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Oscillations near the circular orbits of the helium atom Smith, Harold Duncan 1929

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OSCILLATIONS NEAR THE CIRCULAR ORBITS OF THE HELIUM ATOM. by Harold Duncan Smith A Thesis submitted for the Degree of MASTER OF ARTS in the Department of MATHEMATICS The University of British Columbia April - 1929. ACKNOWLEDGMENTS The writer wishes to thank Dr. D. Buchanan for his valuable help with this problem. He also wishes to thank Mr. M. Nesbitt for his splendid work in plotting the orbit TABLE OF CONTENTS I. Introduction. Part I. Two-dimensional Orbits. 1. The Equations of motion. 2. The Construction of the Solutions. 3. Computation for this ease. Part II. Three dimensional Orbits. 4. The Differential Equations. 5. The Generating Solutions. 6. The Construction of the Solutions. 7. Computation (tables). 8. Graphs of the Orbits. 9. References. INTRODUCTION 1 In 1928, Rawles published a paper entitled "Two Classes of Periodic Orbits with Repelling Forces," in which he considered a system consisting of one body of very great mass which attracts two mutually repellent bodies of very small mass. The forces of attraction and repulsion are assumed to vary inversely as the squares of the distances. Rawles showed that if certain conditions of symmetry are imposed the orbits of the small particles must lie either in two parallel planes or be coplanar. In the former case they are found to be circles with centres on a line drawn perpendicular to the plane of the orbits, and passing through the large body. In the other case the orbits are coplanar "arc orbits" in which the small bodies move back and forth on arcs which are symmetrical with respect to the large body. These orbits were 2 first calculated by Langmuir by numerical integration, and 3 were also discussed by Van Vleck in his work on Quantum Principles and Line Spectra. By assigning suitable values to the constants in Rawles' solutions, we have the case of the helium atom, which consists of the heavy compact nucleus, and two electrons, attracted by the positive nucleus, but repelling each other. It is the purpose of this paper to discuss the near the circular orbits. The second genus orbits near the "arc orbits" have also been considered by D. Buchanan. UBC Scanned by UBC Library On computing the second derivatives, and substituting in the differential equations it is found that -3- . The solutions of these equations which give the (1) circular orbit are : y = [/ / Now let ^ ^ ' where ^ i s a parameter; & a function of ^ dependent variables, and IT the independent variable. When (3) these substitutions are made in^the following differential equations are obtained 1. Rawles ^(/ ^ ^ ^ ^ ^ - - -- ^  jr . ; Where f a = Y f We propose to integrate equations (4) as a power series in Accordingly we let yt- - y L , ^ ^ ^ . - - ^ f ^ ^ / ^ ^ It will be shown that ^ and certain constants of intergration may be chosen so that r and q shall be periodic and satisfy certain initial conditions. 2. CONSTRUCTION OF THE SOLUTIONS. When the values of q, r, and S* are substituted in (4) and the terms independent of ^ considered we get ^ a Integration of which gives where C^ is a constant of integration and - / J Solving we get ^ ^ ^ y In order to have a periodic solution we must put Cp=0. Three arbitrary constants A^, B^, D^, are left. The initial conditions are now chosen such that the particle is projected orthogonally from the y*— axis ^ ^ ^ 3 . ^ ^ ^ These initial conditions are those of orbits known as symmetric orbits. For a third condition we choose / , ^ ^ O, ,, Imposing these conditions we get = / , = ^ , = ^ In order to show that the various differential equations can be integrated and the constants of integration and determined so as to satisfy the periodicity and initial conditions, we proceed to terms of higher order. Terms in ^ . r'J Substitution for r^ and q^ yields Integrating 7(a) we get <7 ^ ^ (y) Substitution in 7(b) yields. The term in d ^ ^ - r in the right member will give rise to a non-periodic term r ^ ^ ^ T r in the solution, and hence for a periodic solution we must put Since /-/- J <3 we have ^ — 3 The solution of (9) is Substitution of (10) in (8) gives ^ ^ ^ ^ i r (/'J r .,. ., and for periodic solutions the constant part must vanish or Integration of (11) gives = - / 7 T " // ^ TT - ^ 2 ^ / / and imposing the initial conditions ^ 3 , jP, - - / — - / z X/ Hence yjX' t ^ ^ ' where A^ and C^ have the values given above. Terms in p Substitution of the values of q^, q^, r^, yields^-11 ^ ^ " (3) (2) The expressions for Q and R may be written as follows: ^ ^ ^ { i ^ T where are the coefficients of the sine and cosine terms in (15) and (16). Integration of (25) gives Substitution of (18) in (16) yields-The term in in the right member upon integration gives rise to a non-periodic term; therefore, for a periodic solution we must put ^ - ^ r ^ -<0 When the values for C^tS), s ^ ) in (15) and (16) are put in (19) and the resulting equation simplified we have Solving 18(b) we get The substitution of (21) in 18(a) gives Since qg Must be periodic, the constant part of qg must vanish f-c.-13 When (22) is integrated we get + C ^ ^ T T + p c Imposing the initial conditions we find J * ^ ^ 3 __ The solutions are therefore (21) and (23) with the above values of , , . 3. COMPUTATION FOR THE FIRST CASE For the helium atom and the parameter must be given the value ^ ^ ^ Therefore m * .5; JT was put equal to .05. and /t = yyye^o/^Tr V- ^^jr-C^^.^.r-The orbit was drawn, but is not shown as it is similar to the Y?Z. plane in the three dimensional case. PART II ORBITS OF THREE DIMENSIONS. 4. The Differential Equations. The problem is one of constructing the three dimensional orbits which are in the vicinity of the circular orbits. The differential equations of motion are the same as before, namely, those given by (l) in Part I. After transferring to rotating axis as in the first case we have In this case we let ^ = t ^ T T T " r . where y* is a parameter; is a function of dependent variables, and c the independent variable. When the substitutions (25) are made in (24) and the factor f taken out, the following differential equations are obtained * < = ^ ^ - - ^ - ' ) where 17 T^ t <3 yt, 3 ^ - 3 t/ ^ 2 The variable ^ enters the f, and to even degrees and the Q, to odd degrees. On integrating 26(b) we have j ^ ^ ^ ^ ^ ^ ^ 4 J ^ r ^ where C, is a constant of integration. Substitution of (37) in 26(c) yields 18 We shall take 26(a), 27, 28, as the three equations defining p, q, r. As in Part I, we shall integrate these equations as power series in ^ and shall show that and other constants of integration can be chosen so that p, q, r will be periodic and satisfy certain initial conditions. We first let ^ ^ f ^ j 7 C = Z ( f y ; J =0 and then substitute these series in 26(a), 27, 28. The resulting equations are denoted by 2^(a), 27^ 28,*** respectively. 5. THE GENERATING SOLUTIONS. We consider first only the terms of 26"**(a), 27^, 28***, which are independent of and thus obtain — o Mgy — i-^st-s!'.'!^-19 These are the equations of variation. Equations 30(a) and 30(c) are independent of 30(b) and will be considered first. Neglecting the right member of 30(c) and denoting by D we obtain (3'J T H E . F(JfNC7-<on(/tt_ IS - n ^ and equating this functional determinant to zero we get P - - Z ^ ^ - ^ -2 The second root of D is always negative, and the first root is negative so long as But for the circular solutions must be less than 1, if they are to be real. That is, both roots are negative. We may therefore let ^ ^ 2, Then the solutions of (31) are M'aere ^ r/,2,, 3,^) are constants of integration.Of the eight constants only four are independent. Upon substituting (33) in (31) we get = S , (J ^ /-Z.; t/ -We have therefore, three sets of generating solutions, R<EfH<7j7 r 2 2 T - R P ^ R ' O J ) - jSLvr __ ^ T^L <71 ' ' ^ The constructions for generating solutions I and II only will be considered. 6. CONSTRUCTION OF THE SOLUTIONS. Since the constructions are the same for generating solutions I and II, we shall drop the subscripts in the following work. We shall now consider the differential equat-ions obtained by equating the coefficients of the various powers of ^ in 26***(a), 27^, 28^. Terms independent of ^ . Ttn complete solutions of 30(a), (c) having the period P A H 6 Substitution of 35(b) in 30(b) yields = # f ' - - 3 (n ) where C^ is a constant of integration 22 (o) Since q^ is to be periodic we put C^ ' 0. Three constants JS*^ remain and we may therefore impose three initial conditions upon the solutions. As in Part I we take the case where the orbits are symmetric viz., ^-(6) - - ^(bj - o ( j This gives B ^ L and o . That is, we have jS^T or arbitrary, and may take "Iwithout loss of generality, since r carries ^ as a factor, and ^ is arbitrary. Therefore 6 s / and ^ = -d ^ ^ <7-r Z - - JL (Tir Terms in ^ . -j - 3 s*, n^^Lu - ^ t f y r ^ J C^rT (T ir 23 -f-- ^ / ^ - ^ From 28(b) we have r - 2 /i, 4-and if we expand we ^ J, -- = - - ^ ^ V- - - ^ % T H E R E F O R E ^ -/-C^ (J^*^ — 3 ^ ^ -^ ^ T^ / - 3-7-1. ^  (TI" <rT7 and Neglecting all terms of the right side of (40) and (42) and solving we find where B's and CMs are constants of integration. The particular integrals for p-]_ and r^ are given by _ - ^ J ? ^ -A- - T-L^y i / The coefficient of C o 5 (TIT in the numerators must vanish since - ^ is a root of the denominator. These coefficients are respectively and these expressions vanish for ^ " 3 O R c/'' - o -The latter are trivial solutions and shall be neglected. Complete solutions are, therefore J- ^ ^ ^ ^ ^ ^ ^ , i ^ f,; _ _ ^ ^ ^ ^ ^ ^ ^ ^ ' W H E R E 26 where ^ C^J' are given in (42) and (44) If we substitute for r^ in (41) and solve we get where Periodicity conditions require the coefficient of t to be zero, hence and from the initial conditions = c The desired solutions at this step are,-G- J <7 ^ -/-L3-W H E R E / s i ^ J ' -" J / I " 71= fLNJ 4--JL. - j r ^ J L 2 -28 (T ^ ^ 29 2 , J Neglecting all terms of the right side of (50) (2) except 2 C^ and solving, we get ^ = ^  ^ ^^ ^^^  - ^ ^^  where ^ constants of integrat-ion. The particular integrals are given by The coefficient of Cos (rr* must vanish at the previous step and therefore ^ The complete solutions for r? and pg are ./ ^ <-4 , — -3 ^ / - TTz'y? J - ry - <3, V- ^ ^ <r y-^-v 3 ^ / -On substituting for r,in 50(b) and integrating and equating the coefficient of "C to zero we get where , , r * -- ^ ^ y , From the initial conditions we get The solutions at the second step are therefore (f2) (11 where r ^ - ^ ^ ^ ^ ^ ^ ' 33 7. COMPUTATION. We have __ r r ^ ^ rr ^ ^ ) 7 and similarly for 7? and ^ we let - , ^ ^ G-E*r ^ - y ^ c r T 2L(7-r- — . <3363 /u — -f<<3*7 7 C<r?6rTT — . -7 d<n? — <<3 <302,7^3 (re-in the above case the subscripts 1 and 2 on <7" and co may be restored, and the two sets of orbits found which have the periods and respectively. The above equations were used in plotting the orbits for the period t = TT Values of t were taken from t=0° to t=2160° at 30° intervals and the values of x, y, z computed, (^ea -r/=te.<-&) 13. The orbits were then plotted in three planes. A check was also made on the work, the vis viva 6 integral being used. 7"HE Expfusyy/o^ f^K r ^ / S c^y/a ; 5 W H E R E . -1 ^ O ^ ^ ^ f - f l ^ T ^ ^ ^ ^ -TABLE I to X y z t° X y z 0 ,560 .00 .95 1026 .458 -.56 .60 30 .555 .54 .77 1080 .440 0 .78 60 .536 .87 .33 1116 .448 .40 .70 90 .515 .87 -.19 1170 .485 .85 .19 120 .488 -.64 -.59 1206 .515 .79 -.35 150 .464 .26 -.78 1260 .552 .10 -.94 180 .450 -.10 -.80 1296 .560 -.47 -.63 210 .440 -.43 -.66 1350 .542 -.92 .14 240 .475 -.65 -.46 1386 .515 -.56 .66 270 .458 -.81 -.14 1440 .470 .17 c<3 CO 300 .480 -.81 .26 1476 .448 .56 .58 330 .506 -.60 .65 1530 .445 .79 .05 360 .530 -.17 .83 1566 .458 .73 -.37 450 .555 .94 -.05 1620 .500 .20 -.87 486 .542 .65 -.66 1656 .530 -.40 -.83 540 .500 -.20 -.87 1710 .555 -.94 -.05 576 .470 -.64 -.56 1746 .557 -.72 .60 630 .455 -.79 .05 1800 .530 .17 .93 666 .443 -.60 .43 1836 .500 .68 .57 720 .470 -.17 .83 1890 .458 .81 -.14 756 .500 .36 .81 1926 .443 .60 -.51 810 .542 .92 .14 1980 .448 .10 -.80 846 .557 .78 -.52 2016 .470 -.36 -.76 900 .552 -.10 -.94 2070 .515 -.87 -.19 936 .530 < CO -.62 2106 .542 -.81 .43 990 .485 -.95 .20 2160 .560 0 .95 Various values were taken for t, x, y, z from the Table I and upon substitution in (51) yielded values for the constant of integration C varying between C= 2.17 and C- 2.31. UBC Scanned by UBC Library 9. REFERENCES. T.H. Rawles - Bull, of the American Math. Society Vol.34, p.618 (1929). Langmuir, - Physical Review Vol.17, (1921), p.339 Van Vleck Quantum Principles and Line Spectra, Bull. Nat. Research Council No.54, p.89. Buchanan, "Periodic Orbits of the Second Genus Near the Straight Line Equilibrium Points in the Problem of Three Bodies D. Buchanan - Presented at the Royal Society in Ottawa, May 1929. Moulton Celestial Mechanics, p.280. 

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