital perturbations. The accuracy of the zeroth-order solutions compares favorably with those obtained by repeated rectification of the short-term solutions. Numerical results successfully assess their relative accuracies. ( i i i ) The results show that the variations in eccentricity and semi-latus rectum are periodic, while the argument of the perigee may show a secular trend in certain cases. The semi-major axis remains constant in the first-order. (iv) Polar plots provide an attractive and concise visualization of the long-term orbital perturbations. Loci of i n i t i a l conditions resulting in specified extremes of eccentricity should prove useful in preliminary mission studies. (v) The effect of the shadow both in the short-term and the long-term context has been assessed. It induces small first-order changes in the semi-major axis, while affecting 83 the already large variations in eccentricity only in a minor way. (vi) An analytical approximation for the long-term behavior of the major axis is derived for near-circular orbits using the two-variable expansion procedure. Unfortunately, its accuracy degenerates after about half a year due to the build-up of second-order effects. (vii) A simple rule linking the long-term perturbations in the semi-major axis to a function depending on eccentricity, solar aspect angle and argument of the perigee is esta-blished, which may be useful for estimating changes in the orbital period. This rule appears to be consistent with the observed satellite motion. 84 3. SOLAR RADIATION INDUCED PERTURBATIONS OF AN ARBITRARY GEOCENTRIC ORBIT The analysis of the previous chapter is now extended to satellites in an arbitrary orbital plane. Another generalization concerns the direction of the solar radiation force: whereas, up to now, this force was taken along the direction of the radiation, in later sections of this chapter, more general configurations are studied, e.g. spacecrafts modelled as a plate in an arbitrary, fixed orientation with respect to the local reference frame. Also the orbital behavior of a satellite in an arbitrary fixed orientation to the radiation or inertial space is explored. The latter situations are of considerable practical interest since they serve as accurate models for satellites with solar arrays (e.g., CTS and SSPS) and instrumentation for deep-space studies (e.g., orbiting telescope). Finally, the analysis is extended to an arbitrarily shaped satellite which may require a number of f l a t plates for accurate modelling. 3.1 Derivation of the Perturbation Equations A researcher in orbital mechanics finds himself surrounded by a multitude of procedures for analyzing perturbations of trajectories. Most of these methods originate with the great mathematicians of the last two centuries like Lagrange, Delaunay, Gauss and Hansen in their analyses of planetary motion. The 'space age' has produced many new and revised techniques for dealing with situations not pre-viously encountered, e.g. air drag. The choice of a particular formulation depends upon the specific nature and objective of the work, the perturbation forces involved and the availability of a digital computer as well as personal preferences. In the present case, a formulation is desired which is suitable for solar radiation forces, remains valid for a l l eccentricities and inclinations, is conducive to geometrical interpretation and, moreover, is capable of producing closed-form long-term solutions or short-term results f i t for rectification and iteration. Probably the most popular approach is the one based on Lagrange's planetary equations using an anomaly, referred to the osculating ellipse, as independent variable. These equations contain singularities for e = 0 and i =0, which can be removed by suitable transformations. Unfortunately, the equations rarely yield closed-form solutions for an orbit of arbitrary eccentricity due to the intricate coupling of the motion of the orbital plane (described by i and ti) and the in-plane perturbations (£,e,co). In his search for an effective algorithm for computing (manually!) planetary ephemerides, Hansen in the previous century employed a frame of 'ideal' coordinate axes fixed to the instantan-127 eous orbital plane . The in-plane equations of perturbed motion in this frame take on a form, identical to the equations for planar perturbations alone, thereby effecting an uncoupling of the motion in the osculating plane from the out-of-plane orbital changes. This approach retains some of the desirable features, like easy geometric visualization, inherent in the osculating elements. Furthermore, a uniquely qualified candidate to serve as independent variable emerges in a natural manner. In order to convey a physical appreciation for the qualitative effects of the components of the solar radiation force, a simple direct derivation of the perturbation equations based on Newton's second law is presented. These equations can also be obtained from Lagrange's planetary equations by introduction of new variables and algebraic manipulations. The motion of a sa t e l l i t e in the inertial X,Y,Z frame, Figure 3-1, under the influence of gravitational attraction of the primary (having radially symmetric mass distribution) and an arbitrary perturbation force £ can be described by Newton's second law (in nondimensional form): r + r / r 3 = F , (3 where the radius vector jr(t) denotes the position of the sat e l l i t e measured from the origin at the center of the primary. It is well-known that in absence of perturbation forces, i.e. when £ = 0 , the resulting motion of the sat e l l i t e _r(t) describes a conic section in a fixed plane formed by the i n i t i a l position £(0) and velocity 87 U) = C J + + , f j = x] -XI . Figure 3-1 General three-dimensional configuration of the earth, s a t e l l i t e and the sun vector r_{0). The five elements a, e, co, ft and i are constants determined by the i n i t i a l conditions, and the true anomaly e is implicitly related to time through Kepler's equation. To study effects of the perturbation force £ , a moving local frame of reference x,y,z is introduced, Figure 3-1. At each instant, the x axis points along the radial direction, the y axis lies in the orbital plane such that the velocity vector has a positive component along this axis, and the z axis is normal to the osculating plane. The force £ is expanded in components ( F x > F , F z) along the local reference frame. The influence of F x and F^ is limited to an in-plane rate of change in velocity and leaves the orientation of the orbital plane unaffected, while F z causes an out-of-plane rotation of the velocity vector with-out affecting its magnitude. The component F z generates a torque _rxF z£ = -rF z_j along the negative y axis causing the vector h_ to rotate in the y,z plane with instantaneous angular rate w = (rF z/h)i_ along the x axis. Thus, the effect of F z is interpreted as imparting a rotation wr of the orbital plane about the instantaneous radial direction (gyroscopic effect). The motion of the local x,y,z frame in the inertial X,Y,Z frame is completely described by the sum, W, of the angular rates wr and v , where v points along the instantaneous z axis and represents the rotation of the radius vector in the osculating plane. It must be emphasized that the angle v is measured from a fixed axis in the instantaneous orbital plane indicated by x n, Figure 3-1. The angular 39 momentuin vector h, defined by r x v, is equal to r x (Wxr) = r vk , which i s , interestingly, of the same form as that for the planar perturbations. The motion of the x,y,z frame can also be described in terms of the Eulerian angles fi, i and $ . The precession fi is taken along the inertial Z-axis, the nutation along the line of nodes, i.e. the intersection of the osculating and the X , Y planes, and the spin '<$> along the z axis. A comparison of the components of the angular velocities along the x,y,z axes leads to: W = rF /h = (i )* cos <j> + fi sin i cos 4> ; A L-Wy = 0 = (i )* si n <J> - fi si n i cos cj) ; W = v = h/r 2 = i + fi cos i . (3.2) The f i r s t two equations yield the standard Lagrange's perturbation equations for the orientation of the orbital plane: (i) = r F z cos (j) / h ; fi = r F sin <J>/(h-sin i) . .....(3.3) Taking into account the motion of the x,y,z frame, described by the rota-tion vector W, with respect to inertial space, the components of Newton's law along the local x,y, and z axes become: 90 r - rv + 1/r1 2 = F x rv + 2 r v = F, y rv W x = F z (3.4) Note that the f i r s t two equations do not contain the out-of-plane compon-ent of the perturbation force F . It is natural to employ the quasi-angle v as independent variable since v = h/r is free of any out-of-plane elements. Using the transformation u = l / r as in the previous chapter and rewriting the out-of-plane Equations (3.3) in terms of the angle v leads to the following complete system of equations: u" (v) +u(v) = l/£ - (F x + F y u'/u)/(u2£) fc'(v) = 2 F y / u 3 ; i'(v) = F z cos(v-^) / Uu 3) ; fi'(v) = F z sin(v - IJJ) / (£u3 sin i) = fi1 cos i t'(v) = l/(uV/2) (3.5) where = v - cp . It may be noted that the role of angular momentum is 2 taken over by the semi-latus rectum £ = h . The elements I, i and fi 91 correspond to Lagrange's oscu lat ing elements and can be in terpreted as such. As in the planar case, u is wr i t ten as (1 + p cos v + q s in v ) / £ where p, q and I are slowly varying oscu la t ing elements. The dependence of £ upon v i s given in the second re l a t i on of Equations (3.5) while the condi t ion of o s cu l a t i on , i . e . , u'(v) = (-p s in v + q cos v ) / £ , leads to a set of f i r s t - o r d e r equations fo r p(v) and q(v) rep lac ing the equation fo r u" in Equations (3.5) , with u and u' to be expressed in terms of p, q, £ and v . The usual form u = (1 + e cos 9)/£ , where 9 i s the true anomaly, i s reta ined s ince 8 = <j>-co = v- to with co denoting O J + I J J . I t can now be seen that p = e cos oj and q = e s in OJ . (Note that fo r an e c l i p t i c o r b i t ( i = 0) , there i s no d i s t i n c t i o n between to and to nor between <j> and v . ) Conse-quent ly , the f a m i l i a r o r b i t a l elements e and to can be der ived read i l y 2 2 1/2 from the formulat ion above: e= (p +q ) and to = arctan(q/p) - \b . The a u x i l i a r y o r b i t a l elements p and q can be in terpreted geometric-a l l y as the project ions o f the e c c e n t r i c i t y vector e_ (po int ing towards the instantaneous perigee pos i t ion) upon the XQ and coordinate axes, Figure 3-1. (3.6) 3.2 Determination of the Solar Radiation Force 92 The perturbation force _F with its components F^, F y and F z along the local frame of reference is evaluated next for a f a i r l y general sa t e l l i t e configuration consisting of n components. Each of these has its own material properties o ^ , G 2 ( < , P k , k=l,2,...,n defined in Equations (2.2), i t s orientation designated by the normal u£ and an effective f l a t surface area A^. A curved surface component may be replaced by an equivalent f l a t area with material characteristics determined by integration. The total (nondimensional) solar radiation force acting upon the satellite becomes (Equation 2.1): (3.7) Two Eulerian rotations and 3^ are sufficient to describe an arbitrary spatial orientation of the surface element A^ with respect to the osculating plane, Figure 3-2b. The normal u£ points along the negative x axis when =3^=0 . The rotation about the z axis takes A^ to the required line of intersection with the orbital plane and, sub-sequently, 3^ along the y-j axis adjusts the surface element A^ to the desired inclination with the orbital plane so that u£ points along the x 2 axis. The components of the vectors u£ in the x,y,z frame are written symbolically as u£ = u£x_i_ + u£ i+uj^Jc. Also the direction of the radiation u^  is expanded along the local coordinate frame: u s = u x ( v ) l + ^ ( v ) j+ u z ( v ) k . The components of the solar radiation force can now be written as (a) (b) Figure 3-2 Idealized s a t e l l i t e configurations: (a) sphere (b) f l a t plate or surface component in arbitrary orientation to orbital plane GO 94 Fx = % n U k l { a l k u x + ^ 2 k + P k U k ] u k x } A k • Fy = £ j / k l j ^ l k uy + ^2k + P k V uky } Ak > Fz = e j T l U k l { a l k Uz + ^ 2 k + P kU kl \z } \ , (3.8) where A k has been nondimensionalized by A, the sum of all surface elements illuminated by the sun, and Uk denotes the dot-product (_u£'jJS) . It should be noted that, in general, u£ are functions of v since the sate l l i t e may experience librational motion in the local reference frame. 3.3 Plate Normal to Radiation In many present and proposed applications, large solar panels are employed for power production either for on-board requirements (e.g. SEPS) or for external needs (e.g. SSPS). The efficiency in terms of power production per unit area of solar cells will be largest i f the panels are kept normal to the radiation. This orientation is n s achieved when u = u or in terms of the Eulerian control angles a and 3 : a(v) = -v + ^  + arctan [cos i tan n ] 3 = a rcs in [s in i s in n ] , (3 where the modif ied so lar aspect angle n stands f o r n - ft . The r e s u l t -ing rad ia t ion force becomes £ = eu f o r th i s case (taking o = 1). As mentioned before, th i s model can also serve fo r c a l c u l a t i n g per tur -bations o f a spher ica l s a t e l l i t e with homogeneous surface p roper t i e s , Figure 3-2a. Since the perturbat ion equations are wr i t ten in terms of the independent var iab le v , the e x p l i c i t dependence of the components of £ and thus u^ s upon v i s needed: uj(v) = - [ c o s 2 ( i / 2 ) c o s ( v - ^  - n) + s i n 2 ( i / 2 ) c o s ( v - + n) ] ; x s 2 ^ 2 ~ Uy(v) = cos ( i / 2 ) s i n ( v - ty - n) + s in ( i / 2 ) s i n ( v - ty - n) ; s ^ u z = s in i s in n " (3 The complete set of equations, inc lud ing the motion of the sun ( repre-sented by the angular rate 6) can be found from the pre l im inar ie s in Sect ion 3.1: £ ' ( v ) = 2 e r 3 uJ ; 2 s s p '(v) = e r { u x s i n v + u ~ [cos v+(p + cos v) r/2,]} ; . z s s q'(v) - er {-u xcosv + u y [sin v +(q + sin v) r/£] } ; 3 s i 1 (v) = e r u 2 cos(v - ty) / £ ; 3 s fi'(v) = e r u z sin(v - ty) / (£ sin i) ; ty' (v) = fi' (v) cos i ; n'(v) = 6 r 2 / £ 1 / 2 . (3. Here the radius r stands for r(v) = £/(l + p cos v + q sin v). It must be emphasized that the singularities in the equations for fi and for i =0 cancel out since u^ also contains a term sin i . In this chapter e will be taken of the same order of magnitude as 6 and the system of Equations (3.11) will be referred to .as a'(v) = e f (a,v). 3.3.1 Short-term analysis As in Section 2.3.1 for the e c l i p t i c case, i n i t i a l l y valid approximations can be obtained by expanding the elements in simple perturbation series, Equation (2.8). After substitution of these series into the system of Equations (3.11) i t follows that a^(v) = a_QQ and the following first-order results are found upon integration: £-,(v) + P 0 0A 3 1(v)+A 3 Q(v)/2 + A 3 2(v)/2]} ; q l ( v ) = £ 0 0 { K 1 0 [ A 2 0 ( v ) + q 0 0 B 3 1 ( v ) + A 3 0 ( v ) / 2 " A 3 2 ( v ) / 2 ] - K 2 0 [ q 0 0 A 3 1 ( v ) + B 3 2(v)/2] } ; . 2 ~ = £ 0 0sin i Q 0 sin n 0 0 [A 3 1 (v)cos ^ 0 Q+ B 3 ] (v)simjj 0 0] ; 2 ~ ft-j(v) = £ Q 0 sin n 0 0[B 3 1 (v)cos i | ; 0 0 - A 3 1 (v)sin ^ 0 Q ] ; ^ ( v ) = fi^v) cos i Q 0 n^v) = £ 3 / 2 A 2 Q(v) / c £ . (3. Here the auxiliary constants K^g and K 2Q depend upon FIQQ» igg, *Q Q and the integrals A n k(v) and B ^ v ) contain Pgg and qgg. The similarity between the results for p^, q-j, H-., and n-j found here and the corres-ponding results of the previous chapter (Equations 2.9) is evident, hence the explicit results of Equations (2.10) for the in-plane 98 perturbations remain valid here provided cos HQQ and sinngg are re-placed by K^Q and K^Q, respectively. Of special importance are the results for v = 2TT where the short-period terms vanish; they may serve as a basis for obtaining long-term valid solutions. As in the planar case, i t follows that Aa = 0, so that the major axis remains constant in the long run. Similarly, the remaining independent elements can be written as: Aco = STTC a 2 0 ( l - e 2 0 ) 1 / 2 ( p 0 0 K 1 0 + q 0 0 K 2 0 ) / e 2 0 - cos i Q 0 Afi ; Ai = -3Tre e 0 0 a 2 0 s i n n 0 0 c o s ( , 0 0 / ( l - e 2 Q ) 1 / 2 ; Afi = -3TT£ e 0 0 a 2 0 s i n n 0 0 s i n W q o / (1 - e 2 Q ) 1 / 2 ; At = STTC a ^ 2 {(4 + P 0 0)K 1 0 + 6q 0 0K 2 0} + O(e 2 0) (3.13) It may be noted that the result for At can be used for calculating the change in overhead position of a communications sat e l l i t e after one —f\ revolution: e.g., for CTS (e = 1.37 x 10 ) i t follows that the s a t e l l i t e may d r i f t as much as 330 meters per day. Note also that the possible existence of a shadow region is overlooked here. For a geosynchronous equatorial orbit, there is no eclipse by the earth during about 9 months of the year. Only when the sun is near one of the equinox positions will there be a shadow interval with duration varying from a maximum of 70 minutes when the sun is on the equinox axis to zero about 22 days before and after that epoch. It should be emphasized that the effect of the earth's shadow upon an equatorial sat e l l i t e i s , quantitatively, less pronounced than that for a space probe in the ec l i p t i c plane analyzed in the previous chapter. Points of entry into and exit from the shadow region as well as their long-term effects upon the orbital elements can best be studied numeric-ally for this arbitrary case. 3.3.2 Long-term approximations As in the previous chapter, Section 2.3.2, long-term results can be derived from the short-term analysis by repeated rectification and iteration of the i n i t i a l conditions. This approach has indeed been followed in the present case and the results will be discussed in the next section. Here, analytical closed-form approximate solutions are explored by means of the two-variable expansion method. Following the procedure outlined in Section 2.4.1, the equations for the zeroth-order approximations become: £ Q . ( v ) = £ 3 JK.,B31(2Tr) - K 2A 3 1(2TT) j / TT ; P 0(v) = { K l [ p0 B31 ( 2 ^ ) + B32(2TT)/2] - K2[A20(2TT) + p nA^(2TT)+A q n(2Tr)/2 + A „ ( 2 T r ) / 2 ] J/(2TT) ; 100 q 0(v) = i K-,[A 2 0(2Tr)+q 0B 3 1(27T)+A 3 0(2rr)/2-A 3 2(2TT)/2] K 2[q 0A 3 1 (2Tr) +B 3 2(2TT)/2] / (2TT) i Q ( v ) = £ Q sin i Q sin n 0[A 3 1 (2TT) cos + B 3 1 (2TT) sin ^ 0 ] / (2TT) ; 9 ftQ(v) = £Q sinn 0[B 3 1(2Tr) cos ^ 0 - A 3 1(2TT) sin 4^] / (2TT) ; 4>Q{V) = QQ[V) cos i Q ; n 0(v) = i3Q/2 A 2 0 (2TT) / (2Trc £ ) . .(3.14) Here the integrals A n k(2Tr) and B^^TT) depend upon PQ(V) and qg(v) and are evaluated in Appendix I. The slowly varying functions K-j(v) and K 2(v) stand for 2 ^ 2 /\ K ](v) = cos (i 0/2)cos(n 0+ IJJQ) + sin (i 0 /2)cos(n 0 - ^ Q) , r\ ^ K 2(v) = cos (i 0 / 2)sin(n 0 + ^ 0) - sin^(i 0/2)sin(n Q- I|J 0) (3.15) The f i r s t integral of the system of Equations (3.14) can readily be found as 101 a 0 ^ ) = ^ 0(v) / [1 - p^(v) - q2(v) ] = a Q 0 . (3.16) Thus, the major axis and total energy of the sa t e l l i t e are conserved in the long run and the motion of the sun does not alter the conclusion reached from the short-term analysis. On substituting the explicit results for the integrals A n ^ (2Tr ) and B n k(2Tr), evaluated in Appendix I, into Equations (3.14) and performing some algebraic manipulations, the system can be written in a f a i r l y compact form as: U j / 2 ) ' = 3 a ^ 2 / 2 [ j Q cos i Q sin nQ - k Q cos nQ ] ; 3 / 2 1 / 2 ~ J'o = " ( 3 / 2 ) a o o £o c o s 1 - o s i n ^ 0 + k o c o s ^0Qo c Q = (3/2) a 3 / 2 i]Q/Z cos f|Q - J 0 c o s i 0 fiQ' i Q = -(3/2) a ^ 2 j Q sin i Q sin ^ / A J / 2 fiQ = -(3/2) a ^ 2 k Q sin nQ / l]Q/2 ; n0 = c - fii . (3.17) Here, the auxiliary elements n = n - fi > j - e cos oo and k = e sin to have been introduced for convenience. Note that j = p c o s ^ + qsinip and k =q cos ty - p si n ty . The simp!ici ty of the equation for Pg(v) i s a direct consequence of the fact that ag(v) = a^ so that the orbital period remains constant in the long run. Integration yields H Q ( V ) = PQQ+cv indicating that (in this order of approximation) the motion of the sun is proportional to v , relegating the non-uniformity of the sun's motion (with respect to v) to higher-orders. Apart from ag(v) = 3 Q Q > at least three additional integrals can be constructed, namely n -2x1/2 . . _ (1 - J 0) sin i Q = D1 2 1/2 ^ [kg-d(l -e Q) sin f^] sin i Q = D2 2 1/2 ^ y\ [(1 - e Q) + d k Q sin iig] cos i Q + d j cos n 0 = D3 , (3 1/2 where d stands for 3eag0 / (26) . For the orbit to remain closed, i.e. eg(v) < 1 , the constant d must be less than unity or in terms of e : e < 0.0018 (for a^=1), which is true for virtually a ll practical cases It is interesting that by the f i r s t and second relations of Equations (3.18), the orientation of the orbital plane, described by elements I'Q(V) and fig(v), can be expressed in terms of the in-plane perturbations represented by jg(v) and kg(v). Note also that the f i r s t relation in Equations (3.18) yields the obvious result that an orbit i n i t i a l l y lying in the ec l i p t i c plane, i.e. iQQ = ^' W 1'^ remain in that plane, iQ(v)=0. The constants , i= 1,2,3 are determined from the 103 i n i t i a l conditions. Due to the fact that the last relation of Equations (3.18) admits, in principle, that be eliminated in favor of j Q , only one remaining equation of the system (Equations 3.17) need to be integrated for obtaining a complete analytical long-term repre-sentation for the orbital elements. For the general case, an un-coupled and tractable equation has not been found. Fortunately, special situations for which closed-form solutions can be derived more readily exist. In case the sun's position i n i t i a l l y lies on the line of nodes and either the i n i t i a l orbit is circular or has the perigee lying on the line of nodes, i t follows that ^ = 0 or TT and k 0 Q = 0, implying that the constant D^  vanishes. In that case n0(v) is readily integrable, yielding the result: fi0(v) = n Q(v) - arctan [tan(bv) / (1 + d 2 ) 1 / 2 ]•• (3.19) Employing the integrals of Equations (3.18), the remaining elements can now be derived: J 0 ( v ) = (1 - D 2 ) 1 / 2 G ( v ) / [ l + d 2 c o s 2 ( b v ) ] 1 / 2 ; k Q(v) = dsin (bv)|l - (1 -D 2)G 2(v)/[l+d 2cos 2(bv)]} 1 / 2/(l+ d 2 ) 1 / 2 e Q ( v ) = { d 2 s i n 2 ( b v ) + (l - D 2 ) G 2 ( v ) } 1 / 2 / ( l + d 2 ) 1 / 2 ; dsin(bv)[l +d 2cos 2(bv)-(l - D 2 ) G 2 ( v ) ] 1 / 2 ^n(^) = arcsin 1 =, ^ TTO O O O O TTO 0 [1 +d^cos^(bv ) ] l / ' ; [dS in2(bv) + (l - D 2 ) G 2 ( v ) ] 1 / 2 104 i n ( v ) = arcsin ( D,[l + d 2 c o s 2 ( b v ) ] 1 / 2 / [ l +d 2cos 2(bv) (3.20) with appropriate branches of the arcsin functions determined from i n i t i a l conditions and by continuity. The auxiliary function G(v) stands for It is noteworthy that a l l of the functions nn, kg, J Q , eg, ojg and I'Q 2 1/2 are periodic with a period of 1/(1 + d ) ' year, so that the elements return to their original values just before the sun has made a f u l l revolution. These results are consistent with those obtained for the ecl i p t i c orbit in the previous chapter. Note also that fig, ipg, COQ, p and qg contain terms of two different but close periods, namely one 2 1/2 year and 1/(1 + d ) year, resulting in a slow secular trend in the long run. The higher-order terms can readily be determined by formal integration, yielding similar (when cos P Q 5 sin n.Q are replaced by K-|(v) and ^ ( v ) , respectively) expressions as in Equations (2.31) for the in-plane orbital elements £-|, a-|, p-j and q-j, while the out-of-plane results are given by G(v) ) = d E cos(bv) - sign(cos i n n ) (1 - E ) 2,1/2 (.3.21) i-|(v,v) = £ Q sin i Q sinrig j [ a ^ cos tyQ + bL s i n i p j sin jv 105 + [c--, cos +d 31 sin ij/ n](l - cos jv) [ + i , (v) ft-, (v,v) = a sin n n {[b 31 cos \pn - a~, sinijj n] sin jv+ [d 31 cos \\J 0 (3.22) with the slow functions i-j and ft^ determined from the boundedness constraint upon the second-order terms (Appendix III). However, the complexity of the equations involved precludes any possibility of extracting, analytically, information on the long-range trends of these terms. 3.3.3 Discussion of results Since an analytical long-term solution has been found only for the case of D^  = 0, the rectification/iteration procedure needs to be employed for cases where the i n i t i a l conditions are different. As an example, a sa t e l l i t e in a geosynchronous equatorial orbit with e = 0.0002 is consi-dered. A rectification interval v^ = 2TT is chosen yielding sufficiently accurate results over a 1200 day span of time. As the major axis remains constant in the long run, the in-plane per-turbations are fu l l y described by the eccentricity vector e_ or its Cartesian components p and q. Concise representation of the in-plane changes can be provided by polar plots for e_ in the x^y^ plane. From the zeroth-order long-term results, the slope of the polar plot at any instant is given by (Equations 3.14), 106 dq 0 Kn (v) (3.23) dp 0 P 0 ( v ) For an equatorial orbit, cos (in/2) is about 25 times sin (ig/2) so that the polar plots should be similar in shape to those obtained for ecliptic orbits in the previous chapter, Figure 2-7 a-d. Figures 3-3 a,b show the resulting polar plots for i n i t i a l l y circular as well as highly eccentric orbits ( e ^ = 0.5). The eccentricity is periodic with a period of about 363 days, and the orientation of the major axis depends c r i t i c a l l y upon the i n i t i a l eccentricity: for sufficiently large, the orbit remains e l l i p -t i c with its axis exhibiting periodic oscillations (amplitude of about 12° for e n n = 0.5) as well as a slow clockwise rotation (about 2° per year, Figure 3-3b). For an i n i t i a l l y circular orbit, Figure 3-3a, the behavior of OJ is completely different, showing an increase of 180° over one year followed by an instantaneous jump of 180° when the eccentricity passes through the origin again. Also the slow clockwise rotation is apparent. The small differences in the in-plane perturbations for an arbitrary as compared to an ecliptic orbit are due to the fact that the magnitude of the in-plane component of the solar radiation force varies slightly with the position of the sun in the former case. The behavior of an i n i t i a l l y circular near-ecliptic orbit may be visualized as follows: the solar radia-tion force changes the circular orbit into an ellipse with increasing eccen-t r i c i t y and perigee at 90° ahead of the projection of the sun-earth line. Subsequently, the major axis tries to maintain the 90° lead over the moving sun, but as the eccentricity increases the orientation of the major axis be-comes more rigid and the sun overtakes the perigee after about half a year. p Figure 3-3 Polar plots, illustrating long-term behavior of the eccentricity vector e: (a) e Q 0 = 0 ; (b) e Q 0 = 0.5 108 At this point, the eccentricity has reached its maximum value and will start to decline while the sun moves ahead of the perigee position. After almost a year the perigee is about 90° behind the sun and, while the eccentricity vanishes again, makes a jump of 180°. The original situation is now re-established and the cycle repeats i t s e l f . According to the results of the two-variable expansion procedure, Equations (3.20), the auxiliary elements J Q ( V ) and k n(v) are periodic with 2 1/2 a period of 1/(1 + d ) year, i.e. about 363 days in the present example. Their behavior can be visualized through polar plots as in Figures 3-3 without the slow clockwise rotation. Since the j,k axes are obtained from the p,q axes by rotation through the angle IJJ , the slow clockwise trends in the polar plots of Figures 3-3 can be interpreted as the negative secular growth of the angle ij;. In Figure 3-4, the behavior of the longitude of the ascending node fi is depicted for an equatorial orbit and a few values of the i n i t i a l solar aspect angle P Q Q. Of particular interest is the insensitivity of its beha-vior to different values of the i n i t i a l eccentricity in the range 0 - 0.5. Results of the two-variable expansion procedure indicate quite close cor-respondence with those obtained by rectification/iteration. As seen in Equation (3.19), the long-term behavior of the longitude of nodes is inde-pendent of i n i t i a l eccentricity. The qualitative behavior of fig(v) may be visualized by considering ftg(v), Equations (3.17): the rate of change of SQ(V) is proportional to -k n sinn^, which remains negative in case HQQ = 0 or TT due to the nature of the polar plot for JQ>I<Q. By following the beha-vior of kg in conjunction with that of T\Q > i t becomes evident that the total regression of fin after one revolution should be essentially independent 109 8 1 i i r 0 200 400 600 800 1000 1200 days Figure 3-4 Typical long-term behavior of the longitude of nodes as affected by the i n i t i a l solar aspect angle 1 1 0 of the i n i t i a l eccentricity and solar aspect angle. Figure 3 - 5 shows the long-term behavior of the inclination of the orbital plane for a few values of the solar aspect angle. The results of the two-variable expansion method match quite well with those obtained by rectification and iteration. The fact that the period of oscillation for PQQ = TT/2 and 3TT/2 is half that for = 0 and TT can be understood from the f i r s t relation in Equations ( 3 . 1 8 ) in conjunction with the in-plane 2 perturbations of J Q. The polar plots indicate that the behavior of J Q is o the same for r\ = 0 and TT : amplitude (e n m ) ' and period 3 6 3 days, uu u ,ma x 2 On the other hand, when n 0 0 = TT/2 and 3TT/2, j Q oscillates with period of 2 1 8 1 . 5 days and an amplitude of (e n / 2 ) . This explains the occurence U ,max of two different frequencies as well as the dependence of the amplitude of variation of ig on the i n i t i a l solar aspect angle. The behavior of the in-clination for an i n i t i a l l y eccentric orbit, egg = 0 . 1 , is both quantita-tively and qualitatively different from an i n i t i a l l y circular orbit, Figure 3 - 6 . The relatively large differences in amplitude depending on the solar 2 aspect angle can be understood by visualizing the behavior of j'g in the cor-responding polar plots. Figure 3 - 7 shows the variation of the inclination for a very eccentric orbit, egg = 0 . 5 . The differences in amplitudes for various values of rigg are much smaller than those in Figures 3 - 5 and 3 - 6 , although the magnitudes themselves are much larger (note the differences in scale). Finally, i t may be mentioned that only when igg = 0 or TT the inclination remains constant throughout in the long run. i n 24 a0o=1 u)00= J100=0 noo- 0 , n e 0 0=0 8 =0.0002 n0o= n/ 2 , 3ry2 I i i o o t w o - v a r i a b l e l e x p a n s i o n I days Figure 3-5 Long-term behavior of the inclination for i n i t i a l l y circular equatorial orbit 112 Figure 3-6 Variations in orbital inclination for i n i t i a l l y equatorial orbit of eccentricity 0.1 113 Figure 3-7 Behavior of orbital inclination for i n i t i a l l y equatorial orbit of eccentricity 0.5 114 3.4 Satellite in Arbitrary Fixed Orientation to Radiation The case considered here constitutes a generalization of the analysis of the previous sections in the sense that now the plate is kept at a fixed but arbitrary angle to the incident radiation. It serves as a f a i r l y accurate model for communications and other satellites having one or two-axis attitude control. The CTS s a t e l l i t e with solar arrays which can be rotated about an axis normal to the orbital plane for maximization of the amount of solar energy intercepted is an obvious example. Further-more, the analysis is relevant to the solar radiation induced orbital per-turbations of a s a t e l l i t e with a fixed orientation in the inertia! space, e.g. a platform for deep-space studies. It is noteworthy that the analysis can be extended quite readily to an arbitrarily shaped sa t e l l i t e modelled by n f l a t surfaces A^ each with its own characteristic material parameters represented by a ^ , and p^. The orientation of the surface A^ with respect to the instantaneous orbital plane is determined by the two Eulerian rotations, expand 3^. A fixed orientation of the s a t e l l i t e with respect to inertial space or the radiation (in a short-term sense) is maintained i f = - v + cv.^  ,• k = l,2,...n, with arbitrary fixed angles and 3 ^ . For instance, the surface element A^ is normal to the radiation i f = ^ + arctan[cos(i) tanp] and 3^ = arcsin[sin(i) sinp], which corresponds to the control law of Equations (3.9). Replacing the elements i , ty and n by their respective i n i t i a l (or mean) values igg, I J J Q Q and X]QQ , writing £ S = 1^ + l! c o s v + !L s i n v a r ,d u£ = HJ^  + c o s v + sinv , the force expression of Equations (3.8) can be rewritten in the following compact form: 115 F = e J '{C? + cj cosv + C2 sinv} , k=l -* -K "* where CJ[ = |UR | k s j + [ a ^ + P | < njj} A k > j = 0,1,2; Uk = cos3 k[K 1 0 cosa k + K 2 0 sina k] + sin3 k s i n ( i n n ) s i n n 0 0 . (3.24) It may be noted that Uk = (u£ • u_S) is constant in the short-term analysis since the sa t e l l i t e as well as the sun maintain a fixed orientation in space. In practise, one needs to update the solar aspect angle PQQ and the control angle ak to account for the slow motion of the sun. Naturally, also the orbital elements change continually and need rectification after a certain time. Employing the usual perturbation Equations (3.5) and (3.6), integra-tion over a short-term interval (vpn,v) yields the following first-order changes in the orbital elements: V") = 2 e 4 ^ { Cky A31 + C k y B 3 1 } ' a i ( v ) = 2e a 2 Q * o n I {C\X[BU - B 2 1 - q Q 0 A 2 0] + c j y A „ + Cky B l l " C k x [ A l l " A21 " p00 A 2 0 ] } ; P i » = e 4 j ; (C k x B 2 2 + C k y [ A 2 0 + A 2 2 + A 3 Q + A 3 2 + 2p Q 0 A 3 ] ] k=l + Ckx ( A20 " A22> + Ckyt B32 + B22 + 2P 00 B 3 1 ^ } / 2 ' 116 Qi (v) " Ckx ( A20 + A22) + Ckyt B32 + B22 + 2^00 A 3 l ] } / 2 ' 0 kz n, (v) ^ (v) c o s ( i 0 0 ) n. T l i ( v ) = I. 3/2 (3.25) 00 Here, the abbreviations A ^ and B ^ stand for A n k(v) - A ^ ^ Q Q ) and B n k ^ " Bnk^ v00^' respectively, and depend upon the elements PQQ and (Appendix I). It may be noted that for iQQ = 0, fi and ty lose their meaning and the angles v, co, measured from a fixed axis in the orbital plane, coin-cide with <j> and to, respectively. After one revolution, the reduced expressions of the integrals for v = 2TT may be substituted. The result for a^(2ir) vanishes after one revo-lution so that, also in the present situation, first-order secular changes in the major axis are absent. By rectification of the orbital elements, as well as the force expression of Equations (3.24) at v = 2TT and iteration of the results of Equations (3.25), a good approximation for the long-term perturbations may be obtained. This process has been executed for a variety of plate orientations, r e f l e c t i v i t i e s , and i n i t i a l conditions leading to the following general conclusions. In case 3=0, i.e. when the plate is kept normal to the orbital plane, the long-term changes in eccentricity, position of the perigee and inclination are periodic with period of about one year 117 regardless of the (specular) r e f l e c t i v i t y p. In the examples cosidered, the fluctuations in inclination range from zero when p = 1 (resulting in a force in the orbital plane) to about 0.5° when p = 0. Typical long-term in-plane perturbations for a few values of ref l e c t i v i t y are presented in Figure 3-8b. Here the plate is in an equatorial orbit and a = ty + arctan[cos(i) tann] > aligning the plate-normal with the projection of the radiation in the orbital plane. When the second rotation 3 ^ 0 is imposed, and part of the radiation is absorbed, the qualitative nature of the in-plane perturbations changes drastically as shown in Figure 3-8a. Note that the polar plots show severe secular perturbations in the in-plane elements e and u>. The orbital inclination may also exhibit long-term secular changes up to about 0.7° per year in the examples considered. The variations in the longitude of nodes are of a long-term secular nature with a rate of regression between 0 and -0.5 degrees per year in the case of 6=0. The smaller the refl e c t i v i t y the higher this rate. When the plate is not normal to the orbital plane ( 3 ^ 0 ) , higher rates of precession have been observed, e.g. -2.4° per year for a= n , 3 = 23° and p_ = 1. 0.1 q 0.05 0 -0.05 -0.1 /3 = 23 ^00=0 a 0 0=1, i00=23.45 Jfl 0 0= co 0 0=0 e00=0.1 E =0-0002 fa) -0.1 -0.05 Figure 3-8 0 0.05 P 0.1 ^ 0 0 = TT/2 /3 = 0 (bj 0 q -0.05 -0.1 -0.15 -0.2 0 0.05 0.1 0.15 0.2 P Typical polar plots for plate maintaining fixed orientation to radiation, a = \p + arctan{cos(i) tan fi } and: (a) B = 23°; (b) 3 = 0 co 119 3.5 Plate in Arbitrary Fixed Orientation to Local Frame In this section a sa t e l l i t e modelled as a f l a t plate, with arbitrary but homogeneous material characteristics, kept in a general fixed orienta-tion with respect to the local coordinate frame is studied. The analysis is relevant to orbiting platforms or mirrors in a fixed orientation with respect to the earth. In particular, large space structures aligned with the local vertical due to gravity-gradient torques represent an obvious example. The major distinction between the present situation and those en-countered previously, is the fact that now both sides of the plate will be exposed to the solar radiation, each during about half a revolution. The change-over occurs when the plate is aligned with the sun-earth line. This raises an interesting question as to the orbital perturbations for a plate with different material properties on the front and back sides. The orientation of the plate i s , as usual, described by the two Eulerian rotations a and 3, which are arbitrary constants now. In a short-term analysis, the orbital elements are considered constant (i.e. equal to their i n i t i a l or mean values) during integration of the perturbation equa-" s tions and in the evaluation of the force expression. Writing u_ (v) = s_^  + sj cosv + s 2 sinv and U(v) = (u_n • u_S) = UQ + U-jCOSV + U^sinv , the force can be expressed in the form: L = e s u ^ ° + pjcosv + D 2 sinv + D3cos(2v) + D_4sin(2v)} , with: D° = a-jfUg s° + (U] s 1 + U 2 s 2)/2] + [a 2 U Q + P U 2 + p(U 2 + U2)/2] u n ; D1 = ^(Ug s 1 + U1 1°) + (a 2 U1 + 2pUg ) u n ; 120 J l ( u 0 I* + U 2 i - U ) + ( a 2 U 2 + 2 p U 0 U 2 ) - n ; D3 = ^ (U1 s 1 - U 2 s 2 ) / 2 + p(U 2 - U 2)/2 u n ; D 4 = a 1 (U] s 2 + U 2 s_])/2 + PU-| U 2 u n , (3.26) It may be noted that the vector u_n is fixed in the local x,y,z frame and is determined by the angles a and 3 in such a manner that i t points towards the earth for a = 3 = 0. When the plate has different material' properties on either side, care must be taken to identify the side facing the sun i n i t i a l l y and make necessary adjustments after about half a revolution when the other side becomes illuminated. The side facing the earth for a = 3 = 0 will be designated as the back side of the plate and its material properties will carry the subscript b. Similarly, the front side will be recognized by the subscript f. Consequently, the value of s u equals 1 i f the front side is illuminated and -1 for the back. The switch-over points. v-j, v 2 are deter-minated by U(v) = 0 and can be written in the following form: Vj = TT/2 - a + n + ty + 2TT(k-1) + 6-j , v = 3TT/2 - a + ri + ty + 2Tr(k-l ) - « 2 , (3.27) where the index k designates the appropriate revolution. The angles 5^  and 6 2 vanish in the case i = 0 and for arbitrary inclination (< TT/2) can be found from the following iteration scheme: 6 ^ = 6 ^ = 0 and 6 J N ) = arcsin {tan(i/2)[2 tang sinfj - tan(i/2) s i n ( 6 - [ N - 1 ^ + 2f i ) ] } , 6 ^ = arcsin {tan(i/2)[2 tan3 sinn - tan(i/2) s i n ( 6 2 n _ 1 ^ - 2 n ) ] } , (3.28) n = 1,2,3,... . This procedure converges very rapidly for small inclination 1 2 1 since the angles 6-j and 6^ a r e small in that case. Retrograde orbits may be accommodated by reversing the sun's motion. The short-term perturbations in the orbital elements can be obtained analytically by integration of the perturbation equations over the interval ( VQQ» v) • Provided the interval does not contain any switch-over points, the integration yields: M v> = 2 s u 4 < DJ A 3 0 + Dy A 3 1 + Dy B 3 1 + Dy A 3 2 + Dy B 3 2 } ' a l ( v ) = Su a00 * 0 0 { P 0 0 [ D X A 2 0 + Dx A 2 1 + <2 Dx " Dx> B 2 1 " Dx A 2 2 + Dx B 2 2 " Dx A 2 3 + Dx B 2 3 ] " Q 0 0 K A 2 0 + <2 Dx + Dx A 2 1 + D x B 2 1 + Dx A 2 2 + Dx B 2 2 + Dx A 2 3 + Dx B 2 3 ] + 2 [ D y A 1 0 + D y A l l + D y B l l + Dy A l 2 + Dy B l 2 ] } ' h ( v ) = su 4 { <Dx + Dl] A 2 0 + K + 2 Dy + Dy) A 2 1 + <2 Dx " Dx + Dy> B 2 1 + <Dy " Dx> A 2 2 + (°x + Dy) B 2 2 + <Dy " Dx> A 2 3 + <Dx + • B 2 3 + + 2 P 0 0 Dy> A 3 0 + ^ 0 0 Dy + 2 D J + Dy) * * A 3 1 + ( 2 P 0 0 Dy + DJ> B 3 1 + ^ 0 0 Dy + Dy> A 3 2 + ( 2 P Q O Dy + . + Dy) B 3 2 + Dy A 3 3 + Dy B 3 3 } / 2 > ^ = Su 4 - { (°y " Dx> A 2 0 + <Dy " 2 Dx " Dx) A 2 1 + ( 2 Dy " Dy " Dx> * 122 * B21 " < Dy + Dx) A22 + <Dy " Dx> B22 " <Dy + Dx> A23 + (°y " • B23 + ( D y + 2 % ) D y } A30 + ( D y + 2%0 Dy> A31 + (2fl 0 0 Dy + 2 D5 - °y} B31 + ( 2%0 Dy " Dy> A32 + ( 2P 00 Dy + Dy> B32 " Dy A33 + Dy B33 } / 2 ; i l ( v ) = s u £ 2 0 { cos ^ [D z A 3 0 + (2 D°z + D3z) A 3 1 + D 4 B 3 1 + »\ ^ + D z B32 + D z A33 + Dz. B33^ + s i n % ) [ Dz 2 A30 +. Dz A31 + ( 2 Dz " Dz> B31 " Dz A32 + D z B32 " Dz A33 + Dz B33 ] } / 2 ; ^ ( v ) - s u 4 { cos % Q [D 2 A 3 Q + D 4 A 3 1 + (2 D° - D 3) B 3 1 - D 2 A 3 2 + Dz B32 - Dz A33 + Dz B33 ] } / 2 1 - s i n % ) t D z A30 + <2 D z + Dz> A31 + Dz B31 + Dz A32 + D z B32 + D z A33 + D z B3 3] >/[2 s i n ( i Q 0 ) ] ; (3.29) and I/J-J (v) = n - j(v) cos ( i ' 0 0 ) ; n-j (v) = £ Q Q 2 A 2 0 ^ v ^ C e ' T h e a b b r e v i a ~ tions A n k and B n k stand for A n k(v) - A n k ( v n Q ) and B n R ( v ) - B n k ( v Q 0 ) , respectively, and are functions of the i n i t i a l or rectified elements p Qg, q n n (Appendix I). The coefficients fJJ , j = 0,1,....,4 were given in Equations 123 (3.26) and is either +1 or -1 over the interval ( V QQ> V ) -The more interesting long-term perturbations are obtained by repeated rectification of the i n i t i a l conditions, i.e. iteration of the short-term results of Equations (3.29). The switch-over points v-j and of Equations (3.27) are the appropriate locations for rectification since s y needs to be updated at those points in any case. This rectification/iteration procedure was carried out numerically for a few examples representing satellites model-led as a plate in a geosynchronous equatorial orbit. The plate surface was taken as perfectly specular reflective either on both sides ( p ^ = = 1) or on one side with the other side perfectly absorptive ( p ^ or p^ = 0). A few representative results are shown in Figure 3-9. Under the influence of the gravity-gradient torque, a plate in a near-circular orbit will tend to be oriented along the local vertical which is the stable equilibrium position. When the plate has different reflecting properties on either side, a gradual increase or decrease in major axis is obtained. For instance, i f the plate is kept normal to the orbital plane (B = 0) i t is obvious that more energy is transferred during the phase when the sunlight strikes the reflecting rather than the absorbing side, since the magnitude of the force is larger and its direction (in an averaged sense) is closer to the instantaneous velocity vector in the former case. This differential in energy transfer results in a continuously growing major axis when the reflecting side is illuminated with the sat e l l i t e moving away from the sun (curve 1) and a decreasing major axis when the absorbing side is facing the sun during that phase (not shown). The polar plot belonging to case 1, Figure 3-9b, is qualitatively similar to the ones for the case when the plate is kept normal to the radiation (Figure 3-3) except that the 124 area enclosed by the eccentricity vector after one year is much smaller here. If both sides of the plate have the same ref l e c t i v i t y , the long-term changes in major axis will be relatively small (curves 2 and 4). Also when the plate is kept along the local horizontal (e.g., a reflecting mirror in orbit), the semi-major axis remains virtually constant regardless of the re f l e c t i v i t i e s on both sides, since the net effect over one half revolution tends to vanish. Curve 3 is a representative example for this case. Note that the polar plot is quite different from the others. Curve 5 illustrates the behavior of a plate along the local vertical with its normal inclined by 30° to the orbital plane and different reflecting properties on either side. It should be noted that the polar plot for this case shows a long-term secular trend. In gene-r a l , one should expect a closed polar plot only in case = p f or when 6 = 0 (regardless of the values for p^ and p ^ ) . As to the perturbations of the orbital plane, i t is found that no changes in its orientation take place when = = 1 and 6 = 0, since the force has no component normal to the orbital plane. Otherwise, widely varying perturbations in i and Q, occur with the inclination staying within 1.2 degree of the equatorial plane. On the other hand, perturbations in the longitude of nodes may be oscillatory (within 1.5 degrees) or secular (less than 2 degrees per year) in the long run for the examples considered. " 1 1 I I. 0 100 200 300 400 days Figure 3-9 Long-term in-plane perturbations for plate fixed to local frame: (a) semi-major axis; (b) polar plot for eccentricity vector e 126 3.6 Concluding Remarks The important aspects of the investigation and resulting conclusions may be summarized as follows: (i) A formulation in which the out-of-plane perturbations are uncoupled from the in-plane variations, while retaining a geometric inter-pretation in terms of osculating elements, is found attractive for studying the influence of the solar radiation force upon the orbital geometry. ( i i ) For a plate normal to the radiation, short- and long-term analytical solutions have been formulated using a straightforv/ard and a two-variable expansion perturbation methods. The in-plane changes are illustrated by means of polar plots. The long-term behavior of the orbital inclination can be interpreted in terms'of the in-plane perturbations. ( i i i ) A short-term analytical solution is presented for the case where a satellite is kept in a fixed arbitrary orientation with respect to the radiation or inertial space. The long-term perturbations in eccentricity and argument of the perigee may be of a secular nature when part of the radiation is absorbed. (iv) Solutions for a satellite modelled as a plate in a fixed orientation with respect to the local reference frame are obtained. Relatively large perturbations in the semi-major axis are observed when the reflecting properties on the two sides are not the same. 127 4. GEOCENTRIC ORBITAL CONTROL USING SOLAR RADIATION FORCES 4.1 Preliminary Remarks In many situations, the perturbing effects of the solar radiation force as assessed in the previous two chapters are detrimental in nature, e.g. a communications sa t e l l i t e drifts away from its desired overhead po-sition. On the other hand, these forces have a potential for effecting desired changes in sat e l l i t e orbits as demonstrated most dramatically by the concept of a solar s a i l . The transfer of the sail from a low or inter-mediate orbit around the earth into a heliocentric orbit forms an important and time-consuming phase of the mission. Therefore, strategies for raising the orbit of a spacecraft by means of solar radiation would represent an important aspect of this maneuver. In particular, the chapter studies optimal sail settings for maximum increase in major axis over one revolution. However, the analysis has a wider range of applicability since i t also pro-vides ways for orbital correction of a sat e l l i t e with controllable solar arrays. The proper orientation of these panels can be maintained by means of small solar-electric servomotors. While an interesting procedure for increasing the total energy (plate with different r e f l e c t i v i t i e s on either side) was considered in Section 3.5, a more effective on-off switching strategy is studied here. During the off-phase when the sa t e l l i t e moves towards the sun, the plates are aligned with the radiation, while in the on-phase (when moving away from the sun) the arrays are kept normal to the radiation for generating the maximum force. The most effective switching points for correction of the orbital elements are assessed and their res-1 2 8 ponses evaluated. Another interesting application for which the analysis presented here would be relevant consists of a mylar-coated plastic sphere with a pumping device which inflates and deflates the balloon at prescribed instants. This concept has the advantage that i t does not require the continuous orienta-tion control of a solar s a i l . For convenience the switchings are assumed to take place instanta-neously since the time needed for completion of the operation would usually represent a negligible fraction of the orbital period. For a particular strategy, the first-order changes in the orbital elements after one revolu-tion are evaluated by integration of the perturbation Equations (3.11) over the appropriate on-interval I Q n = (v , v 0 f f ) > while keeping the orbital elements on the right-hand-side constant. 4.2 Switching at Perigee and Apogee An obvious switching strategy would be to switch off at apogee and on again at the subsequent perigee, Figure 4-1. Upon integration over the in-terval I Q n = (WQQ, OJQQ + TT), the following explicit results are found, using the integrals of Appendix I: A£ = 3TT e a 3 ( l - e 2 ) 1 / 2 [pK 2 0 - qK 1 Q]/e + 4e a 3 ( l - e 2) [pK 1 Q + qK 2 0]/e ; Aa = 4e a 3(pK ] 0 + qK 2 Q)/e ; Ae = 3rr e a 2 ( l - e 2 ) 1 / 2 [pK ] 0 + qK 2 Q]/(2e) + 2e a 2[pK 2 Q - qK 1 Q ; M2 = e a 2 sinn[2 cosco - (3TT/2) sinoj/O - e 2 ) 1 / 2 ] ; Ai = -e a 2 sin(i) sinn[2 sinu + (3TT/2) COSOJ/(1 - e 2 ) 1 / 2 ] . (4.1) project ion of sun - e a r t h l ine Figure 4-1 Configuration of switching points for controlled orbital change 130 Here, i t is assumed that e does not vanish. It must be mentioned that the subscripts 00 are omitted in the present chapter for brevity. It is in-teresting that the change in eccentricity is exactly half of the amount ob-tained when the force acts continuously, Equations (3.13). For near-ecliptic orbits, the expressions pK^ + qK^ and pK^ - qK-|Q' may be replaced by cosx and sinx respectively, where x "is the angle between the projection of the sun-earth line and the major axis, Figure 4-1. It is seen that i f 0 < x < TT the sa t e l l i t e moves against the direction of radiation and looses energy, while i f TT < x < 2TT the major axis increases. Obviously, the expressions of Equations (4.1) are only valid for one revolution. Long-term results are derived by rectification and iteration of the short-term orbital changes. Figure 4-2 shows the resulting long-term response for the particular case of an orbit in the ecliptic plane. There is a wide range of variation in the behavior of the semi-latus rectum de-pending on the i n i t i a l solar aspect angle. In the case that the major axis follows the motion of the sun, a favorable situation is maintained leading to a continuous increase in JI: (curve 1). However, in most cases, especially for larger e^, the axis rotation f a i l s to keep up with the sun (Chapter II), so that in the long run no systematic build-up in the latus rectum or the major axis occurs as shown by the other curves. The long-term variations in eccentricity are found to be of approximately half the amplitude as compared to the case of a continuously acting force. 4.3 Systematic Increase in Angular Momentum Since the nature of the response in the previous on-off switching strategy seems to be strongly dependent upon the i n i t i a l conditions, a more 132 systematic approach is needed to generate a certain prescribed trend. In this section, a switching control strategy with the objective to change the orbital size by increasing the semi-latus rectum as much as possible is explored. The most effective switching instants are the points where £'(v) vanishes: v-j and v 2, Figure 4-1. The on-phase (v-j, v^) coincides with £' (v) > 0 and the off-phase (v^. v-j + 2TT) with £'(v) < 0. The points v-j and x)^ satisfy the equation u^ = 0 and represent, geometrically, the points of intersection of the orbit with the projection of the sun-earth line into the orbital plane: v-j = ty + arctan[cos(i) tann.]. and = v-j + TT. During the on-phase the force has a positive component along the circumferen-t i a l direction and produces a torque r. * F_ adding to the magnitude of the 2 angular momentum vector h_ and the semi-latus rectum £ = h . While the orbital changes can be determined readily by means of a digital computer, analytical results are established for an orbit in the ecliptic plane where I = (n> p + TT): A£ = 2e a2£T_3 - (1 - e 2 ) / ( l - e 2 cos 2 X) + 3e F^e.x) sin X] ; Aa = 4e a2£/(l - e 2 cos 2x) ; Ap = -3e a£ F 2(e,x) sinp ; Aq = 3e a£ F 2(e,x) cosn ; Ae = -3e a£ F 2(e,x) sinx ; Aw = 3e a£ F 2(e,x) cosx (4.2) The functions F^(e,x) and F 2(e,x) are defined by F^e.x) = (TT/2 + arctanle s i n X / ( l - e 2 ) 1 / 2 ] } / ( l - e 2) 1 / 2 ; 133 F2(e,x) = e sinx/(l - e 2 cos 2 X) + F^e.x) • (4.3) The most favorable position of the sun for the increase A£. occurs when the sun-earth line is normal to the major axis. While the expressions in Equations (4.2) designate the changes in the orbital elements after one revolution, the long-term behavior is determined by repeated rectification and iteration of these results. Figure 4-3 shows the long-term implications of this switching strategy: taking a s a t e l l i t e with the parameter e = 0.0002, i.e. A/m = 5 m /kg, the semi-latus rectum increases ten-fold in less than five years when starting out from geosynchronous altitude. The response is almost insensitive to changes in i n i t i a l eccentricity and solar aspect angle. 4.4 Systematic Increase in Total Energy While the strategy proposed in the previous section is the most ef-fective on-off switching control for increasing the angular momentum, this policy is (in the case of a non-circular orbit) not the most favorable one for increasing the total energy of the s a t e l l i t e . The on-off switching points and v^ representing the zeros of a'(v) = 0 correspond to the instants at which the in-plane component of the solar radiation force is normal to the instantaneous velocity vector, i.e. the tangent to the osculating ellipse: a'(v) = 2a 2 (F • r)/v = 2 s s 2 = 2e ar {u (p sinv - q cosv) + u (1 + p cosv + q sinv)}/(l-e ). x y (4.4) 134 12 £=0.0002 L CJ 0 0=J0. 0 0=0 a..=1 00 0 10 20 e00=0.5 , rj 0 0=0 30 40 months 50 60 Figure 4-3 Behavior of semi-latus rectum in (v-| , v 2 ) switching program 135 Locations of the switching points v 3 and are indicated in Figure 4-1 and can be expressed in the following form (for inclination less than 90°), v 3 = n + ty + 6 3 + 2 i r ( k - l ) , V 4 = TT + TI + ^ - 6 4 + 2ir(k-l ) , .. (4.5) where k denotes the appropriate revolution and 6 3 and 6^ are to be deter-(0) (0) mined by iteration: <$3 = 64 a] = arcsin[e sin(n - OJ)] and 6 ^ = arcsin{e sin(n - OJ) + t a n ( i 2 / 2 ) [ s i n ( 6 ^ n ~ 1 ^ + 2n) + e sin(n + 00)]}, 6 ^ = arcsin{e sin(n - OJ) + t a n ( i 2 / 2 ) [ s i n ( 6 | n _ 1 ^ - 2n) + e sin(n + O J ) ] } , (4.6) for n = 1,2,3,.... This process converges very rapidly (only four iterations are needed for accuracy to four significant decimal places) for an equatorial orbit. While the resulting orbital changes for this switching strategy are determined numerically in the case of an arbitrary orbit, analytical ex-pressions can be obtained for an orbit in the ecliptic plane. In that case, n - OJ equals x = n - to and represents the angle between the sun-earth line and the major axis, Figure 4-1. Writing a-j = arcsin(e sinx) , i t follows that v 3 = p + a-j and v^ = n + TT - aj . The on-phase (v 3,v 4) is less than TT radians i f x lies in (0,TT) and more than TT i f x is in (TT,2TT). If e = 0, i t follows that a-j = 0 and the present control strategy, obviously, coin-cides with the one of the previous section. The response of the orbital elements is obtained by substitution of the limits of integration v 3 and v^ into the integrated results of Equations (3.11). After considerable alge-braic simplification, the changes in the elements can be written as, 136 Aa = 4e a 3 ( l - e 2 s i n 2 X ) 1 / 2 , A£ = 3TT e a 3 e s i n x (1 - e 2 ) 1 / 2 + 4e a2£(l + e 2 s i n 2 X ) / ( l - e 2 s i n 2 x ) 1 / 2 , Ap = - e a 2 G(e,x)siwi - e a 2e 2q sin(2 x)/(l - e 2 s i n 2 x ) 1 / 2 , 2 22 2 2 1 / 2 Aq = e a H(e, x)cosn + e a e p sin(2 x)/(l - e sin x) , 2 Ae = - e a H(e, x)sin X , Aco = e a 2 H(e, X)cos x/e + e a 2 e 2 sin(2 x)/(l - e 2 s i n 2 x ) 1 / 2 , .....(4.7) with the auxiliary function H(e, x) defined by H(e,x) = 3TT(1 - e 2 ) 1 / 2 / 2 + 4 e(l - e 2 ) s i n x / ( l - e 2 s i n 2 x ) 1 / 2 (4.8) It can be shown that Aa is larger while A£ is smaller than the corres-ponding expressions of the previous section when e > 0 and that the results coincide for e = 0. Furthermore, the two switching policies are identical when X = 0 or TT for any eccentricity. While the long-term implications of the present switching strategy can be assessed by repeated rectification and iteration of the results of Equations (4.7), an additional insight into the long-term orbital behavior can be obtained by means of the two-variable expansion procedure. The system of equations considered here has a (partly) discontinuous right-hand-side and is written symbolically as 137 a'(v) e f (a_,v) 0 , v in I , v in I on ' off ' n'(v) = 6 r 2 / £ 1 / 2 , (4.9) where the vector a stands for the set of usual orbital elements, excluding the solar aspect angle p. As usual, the zeroth-order two-variable expan-sion results yield a^ = a^(v) with a^O) = a^Q. The first-order equations are found to be of the form: 9\T -IflCv) + fTa^ v ) , v] -Ii(v) , v in I Q f f ( v ) , 3n-| 9v" •n0(v) + £ 3 / 2/[c e(l + p Q cosv + o 0 sinv) 2] , (4.10) where i t should be emphasized that the limits of the intervals I and ' on I n f f are functions of the slow variable v. In order to eliminate secular contributions to the first-order terms a-|(v,v), average values of the right-hand-sides are required to vanish, yielding f§0 9v I (v) onv ' fLin/v), T] dT/(2ir) 9n 9v 0 a3/2/- w = a n' (v)/c (4.11) The first-order equations for a^ may be determined by a Fourier expansion of the discontinuous, yet periodic, function on the right-hand-side of 138 Equations (4.10), although convergence of the series is expected to be slow. This scheme leads to the following set of equations in a^, x Q = e Q sin(n 0 - aj Q) and y Q = e 0 cos(nQ - con) : a Q(v) = 2a 3(l - X 2 ) 1 / 2 / T T ; x Q(v) = y 0 a 3 / 2 / c £ - x 0 a 2 [ l + (1 - x 2 - y 2 ) / ( l - X 2 ) 1 / 2 ] / T T ; y Q(v) = -x Q a 3 / 2/c e+ y 0 a 2 x 2/ [rr(l - x 2 ) 1 / 2 ] . (4.12) The system of Equations (4.12) was integrated numerically using a double-precision Runge Kutta routine with error control. The solution was found to be in good agreement with the one from the rectification and iteration method: over approximately four years, the results are consistent up to the f i r s t decimal place, Figure 4-4. Eventually, however, they d i -verge. Also shown is the response to the switching in case of a lower orbit, = 0.34, i.e. about 8000 km above the earth. As the gravity force is more dominant here, the advance to higher orbits is much slower. Neverthe-less, geosynchronous altitude can be reached within five years. This would be of interest for future space stations like the SSPS, which are to be constructed in a low-altitude orbit: employing the present switching stra-tegy, these structures could propel themselves to a geosynchronous location. An analytical estimate (i.e. upper bound) for a^iv) is readily ob-tained from Equations (4.12), a n f r ) ' < W O " 4 * a n r A ) V 2 . (4.13) 139 Figure 4-4 Controlled variation of the semi-major axis for ( v ~ , v . ) switching program and optimal control strategy 140 0.6 'oo = 0 £ =0.0002 kJoo= •^ •oo=0 n00=o noo = TT/ n0o=TT fIoo=3TT/2 30 40 months Figure 4-5 Long-term variations in eccentricity during ( v 3 , v 4 ) switching program 60 141 It predicts that an escape trajectory would not be reached before 2 v = TT/(4E 3Q Q ) , i.e. about 625 revolutions or 7 years in the present example, e = 0.0002, anQ = 1 . This crude approximation yields remarkably good values (identical in f i r s t decimal) for the semi-major axis over the f i r s t 3 years (or 550 revolutions) for e n n = 0.1 and even over 4.5 years i f e00 ~ ®' T ^ e s a m e ^ o r m u ^ a c a n a ^ s o b e u s e d f° r predicting the major axis when the orbit is out of the e c l i p t i c , provided that an adjustment is made for the average effective in-plane component of the force, which is accom-plished by multiplying e by the factor 1 - (sin i)/4. For an equatorial orbit this factor amounts to about 0.96 and the approximate formula predicts the semi-major axis correctly up to the f i r s t decimal over the f i r s t 450 revolutions (about two years) for e ng = 0.1 and HQQ = TT/2 as compared to the results of the rectification/iteration procedure shown in Figure 4-4. Figure 4-5 shows the behavior of the eccentricity under the influence of the present switching policy. In general, i t can be concluded that for small eccentricity, the resulting orbit remains near-circular, whereas for an i n i t i a l l y highly e l l i p t i c orbit the eccentricity decreases in the long run. 4.5 Optimal Orbit Raising Although the on-off switching strategy of the previous section proves to be very effective for increasing the total energy.of a solar s a i l , i t is obvious that a judiciously chosen, continually varying, sail setting could be even more effective. Therefore, in the present section, the optimal control strategy yielding the maximum possible increase in total energy per revolution is determined. This is done by means of a numerical steepest-142 1 -j g_ 2Q ascent iteration procedure To save some computational effort, the system of equations is transformed to anautonomous form by introducing the auxiliary elements $, y, K, L and M: cosv sinv sinv -cosv • p • > K = cos(i), L = sin(i) sin(v - ty), M = sin(i) cos(v - ty), The variations in the elements are now described by the system: (4.14) a'(v) = = 2a 2 £{F(a) + $ ) 2 + F v(a)/(1 + *)} ; *'(v) = = -V + 2£2 F y(a)/(1 + <D)2 ; H"(v) : = $ + £ 2{F Y(a) + F (a) + $)}/(l x y + $) 2 ; £'(v) = = 2£3 F y(a)/(1 + $ ) 3 ; f ( v ) -= * 3 / 2 / 0 + * ) 2 ; K'(v) = = -F z(a) £2L/(1 + $ ) 2 ; L'(v) -= -M + F z(a) £2K/(1 + $ ) 3 ; M'(v) = = L . (4.15) The dependence of £ upon the control angles a = (a,3) is due to the fact that the normal to the plate is a function of a: _un = u.n(a). The system of Equations (4.15) is denoted by a_' (v) = e S_(a_,a) for convenience. 1 4 3 The various steps involved in determining the optimal control function a* yielding the maximum value for a(2Tr) may be briefly described as follows. Fi r s t , a reasonable starting control function OQ ( V ) is chosen and the cor-responding response vector a_(a_Q,v) is calculated by means of (Runge-Kutta) integration of Equations ( 4 . 1 5 ) with i n i t i a l conditions a_(0) = a ^ . The results are stored in a two-dimensional array containing the elements a.(aQ,v), j = 1 , 2 , . . . , 8 at v = 2TT k/n, k = l,2,...,n with n taken as 3 6 0 to start with. It may be noted that i t is not necessary to take n very large or to perform a highly precise integration in this f i r s t run for OQ ( V) is usually not near the optimal control. Since the objective is to determine a more effective strategy than O Q ( V ) , the influence of small variations in OQ ( V) is studied. The near-by control a(v) = OQ ( V ) + <5a(v), with the norm ||<5a|| (defined as the integral over (0,2TT) of the dot-product of 6a(v) with i t s e l f ) small and prescribed, is con-sidered. An estimate for the difference in the final value of the semi-major axis for this new control function as compared to the final response for OQ is found by means of a first-term Taylor expansion of Equations ( 4 . 1 5 ) around a = a^, yielding: (2TT 2 . 6a(2ir) = • [ I A.(T) SCX.(T)] dr , ( 4 . 1 6 ) JO j=l J J where the influence functions A-(v) are defined by 8 A,(v) = I U v ) 3 k=l K 9g k J j = 1,2, (4.17) a j a ^ v ) with the vector of adjoint variables X(v) determined from the system of equations: 144 8 39, X k (v ) = - J X.(v) i=l 9a k Ja_(an,v) k = 1 , 2 , . . . , 8 , ( 4 . 18 ) and f i n a l condit ions A-|(2TT) = 1, AJ(2TT) = 0 , j = 2 , 3 , . . . , 8 . One would l i k e to know: which v a r i a t i o n of the c o n t r o l , 6 a ( v ) , leads to the maximum poss ib le change in response, 6 a ( 2 i r ) , given in Equations ( 4 . 1 6 ) , under the const ra int that the s teps ize | | 6 a | | i s prescribed? The answer is obtained through Lagrange m u l t i p l i e r s , y i e l d i n g the fo l lowing control s t rategy: Upon subs t i tu t i on of th i s r e s u l t into Equation ( 4 . 1 6 ) , 6a_(.2Tr) i s wr i t ten in terms of the norms ||<5a|| and ||A||. Subsequently, ||6a|| can be el iminated from Equation ( 4 . 1 9 ) , expressing the va r i a t i on of the control angles e x p l i c i t l y in terms of the prescr ibed increase in the semi-major ax i s . For the c a l c u l a t i o n of the in f luence funct ions , Equation ( 4 . 1 7 ) , the der i va t i ves of the r i ght -hand-s ide of Equations ( 4 . 15 ) with respect to a l l s tate var iab les as well as the control angles are needed. This i s a s t ra ight forward, though very ted ious , process. With these re su l t s in hand, the equations for X_ are known, Equation ( 4 . 1 8 ) , and these are integrated backwards by means of the Runge-Kutta rou t ine , using a piecewise constant approximation fo r the state var iab les stored in the array mentioned before. Now, the in f luence funct ions are also known and the new contro l f unc t i on is determined from Equation ( 4 . 2 0 ) . Subsequently, the whole procedure is repeated. While th i s process read i l y leads to a near-optimal c o n t r o l , con-6a ( v ) . = {|16a|| / | |A| \] ' A(v) . ( 4 . 19 ) ( 4 .20 ) vergence becomes progressively slower near the optimum and special care must be taken in this region. It was found that by coupling the stepsize and the error parameter of the integration to the length of the 'gradient' ||A||, reasonably accurate results could be obtained within about 40 ite-rations, which amounted to less than a minute of the computer time. Note that ||A|| approaches zero as a - * a* . The results of the iteration program for an orbit in the ecliptic plane (3 is taken zero, here) and a solar sail with perfect specular re-flection are shown in Figure 4-6. Starting out with the control function indicated by N = 1, the response a(2TT) grows rapidly during the f i r s t few iterations while the control program approaches the optimal strategy. The convergence is notably faster in the highly sensitive region near v = TT/2, even though the changes in a have been subdued here (by means of a weigh-ting function) in favor of those near v = 3TT/2. Figure 4-6b shows the optimal orientation of the sail at a few points in the orbit. In the exe-cution of the program, the two sides of the sail are taken to be identical. In case the properties on the two sides are different, a rotation over 180° of the sail will be required at v = 3TT/2. It is interesting to compare the effectiveness of the on-off swit-ching policy, Section 4.4, and the plate having different r e f l e c t i v i t i e s on either side, Section 3.5, with that of the optimal control strategy esta-bished here. Taking e = 0.0002, the increase in semi-major axis after one revolution for the various controls is summarized in Table 4.1. V Figure 4-6 (a) Optimal control strategy for maximization of Aa; (b) Corresponding optimal orientation of solar sail 147 Table 4.1 Comparison of Control Strategies (e = 0.0002) Strategy e00 p f pb Aa x 104 % of optimal Optimal Control 0 1 11 0 100 On-Off Switching (vg, v^) 0 1 8 0 73 a = 90° (Local Vertical), On-Off 0.1 1 6 2 56 a = 45°, On-Off 0.1 1 4 9 45 a = 90°, Different Reflectivity 0.1 1,0 3 1 28 Judging from Figure 4-6, a linear approximation to the optimal control would be given by a(v) = (v - n - TT/2)/2, which remains within 10 degrees of the optimal angle at all times. The response for this steering program was cal--4 culated and an increase in the semi-major axis of 10.7 * 10 was obtained, amounting to about 97% of the optimal value. Finally, i t must be emphasized that the optimal control strategy de-termined here is valid for one revolution. For the following orbits, the best control would have to be determined using the particular i n i t i a l condi-tions involved. Naturally, to obtain a long-term valid optimal control and corresponding response would take considerable amount of computer time. Fortunately, some idea about the long-term effectiveness of the optimal strategy may be obtained by resorting to the approximate result of Section 4.4. Presuming that the ratio of 100/73, for the increase in semi-major axis of the optimal as compared to the (vg, v^) on-off switching strategy, will be maintained throughout, Eauation (4.14) with e adjusted accordingly yields an estimate for the long-term effectiveness of the optimal control 148 program. The result is depicted in Figure 4-4 along with those of the on-off switching trajectories. 4.6 Orientation Control of the Orbital Plane In this section, the f e a s i b i l i t y of controlling orientation of the orbital plane by an on-off switching strategy is investigated. In the beginning, results for the case where the force is acting continually are interpreted so as to obtain a physical appreciation as to the nature of the solar radi-ation effects upon the orientation of the orbital plane. Note that pertur-bations of the osculating plane can be visualized by means of the rotation vector wr = e(u^/!?^)r_, affecting the direction of h_ through h_ = wr* h_, so that h_ rotates instantaneously in a plane normal to the radius vector. In terms of the independent variable v, the rate of change of the vector h_ is written as h'(v) = fi/v = e r 2 u^(r x h)/l = e r 3 u^ (sin<J> i - cost}) j j / £ 1 / 2 , — — z — — z —n n (4.21) where i ^ and j ^ are unit vectors along and normal to the line of nodes in the osculating plane, respectively. While Equation (4.21) represents the instantaneous rate of change of h_, i t is interesting to calculate the total variation in Rafter one f u l l revolution of the s a t e l l i t e . A first-order approximation Ah_ is obtained by integrating the right-hand-side of Equation (4.21) from v = 0 to 2TT keeping the slowly changing orbital elements constant. The vector Ah_ is expanded in its components along the x n and y n axes: Ah_ = Ah-, i ^ + Ah ? j_ n , yielding: 149 Ah1 = e u sz I2 (•2TT sin(v - ty)dv 0 (1 + p cosv + q sinv)' , s 5/2 -3TT e u z a e since Ah2 = -e u z I2 2TT cos(v - ip)dv 0 (1 + p cosv + q sinv) o s 5/2 = 3TT e u a e cosco . (4.22) The changes in orientation of the orbital plane can be visualized in terms of the vector Ah_ (Figure 3-1). Also, perturbations in the orbital elements 1 /2 i and Q can be expressed in terms of Ajr_: Ai = - Ah,,/£ and sin(i) AQ = 1/2 Ah-j/£ , where Ai and Afi are treated as infinitesimal angles. It is evident that, in case of a circular orbit, the net effect of the solar radiation torque on the direction of h_ must vanish after one revolution, since the effective component of the torque at any position v is equal in magnitude but opposite in sign to the one at v + TT (in the first-order ap-proximation). For an e l l i p t i c orbit, variation in the orientation of the orbital plane depends upon the argument of the perigee with respect to the line of nodes: e.g. i f to = 0 or TT, only the inclination will be affected (provided that n and i are not 0. or TT), whereas for oo = TT/2 3TT/2, the resulting perturbation consists of a pure precession (or regression) of the 1i ne of nodes. The changes Ai and Afi obtained by continuous exposure to sunlight pressure are small in the long run, especially for near-circular orbits (of the order e e). In order to obtain more significant changes in i and Q , two on-off switching strategies are proposed and their effectiveness as to the nature and magnitude of the variations in the orbital elements is assesed and interpreted. 150 4.6.1 Control of the inclination 1/2 Since Ai = -hh^/l , i t is seen from Equations (4.22) that the inclination would increase continually i f the following switching strategy is adopted: i f u z > 0 : on i f ty - TT/2 < v < ty + TT/2 ; i f u^ < 0 : on i f ty + TT/2 < v < ty + 3TT/2 . (4.23) The condition for the sign of u z is easily translated in terms of the quadrants of the angles i and p . The resulting changes in i and ft after one on-off cycle can be determined using the integrals of Appendix I evaluated over the on-interval. In terms of j = e cosco and k = e sinoj, the results can be written in a compact form as follows: Ai = e a 2 |uz| {3 - (1 - e 2 ) / ( l - k 2) - 3J[TT/2 - arctan{j/(l - e 2 ) 1 / 2 } ] / ( l - e 2 ) 1 / 2 }; sin(i) Aft = e a 2 |u*| k {j[3 + 2(1 - e 2 ) / ( l - k 2 ) ] / ( l - k 2) -3[TT/2 - arctan{j/(l - e 2 ) 1 / 2 } ] / ( l - e 2 ) 1 / 2 }. (4.24) This particular control program changes the inclination appreciably, while leaving the longitude of nodes virtually untouched for near-circular orbits. Note that for near-circular orbits, the change Aft is half of that obtained for the case of continuous exposure (Equations 4.22). It is evident that by taking the opposite strategy of Equations (4.23), i.e. replacing the on-phase by the off-interval,the results of Equations (4.24) would change sign and the inclination decreases. 151 Figure 4 - 7 illustrates the long-term effectiveness of the proposed control strategy as found by repeated rectification and iteration of the results in Equations ( 4 . 2 4 ) . Starting out from the equatorial plane (I'QQ = 2 3 . 4 5 ° ) , about two degrees per year may be added to the inclination for an Echo-type sat e l l i t e in an i n i t i a l l y circular orbit. For orbits with large eccentricity, the rate of change of inclination is much higher (about 1 0 degrees per year for eg n = 0 . 5 ) . Also, with an increase in iQ Q (up to iQQ = 9 0 ° ) , the increase in inclination becomes larger as exemplified by the curve for i n n = 6 8 . 4 5 ° , i.e. the i n i t i a l orbital plane is 4 5 ° above the equator. It is of interest to assess the changes in the other elements under this control strategy. The resulting behavior of the eccentricity is shown in Figure 4 - 8 . As a general rule, i t may be concluded that the eccentricity increases steadily until the orbit is normal to the ecliptic plane when i t starts declining. The major axis (not shown) decreases at a rate between 0 . 1 and 0 . 1 5 per year and the smaller the i n i t i a l eccentricity, the larger the decline in the semi-major axis. 4 . 6 . 2 Control of the line of nodes For the line of nodes to exhibit a steady precession, i t is neces-sary that Ah-|/sin(i) > 0 leading to the proposed switching strategy: i f sin(p - ft) > 0 : ' on i f ifj < v < IJJ + TT ; i f sin(p - Q) < 0 : on i f <|; + T T < V < ^ + 2 T T ( 4 . 2 5 ) The changes in the elements after one revolution under this control strategy are determined using the integrals of Appendix I evaluated over the relevant interval: 152 1 5 3 154 Aft = e a 2 |sinn| {3 - (1 - e 2 ) / ( l - k 2) -3k[Tr/2 - arctan{k/(l - e 2 ) 1 / 2 } J / ( l - e 2 ) 1 / 2 } ; Ai = e a 2 sin(i) |sinn| j{k[3 + 2 (1 - e 2 ) / ( l - k 2 ) ] / ( l - k 2) -3[TT/2 - arctan{k/(l - e 2 ) 1 / 2 } ] / ( l - e 2 ) 1 / 2 } . (4.26) This strategy produces a substantial change in the longitude of nodes, while the changes in inclination are relatively small. Also for near-cir-cular orbits, the change in inclination is only half that of the continuous exposure. Figure 4-9b shows the effectiveness of the proposed switching stra-tegy: for an Echo-type s a t e l l i t e , the line of nodes may precess by as much as five degrees per year, double the amount of the natural perturbations, Section 3.3.3. On the other hand, the behavior is not very sensitive to changes in the i n i t i a l eccentricity or inclination. Figure 4-9a illustrates the accompanying variations in the eccentricity: in general, the eccentricity decreases for highly e l l i p t i c orbits but increases for i n i t i a l l y near-circu-lar trajectories. By following the opposite strategy of Equations (4.25), the line of nodes could be made to regress instead of advance. 4.7 'Half-Yearly Switching Another interesting strategy for achieving f a i r l y large changes in the orbital elements is by switching off after a half-year instead of a half-period. The eccentricity and inclination are essentially periodic functions with a half-year period. By switching off just when an element has reached its maximum and subsequently, switching on a half-year later just when the 155 0.6 - ^ 0 0 = 0 a00=1 " f l 0 o= 1V2 u)oo=A00=0 - T l o o = TT £ = 0 . 0 0 0 2 /i0o=68.4 Figure 4-9 (a) Behavior of eccentricity for switching program of Equations (4.25) 20 o £1 15 10 5h n0=o Tloo =TT/2 a o o = 1 i =0.0002 co00=n00=o 00 68.4 o e00=0 7 e00=0.5»i = 23.4 © 0 0 = 0 O -J 2 0 0 4 0 0 6 0 0 8 0 0 1000 1200 Days Figure 4-9 (b) Controlled change in position of line of nodes 156 up-hill phase starts again, sizable orbital changes can be achieved. Figure 4-10 illustrates this concept for an i n i t i a l l y circular, equatorial orbit. The average rate of increase of inclination is approximately half of that attained by the switching strategy of Section 4.6.1, while the in-crease in eccentricity is approximately the same in the long run. As be-fore, the changes in inclination and eccentricity increase for larger i n i t i a l eccentricity. Whereas the effectiveness of this approach upon changes in inclina-tion is undoubtedly inferior to the strategy described in Equations (4.23), the benefit of a much lower frequency of switching (2 vs. 365 per year) could become a decisive factor in a practical situation. 4.8 Concluding Remarks Important aspects of the analyses presented in this chapter may be summarized as follows: (i) A few switching programs are explored and their effectiveness in achieving orbital changes established. ( i i ) Whereas apogee-perigee switching does not lead to readily predic-table results, the sun-earth line switching achieves a rapid in-crease in angular momentum and thus the semi-latus rectum. ( i i i ) Switching when the velocity is normal to the direction of radiation is particularly effective, since i t is the best on-off switching strategy in terms of adding energy and, consequently, increasing the major axis. Under this strategy, an Echo-type s a t e l l i t e may in-crease its major axis by a factor of ten in five years, starting 158 from geosynchronous altitude. (iv) The optimal time-varying orientation of a solar sail for maximum increase in semi-major axis per revolution is established. This should be of importance for raising a solar sail into a helio-centric orbit. (v) Two switching programs for controlling orientation of the orbital plane are proposed and analysed. One strategy leads to appreciable changes in the inclination, while the other produces a precession of the line of nodes. (vi) Changes in eccentricity and inclination in a half-yearly switching policy are relatively less pronounced, but the benefit of a much lower number of switching points could be attractive. 159 5. HELIOCENTRIC SOLAR SAILING WITH ARBITRARY FIXED SAIL SETTING 5.1 Preliminary Remarks Whereas up to now the effects of solar radiation forces upon geocen-tr i c orbits were studied, in this and the following chapter, the attention is focused on heliocentric orbits. For many deep-space missions, the solar sail constitutes a viable option since i t derives its motive power from an unremitting source of energy. The combination of useful payload and solar 2 sail leads to an area over mass ratio in the range of 50 to 200 m /kg with 2 characteristic accelerations between about 0.5 and 2 mm/sec . This chapter studies the solar radiation effects on the orbital be-havior of an arbitrarily shaped spacecraft (or a solar sail in particular) in a general fixed orientation with respect to the local coordinate frame. While a constant orientation is not necessarily the best possible setting in an actual mission, a thorough understanding of its response would f a c i -l i t a t e the assessment of its potential as a function of area/mass ratio and i n i t i a l conditions. While exact solutions in the form of logarithmic spi-51 52 rals have been established in the literature ' for planar orbits, the analysis presented here is extended to general three-dimensional trajecto-ries. Moreover, the parameters of the trajectories are expressed analyti-cally by means of asymptotic series in terms of the solar parameter and a spacecraft with arbitrary material characteristics is considered. When the out-of-plane component of the thrust is kept constant, the orbital plane i t s e l f exhibits a precessional motion, returning to its original orientation after l i t t l e less than one revolution. An effective out-of-plane spiral 160 transfer trajectory is obtained by reversing the force component normal to the orbital plane at specified positions in the orbit. By choosing the appropriate control angles for the sail orientation, any point in space can be reached eventually by this three-dimensional spiral trajectory. Whereas a very specific i n i t i a l velocity vector is required for em-barking upon the spiral trajectory, other orbits emanating from different i n i t i a l conditions may also be of interest. Hence, a three-dimensional short-term solution is presented for arbitrary i n i t i a l conditions. Subse-quently, the long-term behavior is analysed by means of the two-variable expansion procedure yielding an implicit expression for the eccentricity. By iteration, the solution can be determined up to the desired accuracy. For not too large values of the i n i t i a l eccentricity, asymptotic expansions 5 up to the order e are derived. The other orbital elements are expressed in terms of the eccentricity and can be evaluated up to the desired accuracy. Higher-order terms may become important when the area/mass ratio is large. Equations for the higher-order terms can be derived. While the periodic part of the solution can be evaluated readily, secular terms can be deter-mined analytically only for a circular i n i t i a l orbit. 5.2 Formulation of the Problem An inertia! X,Y,Z reference frame with origin at the centre of the sun is introduced in Figure 5-1 a where the X axis points to the i n i t i a l position of the spacecraft and the X,Y plane constitutes the i n i t i a l oscu-lating plane, usually the ec l i p t i c . The Z axis is aligned with the i n i t i a l angular momentum vector. In addition, a local ?Q»HQ>CQ reference frame moving along with the spacecraft is introduced: the nn a r |d Kn a x e s v = 2 n k [a] v = * + i|i Figure 5-1 (a) Configuration of the sun and solar sail in a heliocentric trajectory; (b) Successive rotations a , 3 (and y) for defining arbitrary orientation of solar sail 162 point along the local v e r t i c a l , local horizontal and orbit-normal directions, respectively. Any desired orientation of the solar sail in the g^'^ O'^ O can be described by three successive Eulerian rotations (Figure 5-1b). Taking, i n i t i a l l y , the outward normal to the sail to be directed along the CQ axis, a f i r s t rotation a about the ?g axis produces the ^ ,n-j ,?•] frame and brings the solar sail to the required line of intersection with the or-bital plane. A subsequent rotation 3 about the n-| axis yields the £,n>C frame and moves the normal to the sail out of the orbital plane to its pre-scribed orientation. A final rotation y about the normal (£ axis) could be performed for attaining the proper attitude of the sail in its r\,z, plane without affecting the resulting solar radiation force. The components of u_n taken along the local o^'^ O'^ O a x e s depend o n a a n c* $ only: u_n = (cosa cos3» sina cos3, - sin3). (5.1) For many sat e l l i t e s , solar panels form a substantial portion of the total surface area. This would particularly be so for a spacecraft designed to be propelled by solar radiation pressure. Hence, in these situations, only the area of solar panels or sails needs to be considered. In general, the spacecraft is modelled by a number of surface components, characterized by their own material parameters and orientation. In nondimensional form (unit of length equals a g = 1 A.U. and unit of time is 1/(2TT) year), the solar radiation force upon an arbitrary space structure of n homogeneous, illuminated surface components A^ in an heliocentric orbit is written as - • es j , K • M_S I f l k u« • [ a 2 k • pk(u£ • u5)] u£} A k / r 2 , (5.2) 163 where the physical force is nondimensionalized through multiplication by a e/(y sm). The small parameter eg denotes the ratio of solar radiation and attraction forces, es = 2S«(A/m) (a 2/y s) = 1.57 x 10"3 (A/m) . (5.3) This illustrates that the parameter es is about 200 times as large as the parameter e in Equations (2.5) for geocentric orbits. It may be mentioned that the solar constant and hence the radiation force in Equation (5.2) varies inversely as square of the distance from the sun. This is because the total radiant energy emitted by the sun in a given time equals that passing through any concentric spherical surface around the sun in that time 2 (taking the rate of energy output constant). Writing F_ = esR/r with auxi-liary vector R = (R,S,T) the components of R_ can be evaluated for an arbitra-r i l y shaped spacecraft using Equation (5.2): R = ][ cosa k cosB k | a l k + [o^ + P k cosa k cosB k] cosa k cosB k| A k ; n 2 S = I sina k cosa k cos Bk I a 2 k + p k cosa k cosB k] A k ; k 1 n T = ~l cosa k cosB k sinB k I c ^ + p k cosa^ cosB kJ A k . (5.4) k 1 The normal u^ to a surface element A k, k = l,2,...,n is taken in such a manner that its projection along the radiation is always positive. In general, when T ^ 0, the plane of the orbit will be subjected to changes in its orientation. The motion of the local 'n0'^0 ^ r m e relative to the inertial X,Y,Z frame is described in terms of the rotation vector W = wr + v , effecting an uncoupling of the in-plane and the out-of-plane perturbations. The equations of motion are similar to those obtained earlier (Equations 3.5) with F_ to be replaced by e g u R_. Since the equation for fi'(v) contains a singularity for i = 0, a formulation in terms of the unit vector K_ directed along the inertial Z axis with components M = sin(i) sin(v - ty), I = sin(i) cos(v - ty) and K = cos(i) along the local O^'^ O'^ O a x e s 1 S f a v o r e d - As t n e l° c al O^'^ O'^ O ^ r a m e m o v e s along with the spacecraft in its orbit, the vector _K(v) traces a path upon the sphere (_K • K) = 1 in the ?Q,n0'^0 f 1" 3 1 1 1 6- T n e o r D i t a l elements i and ty can be determined quite readily from the vector _K. The complete system of equa-tions is written as: u"(v) + u(v) = (1 - esR)/£(v) - S u'(v)/[u(v) £(v)] ; £'(v) = 2esS/u(v) ; M"(v) + M(v) = e s T K(v)/[u(v) l(v)] ; K'(v) = -e s T M'(v)/[u(v) l(v)] . (5.5) The f i r s t two equations f u l l y describe the in-plane perturbations and the latter two equations define the orientation of the osculating plane. The component L(v) can be shown to be equal to M'(v). The i n i t i a l conditions for the system of Equations (5.5) are written as £(0) =• £ n n ; u(0) = (1 + e Q 0 cosoi 0 0)/£ 0 0; u'(0) = ( e Q n sinu 0 0)/£ n o; K(0) = 1 and M(0) = M'(0) = 0. In a few particular situations, exact solutions for the system of Equations (5.5) can be established: in the case where the component S vanishes (e.g. when the normal to the solar sail lies in the Sn,Cn plane 165 or when all of the radiation is absorbed), solutions for the orbital motion can be obtained using the classical Keplerian procedure. After modification of the sun's gravitational parameter to account for the apparent reduction in attraction because of the solar radiation force, the trajectory for the case S = 0 can be written as u(v) = [1 + e cos(v - co )]/SL with modified p p p orbital elements £ p = aQQ/{1 - ^ R ) , e p = [ e ^ + 2es p 0 QR + e|:R ] / (1 - egR) and co = arctan[qQp/(PQQ + £SR)]> where a l l angles are measured in the osculating plane. Another, more interesting, exact solution arises when the i n i t i a l velocity vector satisfies a prescribed condition leading to a trajectory in the shape of a logarithmic spiral. 5.3 Three Dimensional Spiral Trajectories The spiral trajectory of the form r(v) = r ^ exp(c sv) emerges from Equations (5.5) when one looks for solutions having the properties that the product u(v) £(v) remains constant, say C, and u'(v) = -c u(v) at a l l times. The constants c g and C can be evaluated from Equations (5.5) after substitution of these two relations: c s = {(1 - esR) - [(1 - e sR) 2 - 8 s 2 S 2 ] 1 / 2 } /(2e s S) = 2es S | l + e s R + e 2(R 2 + 2S2) + e 3 R(R2 + 6S 2)} + O(e^) ; C = 2es S/cs = (1 - esR) | l - 2e 2 S 2 [1 + esR + e 2 (R2 + 2S 2)]| + O(e^) • (5.6) Taking r(0) = r n n , the complete in-plane and out-of-plane solutions of 166 Equations (5.5) can be expressed in terms of c g and C: r(v) = r Q Oexp(c sv) ; i{v) = C r(v) = C exp(c sv)/r 0 Q ; M(v) = B {1 - cos (1 + B 2 ) 1 / 2 v }/(l + B2) ; L(v) = B sin[(l + B 2 ) 1 / 2 v ] / ( l + B 2 ) 1 / 2 ; K(v) = {1 + B 2cos[(l + B 2 ) 1 / 2 v ] } / ( l + B2) . (5.7) Here the constant B stands for e T/C. It is seen that the radial distance s takes the form of a logarithmic spiral (outward i f S > 0 and inward for S < 0), while the orbital plane exhibits a periodic wobbling motion with 2 1/2 maximum inclination at v = TT/(1 .+ B ) . This is of practical interest for a solar sail since i t predicts that no secular changes in the orbital orientation are induced by a constant force component normal to the plane of the orbit. It must be emphasized that the spiral trajectory arises only when the spacecraft possesses the right velocity vector at injection. Its radial and circumferential components are given by r = c ( C / r ) ^ 2 , rv = ( C / r ) ^ 2 , and the spiral angle a g equals arctan(c s). Additional insight into the nature of the trajectory is provided by studying the osculating ellipses of the spiral. The eccentricity and perigee position at any point v-j are given by e 1 = i c 2 exp(-2c sv 1) + (1 - C ) 2 ] 1 / 2 , co-j = v-j - T + arctan'fc exp(-c v-j )/(l - C)] , (5.8) 167 so that the equation for the osculating ellipse at v = can be written as r(v-j) = C exp(c sv-|)/[l + e^  cos(v-j - to-|)]. It is of interest to note that the eccentricity which is of the order e g to start with, decreases slowly attaining the limiting value egR + 0( e ) as v-| -> °°. This is of considera-ble importance since i t predicts that a spaceprobe may be released from a spiral solar sail trajectory into a near-circular heliocentric orbit at any time. As to the position of the perigee of the osculating ellipses, i t f o l -lows that to-j follows v-j steadily, lagging behind by an angle of between TT/2 and TT radians in case S > 0 and between TT and 3TT/2 radians i f S < 0. As the spacecraft moves along its trajectory, the angle between the radiusvector and osculating perigee will increase (for S > 0) or decrease (S < 0) slowly unt i l , f i n a l l y , -> v-j - TT for v-j -> °°. An explicit expression in terms of the solar sail parameters can be obtained for the time history in the spiral trajectory, t(v) = ^ [ r 2 ( T ) / £ 1 / 2 ( x ) ] dt = r ^ 2 [exp(3csv/2) - l ] / c t , wi th c t = 3 sign(S)/2 {(1 - e sR) - [(1 - e s R ) 2 - 8e 2 S 2 ] 1 / 2 } 1 / 2 . (5.9) The radial distance as a function of time follows by combining Equations (5.7) and (5.9), r(t) = r o n (1 + c t t / r ^ 2 ) 2 7 3 . (5.10) This result is valid for both outward (S > 0) and inward (S < 0) spirals. To obtain the most favorable sail setting for reaching the maximum radial distance at any time t, the coefficient is maximized as a function of the rotation angles a and 6 . It follows that the maximum occurs when 6 168 vanishes, producing a planar trajectory. The value of a is determined from 8 E S S l l + - £ s R > - fC " S R ' 2 " 8 e s s 2 l 1 / 2 > If- 0- — . ( 5 . 1 1 ) Since the exact solution of this implicit equation for a can not easily be found, i t is useful to determine subsequent levels of approximation for a written as an asymptotic series in the small parameter e g : a = + e a-, + 2 e oi^ + ... . The equations for a . , i = 0 , 1 , 2 , . . . , can be derived by sub-stituting the series into Equation ( 5 . 1 1 ) , developing the relation in terms of a Taylor expansion around a = a p and requiring that a l l coefficients of e " , n = 0 , 1 , 2 , v a n i s h . After a considerable amount of algebra, the f o l -lowing asymptotic representation for a is found (taking a sail with - 0 ) : a = a r c s i n ( 3 " 1 / 2 ) - e s 3 1 / 2 ( a - , + 2 p ) / 3 6 - e 2 2 1 / 2 ( a - , + 2 p ) ( 5 p + 4 3 a / 6 ) / 2 8 8 - e 3 3 1 / 2 ( a ] + 2 p ) / 2 {1261 a 2 + 1 252 p 2 + 1 5 8 8 pa-, + 72 a ] + 144 p ) / ( 3 6 ) 3 + 0(e*). . . . . . ( 5 . 1 2 ) Subsequently, an explicit relation for the spiral angle a g corresponding to the optimal orientation is found by substituting the optimal angle into c , Equations ( 5 . 6 ) . Expansion for small e $ yields: a $ = 4 p e s 3 1 / 2 / 9 {1 + 6 1 / 2 e ^ a - , + 2 p / 3 ) / 3 + el[(o} + 2 p ) 2 / 4 8 + 2 ( 0 ] + 2 p / 3 ) 2 / 3 + 8 p 2 / 8 1 ] } + O ( e ^ ) ( 5 . 1 3 ) In Figure 5-2a, the optimal orientation of the solar sail as well as the corresponding spiral angle have been plotted for various values of the cm/ 2 ,_2 /sec (b) Actual planar spiral trajectory for e g = 0.15 and p = 1 170 r e f l e c t i v i t y p. For low values of e^, the optimal orientation can be taken as 35.26°. It is evident that the spiral angle approaches zero for p •+ 0 since the case p = 0 corresponds to a closed trajectory. Figure 5-2b illustrates an example of a planar spiral trajectory, showing the spi-ral angle and the orientation of the s a i l . The value of es taken here (0.15) would correspond to A/m of about 100 m /kg. It is interesting to calculate the optimal radial distance over a long duration of time, showing the effectiveness of the spiral trajectory in near-circular orbital transfer, for a few values of the solar parameter e . The results are summarized in Figure 5-3 for both inward and outward 2 spirals. In case e s = 0.015, i.e. A/m = 10 m /kg, the orbit of Mars could be reached within 9 years and Venus in 4 years. For higher values of e g the opportunities increase rapidly: even a long journey to the distant pla-2 net Uranus may be feasible i f a solar sail with A/m of the order of 400 m /kg could be constructed. The analysis remains valid when the component T of the force is non-zero: the position and velocity vectors of the spiral trajectory l i e in the osculating plane in that case. The orientation of the orbital plane descri-bed by the angles i and ty follows from Equations (5.7): i(v) = arccos {1 - 2 B 2 s i n 2 [ ( l + B 2 ) 1 / 2 v / 2 ] / ( 1 + B2)} ; ty(v) = v - arctan {tan[(l + B 2 ) 1 / 2 v / 2 ] / ( l + B 2) 1 / 2} . (5.14) Expansion of ty[v) for small e g leads to ty{v) = v/2 + 0 ( e ) , 0 £ v <_ 2TT, so that the line of nodes precesses at approximately half the orbital rate. years Figure 5-3 Potential for near-circular interplanetary transfer by solar sail for a few values of e 172 2 1/2 The inclination reaches its maximum at v = TT/(1 + B ) and returns to 2 1/2 zero at 2TT/(1 + B ) , while T|;(V) shows a discontinuity of .TT radians 2 1/2 at v = 2TT/(1 + B ) . Figure 5-4a shows the orientation of the osculating plane at a few points in the orbit. 5.4 Out-of-Plane Spiral Transfer In order to obtain a net increase in inclination after one revolution, the orientation of the sail would have to be changed during the orbit. Ob-vious switching points would be the instants when i(v) is stationary, i.e. 2 i / 2 at v-| = TT/(1 + B ) and - 2v-|. Assuming the switching to take place instantaneously from -3 to +3 (without affecting the control angle a) and repeating the procedure during each successive revolution, the out-of-plane component T becomes T |T| , 3 < 0; v?_. < v < v 2 j + 1 ; l T l ' 3 > ° ; V2j+1 < V < v 2 j + 2 ! (5.15) for j = 0,1,2,..., and the switching points v k = k-ir/O + B 2 ) 1 / / 2 , k = 0,1,2,... .. Since the operation takes place instantaneously, the force components S and R remain unchanged throughout. Writing = M(v^) and = K(v^), etc., k =' 0,1,2,..., the solution i(v) is found by repeated application of the results in Equations (5.7): pjrccos{K2j + |B|(M2j - |B|K 2 j)(l - cos[(l + B 2 ) 1 / 2 v ] ) / ( l + B2)} i ( v ) = arccos{K 2 j + 2|B|(M2j - |B|K 2 j)/(l + B2) - |B|[(3B2 - 1 )M y + |B|(3 - B 2 ) K 2 j ] ( l + cos[(l + B 2 ) 1 / 2 v ] ) / ( l + B 2) 2} , (5.16) fi < 0 , T > 0 s w i t c h i n g ec l ipt ic p lane osculat ing planes Figure 5-4 (a) Orientation of the osculating plane as affected by a constant force normal to i t ; (b) Switching strategy leading to a systematic increase in orbital inclination 174 where the former r e l a t i o n holds for v ^ j < v '< v 2j+ l sind. the l a t t e r for v 2j+ l < v < v 2j+2 ' ' ' 0 W 1 ' n 9 recurrence re l a t i on s for K 2 j and M^j can be es tab l i shed: M 2 j = -{4|B|(1 - B 2 ) K 2 j _ 2 + [4B 2 - (1 - B 2 ) 2 ]M 2 j _ 2 ) / ( 1 + B 2 ) 2 , K 2 j = " { [ 4 b 2 " ( 1 " B 2 ) 2 ] K 2 j - 2 " 4 | B I ( 1 " b 2 ) M 2 J - 2 } / ( 1 + 8 2 ) 2 ' (5.17) with j = 1,2,3,. . . and MQ = 0, = 1. A long-term l i n e a r approximation O "1/9 fo r i ( v ) , i ( v ) = 2e |T|(1 + B ) ' V / T T , provides a good estimate as long as e g i s s u f f i c i e n t l y smal l . The l i n e of nodes, i . e . the i n te r sec t i on of the instantaneous o r b i t a l plane and the X,Y plane is located at v = v-| - TT/2 = TT/2 + 0 ( e s ) when the f i r s t switching takes p lace. It returns to th i s po-s i t i o n at a l l switching points while s l i g h t l y dev iat ing from th i s l i n e in between. Through Equation (5.9), the switching instants are also known in terms of time. The foregoing ana lys i s i s v a l i d for any f i xed s a i l o r i en ta t i on de-signated by the contro l angles a and B. Since the rate o f increase in i n -c l i n a t i o n i s proport ional to the magnitude of the force component |T|, the most e f f e c t i v e ( f ixed angle) strategy i s the one which maximizes |T|, i . e . a = 0 and .|3j = a r c s i n ( 3 " 1 / / 2 ) = 35 .26° . In th i s case S = 0 and the t r a -j ec to ry i s a degenerate sp i ra l maintaining a constant distance from the sun ( so -ca l led 'cranking o r b i t 1 ) . The behavior of the i n c l i n a t i o n f o r th i s case is i l l u s t r a t e d in Figure 5-5c for a few values of e . While i t would take 3 s about 14 years to make a f u l l 180° swing through space at 1 A.U. from the sun, the durat ion would be less than 5 years at 0.5 A.U. (taking e $ = 0.15). An obvious app l i ca t i on of three-dimensional sp i ra l t r a j e c t o r i e s in 175 conjunction with switching would be in a transfer mission where both i n c l i -nation and radial distance are to be changed. From this consideration, i t would be interesting to determine the most efficient orientation of the sail for a near-circular out-of-plane transfer with the final radial distance prescribed and the inclination to be maximized or, vice versa, the final inclination is predetermined while the distance is to be maximized (mini-mized). Since only constant control angles are considered, the problem may be stated mathematically as maximizing the force component |S| as a function of a and 3 under the constraint that |T| is constant and vice versa . Using Lagrange multipliers, the best control program in both cases is 2 2 found to satisfy the relation cos a cos 3 = 2/3. The range of inclinations and distances which can be reached within a given time by these strategies is shown in Figure 5-5a. Here the solar parameter e s is taken to be 0.15 (A/m = 100 m /kg) and the results are valid for any starting radius and for outward ( a > 0) as well as inward ( a < 0) trajectories. The plot is derived from the analytical values for i ( v ) , r ( v ) and t ( v ) involving deter-1/2 mination of the response for various values of a and 3=±arccos[6 /(3cosa)]. The arrows in Figure 5-5a indicate the direction in the r , i plane taken by a particular control strategy a , 3 . In the case where the radial distance is prescribed at some final time, the required ratio |S|/|S| for a given value of e g may be established in conjunction with Figure 5-4, showing the response for the strategy with |S| = |S| m a x (i.e. | a | = 35.26° and 3 = 0). The ratio of the value for e g corresponding to the desired response and the actual e s determines the required -|S|/|S| with sufficient accuracy. The sail setting a , 3 yielding the maximum inclination is given by the point of intersection of this particular value of |S|/|S| and the curve Figure 5-5 (a) Combinations of inclination and radial distance attainable after a given ( b ) Levelcurves for constant |S| and | T | ; (c) Growth of inclination for pure out-of-plane transfer 177 2 2 cos a cos 3 = 2/3 (i.e. the solid curve in Figure 5-5b). Conversely,, i f the final inclination is prescribed, the corresponding optimal control program can be determined as follows. For a given e^, the required value for |T|/ | T l m a x may be taken equal to the ratio of the desired final inclination and the one obtained under the control program corresponding to |T|max> i-e. a = 0, 3 = ±35.26°. (The behavior of the inclination under the latter control strate-gy is shown in Figure 5-5c for a few values of e ). The optimal sail setting follows readily from Figure 5-5b as the intersection of this value of |T| and the sol id.curve. 5.5 Arbitrary Initial Conditions In this section, approximate analytical solutions for solar sail tra-jectories with an arbitrary but fixed sail setting and general i n i t i a l con-ditions are developed. 5.5.1 Short-term approximate solution By expanding the variables u, £, and J< in terms of a straightforward perturbation series in the small parameter e , an i n i t i a l l y valid approxi-mate solution is obtained with the zeroth-order solution representing the un-perturbed Kepler ellipse with parameters £QQ, PQQ and q ^ . The first-order equations are solved, yielding the expressions for in-plane perturbations as: £.,(v) = 2£ 0 QS A l p(v) ; u^v) - R(cosv - 1)/S.00 + S{cosv[q 0 QB 1 2 + p Q 0 (A ] 2 - A 1 Q) + 4A].|] + s i n v [ P ( ) 0 B 1 2 - q Q 0 ( A 1 2 + A 1 Q) + 4B-, ] - 4A 1 Q}/(2il 0 0) (5.18) 178 It may be noted that for an i n i t i a l l y circular orbit, the changes in a, £ and r after one revolution are all equal to 4TT3QQS . The short-term behavior of the orbital plane expressed in terms of i and ty is given by ty(v) - v - arctan T[A-,i (v)sinv - -, (v)cosv] T[A-|., (v)cosv + B-|(v)si nv] •0<# i 1(v) = |T|[A,2,(v) + B ^ f v ) ] 1 ' 2 + 0(e 2). (5.19) This result indicates that after one revolution the position of the ascending node is at COQQ+TT+0(es), i.e. near the aphelion, i f T > 0 and at CO 00 + 0 (£s ) (flear perihelion) for T < 0. This result can be understood 112 physically: although the angular rate of the orbital plane, W = e Tu/£ ' , c, o is smaller near aphelion than that near perihelion, the angular change per radian traversed by the sa t e l l i t e is larger near aphelion since 1/v is 2 proportional to r . Hence i t is also evident that for an i n i t i a l l y circular orbit, the orbital plane returns to its original position after one revolu-tion (in the first-order approximation). 5.5.2 Long-term behavior of the elements A long-term approximate solution for orbital elements of the solar sail trajectory with fixed sail setting and arbitrary i n i t i a l conditions can be derived by means'of the two-variable expansion procedure. Thereto, a new independent slow variable v = ev is introduced and the variables u, £ and K_ are expanded in asymptotic series: "(v) = Y ^ %(v,v) + 0( eJ) ; n=0 s n s 179 * ( v ) '= Y e " £ ( v , v ) + 0 ( e J ) ; n=0 s n s N-l w i<(v) = I e " K ( v , v ) + 0 ( e * ) . ( 5 . 20 ) n=0 5 ~^ s Substituting these series into Equations ( 5 . 5 ) , using d/dv = 3/3v + e s 3 / 3 v and d 2/dv 2 = 3 2 / 3 v 2 + 2 e s 3 2 / ( 3 v 3v) + e 2 3 2 / 3 v 2 , and collecting terms of like powers in e s , leads to equations for the subsequent levels of approxi-mation. The zeroth-order equations admit solutions, written as follows: U Q ( V , V ) = [1 + p Q(v)cosv + q n(v)sinv]/£ n(v), p Q ( 0 ) = p Q 0 P 0(0) = q 0 Q iQ(v,v) = £ Q(v) , £ Q ( 0 ) = £ 0 Q Mn(v,v) = A Q(v)cosv + B Q(v)sinv , A Q ( 0 ) = B Q(0) = 0 ; KQ(v,v) = K Q(v) , K Q(0) = 1 . ( 5 . 21 ) Physically, one can interpret the expression for UQ as a trajectory tangent to osculating ellipses with slowly varying mean elements. These averaged orbital elements differ from the usual osculating parameters in the sense that short-term periodic variations are disregarded. The functions p^, qg, £ n, A n, B n and KQ of the slow variable v are determined from constraints imposed upon the first-order contributions. The equations for the first-order terms become: 3 2u 1 3 2 u n „ 3 u n r r + u i = - 2 r^ " V £o ~ (R + s ~~~ i u 0)/£ 0 , 3v 3v3v 3v 3U-, 3 u n MO) = 0 ; - 1 ( 0 ) = - - I (0) ; 3v 3v 180 3£, d£„ __L = u + 2 S / ^ % ( Q ) = 0. 3v dv 0 1 32M, 9 2M n 3M, 3M n ' + M = - 2 ^+ TK /(u £ ), M,(0) = 0; —^(0) = - —^(0) ; 9v 3v3v u u u i 3v . 3v - ± / ( u 0 O , K (0) = 0; (5.22) 3v In order that the zeroth-order terms remain a valid approximation over a long duration, i t is required that the first-order terms do not contain un-bounded contributions (in the variable v). Therefore, the right-hand-sides of Equations (5.22) are developed in Fourier series with slowly varying co-efficients. To eliminate (mixed) secular terms in the solutions for u-, and M-,, the coefficients of sinv and cosv need to vanish, while for suppressing unbounded contributions in £-, and Ky, the non-harmonic terms must be set equal to zero. This leads to the system of equations: PQ(V) : = S p Q [ l - (1 - e 2 ) 1 / 2 ] / e 2 P 0(0) = poo q 0 (v) = = S q Q [ l - (1 - e 2 ) 1 / 2 ] / e 2 q 0(0) = qoo £ 0 (v ) = = 2S£0/(1 - e 2 ) 1 / 2 £Q(0) = 5 00 AJ(v) = = T KQ q 0 [ l / ( l - e 2 ) 1 / 2 - l ] / e 2 A Q(0) .= o BJ(v) = = -T KQ p 0 [ l / ( l - e 2 ) 1 / 2 - l ] / e 2 BQ(0) = 0 K^(v) : = -T[p QB 0 - q 0 A 0 ] H / ( l - e 2 ) 1 / 2 - 1]/ e 2, . KQ(0) = 1 (5.23) It follows from Equations (5.23) that co0(v) = UJQ 0 is a constant so that the orientation of the major axis remains fixed in the long run. To analyse the 181 2 - 1 / 2 behavior of the eccentricity, the auxiliary element w(v) = 1 - [1-e ( v ) ] is introduced and the following equation for WQ is found from Equations (5.23) , w 0 ( v ) = S w 0 / ( l - w 0 ) , wQ(0) = w00 . (5.24) If WQQ = 0, i.e. i n i t i a l orbit is circular, i t follows that the orbit will remain circular in the long run: Wg (v) = e n ( v ) = 0. It may be noted that UQ(V,V)£Q(V) = 1 and £ n ( v ) = £ ng exp(2 Sv) when e Qg = 0 in accordance with the exact spiral solution discussed in Section 5.2. For Wg n f 0, integration of Equation (5.24) leads to the following implicit equation for Wg ( v ) , WQ(V) = WQQ eXp[SV + WQ(V) - WQQ] . (5.25) Quite accurate representations for Wg (v ) can be established through a pro-cess of successive substitution. Initiating the procedure by replacing WQ(V) with Wg^= WQQ in the r i g h t - h a nd - s i d e of Equation (5.25), subse-quent more accurate approximations for WQ(V) follow from: W ^ ( V ) = WQQ e X p[SV + W^"^- WQQ] , (5.26) for n = 1,2,3,... . This iteration scheme converges very rapidly as long as egg is not too close to unity. For small egg, an asymptotic series in terms of powers of Wgg can be established from the scheme in Equation (5.26). It can be shown that the errorterm in Wg n ^ ( v ) as an approximation for Wg (v) is of the order Wgg^ for Wgg ->- 0. For most purposes, the asymptotic ex-(3) -pansion of Wg '(x>) for Wgg -> 0 would provide sufficiently accurate results: 182 (3) - - 2 WQ ;(V) = wQ0 exp(Sv) + wQ0[exp(2Sv) - exp(Sv)] + WQ Q[3 exp(3Sv) - 4 exp(2Sv) + exp(Sv)]/2 + 0(wJQ) . .....(5.27) It should be emphasized that a series in terms of powers of Wgg is more use-2 4 ful than the one in powers of egg for small egg, since Wgg = egg/2 + O(egg) I n ) for e^ Q 0. From the results for Wg '(V), Equation (5.26), the corresponding eccentricity eg n^(v) can readily be evaluated from the relation, e j n ) ( v ) = {1 - [1 - w ^ n ) ( v ) ] 2 } 1 / 2 , (5.28) to any desired accuracy by taking n sufficiently large. For small egg, asymptotic series in terms of powers of Wgg can be derived. The expansion (3) -of e^ '(v) would serve most needs: e^ 3 )(v) = e 0 0 exp(Sv/2){l + w0g[exp(Sv) - l]/4 + W2Q[3 - 10 exp(Sv) + 7 exp(2Sv)]/32 + O(Wgg)} (5.29) The long-term solutions for PQ(V) and qg(v) are readily expressed in terms of eg(v), P^ n )(v) = pgg e^ n )(v)/e 0g , q£ n )(v) = % Q (v)/eQQ , (5.30) and asymptotic series are established using Equation (5.29). The attention is focused on the behavior of the semi-latus rectum. Through Equations (5.23), £g(v) can be expressed in terms of Wg(v): r r V dT ) £ 0(v) = £g 0 exp |2S . (5.31) o 1 " W O ( T ) 183 For the f i r s t few approximations of Wg (v ) , the integral can be evaluated expli ci t l y : $h\>) = £ 0 0(1 - e 2 Q) exp(2Sv)/[l - wQ0 exp(Sv)] c ; (1 - w Q 0) exp(2Sv) 1 - wQ0 exp(Sv) + w n n[exp(Sv) - exp(2Sv)] 2 + wno[(wgQ - 2w0Q + 5 ) V 2 + w0Q - 1]exp(Sv)  2 " w00 [ ( w00 " 2 w00 + " 5 ) V 2 - w00 + ^ P ^ ) (l-w 0 Q) ( w o o - 2 w o o + 5 ) 1 / 2 , (5.32) However, the following asymptotic representation is more useful for small e00 : S-Q2h\>) = exp(2Sv){l + 2w0Q[exp(Sv) - 1] + w 2 Q[4 exp(2Sv) - 6 exp(Sv) + 2] + 0(WQ Q ) } (5.33) A long-term approximation for the radial distance r = 1/u is given by r£n)(v,v) = 4n)(v)/[l + e< n )(v) cos( v - S 0 0 ) ] , (5.34) where the desired representations for and e ^ need to be substituted, Also, a long-term approximation for the semi-major axis S Q ( V ) is known, a < n ) ( v ) =• ^ n ) ( v ) / [ l - w < n W (5.35) Next, the time history of the s a t e l l i t e in its trajectory is studied, 2 1/2 Since t'(v) = r /I 1 , i t is obvious that t(v) 184 V £ 3 / 2 ( T ) dx/[l + e(x) C O S(T - C3 n o)] 2 . (5.36) 0 • . Through substitution of and e ^ into the integrand, a long-term valid explicit approximation for t(v) may.be derived. It is more convenient, how-ever, to determine asymptotic series for t(v). In this regard, i t must be emphasized that, due to the integration of terms depending upon v, a consis-tent asymptotic series of t(v) should be of the form: t(v) = t_^{v)/es + t Q(v,v) + e^^v.v) + 0(e 2) (5.37) Substitution of and e ^ into Equation (5.36) and integration leads to the following approximation for t -,(v): t[]Hv) = a 3 / 2 {[exp(3Sv) - l]/3 + e 2 Q[3 exp(4Sv) - 4 exp(3Sv) + 1] /4 + 0(e 3 0)} /S (5.38) It is interesting to note that this result is consistent with the exact spiral solution of Equation (5.9) when egg = 0. Turning to the long-term behavior of the orbital plane, i t can be seen (from Equations 5.23) that the vector Kg(v) = ( M Q , L Q , K Q ) traces a path '00 and 2 2 2 upon a spherical surface : Ag + Bg + Kg = 1. Writing Ag = CQ sin Bg = - Cg cos ojgg, an equation for CQ can be derived and solved, Cg(v) = sin{ T[arcsin e Q(v) - arcsin e0g]/S } (5.39) Through this expression, a l l of Mg(v), Lg(v) and Kg(v) can now be written in terms of eg and are thus determined up to the required accuracy by substi-tuting the appropriate approximation eg n^(v) or its expansions for small egg. The orientation of the orbital plane in terms of the angles ^ Q and i Q is given by: 185 0 J Q 0 + TT , T > 0 ; %0 , T < 0 ; i n ( v ) = |T| [arcsin e n(v) - arcsin egg] /S. (5.40) 5.5.3 Higher-order contributions It may be noted that the maximum deviation of the zeroth-order solu-tion from the actual solution is of the order e s only for v up to about l / e s - Thus for large values of A/m, higher-order terms may be needed to establish sufficiently accurate long-term approximations. After incorporating the zeroth-order solutions, the remainder of Equations (5.22) can be integrated formally, yielding the first-order results: u-|(v,v) =- [ R + A 3 ( v ) ] ( l - cosv)/£Q(v) + A 2(v) cosv + B 2(v) sinv + S/£n(v) J{[2 a3Q/j - p Q d ^ +'qQ c ^ ] sin(jv) - [2 c j 0 / j + p Q b ^ - q Q a^] cos(jv)}/(j 2 - 1) ; A-|(v,v) = £ Q(v) | 2S j {a3Q sin(jv) + c^"0 [1 - cos(jv)]} / j + A 3(v) j ; M-|(v,v) = T K Q(v) (1 - cosv)/(l - e 2 Q ) 1 / 2 + A 4(v) cosv + B 4(v) sinv 00 • • o - T K Q(v) I ' {a^ Q cos(jv) + c3Q sin(jv)} / ( j - 1) ; 3 ^ 0 0 . _ _ . -K (v,v) = T/2 I {[c3^ - c 3 ' 1 ] A Q(v) sin(jv) - [a3+Ql - a ^ 1 ] . A Q(v)* j=l *[1 - cos(jv)] - [a^J 1 + a^"1] B Q(v) sin(jv) - [c3+Q] + c ^ ' 1 ] * 186 *B Q(v) [1 - c o s ( j v ) ] } / j + A 5(v) . (5.41) The Fourier coefficients a ^ , b ^ , etc., depend on the slow functions P Q ( V ) and q n(v) and are evaluated in Appendix I I . The functions A.(v), B.(v), j = 2,3,4,5, are to be determined, as usual, from constraints imposed upon the behavior of the second-order terms. Equations for these terms can rea-dily be obtained (Appendix I I I ) , leading to lengthy equations for the func-tions A., B. when eliminating the secular contributions to Up, JU' E T C -For instance, the least complicated one is given by, A3'(v) = - A 3 £ 0/£ Q - S(R + A 3) [a° Q - ] + SlQ [A 2 a ^ + B 2 b ^ ] + S y {c J 2 0 [2 a ^ / j - p Q d ^ + q Q c ^ ] - a J 2 Q * 3 ^— *[2 c ^ 0 / j + p 0 b ^ - q Q a^]} , with a l l Fourier coefficients depending on v . While analytical solutions have not been found for general eccentricity, in the special case of - 0 i t follows that e n(v) = 0 and the equations for A. and B. can be integrated yielding the following complete first-order solutions: £ 1 (v) = 2£,Q0 RSv exp(2Sv) ; e-|(v,v) = {(R2 + 4S2)[1 + exp(Sv) - 2 exp(Sv/2) cosv]} 1 / 2 ; r. 2S[exp(Sv/2) - cosv] - R sinv O J - , ( V , V ) = arctan j t R[exp(Sv/2) - cosv] + 2S sinv r(v,v) = £ Q 0 exp(2Sv){l + e g R[l - exp(Sv/2) cosv] -2 e s S exp(Sv/2) sinv} + 0(e 2 ) . .....(5.42) 187 It is seen that the radial distance oscillates around the spiral solution r = £QQ exp(2Sv) with slowly increasing amplitude of oscillation. As to the orientation of the orbital plane, i t follows that r S(l - cosv) + [2S cosv - R sinv][exp(Sv/2) - 1] <Mv,v) = v - arctan I <• S sinv - [2S sinv + R cosv] [exp(Sv/2) - 1] + 0 U 2 ) ; i(v,v) = e Q |T|/S{(4S2 + R2)[exp(Sv/2) - I ] 2 + 4S 2(cosv - 1)* *[exp(Sv/2) - 3/2] - 2RS sinv[exp(Sv/2) - 1.]} + 0(e 2) (5.43) These results illustrate that the amplitude of the perturbations grows slowly. 5.5.4 Discussion of results In order to assess the relative accuracies of the approximate results, comparisons are made with a numerical solution of the exact Equations (5.5) using a double-precision Runge-Kutta integration routine. The high value of i n i t i a l eccentricity (egg = 0.6) is chosen to illustrate a rather extreme situation, while e s is taken to be 0.015. Figure 5-6 shows the various approximations for the semi-latus rectum: obviously, the short-term solution has a limited range of validity, while the near-circular expansions of I^^ (2) and £g ; may give f a i r l y accurate long-term approximations provided a suf-ficient number of terms are retained for high values of egg (curve a). (2) The solution £g ' is more accurate, naturally, and would be the most appro-priate candidate for predicting long-term, high-eccentricity trends. The effect of the first-order contributions, £-|(v,v) from Equations (5.41) is 188 revolutions Figure 5-6 Comparison of the analytical results for the long-term behavior of the semi-latus rectum Figure 5-7 Long-term behavior of semi-major axis and eccentricity as predicted by the zeroth-order solution 190 (2) added to ' illustrating the small-amp!itude oscillations around the (2) mean trend designated by ' i t s e l f . The slow function Ag (v) was taken to be zero throughout. The discrepancy between the numerical solution and this (best) analytical approximation is largely due to the effect of Ag ( v ) . (?) Other contributions to the error may be attibuted to the fact that £Q represents an approximation for £ n ( v ) and the higher-order terms are neglected. Figure 5-7a shows the long-term trend of the semi-major axis for a (2) -few sail settings. The approximation a^ ; ( v ) compares quite well with the exact numerical solution: at least to two significant digits over the f i r s t 12 revolutions. The long-term behavior of eccentricity is depicted in (2) -Figure 5-7b, where the approximation e^ ' ( v ) was used. Relatively large first-order contributions separate the zeroth-order approximations from the exact solutions in this case. Nevertheless, the qualitative trend of the long-term behavior of the eccentricity is predicted correctly. 5.6 Concluding Remarks The results of the present chapter can be summarized in the form of the following general conclusions: (i) An exact three-dimensional solution in the form of a logarithmic spi-ral is presented for certain specific i n i t i a l conditions by separating the out-of-plane and in-plane motions. ( i i ) An effective near-circular, out-of-plane spiral transfer trajectory has been explored in detail permitting any combination of final radial distance and orbital inclination. 191 ( i i i ) Short- as well as long-term approximate solutions have been established for arbitrary i n i t i a l conditions. For small i n i t i a l eccentricity, asymptotic series for the orbital elements should prove useful for long-term trajectory evaluation. 192 6. DETERMINATION OF OPTIMAL CONTROL STRATEGIES IN HELIOCENTRIC ORBITS 6.1 Preliminary Remarks Although i t is evident from the results of the previous chapter that fixed sail settings can produce effective transfer trajectories, especially i f the best possible orientation of the sail is chosen, time-varying control strategies are l i k e l y to be more efficient. Therefore, the attention is focused upon the determination of optimal control strategies in this chapter. In many missions, e.g. rendezvous- with a distant planet or escape from the planetary system, i t is important to increase the size of the orbit in the most efficient manner. A specific optimization criterion must be formulated according to the nature and objective of the actual mission involved. Here, two particular c r i t e r i a with general applicability are selected: f i r s t , the optimal steering program of the orientation of the solar sail for maximum increase in total energy (and thus semi-major axis) after one revo-lution is determined. Next, the best steering program for maximum increase in angular momentum (and thus semi-latus rectum) after one revolution is derived. While the control strategy which directs the thrust along the instantaneous velocity vector at a l l times would likely be very effective as to the f i r s t objective, especially for near-circular orbits, a formula-tion in terms of optimal control theory would evaluate, for instance, the effect of steering the spacecraft relatively closer to the sun i n i t i a l l y in order to take advantage of the larger magnitude of the force there. The solutions are found in an implicit form in terms of state and adjoint variables by means of Pontryagin's 'maximum principle'. Approximate 193 explicit representations can, subsequently, be determined in asymptotic series containing the small parameter e g denoting the ratio of solar radiation and gravity forces. In general, only the f i r s t few terms of these series can be evaluated. These approximate analytical results have been substantiated by means of a numerical iterative procedure based on the steepest-ascent method. No restrictions are placed on the position of the sat e l l i t e in the i n i t i a l orbit nor on the nature of the i n i t i a l and ensuing osculating ellipses. 6.2 Formulation of the Problem The governing equations of motion for the solar sail are essentially similar to Equations (5.5) except for the fact that the force components R_ depend on the independent variable since R_ = R_[a(v)] here. For convenience, the solar sail is represented by a f l a t plate of homogeneous surface characteristics and the parameter is neglected. Note that for a r e a l i s t i c solar sail surface, the magnitude of amounts to about two percent of the re f l e c t i v i t y p, Table 2.1 . The components of JR. can be written as: 2 2 R(a) = (a-, + pcos a cos B) COS a cos B ; 2 3 S(a) = p sin a cos a cos B ', 2 2 T(a) = -p cos a cos B sin B ; (6.1) The vector a stands for (a, B) and is a function of v. For the analysis of this chapter, a more convenient alternative system of autonomous f i r s t -194 order equations is derived for the in-plane orbital elements by means of Equations (5.5): $'(v) = - Y(v) + 2es. S(a) ; ; $(0) = pQQ ; r(v) = $(v) + e s {R(a) + S(a) V(v)/[1 + $(v)]}; <F(0) = -q Q 0 ; £'(v) = 2e S(a) J2,(v)/[1 +*(v)] . (6.2) The variables <J>(v) and ¥(v) are defined in Equations (4.14). The two problems to be studied here can be stated as follows: (i) which control strategy a(v) leads to the maximum value of the semi-major axis after one revolution ? (ii) which control function a(v) yields the maximum value of semi-latus rectum after one revolution ? These problems are approached using the results of optimal control theory. To minimize algebraic complexity, new variables a = -1/a and £ = ln(£) are introduced and the complete system including the adjoint equations be-comes (note that $'(v) and V1 (v) are also part of this system): a'(v) = 2es exp(-£) {R(a) Y + S(a) (1 + $)} ; a(0) = - l / a Q 0 ; £'(v) = 2es S(a)/(1 + $) ; *(0) = w(lQQ) ; A Q(v) = 0 ; A-j(v) = 2es AQ {R(a) H» + S(a) (1 + $)} exp(-£) ; A 2(v) = - A3 - 2eg S(a) exp(-£) + e s S(a) [2A-j + A^l/O + $) 2 ; A 3(v) = A2 - 2es AQR(a) exp(-£) - e $ Ag S(a)/(1 + $) . (6.3) The out-of-plane equations turn, out to be irrelevant and are omitted here. 1 9 5 6 . 3 Maximization of Total Energy In this section, an approximate analytical representation for the optimal control strategy a ( v ) maximizing the total energy E (and thus major axis a) at v = 2TT is derived. The Hamiltonian for the present pro-blem, Equations ( 6 . 3 ) , becomes: H (a) = X 3 $ - X2V + e sR ( a ) { 2 X 0 ¥ exp( -A) + X 3 > + es S ( a ) { 2 X Q (1 + <J>) exp(-£) + (2X- , + X g + $ ) ' + 2 X 2 > . ( 6 . 4 ) For a ( v ) to be the optimal control vector over the fixed interval (0, 2TT), the following necessary conditions must be satisfied: ; 9a 93 i i ) H(a) = constant ; i i i ) X.(2TT) = 0 , j = 1 , 2 , 3 ; (transversality) iv) X Q ( v ) = 1 ; ( 6 . 5 ) according to Pontryagin's Maximum P r i n c i p l e ^ . From the conditions i i ) , i i i ) and iv) i t follows that H = 2 e s {[R (a) + ( ! + $ ) S ( a ) ] exp(-£)} v = ^ , ( 6 . 6 ) which equals a 1 (2TT), Equations ( 6 . 3 ) . The conditions in i) lead to the 196 following equation for a , [2 exp(-£) + X 3] | | + [2(1 + $) exp(-£) + (2X] + A - ^ / O + $ ) + 2A2] | i , (6.7) and a similar one for 8 . It follows readily that 8(v) =0 is a solution for the out-of-plane rotation confirming that the optimal trajectory is a planar one since the solar radiation force remains in the plane of the orbit. The equation for the control angle a(v) is reduced to the follow.^ ing implicit relation: pcos a (1 - 3 s i n 2 a) = 2 y + 1 X 3 . sin a (c-| + 3 p cos 2 a) 2(1 + $) + (2^ + X 3 $) £/(l + $) + 2£ .X? (6.8) with a in the interval (0, TT/2) on physical grounds. For obtaining approximate solutions for a(v) from Equations (6.8) i t is imperative to assess, carefully, the orders of magnitude of the various terms on the right-hand-side of Equation (6.8). Thereto, the orbital elements and adjoint variables are written as a system of coupled integral equations derived from Equations (6.3) by integration while taking the mixed boundary conditions into account: v ? a(v) = a 0 0 + e s / {a^[R Y + s(l + $)]/£} dx ; v £ ( v ) = £„ + 2e c / {£ S/(l + <D)} dx ; s Q v *(v) = * Q(v) + e /' {2 S COS (T - V) + [R + S + $)] sin(x - v)} dx ; 0 197 v Y(v) = V n(v) + c ( {[R + S Y / 0 + $ ) ] C O S ( T - V ) - 2S sin(x - v)} dx ; u b n 2TT A, (v) = - 2e j {[R V •+ S(l + $ ) ] / £ } dx ; v 2-TT A 2(v) = e s /' {Q sin(x - v) + P C O S ( T - v)} dx ; v. 2TT A 3(v) = e g / {Q C O S ( T - V) - P sin(x - v)} dx . (6.9 ) Here $ Q(v) = p n n cosv + q Q 0 sin v and y Q(v) = p Q 0 sinv - q n Q cos v and the auxiliary functions P and Q stand for: P. = 2 S/S. - S(2A-, + A 3 ¥ ) / ( ! + $ ) 2 ; Q = 2 R/£ + X 3 S/(l + *) . (6.10) An asymptotic series for a(v) in terms of the small parameter e g can now be constructed. By writing a(v) = a Q(v) + e s a-j (v) + O(e^) , developing the left-hand-side of Equation (6.8) in a Taylor series around a Q and expanding the right-hand-side using the results of Equations ( 6 . 9 ) , successive terms in the series for a(v) can be established. The leading term satisfies the implicit relation: 2 p cos a n ( l - 3 sin a n ) . ^n(v) 5 2 — ° - = °- . (6.11) sin QQ(O^ + 3p cos ag) 1 + $Q(V ) A good approximation to the solution of Equation (6.11) may be obtained by 193 successive substitution with a starting value a g ^ ( v ) = 35.26°, (i.e., the solution of Equation (18) for an i n i t i a l l y circular orbit) The (n + l)th approximation is obtained from c tQ n ^(v ) as follows: (n+1) / \ «Q ( v ) = arcsin {1/3 - Y 0 tan a ^ n ) [ ( a ^ p ) + cos 2 c ^ n ) ] / (1 + % ) } ] / 2 (6.12) n = 1,2,3,..., which converges rapidly provided that the i n i t i a l eccentri-city is not too large. Geometrically, the steering angle (XQ(V) in Equa-tion (6.11) makes the resulting solar radiation force aligned with the velocity vector of the unperturbed i n i t i a l osculating ellipse at each instant. Whereas this may serve as a useful guide for very small values of e , i t is evident that higher-order terms relating to the slowly varying geometry of the osculating ellipse must be evaluated when practical values of e s are taken. For the analytical evaluation of the higher-order terms, an explicit relation for O IQ(V) would be needed. In the special case when the reflection is specular (p = 1, a^= 0), a closed-form result for C X Q(V) can be derived from Equation (6.11), CXQ(V) = Tj- arcsin [ 0 + $ 0'){[9 y 2 + 8(1 + $ 0 ) 2 ] 1 / 2 - V ] 3[^ 2 + (1 + $ 0) 2] (6.13) On expanding both sides of Equation (6.8) as a Taylor series in terms of the small parameter e s , the first-order term a-,(v) now becomes , 199 [3 - cos(2a Q)] ^0 £00 [ X1 ^0 41)/2^/(1 + V'" £00 *0 X 2 ] / ( ] + V} »' (6.14) where the superscript (1) denotes the coefficient of e$ in the expressions in Equations (6.9) . The trigonometric terms in Equation (6.14) can be eliminated in favor of the orbital variables $ Q and ¥Q through Equation (6.13). Also the integrands in Equations (6.9) can be expanded for small es and expressed in terms of v. Whereas the resulting integrals are un-wieldy for arbitrary eccentricity, analytical results can be obtained for near-circular orbits. Thereto, expansions of the trigonometric terms for small are needed. These can be derived using the expansion of Equation (6.13) for small and developing R(a) and S(a) around a = a^. With these results, a l l integrands in Equations (6.9) can be evaluated and near-circular approximations for \!^\ etc., in Equation (6.14) are obtained by integration. Finally, the following expression for a-|(v) with 3 an error of the order e^ Q is established: a ^v) = - 3 ~ 3 / 2 {1 - cos v + 3TTO/2 + (4TT - 3v/2)¥Q - (p + * Q) cos v + 2p - 3q/2 sin v - 9(2" 3 / 2) VQ (1 - cos v)} - v {7(3 1 / 2) (p 2 - q 2) sin(2v) - 2(6 1 / 2) e2} - TT {4(6 1 / 2) e 2 + 3 3 / 2 pq}/18 - sin(2v) { (p 2 - q 2 ) [ 6 1 / 2 - 4(3 3 / 2)TT] - pq/2}/18 - [1 - cos(2v)] {(p 2 - q 2) ( 3 " 1 / 2 + 1/4) + 2(6 1 7 2)pq}/18 200 - (1 - cos v) {(p 2 - q 2 ) [ 2 ( 3 ~ 1 / 2 -1/4+4 sin(2v)] - 3e2/4 - (3 3 / 2)Trpq}/18 - sin v {pq - 3 3 / 2 (2p 2 + q2)TT + 3 1 / 2[4pq cosv + (q 2 - 3p 2) sinv] - 4(3 1 / 2) (p 2 - a 2) sin(2v) + 6 1 / 2 [ e 2 - (p 2 - q 2) cos(2v) - 2pq sin(2v)]}/18 - 61//2' { 3 T T q ^ 0 + (8TT - 3v)Tg - 2(p + ^ n.)^ C 0 S V + 4 p ¥ 0 " 3 q l i0 s i n v - ^ n I2$ 0 + 3 ( 2 " 1 / 2 ) ^ 0 ] (1 - cosv)}/144 + 0(e 3 Q) . (6.15) Here the subscripts 00 are omitted for brevity. It follows that the f i r s t -order correction (v) for an i n i t i a l l y circular orbit is at most 22e$ degrees (at v = TT) below the constant a n = 35.26° control program. It is interesting to evaluate the response of the major axis under the optimal control strategy. For a near-circular orbit, a(v). can be written as 2 -3/2 a(v) = a n f ) exp{2es(l + e Q 0) [2(3" ) (v + p Q 0 sinv + q Q 0 - q R 0 cosv) 1/2 2 poo c o s v " qoo s i n v ) / 9 + eoo V / / 2 + (qOQ " Poo) sin(2v)/4 + p o n q 0 0 cos(2v) - p 0 ( ) q 0 p + 0 ( e 3 ) ] + 0(e 2)} . (6.16) If e n n = 0 this result can be reduced considerably yielding 300 C AHL"+.u->uo t -r u v t - s , a(2Tr) = a n n exp[4.8368 e c + 0 ( e 2 ) ] after one revolution. 201 6.4 Maximization of Angular Momentum Here, the optimal control strategy for maximum increase in angular momentum (and thus semi-latus rectum) per revolution is determined. This corresponds with maximization of £(2TT). The system of Equations (6.3) remains valid provided that the equations for a and An are ignored and the equation for is replaced by A^  (v) = 0. Now the Hamiltonian becomes H£(a) = X3 $ - X2 ^ + e s A3 R(a) + eg S(a){(2X1 + A ^ / O + $) + 2A2>.. (6.17) Application of Pontryagin's maximum principle leads to results as in Equations (6.5) with A-j = 1 now. It follows that H£ = £'(2TT) and the out-of-plane rotation 3(v) = 0 while the optimal control angle a(v) is given by the implicit relation, p cos a (1 - 3sin a) 2 sina (a-j + 3p cos a) The orbital elements I, <f> and ¥ can be written in the form "of Equations (6.9) while the adjoint variables A 2(v) and A^(v) become 2TT A 2(v) = e s J {Q£ sin(x - v) + P £ cos(x - v)} dx , v 2TT A3(v) = e s / {Q£ cos( T - v) - P £ s i n ( T - v)} d T , (6.19) v with P and Q defined by A3 (1 + $) 2 + A3 ¥ + 2 A2 (1 + $) .(6.18) 202 P £ = - S(2 +• X 3 + $ ) 2 , Q£ = A3 S/(l + $) . .....(6.20) The right-hand-side of Equations (6.18) is of the order e g so that a(v) 2 -1 /? is written as a(v) = a n + e a, + 0(e ) with a„ = arcsin(3 ") 0 s I • s 0 = 35.26° . The first-order term a-,(v) is determined by expanding both sides of Equations (6.18) in Taylor series for small e g yielding the following explicit result, cc-](v) = - (a-, + 2p) 3" 3 / 2/2 [1 + $ Q(v)] [2 VQ(v) * r 0 - e 2 ) 1 / 2 tan(v/2) , ? , / ? _ * ^ TT - arctan[ ^ ] [ / ( ! - ei , ) 1 '^ 1 1 + Poo + W a n ( v / 2 ) + 1 - [$ 0(v) + cos v]/(l + p Q 0) /(I - e 2 Q) . (6.21) The resulting response £(v) under the optimal sail setting can be approximated by integrating £'(v) in Equations (6.3) (up to order e ), £(v) = £ Q 0 exp s 1 /? -3/2 ^ " e00^ tan(v/2) 9 , ,„ e 0 3 J / z arctan[ ^ ] / ( l - e 2 ) 1 / 2 b 1 L „ . _ 4 . _ / . ; o \ UU 1 + P00 + P 0 0 t a n ^ v / 2 ^ (6.22) Considering an i n i t i a l l y circular orbit, i t follows that £(2TT) = exp{4.8368 p e s + 0 ( e s ) ' J . This result is identical to the one found in the previous section while maximizing the semi-major axis for a near-circular starting orbit. Obviously, the control programs in Equations (6.15) and (6.21) are also identical for e ^ = 0 in the present approximation. 203 6.5 Discussion of Results The accuracy of the analytical solution obtained in Section 6.3 is now assessed by comparison with results from a numerical iteration pro-n g cedure based upon the steepest-ascent method . An arbitrary nominal con-trol strategy is selected and the influence of a small variation in that control program upon the response is investigated. The variation leading to the maximum increase in major axis under a prescribed step-length (i.e., the integral from 0 to 2TT of the square of the variation in the control function) can be determined in terms of the derivatives of the system of Equations (6.3) with respect to the control angle. Thus, a generally more effective new control strategy is obtained and the procedure is repeated. While the algorithm converges rapidly to a near-optimal control strategy, care must be taken in the neighborhood of the optimum due to the weakness of the gradient f i e l d . By making both the step-size and the error para-meter in the Runge-Kutta integration routine proportional to the length of the gradient, satisfactory results are obtained. In the present case, the i n i t i a l control program is taken as a(v) = (2TT - v)/6 and the optimal strategy is established to within, approximately, 0.1 degree-in less than 30 iterations, Figure 6-1. A relatively small value of the solar parameter (based on A/m = 10 m /kg) is taken in this example. The first-order ana-lytic a l result of Equation (6.15) for a near-circular i n i t i a l orbit in con-junction with the exact zeroth-order term in Equation (6.13) yields an ex-tremely close approximation when = 0.2 (Figure 6-1 a): the maximum discrepancy is less than 0.1 degree. On the other hand, i f = 0.4 (Figure 6-1b), the near-circular analytical solution is in error by almost three degrees around v = 270°, while s t i l l providing a valid representa-Figure 6-1 Comparison of analytical and numerical optimal controls for e g = 0.015: (a) e Q 0 = 0.2; (b) e 0 Q = 0.4 205 tion for the optimal strategy in the remaining portion of the orbit. The breakdown in accuracy must be attributed to two reasons: f i r s t , i t should be recognized that the first-order analytical result developed here does 3 not contain terms of order and higher which are li k e l y to be influential when the eccentricity is as high as 0.4. Secondly, the state and adjoint variables are represented as perturbation series in terms of e g and only the first-order solutions are taken into account leading to a rapidly gro-wing error when away from the i n i t i a l and final points. Figure 6-2 shows the results for a higher value of e s , namely 2 e g = 0.09, corresponding to A/m = 60 m /kg. As can be expected, the analytical prediction for the optimal control is most accurate in the case epQ = 0; the maximum discrepancy of about one degree is due to higher-order (in e ) effects. It is interesting to note that i n i t i a l l y the solar radia-tion force points slightly inwards from the velocity vector and its magni-tude is smaller than that for the case where the force is aligned with the velocity. This is true for both the numerical and the first-order analyti-cal results, although the effect is less pronounced in the latter case. This apparent waste of energy is more than recouped during the middle phase of the orbit when the spacecraft is closer to the sun and the force is lar-ger. In this phase, the direction of the force is kept outward from the velocity vector, thus providing an additional boost to its magnitude. In the final phase the force tends to align i t s e l f with the velocity. The os-culating ellipses corresponding to the resulting trajectory show that the eccentricity increases from 0 to a maximum of about 0.2 near v = 190° and decreases to about 0.02 with the position of the perigee at about 70° in the end,' v = 2TT . The analytical result for e n n = 0.2 shows a maxi-206 a n a l y t i c a l , nea r - c i r cu l a r s teepes t - ascent 0 % 2TT/3 n 4 i j / 3 5TJ/3 2n v [radians] Figure 6-2 Optimal sail setting for e g = 0.09, = 0 and a few values of e 0 0 207 mum error of about 2.5 degrees as compared to the steepest-ascent solution. Figure 6-3 shows the optimal steering programs for three different starting points in the same i n i t i a l orbit of eccentricity - 0.2 (e = 0.15) , obtained by the steepest-ascent iteration routine. It is seen that the nature of the control strategy as'well as the resulting final value a(2Tr) vary considerably with the position of the starting point. It is interesting to compare the effectiveness of the optimal strate-gies with that of other near-optimal control programs, in particular the -1/2 constant sail setting a = arcsin(3 ' ) = 35.26°. The latter control is expected to be a very effective strategy for small e g. and small since i t generates the maximum component of the force along the velocity for an unperturbed circular orbit. Table 6.1 gives a comparative overview of the response a(2Ti) for a few values of e s and (UJQQ is taken zero). Table 6.1 Response a(2Tr) for Optimal Control Strategy and -1/2 for a = arcsin(3 ) ^ ^ s 0.015 0.09 0.15 e00 \ \ * 0 1.0761 .1.  590 2.280 1.0760 1.587 2.258 0.2 1.0808 1.668 2.608 1.0796 1.640 2.454 0.4 1.0984 1.962 4.314 1.0922 1 .819 3.202 The upper values correspond to the optimal response while the lower ones represent the results for a = 35.26°. 60 50 o 40 30 20 Ss=0.15 o 0 45 90 0.2 0 a(2n) = 2.297 ' 2.608^/ / / / / 2.751 / I i / / / \ / 0 •0.2 0 -90 60 50 / / A.. N 40 30 20 135 180 225 270 ..o 315 360 Figure 6-3 Optimal control programs for e g = 0.15, e 0 Q = 0.2 and a few values of co Q O ro o oo 209 Although the results seem to be close in most cases, i t must be emphasized that a difference of one digit in the fourth decimal place represents a physical distance of about 15,000 km. On the other hand, i t is evident that a(v) = 35.26° is a very effective control strategy even for eccen-t r i c i t i e s as high <as 0.4. It should be mentioned that the results in Table 6.1 are derived numerically, since the analytical prediction for the response under the optimal control, Equation (6.16), yields useful values for a(2Tr) only for small e $ and e ^ and is not capable of providing accu-racy beyond three significant digits in the most favorable case, while being in error by as much as 0.3 in the most severe situation of Table 6.1. The actual trajectory resulting from the optimal strategy for e s = 0.09 is depicted in Figure 6-4. It is seen that Mars' orbit is inter-cepted at about v = 135° after approximately one year. Also the inward trajectory crossing Venus' orbit is shown. These trajectories are obtained from the steepest-ascent results. It may be mentioned that the leading term in the analytical solution of the optimal strategy for inward trajec-tories is equal to but opposite in sign compared to the one for the outward ones. The first-order (in e ) terms, however, are different and can be readily evaluated by taking AQ = -1 rather'than +1 .These conclusions are substantiated by the numerical results. Finally, the optimal sail settings leading to the maximum increase in angular momentum for a few values of i n i t i a l eccentricity and solar parameter are shown in Figure 6-5. The approximate analytical solution for the present case is likely to be more accurate than the ones presented before (Section 6.4) due to the fact that. a n(v) is obtained for general e n n , Figure 6-4 Actual trajectory under optimal sail setting showing interception with Mars' and Venus' orbits 211 ntrol s t ra teg ies f o r maximization of angular momentum 212 Equation ( 6 . 2 1 ) , leaving only the errors caused by higher-order (in e ) terms. It may be noted that the resulting optimal control for - 0 corresponds identically (up to first-order) to the one which maximizes a ( 2 i r ) , Figure 6 -2 . Compared to the optimal strategy for maximization of a(2n-), the present control programs are closer to the 3 5 . 2 6 ° line, repre-senting the zeroth-order approximation of the optimal control for circular as well as e l l i p t i c orbits. 6.6 Concluding Remarks Important aspects of the analysis and conclusions based on them can be summarized as follows: ,(i) Analytical approximate solutions for the time-dependent optimal sail setting maximizing the total energy (major axis) or the angular momentum (latus rectum) after one revolution are obtained from Pontryagin's maximum principle by means of a straightforward pertur-bation expansion of the state and adjoint variables. ( i i ) The validity of the approximate solution is assessed by means of a numerical iteration procedure based upon the steepest-ascent method. In general, the accuracy of the analytical solution decreases with increasing es and eccentricity. For values of e g as high as 0.1 and e up to 0 . 2 , the maximum deviation in control angle is less than 3° (which is comparable to the expected error in manoevring the s a i l ) . ( i i i ) It is found that the optimal strategy as well as the response may vary considerably depending on the starting point in the orbit. Effectiveness of the optimal sail setting is compared with that of a near-optimal constant sail orientation showing a growing diver-gence in responses for increasing values of e g and egg. The optimal steering program for maximizing angular momentum stays relatively close to the 3 5 . 2 6 ° line and coincides with the optimal sail setting for maximizing a(2ir) (in first-order) when egg = 0 The optimal control strategies developed here should prove useful in planning missions by solar sail to the distant planets and for reaching an escape trajectory from the solar system. 214 7. CLOSING COMMENTS 7.1 Summary of Conclusions The main objective of the study, to gain insight into the long-term evolution of sat e l l i t e orbits under the influence of a r e a l i s t i c a l l y modelled solar radiation force as well as exploring possible control stra-tegies for desired orbital change, is accomplished in some measure. The im-portant aspects and conclusions of the thesis may be summarized as follows: i) The long-term orbital perturbations of satellites modelled as a plate normal to the incident radiation are determined using the two-variable expansion procedure and rectification/iteration of the short-term results. The in-plane orbital changes are easily visualized through polar plots for the eccentricity vector. The long-term periodic variations in the inclination of the orbital plane are explained in terms of the in-plane perturbations, while the line of nodes regresses in a slow secular manner. i i ) Analytical representations for the short-term behavior of arbitrarily shaped space structures pointing in a fixed direction with respect to inertial space or those kept in an arbitrary fixed orientation to the solar radiation are obtained. Subsequently, long-term results are obtained by rectification and iteration. Also the perturbations of a sat e l l i t e modelled as a plate in an arbitrary orientation to the local reference frame with different material properties on both sides are analysed. 2 1 5 i i i ) A few on-off switching strategies are proposed and their effectiveness in changing orbital parameters is explored. While substantial changes in the major axis can be achieved in this manner, the time-dependent optimal control strategy for maximization of total energy is derived by means of a numerical iteration scheme based on the steepest-ascent method. This result should be of interest for raising a solar sail from a geocentric into a heliocentric or escape trajectory. iv) A detailed investigation of the long-term evolution of heliocentric trajectories for arbitrary fixed sail setting is presented which should be useful for evaluating possible solar sail missions depending on sail parameters and i n i t i a l conditions. For specific i n i t i a l conditions, exact three-dimensional solutions in the from of spirals and conic sections are established, while an effective near-circular out-of-plane spiral transfer trajectory is obtained by switching at appropriate locations. v) Optimal time-dependent steering angles for maximization of total energy and angular momentum are determined both by an approximate analytical perturbation method and the numerical steepest-ascent procedure. The results are of interest for designing solar sail missions with the ob-jective to rendezvous with a distant planet or to escape from the pla-netary system. 7.2 Recommendations for Future Work While the thesis may provide an overview of the various aspects of solar radiation effects upon satellite orbits, i t is by no means exhaustive and 216 numerous options for future work are available. An obvious extension may concern the derivation of an analytical prediction for the orbital behavior of an arbitrarily shaped space structure of general material characteristics under a suitable control strategy. An interesting manner to describe a time-dependent control would be by means of a Fourier series. The main d i f f i c u l t y would l i e in keeping track of the continuously changing number of illuminated surface components. Conversely, after the analysis has shown its practical usefulness, i t might be possible to derive conclusions as to the long-term degradation of reflecting properties of surface materials by carefully studying the orbital behavior of the spacecraft. Various possibilities exist for extending the analysis on control strategies using solar radiation forces. For instance, the control stra-tegy developed here for optimal orbit raising might be extended to allow for constraints on the final state and/or for a second component of the control vector. Convergence problems are expected near the optimum and a proper combination of step-size and weighting function needs to be de-veloped for each case. Also other optimization criteria could be investi-gated, e.g. minimum-time transfer problems, for which a formulation in terms of radius and velocity vectors would likely be more expedient than the present one in orbital elements. As to heliocentric solar sail orbits, many topics are s t i l l open for study. Especially a generalization of the spiral out-of-plane transfer trajectory to arbitrary i n i t i a l and final conditions would be of interest. 217 BIBLIOGRAPHY 1. Musen, P., Bryant, R., and Bailie, A., "Perturbations in Perigee Height of Vanguard I," Science, Vol. 131, No. 3404, 25 March 1960, pp. 935-936. 2. Musen, P., "The Influence of Solar Radiation Pressure on the Motion of an A r t i f i c i a l Satellite," Journal of Geophysical Research, Vol. 65, No. 5, May 1960, pp. 1391-1396. 3. 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Campbell, T.K., and Gold, T.T., "Three Dimensional Trajectory Optimi-zation Program for Ascending and Descending Vehicles," Proceedings of  the 16th International Astronautical Congress of the I.A.F., Athens, 1965, Astrodynamics, Editor-in-chief: Lunc, M., Gauthier-Villars-Dunod, Paris, 1966, pp. 161-177. 226 121. Glaser, P.E., "Power from the Sun," Mechanical Engineering, Vol. 91, No. 3, March 1969, pp. 20-24. 122. Glaser, P.E., "Solar Power Via Satellite," Astronautics and Aero- nautics, Vol. 11, No. 8, August 1973, pp. 60-68. 123. Williams, J.R., "Geosynchronous Satellite Solar Power," Astronautics  and Aeronautics, Vol. 13, No. 11, November 1975, pp. 46-52. 124. Raine, H.R., "CTS Flight Performance," Presented at the 27th Interna- tional Astronautical Congress of the I.A.F., Anaheim, California, October 10-16, 1976, Paper 76-223. 125. Wright, J.L., "Physical Principles of Solar Sailing," Presented at the Solar Sail Small Symposium, American Astronautical Society, Jet Pro-pulsion Laboratory, Pasadena, California, April 20-21, 1977. 126. Lubowe, A.G., "Order of a Perturbation Method," AIAA Journal , Vol. 3, No. 3, March 1965, pp. 568-570. 127. Geyling, F.T., and Westerman, H.R., Introduction to Orbital Mechanics, Addison-Wesley, Reading, Massachusetts, 1971, Chapter 9. 227 APPENDIX I EVALUATION OF THE INTEGRALS A . AND B . nk nk The integrals A N K and B^ are defined as Ank<v> n cos(kx) dx/(l + p COST + q sinx) , 0 rv n B n | <(v) = J sin(kx) dx/(l + p cosx + q sinx) , (1.1) for k = (0) ,1,2,...; n = 1,2,3 The parameters p and q represent the i n i t i a l conditions PQQ and or the slow functions PQ(V) and qg(v). While the integral A-,Q can be evaluated by elementary means, the integrals A NQ for higher values of n can be obtained from A-,Q by repeated differen-tiation within the integrand , A l n(v) - 2 arctan { P - e 2 ) 1 / 2 Un(v/2) 1 / ( 1 _ e 2 ) l u ( 1 + p + q tan(v/2) > A 2 0 ( v ) = { A 1 0 ( v ) " y-M/V +$(v)] - q/(l +p) } / (1 - e 2 ) , .2x1/2 A 3 0 ( v ) = 1 {( 2 + e 2)A 1 0(v) - 3Y(v)/[l+*(v)]-3q/(l+p) - (1 -e2)¥(v)/[l + $ ( v ) ] 2 - q ( l - e 2 ) / ( l + p ) 2 }/ (1 _ e 2 ) 2 . (1.2) 228 The integrals A n k(v) and B n k(v) for k>l may be expressed in terms of A n - l , k - l ' Bn-l,k-T An,k-1' a n d Bn,k-1 a c c o r d i n 9 t o t h e following recurrence formulae e V l , k+1 P \ — n ~ Ank " An+1 ,k J " q ( — Bnk 1 [p sin (kv) + q cos (kv)] q ~Bn+l,k j „n _L * / . . \ i n + n [ l + $ ( v ) f n(l+p) n A n + , , k + 1 " - { ^ A n k - V l , J • P { ^ B „ k - > k } [p cos (kv) - q sin (kv) ] n [ l + * ( v ) ] n n(l+p) n for k = 0,1,2,••- and n = 1,2,3,-•• . The following results can be obtained from Equations (1.3): (1.3) A 2 i ( v ) = {"PA 1 0(v) + (q + s i n v ) / [ I + $ ( v ) ] - q / ( l +p)|/ (1 - e 2) B2 1>) = | -q A 1 Q(v) - (p + cosv)/[l +$(v)] + l | / (1 -e 2) ; A 3 1(v) = 1 | -3pA 1 0(v) + [3q + ( l + 2 e 2 ) s i n v ] / [ l + $ ( v ) ] - 3q/(l +p) + (1 -e 2)(q + sinv) / [l + $(v)] 2 q(l - e 2 ) / ( l +p) 2 j / (1 - e 2 ) 2 ; 229 B 3 1(v) = \ | -3q A 1 Q (v) - [3p+ (1 + 2e2) cosv]/[l +$(v)] + (1 + 3p + 2e^)/(l +p) - (1 - V ) ( p + cosv)/ [1 + *(v)]' + 0 - e 2 ) / ( l +p) }/(1 - e 2 ) 2 (1.4) The integrals with n<k can usually be determined quite readily: A n 1(v) p[v-A 1 Q(v)].+q In 1 + $(v) 1 +p / e< B n ( v ) q [ v - A 1 Q ( v ) ] - p £n 1 + $(v) T T p - 7e< A 1 2(v) = -2[q + y(v)]/e 2 - (p 2 - q 2) [(2 - e 2)A 1 Q(v) -2v] / e -4pq£n 1 + $(v) / e 4 ; B 1 2(v) = 2[p-$(v)]/e 2 + 2pq[(2-e 2)A l n(v) - 2v]/e 4 2(p 2-q 2)£n 1 + $(v) 1 +p / e ltV 4 (1.5) These results are not suited for e-*0, and are to be replaced by: A^(v) = - pv/2 +[1+3p 2/4 + q2/4] s i n v - p sin (2v)/4 + pq(l -cos v)/2- q[l -cos(2v)]/4+ (p 2 - q 2) sin(3v)/12 + pq[l - cos(3v)]/6 + 0(e J) ; 230 B n ( v ) = - qv/2 + pq sin v/2 + [1 + p2/4 + 3q 2/4](l - cos v) + q sin (2v)/4 - p[l - cos(2v)]/4 - pq sin(3v)/6 + ( p 2 - q 2 ) [ l - cos(3v)]/12 + 0(e 3) ; A 1 2(v) = sin(2v)/2 - p/2 sin v+ q(l - cos v)/2-p sin (3v)/6 - q [ l -cos(3v)]/6 + 0(e 2) ; $ 1 2(v) = [1 - cos(2v)]/2 - q/2 sin v + p(l - cos v)/2 + q sin (3v)/6 - p[l - cos(3v)]/6+ 0(e 2) . (1.6) In many applications, the values of A n k(v) and B n k(v) for v = 2T7 are required. These can be determined from the integral I n k ( p . q ) 2 l T exp(ikx) dx 0 [l'+p cos x + q sin x ] n n = 1,2,3,-.. ;" k = 0,1,2,-.. ; (1.7) with i = (-1) 1 / 2 and (p 2 + q 2 ) 1 / 2 = e < 1 . The integral I k can be evaluated by means of residues: I n k(p,q) = 2 T r e x p(ik w) e ~ k ( l - e 2 ) - ( n " k ) / 2 Y 2 j " n + 1 * j=0 * ("T1) ( n . ^ 1 ) [ l - ( l - e 2 ) - 1 / 2 ] n + k - J " 1 , (1.8) 231 where to = arctan (q/p). The binomial coefficients are defined by ("") = (-n)(-n-l) [-n- ( j - l ) ] / j ! , j = 1,2, — . (1.9) and ( J) = 1 . The values A n k(2Tr) and B n ( <(2Tr) are simply: A n ( < ( 2 T r ) = Re | I n k(p,q)} = 2ircos ( k o ) ) [ - - . . ] , B n k ( 2T r ) = Im | I n k(p,q) } = 2TT sin (ku>) [• •] ( 1 . 10 ) The following explicit results are obtained: A 1 0 ( 2 T T ) = 2TT/(1 - e 2 ) 1 / 2 ; A 2 0 ( 2 T T ) = 2ir/(l - e 2 ) 3 / 2 ; A 3 Q = rr(2 + e 2 ) / ( l - e 2 ) 5 / 2 ; A n ( 2 r r ) = 2Trp[l - (1 - e 2 ) " 1 / 2 ] / e 2 ; B^(2v) = 27rq[l - (1 - e 2 ) - 1 / 2 ] / e 2 ; A 1 2 ( 2 r r ) = - 2 r r ( p 2 - q 2 ) J 2 - (2 - e 2 ) / ( l - e 2 ) 1 / 2 } / e 4 ; B 1 2(2T T ) = - 4 T r p q { 2 - ( 2 - e 2 ) / ( l - e 2 ) 1 / 2 } . / e 4 ; A 2 1(2T T ) = - 2TT P / ( 1 - e 2 ) 3 / 2 ; B 2 1 (2TT) = -2Trq/(l - e 2 ) 3 / 2 ; A 2 2(2 T T ) = 2 T r ( p 2 - q 2 ) | 2 + ( 3 e 2 - 2 ) / ( l - e 2 ) 3 / 2 } / e 4 ; 232 B 2 2 ( 2 T T ) = 4Trpq | 2 + (3e 2 - 2)/(l - e 2 ) 3 / 2 } / 4 e A 2 3 ( 2 T T ) = -2rrp(p2 - 3q2) j 8 - 3(2 - e 2 ) / ( l - e 2) 2 v l / 2 (3e 2-2)/(l - e 2 ) 3 / 2 } 1 ^ B 2 3 ( 2 T T ) = -2rrq(3p 2-q 2){ 8 - 3 ( 2 - e 2 ) / ( l -e 2) + (3e 2-2)/(l - e 2 ) 3 / 2 } / ^/2 - e 4 A 3 1 ( 2 r r ) = - 3TT P/(1 - e 2 ) 5 / 2 ; B 3 1 ( 2 T T ) = -3rrq/(1 - e 2 ) 5 / 2 ; A 3 2 ( 2 r r ) = 3rr(p 2 - q 2 ) / ( l - e 2 ) 5 / 2 ; B 3 2 ( 2 r r ) = 6i:pq/(1 - e 2 ) 5 / 2 ; A 3 3 ( 2 r r ) = rrp(p 2-3q 2) { 8+(12e 2-8)/(l - e 2 ) 3 / 2 - 3 e 4 / ( l - e 2 ) 5 / 2 l / e 4 ; B 3 3(2TT) = ^q(3p 2-q 2) { 8+(12e 2-8)/(1 - e 2 ) 3 / 2 - 3 e 4 / ( l - e 2 ) 5 / 2 ) / e 4 . (1.11) 233 APPENDIX II EVALUATION OF THE FOURIER COEFFICIENTS a J, , d\ nk' ' nk The Fourier coefficients of the functions cos(kv)/[l + p cos v + q sin v ] n = a n k / 2 + l i a n k c o s j v + c n k s i n j v 3 * sin(kv)/[l + p cos v + q sin v ] n = oo , b n k / 2 + { b n k c o s ^ + d n k s i n J ^ • 3 ' d i . i ) for k = (0),1,2, and n = 1,2,3,... can be expressed in terms of the integrals A n | <(2ff) and B n | < (2Tr ) a s follows: 2TT nk }nk J - 1 'nk nk f cos(kr) cos(jx) d T (1 + p COST + q sinx) n J 0 2TT 2TT . 0 2TT sin(kx) cos(jx) dx (1 + p cosx + q sinx) n cos(kx) sin(jx) dx [An,j+k<2*> + V j - k ^ ^ 2 * ) ' [ B n , j + k ^ " Bn,j-k<2*>J/<2*> •> ' W , n = £Bn i - k ( 2 ^ + Bn i + k ( 2 ^ / ( 2 - ) > (1 + p cosx + q sinx) n n' J K n ' J + K TT I sin(kx) sin(jx) dx (1 + p cosx + q sinx) n [An,J-k<2lf> " A n , j + k ( 2 ^ / ^ > ! (II.2) It may be noted that A . , = A . . and B . . = -B , . . n,j-k n,k-j n,j-k n,k-j 234 By means of the results of Equations(1.1 0), the following explicit expressions for the Fourier coefficients can be derived: 10 '10 cos ( j u ) sin( jco) [ ( 1 - e 2 ) 1 / 2 - l ] j e- J'/(l - e 2 ) 1 / 2 "20 J ' 2 0 = 2 cos(jco) s i n ( j c j ) [(1 - e 2 ) 1 / 2 - l ] j [ l + j ( l - e 2 ) 1 / 2 ] e ^ / ( l - e 2 ) 3 / 2 *30 J ' 3 0 c j f b ^ l [(1 - e 2 ) 1 / 2 - l ] = " e J (1 - * l ) b l T C0S(jco) sin ( j w ) [2 21(1 - e 2 ) 1 7 2 - l ] j { cos (1 - e 2 ) 1 / 2 2[(1 - e 2 ) 1 7 2 - l ] j sin w (i - o 2x1/2 e 2 + 3 j ( l -2x1/2 . e ) +• j 2 ( l - e 2 ) ] COS ( jw) sin (jco) + sinto - "\ s in(jo i ) -cos(jco) cos(jco) + COS co -sinO'co)' sin (-jco) COS(jto) a J ^ a21 c j I 21 ; 2 r d - e 2 ) 1 7 2 -e ^ ( l - e 2 ) 3 7 2 -Iii- | [ e 2 + j ( i - e 2 ) 1 / 2 ]cosco f cos(jco)] sin(jco) bJ 21 d21 + j (1 - e ) s i nco f sin ( jco) ] cos (jco) 2[(1 - e 2 ) 1 7 2 - l ] j j e ^ + 1 ( l - e 2 ) ^ [e 2 + j ( l - e 2) 1 / 2]sino> cos(jco)"] sin ( j w ) 235 + j ( l - e ) cost f-sin(jw)! cos(jco) r 1 -\ l31 I 31 ; I ( L " ^ 'j/ ( P e 2 + j ( l + 2e 2)(1 - e 2 ) 1 / 2 + j 2 ( l - e 2 ) ] , e J + l ( 1 : 7)572 *COSu) f COS(ja))] s i n ( j u ) + j ( l - e 2 ) [ l + j ( l - e 2) 1 / 2]sinco f sin ( j u ) ) ] cos ( j w ) D31 d3 I 31 ; ^ 5 / 2 ^ { + W + 2 e 2 ) ( l - e 2 ) ^ + j 2 ( 1 _ e 2 ) ] , * s i nu cos(ju)] sin(jto) + j ( l - e 2 ) [ l + j ( l - e 2 ) 1 / 2 ] c o s o f - s i n ( j w ) ] COS (jco) ...(II.3) The coefficients , are equal to a ^ , b^ 0 respectively. For values of k larger than 1, the dominant coefficients can be expressed in terms of the results of Equations (II.3): 'nk " d n j ' ; d3 nk d^- ; J =0,1, 2, (II-4) 236 APPENDIX III DERIVATION OF HIGHER-ORDER EQUATIONS The unknown secular terms in the first-order solutions obtained by the two-variable expansion method may become of importance eventually, especial ly in the heliocentric case and when the A/m ratio is large. These terms are to be determined from the boundedness constraint imposed upon the second-order solutions. In assessing the nature of the various contributions, products of (sometimes incomplete) Fourier expansions need to be analysed. Thereto, the following formal result is employed: j ^ [{a}3' cos(jv) + {g} j sin ( j v ) ] } | _f [{ Y) jcos ( j v ) + {6} jsin ( jv ) ] j = (a}° + I [{a}k cos(kv) + {b}k sin(kv)] , wi th: k=l {a}0 = I [{a}"' { Y} n + ( e l " {6) n]/2 ; n-1 {a}1 = [{a}1 { Y} 2 + ( e ) 1 {«)2]/2 + I [ { a ) n '({ Y } n + 1 + { y } " " 1 ) + {6) n ( ( 6 } n + 1 + {6} n" ])]/2 ; n=2 {b}1 - [ { a } 1 {6} 2 - ( B ) 1 {y} 2]/2 + • I H a } " ( { 6 > n + 1 " { 6 } n _ 1 ) " ( B l " ( { Y ) N + 1 - {Y} n- ])]/2 ; n=2 etc. (III.l) 237 Depending upon the nature of the left-hand-side of the differential equations, either the non-harmonic or the first-harmonic terms are reauired to vanish. The equations for the secular first-order as well as for the second-order periodic terms can now be derived readily; these equations are untractable in general. Solutions have been obtained only for circular starting orbits. It may be mentioned that numerical integration of the original equations is preferable in the general case. 

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