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Evolutionary and pulsational models of metal-poor subdwarfs Shkolnik, Evgenya 2000

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EVOLUTIONARY AND PULSATIONAL MODELS OF METAL-POOR SUBDWARFS by EVGENYA SHKOLNTK B.Sc., Dalhousie University, 1998  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES \  Department of Physics and Astronomy  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 2000 © Evgenya Shkolnik, 2000  In presenting degree  at the  this thesis  in partial fulfilment  of the  University of British Columbia, I agree that the  freely available for reference or  by  his  or  for an  advanced  Library shall make it  and study. I further agree that permission for extensive  copying of this thesis for scholarly purposes department  requirements  her  representatives.  may be granted It  is  by the  understood  that  head  of my  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  flcJr.  J 3 .rQfiKD  ABSTRACT  Metal-poor subdwarfs (MPSDs) are probably the Sun's oldest neighbours and are almost certainly low-amplitude p-mode pulsators, like the Sun. One of the goals of the MOST Space Telescope project (due for launch in 2002 - 2003) is to detect MPSD oscillations and apply asteroseismology to refine the calibration of globular cluster isochrone fitting and possibly set a lower limit to the age of the Universe. To be able to interpret the eigenfrequency data, a comprehensive grid of nonadiabatic, nonradial pulsation models for MPSDs was generated for the first time using a code developed by Guenther (1994). Each pulsation model was calculated from a structural evolutionary model using the Yale Stellar Evolution Code with Rotation (YREC) which included up-to-date physics such as both He and heavy element diffusion and the latest EOS and opacity tables. The grid spanned the following ranges: helium abundance Y = 0.235 to 0.255 (consistent with estimates of primordial Y); heavy element abundance Z = 0.0002 to 0.01 (consistent with MPSD spectra); mass M = 0.7 to 1.0 M ; 0  and age A = 5 to 16 Gyr. From the calculated eigenfrequencies of each model, the large and small frequency spacings (A v, £^0,2) were derived as defined by asymptotic pulsation theory (Tassoul 1980). These spacings are sensitive diagnostics of the mass and mainsequence age of the star. Combining MOST's resolution with the uncertainties in the input parameters, MOST data could refine MPSD ages to better than ± 0.056 Gyr (an average error assuming the input physics are correct).  Table of Contents ABSTRACT  ii  LIST OF TABLES  v  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  viii  Chapter 1. INTRODUCTION  1  1.1 In the Beginning  1  1.2 Fundamental Principles of Pulsation  1  1.3 Linear Theory of Non-radial Modes  6  1.4 Driving Mechanisms  15  1.5 Pulsation as a Diagnostic of Stellar Structure  18  1.6 Seismology from Space  23  1.7 The Goal of this Study  25  Chapter 2. P-MODE ASTEROSEISMOLOGY  27  2.1 Asymptotic Pulsation Theory  27  2.2 Verification of Asymptotic Theory  32  Chapter 3. METAL-POOR SUBDWARFS  36  3.1 Observational Properties of MPSDs  36  3.2 Importance of MPSDs  43  3.3 MPSDs as Targets for MOST  48  3.4 History of MPSD Models  48  Chapter 4. EVOLUTIONARY MODELS OF MPSDs  51  4.1 Modeling Procedure  51  4.1.1  Initial Helium Abundance  54  4.1.2  Diffusion  56  iii  4.1.3  Treatment of Convection  56  4.2 The Age and Mass Range of the Grid  58  4.3 Sensitivity of the Models to Input Physics and Parameters . .  60  Chapter 5. NON-RADIAL OSCILLATIONS OF MPSDs  70  5.1 The Code  70  5.2 The Pulsation Models  74  5.3 Diagnostics from Eigenspectra  78  5.4 Sensitivity to Input Physics and Uncertainties  82  Chapter 6. SOLVING FOR THE UNIQUE MODEL  88  Chapter 7. SUMMARY AND CONCLUSIONS  94  7.1 Sensitivities and Uncertainties  95  7.2 Next Steps  97  REFERENCES  99  List of Tables  Table 3.1:  A Sample of Bright MPSDs  42  Table 3.2:  Estimates of the Age of the Universe  47  Table 4.1:  Estimates of Primordial Helium Abundance  55  Table 4.2:  Calibrated Solar Model  58  Table 4.3:  Model Characteristics for Reference Composition  60  Table 6.1:  Interpolated Model Characteristics for the Test Point  91  Tabel7.1:  Errors for A v and Sv  97  v  List of Figures Figure 1.1:  Spherical Harmonics for Various I and m Values  5  Figure 1.2:  Propagation Diagram for jr?-modes and g-modes  9  Figure 1.3:  Eigenfunctions for /?-modes  12  Figure 1.4:  Eigenfunctions for g-modes  13  Figure 1.5:  Schematic Cross-section of Two p-mode Propagation Zones  Figure 1.6:  Solar /?-modes  20  Figure 2.1:  Eigenfrequency Spectrum for a MPSD Model  31  Figure 2.2:  Solar Oscillation Power Spectrum in Integrated Light  33  Figure 3.1:  Ultraviolet Colour Excess Correlation with Stellar Metallicity . . .  39  Figure 3.2:  H-R Diagram of MPSD Main Sequence  40  Figure 3.3:  Observations of Weak Metallic Lines of Typical MPSDs  41  Figure 3.4:  Fiducial Sequence of M92  46  Figure 4.1:  Evolutionary Tracks for Reference Model, (Y, Z) = (0.245, 0.001)  . . .  14"  with Various Masses and a's  61  Figure 4.2:  Evolutionary Tracks Showing Effects of Varying Y  64  Figure 4.3:  Evolutionary Tracks Showing Effects of Varying Z  65  Figure 4.4:  Evolutionary Tracks Showing Effects of Y and Z Diffusion  Figure 4.5:  Evolutionary Tracks Showing Sensitivity from an Expected Error in  . . .  Zofl0% Figure 4.6:  68  Evolutionary Tracks Showing Relative Sensitivity to a, Z, Y, and Diffusion  Figure 5.1:  67  69  Effects of Non-adiabaticity on the Eigenfrequencies and Large Frequency Spacings (A v)  73  Figure 5.2:  Av versus n, the Radial Order  76  Figure 5.3:  Bumped Modes  77  Figure 5.4:  Echelle Plot for the Reference Model at Two Stages of Evolution . . 79  Figure 5.5:  The "Asteroseismic H-R Diagram" or (A v, Sv) Diagram for the Reference Model  ? 80  vi  Figure 5.6:  (A v, Sv) Diagrams for the Four Extreme Compositions of the Pulsation Grid  81  Figure 5.7:  Effects of Variations in Y and Z on the (Av, Sv) Diagram  Figure 5.8:  Effects of Y and Z Diffusion on the (A v, Sv) Diagram  84  Figure 5.9:  Effects of Variations in a on the (A v, Sv) Diagram  86  Figure 5.10:  Effects of an Expected 10% Error in Z on the (A v, Sv) Diagram  Figure 6.1:  Schematic Diagram of Complete Pulsation Grid  Figure 6.2:  (A v, Sv) Diagrams for Four Intermediate Compositions  Figure 6.3:  Plots of log(L/L ) versus Z and log(T ff) versus Z for Test Case @  e  83  . .  87  89 90 . .  93  vii  Acknowledgements There are many people to thank who have assisted me in many ways during the duration of my M.Sc. I would like to thank my supervisor, Professor Jaymie Matthews, for his guidance and enthusiasm. Much of the coherence of this thesis is due to his editing skills. I very much appreciate the involvement of two supportive and knowledgeable advisors, Professors David Guenther and Gordon Walker, who have answered my many questions with their full attention and consideration. Prof. Guenther supplied both the evolutionary and pulsation codes and also arranged for all of the required computing to be done at Saint Mary's University. Thank you. To Kristy Skaret, my office-mate and partner in crime, a hearty thank-you. Thank you to Janet Johnson, the graduate secretary, without whom nothing administrative would ever get done. Thank you to Dr. Patrick Durrell and Dr. Rainer Kuschnig for having their office doors open whenever new astro-jargon needed explaining. I also appreciate the editing efforts of my friend, Ms. Julienne Hills. Finally, to all member of the Geophysics and Astronomy Building Coffee Room Club, thank you for sharing in the adventures of my Astronomy M.Sc.  viii  1.  INTRODUCTION  1.1 In the Beginning ... In August of 1595, David Fabricius, an amateur astronomer and Lutheran pastor, noticed the star o Ceti had disappeared. After a few months, it reappeared and eventually (after five years of observations) an 11-month period of brightness variation was confirmed in o Ceti. Fabricius re-christened it Mira to celebrate this "miraculous" star. It was not until two centuries later, in 1784, that another pulsating variable star was discovered: 8 Cephei, after which all Cepheid variables were named. A 19-year-old, mute, deaf, English amateur astronomer, John Goodricke, measured this star to be 2.3 times brighter at its maximum brightness than at its dimmest. Goodricke noticed that the variation in brightness was cyclic with a period of 5.4 days. For over a hundred years, these rhythmic motions of Cepheids were thought to be due to a number of phenomena; eclipsing binarity, tidal effects in the atmospheres of binary stars, or rotation of dark spots. It was not until 1914 that Harlow Shapley proposed that the stars were "breathing" (expanding and contracting) as they became brighter and dimmer, and in 1926, Sir Arthur Eddington offered the first true theory of stellar pulsation.  1.2 Fundamental Principles of Pulsation All stars are self-gravitating bodies of gas that are maintained by the balance between radiation pressure and gravity. Any short-lived disturbance to the stellar structure will displace the star from equilibrium and trigger a subsequent oscillation. This pulsation  1  manifests itself as changes in a star's brightness or as Doppler shifts in its spectral lines (as well as changes in their strength, width and shape). The simplest way a star can pulsate is in the fundamental mode which propagates through the star along its radius. All radial modes are essentially standing waves, not unlike the standing waves in an organ pipe with one open end. Radial modes cause a star to expand and contract globally maintaining the star's spherical symmetry. The fundamental period of radial pulsation IT depends primarily on the star's density, since it is the inverse travel time of a wave passing across the diameter of the star, and depends on the internal sound speeds and size of the star. The adiabatic sound speed in a gas is  d\np d\np lad  1P  (1.1)  where T is the adiabatic exponent, p is the pressure and p, the mean density. For illustrative purposes, we can start with the naive assumption that the density is constant throughout the star, so the pressure gradient can be simplified to  '4  dp _ dr  GM p r r  2  _  °  ^  -xr p P 2  = -^Gp r, 2  r  1  (1.2)  3  where M is the mass enclosed by a shell of radius r, and G is the gravitational constant. r  Setting the boundary condition of p = 0 at the surface, and integrating,  2  p(r) =  lxGp {R -r ). 2  2  (1.3)  2  Therefore, the period is roughly,  dr ^T7tGp{R  2  -r  1  or n  2VGp  (1.4)  This is similar to the dynamical timescale of the star (the time for a test particle to fall freely through the star due to gravity alone),  t.  (1.5)  Equation (1.4) illustrates the period-mean density relation for pulsating stars. This hints at the potential of aster oseismology, probing the properties of stars through their observed pulsations. An oscillation can be approximated by a small perturbation on a spherically symmetric, static star. If the star's equilibrium state is considered constant (since evolutionary changes are very slow), then the time-dependent eigenfunction describing the perturbation is  3  (1.6)  where Y (0,tp) are the spherical harmonics, m  e  y/"(0,^) = C;'Pj (cos 6)e m|  (1.7)  and C is a normalization constant related to the amplitude of the pulsation. P(x) are the Legendre polynomials of degree £ and azimuthal order m. 6 is the latitudinal angle and <j> is azimuthal angle in spherical polar coordinates, cris the angular frequency of the wave. The harmonic degree £ (= 0, 1,2, ...) is the number of nodal lines on the stellar surface, while the azimuthal order m ( = -£, -£+1, ..., 0, 1,2, ...,£) represents only the number of longitudinal nodal lines. Figure 1.1 shows a pattern of oscillation modes of various £ and m values. A third index, the radial order, n, represents the number of nodes along the radius of the star. The frequencies of the modes are linearly dependent in n and £ but are (2£+l)-fold degenerate in m. A radial oscillation is a special case whose spherical harmonic index is £ - 0 (and thus m = 0). All other oscillations (£ > 0) are nonradial which cause the star to deviate from its spherical shape. The radial modes of oscillation in classical Cepheids were thefirstto be detected because the amplitudes of the waves are large and not diluted by different segments varying in antiphase on a stellar surface. Figure 1.1 (k) and (j) illustrate the high£ spherical harmonics for which this diminished contrast effect is severe.  4  Figure 1.1: Contour plots of the real part of the spherical harmonics (Equation 1.7) Positive contours are represented by solid lines and negative contours, by dashed lines. The equator is shown by "++++++". The cases illustrated are: a) £ = 1, m = 0; b) £ = 1, m = 1; c) £ = 2, m = 0; d) £ = 2, m = 1; e) £ = 2, m = 2; f) £ = 3, m - 0; g) I = 3, m = 1; h) £ = 3,m = 2; i) £ = 3, m = 3; j) £ = 5, m = 5; k) £ = 9, m = 5; 1)€= 10, m = 10 (courtesy of J. Christensen-Dalsgaard 1998).  5  One of two forces, gravity or pressure, will generally act to restore a star to its equilibrium structure after a perturbation is imposed on it. Those wave motions whose restoring force is gravity are termed g-modes and similarly, for pressure, p-modes. Generally, in non-degenerate stars, g-modes are generated in the interior of the star where the gravity gradient exceeds that of pressure. In this region, the amplitudes of the g-modes are greatest. It is possible for the g-modes to be trapped in the interior of the star and never propagate to the stellar surface. P-modes, on the other hand, have maximum amplitude in the envelope of the star, where the pressure gradient is larger than the gravity gradient. In the presence of a global perturbation, say a contraction of the star, the pressure gradient is larger than the gravity gradient and hence the star will rebound and expand beyond its original size. The gravity modes, whose restoration is due to buoyancy, require a variation in the horizontal direction, so there are no purely radial g-modes.  1.3 Linear Theory of Nonradial Modes For small perturbations, the equations are linear and are functions of the radial distance r from the center of the star. The radial displacement £ can be expressed as  (1.8)  In a rapidly rotating sphere, the normal mode is expressed as a sum over an infinite number of terms in the expansion of Y (&,</>). However, when studying single modes of m  f  a slow rotator, the summation sign can be removed and there is only one eigenfunction 6  and one eigenfrequency to consider. In the case of no (or low) rotation and/or no (or weak) magnetic field, there exists a (2^+l)-fold degeneracy in m. The angular frequency a of a particular normal mode is nl  o =2nv nl  (1.9)  M  where vis the eigenfrequency of the mode. Since the eigenfunctions in Equation (1.8) are complex expressions, only the real part is used in the final analysis. Two characteristic frequencies are used to describe the local vibrational properties of a star. Thefirstis the Lamb frequency L which corresponds to the t  reciprocal of the travel time of one horizontal wavelength divided by the local sound speed:  L)=(k cJ h  =  i { i +  l  ) C  r  °  (1.10)  where c is the local adiabatic sound speed given in Equation (1.1). The radial distance r s  is measured from the star's center. The horizontal wave number fa is defined using the horizontal wavelength, Aw, kh = 2n/Ah where Au s 2nr/£. The sound wave travels the distance of Ah in a period of 2n/L . e  The second characteristic frequency is the Brunt-Vaisala frequency N. A bubble of gas may oscillate vertically around an equilibrium position with angular frequency N  under the gravitational restoring force. For this to occur, the local gravity gradient must be greater than the local pressure gradient. The Brunt-Vaisala frequency is written as  | l Jinp N - g\ T, dr  d\np ^  2  where g = GM lr  2  r  (1.11)  dr  is the local acceleration due to gravity.  Figure 1.2 is a propagation diagram showing the angular frequencies against the stellar radius fraction. Representative curves for ofl/andiV map the regions of 2  propagation for the g-,p- and /-modes (or fundamental modes). For high-frequency oscillation, (cr > L , N ), the relative Eulerian pressure perturbationp'/p dominates and 2  2  2  is therefore greater than £ IH , the relative radial displacement. H is the pressure scale P  p  height defined by  dr  H=  — .  "  d\np  (1.12)  This excess in pressure makes pressure the restoring force of the gas packet, creating an oscillation that behaves as an acoustic wave (or p-mode) and propagates closer to the star's surface. The region of propagation nearer to the star's center is governed by oscillations whose frequencies are low (cr < L , N ) where p'/p is less than the relative radial 2  2  2  displacement, £ IH . This deficiency in pressure causes gravity to be the restoring force. P  Therefore, gravity modes tend to be longer in period than the p-modes. There is a third  region of interest, Lf < o < N and Lf > cr > N , in which the amplitudes of the waves 2  2  l  2  decrease exponentially. This wave motion is analogous to the "tunneling" effect in quantum mechanics. This "tunneling" region is referred to as the evanescent zone.  2  2 * i  Figure 1.2: A propagation diagram with variations of N and L with respect to stellar radius fraction of a polytropic model (with polytrope index 3). The horizontal lines represent the p-mode^mode and g-mode propagation zones (see explanation in text). The dashed lines depict the evanescent propagation zone. (This figure was adapted from Unno et al.'s (1989) Figure 4.3.) f  9  The amplitudes of the modes are proportional to ^ and, for the Sun, are illustrated in Figure 1.3 and 1.4 for //-modes and g-modes, respectively. The quantity plotted along the vertical axis is related to the energy density from the radial component of the velocity. For /?-modes, this shows the concentration of energy density is towards the outer regions of the star for high-^ modes. The energy density is almost evenly distributed throughout the star for low-^ modes, essentially allowing the p-modes to penetrate deeper into the star. Defining k = kh + k , then k must decrease as ap-mode, originating at the 2  2  2  r  surface, propagates deeper into the star (encountering an increase in c as the temperature s  increases). Therefore, k decreases more rapidly than k/,. A wave can no longer r  propagate radially inward and is "reflected" at the point along the radius where k  r  vanishes.  This point along the radius is the turning point r,, defined by k = kf, =  ^£(1 + 1)/^ . In terms of sound speed and frequency, the turning point is established by the condition  £^2=- iL_. r  (i.i3)  The arrows, on the bottom two panels of Figure 1.3, identify the turning points of the modes. The turning points define the boundary of the zone in which the p-modes are trapped (referred to as the p-mode propagation zone). Figure 1.5 is a schematic diagram showing the paths of the wave fronts of two p-modes. The turning point for each mode is labeled as the radius of the propagation cavity for the wave. The mode amplitudes  10  decrease rapidly near the surface where the energy decreases exponentially with height for frequencies below the acoustic cut-offfrequency <j defined by ac  a  =  2H  P  (1.14)  2c,  Typically, cr is greater than the Brunt-Vaisala frequency. For T = 5/3, cr = 1.02 N. ac  ac  For the Sun, the cut-off frequency is about 5000 juliz (period « 185 seconds). Conversely, g-modes in a solar-type star would be concentrated in the deep interior where their energies and consequently their amplitudes are at a maximum. The number of times £ crosses the zero-line increases dramatically with £ but the energy distribution stays reasonably constant causing the modes to stay within the same radius fraction of the star. In both figures, the number of zero-crossings of the eigenfunction £ is close to the radial order n of the mode.  11  '  • 1 •  •  •  1  ••—'••••'1  • . • I , . . . . . , . - . , — • , — — — • •r—  .  a)  ^-^  .  ^\  ^ .  .  0.0  1  .  .  r\ r\ AAA/lfl  r\  w \j \J \JVVVIII .  0.2  1 1,1 *  0.4  •»  •  r/R  1  .  0.6  1 .•  .  0.8  1.0  b) s\  /-x  1  0.0  0.2  -  -  -  1  -  -  0.4  -  1  AA  A  -  1  0.6  All  -  0.8  1.0  r/R  '  0.0  •  '  1  0.2  •' •T ••  0.4  1  •  •  1  *  -  c)  0.6  0.8  1.0  r/R  Figure 1.3: Eigenfunctions of selected /?-modes in a solar model with a) £ = 0, n - 23, v = 3310 //Hz; b) £ = 20, n = 17, v= 3375 /Mz; and c) £ = 60, n=\0,v= 3234 {Hz. (Christensen-Dalsgaard 1998, p. 75; reproduced with permission). The arrows coincide with the turning point defined by Equation (1.13).  12  0.0  0.2  0.4  0.6  0.8  1.0  r/R  -a  B  Figure 1.4: Eigenfunctions for selected g-modes in a solar model with a) £ = \,n = - 5 , v = 110 //Hz; b) £ = 2, n = -10, v= 104 [Mz; and c) £ = 4, n = -18, v = 106 //Hz (Christensen-Dalsgaard 1998, p.78; reproduced with permission).  13  Figure 1.5: Schematic cross-section of p-mode propagation zones. Acoustic waves with £ = 30, v= 3000 /Mz penetrate deeper into the star having a lower turning point r, than the more shallow penetration of the £ = 100, v= 3000 /Mz pmode. The lines orthogonal to the deeper path of propagation illustrate the wave fronts. (Diagram courtesy of J. Christensen-Dalsgaard (1998).)  14  1.4  Driving Mechanisms  Any star has a very large number of natural resonances but most stars show only a few observable pulsation frequencies. This begs the questions: Which modes are excited and how? Eddington (1926) proposed that the mechanism driving standing waves in stars must be a thermodynamic heat engine. The imbalance between positive and negative work, pdV, on the gases either amplifies or damps an oscillation, respectively. Therefore, if the cumulative contribution to the work (i.e., the work integral) from all the layers of a star is positive, then the pulsation amplitude will grow. Negative work will cause the amplitude to decay. The vertical displacement of a pocket of gas will oscillate but over time will settle at the equilibrium value for which the total work is zero. In order for oscillations to be amplified, the driving layers of a pulsating star must absorb heat at the time of their maximum compression at which time the layers will feel maximum pressure. Eddington identified one obvious region for maximum compression, the star's centre. At this location, the condition of maximum pressure results in an increase in the thermonuclear reaction rates e. This energy mechanism, called the emechanism, probably exists in the centre of some or all stars and may drive some oscillations. However, the centre of the star is near a node of any standing wave and therefore will have minimal radial displacement, making it difficult to detect. Eddington also proposed an alternative called the K-mechanism. The Kmechanism works like a valve:a particular layer of the star becomes more opaque when compressed and will "dam up" the energy before releasing it to the surface. This increased pressure pushes the entire layer towards the surface where it cools and  15  becomes more transparent, releasing the built-up energy and then returning the layer deeper into the star. In this scenario, the opacity must actually increase with compression. The opacity K of gas in a star generally follows Kramers law (see, e.g., Hansen and Kawaler 1994) describing its dependence on the local temperature and density:  ^ o c A .  (1.15)  2^3.5  In most regions of a star, where there is a compression of the gas (increase in temperature as well as density), the opacity will decrease. It is therefore necessary to identify a layer in the star where compression will yield a much larger relative increase in density than in temperature. These special conditions occur in the partial ionization zones. These zones werefirstidentified by Zhevakin (1963) and later calculated by Kippenhahn and Baker (1965) and John Cox (1968). They found that in a layer where the atoms are partially ionized, part of the work done on the gas as it is compressed causes further ionization rather than increasing the temperature. This smaller increase in temperature leaves the increase in density to dominate, causing the opacity to increase. Similarly, during expansion, the temperature does not decrease as much as expected due to the energy released as the electrons recombine with the ions. Again the density change dominates Kramers law and the opacity decreases with it. This cycle is repeated creating a continuous amplification of the pulsation modes. Stellar pulsation models that include partial ionization zones coincide with the observed instability strip on the H-R diagram (a narrow, almost vertical band spanning  16  temperatures of ~ 600 to 1100 K) where the Cepheids are found. Calculations show that most stars on this strip have variations due to the He II partial ionization zone (He II — > He III). This zone is deeper within the star, with a characteristic temperature of 4 x 10  4  K, than the hydrogen partial ionization zone (both H I -> H II and He I —» He II) whose characteristic temperature is 1 to 1.5 x 10 K. 4  According to the /r-mechanism, the luminosity incident at the bottom of the hydrogen ionization zone is at a maximum when the zone is at its minimum radius. This merely propels the zone outwards through mass towards the stellar surface at its most rapid pace, and forces the emergent luminosity to be at its maximum just after the star is at its minimum radius. This delaying motion of the hydrogen ionization zone explains the phase lag of maximum luminosity behind minimum radius that is observed for classical Cepheids and RR Lyrae stars (Simon 1984). Surely, pulsation amplitudes do not grow to infinity or all pulsating stars would blow apart. Accordingly, the fate of a perturbation from equilibrium depends on the "uncertain competition" between driving and damping mechanisms within the star (Christensen-Dalsgaard et al. 1989). The finite amplitude of a mode is maintained partly by energy dissipation due to larger radiation loss from positions of higher temperatures. Coupling with other modes can also reduce the amplitude of the wave. (See the discussions on three-mode coupling by Vandakurov (1979) and Dziembowski (1982)). The finiteness of the oscillation amplitudes and the observed lifetimes of the solar pmodes (on the order of 10 hours to 10 days for high and low £, respectively; Chen et al. 1996) are the results of an uncertain form of damping intrinsic to stellar structure.  17  1.5 Pulsations as a Diagnostic of Stellar Structure Stobie (1969)firstsuggested that Cepheids might be able to pulsate not only radially in their fundamental modes (n = 0, £ = 0) but also in theirfirst(n - 1) and second (n = 2) overtones. Such double-mode Cepheids were observed (see Fitch 1970) and recently, Welch et al. (1994) discovered 43 Cepheids beating in theirfirstand second overtones in the Large Magellanic Cloud. The period ratio of a double-mode Cepheid can be used to estimate the radius of a Cepheid. Combined with T ff, this solves for the luminosity and e  the star's location on the H-R diagram. However, to obtain more information, one requires many frequencies, each having its own turning point (Equation (1.13)) and therefore each sampling the a different depth in the star (see Figure 1.5). Thus, the greater the number of pulsations observed in a star, the more information can be derived from the pulsations. This leads us deeper into the stars as well as deeper into the study of asteroseismology. Below the Cepheid region of the instability strip lie the 5 Scuti stars. They are late-type stars close the main sequence which are known to be pulsate in radial and nonradial modes with periods of 0.5 to 6 hours (see e.g., Breger 1995). Even though there are many resonant frequencies, only a few low-overtone modes are excited in the 5 Scuti stars with no regularity in their frequencies (i.e., they do not fall in the asymptotic regime discussed in §2.1). Stellar models produce so many resonant frequencies, that it is difficult to match an observed frequency with a specific predicted mode (Matthews 1993) unless the mode values (n, £, m) are known already. Two promising mode discriminants are rotational splitting and amplitude ratios with phase lags.  18  The most successful case of stellar seismology has been the five-minute oscillations of the Sun. First recognized in 1960 by Leighton et al. (1962), these oscillations have periods nearfiveminutes and consist of tens of millions of p-modes. They are observed to have local velocity amplitudes that are on the order of 10 cm/s and when modes add in phase, the combined amplitude can be as large as 500 m/s. Figure 1.6 is a plot of mode frequency v versus the spherical harmonic I for the intermediate-^ p-modes in the Sun. The ridges of power are consistent with p-mode oscillations of high overtone n propagating in acoustic cavities of various depths (Ulrich 1970; Toomre 1970). Thefive-minutemodes travel through the evanescent zone of the propagation diagram (Figure 1.2). These standing /?-modes are trapped between two reflective boundaries in the subphotospheric regions of the Sun. The observed evanescent waves are the "tunneling" or escaping waves from the upper boundary. Solar/>-modes have been observed with I = 0 up to at least 3000 and n < 26 with corresponding frequencies from ~ 1.0 to 6.0 mHz (Libbrecht, Woodward and Kaufman 1990). There are two approaches that make use of the tremendous number of solar eigenmodes as diagnostics of the internal structure of the Sun. Thefirstis the forward problem in which theoretically obtained eigenfrequencies are compared with observations such that a best-fit structural model emerges from a series of models with varying parameters (e.g., Ando and Osaki 1975). An example of the forward problem concerns the determination of the depth of the solar convection zone. By calculating the turning points of the high-^ five-minute/?-modes that extend down to the superadiabatic 1  boundary layer, one can calculate the depth of the convection zone. One of the latest  19  results of this analysis indicates that the base of the convection zone is at 0.713 ± 0.003 R (Guenther and Demarque 1997). @  too Degree (  Figure 1.6: Solarp-modes shown in an f-v diagram. South Pole frequency measurements are indicated by crosses, and Big Bear Solar Observatory measurements are shown by plus signs. The lowest ridge is of modes with n = 3 and n increases by one for each higher ridge (from Duvall et al. 1988).  20  The second approach is known as the inversion problemfirstapplied to the Sun by Gough (1984) and Shibahashi (1988) and others. In this case, functional forms of known physical quantities such as the sound speed distribution can be directly determined from integral equations provided by observed eigenfrequencies. Two theories have been proposed for the excitation mechanisms for the fiveminute solar oscillations. Thefirsttheory (Ando and Osaki 1975) describes linear overstability of the eigenmodes due to the /^-mechanism of the hydrogen ionization zone. This theory, however, neglects the effects of turbulent convection on the acoustic modes. The second, and more widely convincing theory wasfirstintroduced by Goldreich and Keeley (1977) who proposed that thefive-minuteoscillations are stochastically excited by the turbulent convection itself. The convection excites the modes at some resonant frequency which is maintained by the Brownian motion of the particles of the system. Goldriech and Kuman (1988) attempt to explain the velocity amplitudes of the observed p-modes by stochastic excitation as well, but this requires knowing the velocity of the convective eddies in order to know the exact energy being transferred to the eigenmodes. However, it has been shown from observations that the dependence of mode lifetimes on mode energy is consistent with this theory of acoustic power radiated by convective turbulence. (See Stein and Leibacher 1981; Goldreich and Kumar 1988; Libbrecht 1988; Christensen-Dalsgaard et al. 1989.) Helioseismology of the Sun's interior has lead to strong tests of the Standard Solar Model (SSM). Helioseismology has also shown that it is almost impossible to change the SSM enough to account for the deficit of observed solar neutrinos (e.g.,  21  Guenther and Demarque 1997), strongly suggesting the solution to the "neutrino paradox" lies in neutrino, not solar, physics. Encouraged by the success of helioseismology, astronomers would like to apply the approach to other stars. The close proximity of the Sun allows helioseismologists to resolve the solar disk such that high-^ modes can be detected and horizontal wave numbers kh can be directly measured. Most stars must be observed in unresolved integrated light. This limits observations to modes of £ < 4 (except through Doppler imaging; e.g., Kennelly et al. 1999). Higher-^ modes will sustain cancellation effects due to so many segments of the stellar disk varying in anti-phase to each other. Fortunately, the regime of low £, high n modes allows them to be described by asymptotic pulsation theory which has significant diagnostic power (see §2; Tassoul 1980). Low-^ modes have the deepest inner turning points and are therefore maximally sensitive to the internal structure of the star. In 1978, Kurtz (1978) first found short period, nonradial oscillation in spectral type Ap stars. They are cool, highly magnetic stars considered spectroscopically peculiar due to their SrCrEu line-strength oddities. This class of stars, appropriately named rapidly oscillating Ap (roAp) stars, have oscillations which are consistent with high-rc and low-£p-modes, but with amplitudes far larger than the Sun's/7-modes. The roAp stars are discussed further in §2.3. Cepheids, 8 Scuti stars and roAp stars have large enough amplitudes such that they can be observed from the ground. The Sun is uniquely close and resolvable and therefore its much smaller amplitudes are also observable by ground-based telescopes.  22  However, the challenge of detecting and resolving Sun-like pulsations in nearby stars is leading asteroseismologists to observe from space.  1.6  Seismology from Space  Solar-type oscillations in Sun-like stars have not yet been convincingly detected due to their small amplitudes and the limitations of ground-based observations. However, claims of such detections have been reported for Procyon A (e.g., Barban et al. 1999) and n Boo (Kjeldsen et al. 1995). The Sun's oscillation amplitudes in integrated light are a few micromagnitudes in brightness and a few cm/s in radial velocity, which are about 10 - 100 times below current ground-based detection limits. The photometric limits are mainly set by the scintillation noise of the Earth's atmosphere. Even the Sun's oscillations in brightness are only detectable from space; e.g., the ACRIM bolometer aboard the Solar Maximum Mission (Woodward and Hudson 1983), the IPHIR experiment (Froelich et al. 1988) and the irradiance instrument on the SOHO satellite (Roviraetal. 1998). Nevertheless, asteroseismology may take a great leap forward within the next two years as a result of plans for ultra-precise photometry of stars from space. The first attempted leap was, regrettably, a very short one. In 1997, a 9-cm optical telescope and photoelectric photometer called EVRIS (Etude de la Variabilite et de la Rotation des Interieurs Stellaire; Buey et al. 1997) was launched aboard the Russian MARS-96 probe en route to Mars. A French-Russian-Austrian collaboration, EVRIS was intended to measure photometric microvariability of a few bright stars during the cruise phase of the mission. Unfortunately, the MARS-96 crashed soon after launch.  23  Currently, there are at least three asteroseismology satellites in advanced stages of planning or construction whose primary goals are to study solar-type p-mode pulsation. MOST (Microvariability and Oscillations of STars) is now on track to be the first asteroseismic space mission into orbit with a projected launch date of 2002 - 2003 ( The baseline orbit is a Sun-synchronous, circular orbit at an inclination of 98.6° and an altitude of 785 km. This orbit gives MOST a Continuous Viewing Zone (CVZ) of ~ 54° (corresponding to declination limits of-18 to +36°) allowing the telescope to observe a star continuously for up to ~ 40 days. This modest-sized telescope, with an aperture of only 15 cm, will be capable of unprecedented photometric precision which even the largest ground-based telescopes cannot match. MOST will be able to detect variations as small as few parts per million (ppm) for clear detections of the oscillation eigenspectrum with good resolution in relatively bright stars. There are also two European asteroseismology space missions aiming for launch in the next few years. COROT (Convection, Rotation and planetary Transits), a mainly French space mission in Phase C (flight design), is aiming for launch in 2004 (Catala et al. 1995). COROT will be in an inertial low-Earth orbit allowing it to monitor two small (-12° wide) starfields for ~ 5 months each. COROT's telescope, with an aperture of about 25 cm, will have high photometric sensitivity and very high frequency resolution. In addition to seismology, COROT will search for extra-solar planets. MONS (Measuring Oscillations in Nearby Stars) is a Danish project awaiting funding. Its proposed 40-cm telescope would be in a highly-elliptical geostationary transfer orbit giving it access to stars across almost the entire sky long-term (MONS Mission Outline Document MONS-98/05;  24  Further into the future, there are even more ambitious (and expensive) asteroseismic missions such as the Eddington (Penny et al. 2000) and Kepler (Borucki et al. 2000) projects.  1.7  The Goal of this Study  Although "normal" Sun-like stars are the prime targets for these space missions, another high priority is the class of metal-poor subdwarfs (MPSDs). MPSDs are old, halo field stars that stand out due to their high-velocity proper motions as they pass through the solar neighbourhood. Like the Sun, MPSDs have outer convection zones and similar masses, but significantly lower metal (i.e., heavy element) abundances. The resemblance of MPSD structure to the Sun's provides a promising opportunity to apply />-mode asteroseismology to MPSDs. The goal of this thesis was to develop a grid of MPSD evolutionary models from which pulsation eigenspectra could be derived. The parameters of these eigenspectra correlate with physical characteristics of the stars (e.g., density and age). Specifically, this grid is geared toward interpreting the eigenfrequency data of very old MPSDs to be returned by space photometry missions, specifically MOST. In particular, the information could independently set a lower limit to the age of the Universe, refine globular cluster isochronefitting,and improve our understanding of the correlations between stellar ages and kinematics. The following section (§2) introduces the asymptotic theory of asteroseismology including definitions for the characteristic large and small frequency spacings, A v and Sv, and their dependence on mass and age, respectively. The known characteristics of  25  MPSDs are discussed in §3 along with a history of observational and theoretical studies. The grid of evolutionary models is presented in §4 with basics of the stellar evolution code used. The grid samples in four fundamental stellar parameters: mass M , age A , helium abundance Y , and heavy-element abundance Z. For each model in the grid, a pulsation spectrum was calculated. The results are compiled in the form of diagnostic (A v, Sv) diagrams and presented in §5. The grid is intended that future observations of pulsation frequencies and their spacings can be matched to the best model for the target star. A method to isolate one model from such an extensive grid is outlined in §6. A summary of results and conclusions is in §7.  26  2  P-MODE  ASTEROSEISMOLOGY  Numerical descriptions of nonradial oscillations of stars were pioneered by Pekeris (1938) and Cowling (1941) who showed that the classifications of the p-, g-, and /-modes (described in §1.3) were not nearly as clearly defined as initially believed. Their theories showed that, in pulsation models that were highly condensed (i.e., unevolved stars), there were eigenfunctions that corresponded to modes that propagated through both the pmode and the g-mode propagation zones. Scuflaire (1974) described them as "mixedmodes" or "bumped modes", and found they were typically of low order modes in latetype dwarf stars. This ambiguity is not apparent in higher-order modes where the characteristic frequency spacings increase for p-modes and decrease for g-modes as n increases. The overlap between the two propagation zones in the low-« regime instigated the investigation of asymptotic methods.  2.1  Asymptotic Pulsation Theory  To describe the nonradial pulsations of a spherically symmetric star in hydrostatic equilibrium, Pekeris (1938) developed a fourth-order set of linearized equations. The perturbations % were assumed to obey Equation (1.6). Tassoul (1980) eventually r  reduced the equations to a second-order set of differential equations that are valid in the asymptotic regime of high n and low £ (i.e., n » £). This approximation can be used with high confidence since it has been shown that for the higher-order modes of oscillation, the Eulerian variation of the gravitational potential can be neglected (e.g.,  27  Vandakurov 1967; Tassoul and Tassoul 1968). The second-order system consists of the following two equations: (2.12)  ( s: \  ( s + A  P  \  p  IP)  (2.13)  with  p  r,/?  (2.14)  (2.15) p  x  Here, x is the normalized distance r/R, L and N are the Lamb and Brunt-Vaisala frequencies respectively (Equations (1.10) and (1.11)). In order to solve the basic equations to identify possible p-modes and g-modes, the asymptotic theory requires that the stellar model be divided into several domains such that each contains only one transition point. In general, transition points include all the positions within the star that bound the p-mode and g-mode propagation zones and where N - L =0. For higher order modes, there are only two fixed transition points, the center and the surface. The asymptotic equations are solved for each domain and through matching of the separate solutions the characteristic frequencies emerge as the eigenvalues. The resultant p-modes frequencies of high n and low I are  v  n(  « Av n + — + 2  y  (2.16)  28  where y is a small phase constant and A v, the characteristic large frequency spacing, is  -1-1  (2.17)  Av =  The small frequency spacing 8v„ is a second order term: f  5v  - ( 4 ^ + 6)  Av 4n v  •fi  2  5  dc, dr dr r  (2.18)  In Figure 2.1, the pulsation frequencies of a stellar model (restricted to low £ and high n) exhibit the roughly equal frequency spacing characteristic of asymptotic behavior. A v is the inverse of the travel time of a sound wave across the stellar diameter and is therefore sensitive to the density and hence to the mass if the radius is known. It approximately scales as (M/ R ) 3  1/2  (Gabriel et al. 1985). Sv is most sensitive to the conditions in the  stellar core due to its dependence on Mr. As the star evolves, the isothermal core undergoes nuclear fusion where there is a constant depletion of hydrogen, thus increasing the mean molecular weight ju. Since the sound speed c for an ideal gas obeys the s  relation  (2.19)  pjn  H  29  then in the conditions of the core, the increase in fx with relatively constant T, decreases the sound speed. Since the main contribution to the sound speed gradient in the core comes from the gradient of the mean molecular weight, then Sv is a direct measurement of the amount of hydrogen burning that has taken place and hence of the age of the star. In the nuclear burning core, dcjdr is positive and increases with stellar age causing Sv to decrease with increasing age. By contrast, g-modes are characterized by their period spacings:  (2.20)  for which the angular frequencies are  (2.21)  y is a small phase constant and the turning point r, in a Sun-like star is usually taken as the base of the convection zone. Even though g-modes have not yet been convincingly detected in the Sun, they have been observed in white dwarfs and used for asteroseismology of those degenerate stars (e.g., Fontaine and Brassard 1994).  30  2  ^ ©g  ©  +  II «v o o  I  a  00 CN  C I  1 H  O O  sf  cu CU  co  CN  ^  5k >>  T3  00 6 .S CU s 03 u oS O  O O CD CM  ex  CU  CU  11  03  cr JS  O O  ,o  CN  T3  3  CD  C3  O O CNI  IN  X  5*.  <£3  sa  3 u P3 cu  N  3 C3  c ry.  —  cu IH  ^  rt  «r,  O ^  ^  H  5^3  O  o  CO CN  CU  CU  CO •3  •  X)  C3 c  o o  CN CN  CO  00 <U "8 c e -a 93 3  O O  o  ao 93 -3 CM  c u  CN  CO  o o  o o  o CO  o co  o -3-  o  CN  o o  o CO  o CD  eprmidwv Ajej}!qjv  o  o CM  8  S  O  CO  _cu  -C 3  *H N  3  OX)  1 I co O  C 3  1>  !—i r-  1  ^  O"  O  00  CM  <L> CU  W  'S "2 U 09  3  « cn E  2.2 Verification of Asymptotic Theory Brookes, Isaak and ven der Raay (1978) observed the Sun using integrated sunlight over the whole disk in an attempt to measure the gravitational redshift of the Sun. In the residuals of their data, they noticed a variation with an amplitude less than 1 m/s and with a period of about five minutes. The power spectrum of these fluctuations revealed equally spaced peaks with a separation of 67 //Hz. The power spectrum of the solar fiveminute /?-modes seen in integrated light is shown in Figure 2.2. Since integrated light observations cancel out high-degree modes, only low-^ modes remain. The short oscillation periods near five minutes are much shorter than the Sun's fundamental mode (~ 1/2 hour) which implies that their radial order must be high. This scenario of n » £ coincides with the asymptotic theory which predicts a characteristic frequency spacing of A v= 134 //Hz for the Sun. The first photometric detection of solar p-modes was made by the A C R I M bolometer aboard the Solar Maximum Mission (Woodward and Hudson 1983). The pulsations of rapidly oscillating Ap stars (roAp) also appear to be described by asymptotic theory. The roAp stars (see §1.5) are observed to have oscillation periods of 4 to 15 minutes (Kurtz 1990) which are short relative to the expected periods of their fundamental modes (~ 2 hours; comparable to 8 Scuti stars). This implies the stars are oscillating in high overtone (n ~ 25 to 40). The oscillation amplitude is usually a few millimagnitudes, which is small, but much larger than for the Sun (in integrated light).  32  X  a. a  150  5T  100  h  2000  2500  3000 v (/uHz)  3500  4000  4500  Figure 2.2: Power spectrum of solar oscillations obtained from Doppler observations in light integrated over the disk of the Sun. The vertical axis is normalized to show velocity power per frequency bin. The horizontal axis is centered on the dominantfive-minuteoscillations. (See Elsworth et al. 1995.) (Thisfigureis courtesy of J. Christen-Dalsgaard.)  Oscillations in roAp stars werefirstdetected by photometric variations (Kurtz 1978) and later as Doppler shifts in spectroscopic lines (Matthews et al. 1988; Libbrecht 1988). Their pulsation amplitudes are directly correlated to the observed magnetic field strength, which varies along the line of sight with the star's rotation. A closer look at the Fourier amplitude spectrum of some roAp star eigenspectra reveals a tripletfinestructure splitting of the dominant frequency. The side lobes are nearly symmetrically separated from the central peak by the frequency of the stellar rotation. Kurtz (1982) explains this splitting with his Oblique Pulsator Model which consists of a rotating star with an oblique magnetic axis whose pulsation modes are axisymmetric (m = 0) to the symmetry axis of the magnetic axis. The modes are interpreted as nonradial oscillations with primarily (£, m ) = (1, 0). Using Tassoul's (1980) asymptotic theory outlined above, the eigenfrequencies can be approximated by Equation (2.16). This equation predicts that p-modes of even and odd £ alternate with a frequency spacing of A v/2 (ignoring the second-order small spacing 8v). Kurtz and Seeman (1983) demonstrated the roAp star HD 24712 (HR 1217) has such a frequency spacing of 33 //Hz implying A v ~ 66 //Hz. This is consistent with the predicted value of A v for models of similar mass and evolutionary stage (see Shibahashi and Saio 1985 for details). Using these frequencies, the inferred value of A v and assuming 0 < £ < 2, Equation (2.16) predicts overtones to be near n ~ 30. But a later campaign by Kurtz, Matthews et al. (1989) revealed another frequency which may not match this interpretation. Another example is the 50 - 55 //Hz equally spacing of frequencies from 12 to 20 minutes resolved by Matthews et al. (1987) in the roAp star,  34  HD 60435. These observations match the model predictions for the p-mode spectrum of a slightly evolved A star. A key test of asteroseismology involved the comparison of Hipparcos parallaxes (and the resulting luminosities) of roAp stars with predictions from asymptotic pulsation theory (Matthews, Kurtz and Martinez 1999). Radii were seismically determined from observations of A v for a sample of 12 stars showing asymptotic spacings. Coupled with independent estimates of their effective temperatures through H(3 photometry, luminosities and subsequently, parallaxes were determined. There is remarkable agreement with Hipparcos parallaxes for most of the roAp stars however the few discrepancies point to possible inaccuracies in the stellar physics or even systematic errors in the Hipparcos data (Matthews et al. 1999). The potential of asymptotic pulsation theory and asteroseismology has led to many efforts to detect and model p-modes in other Sun-like stars. Only recently have astronomers seriously considered applying these techniques to the metal-poor subdwarfs. They are Sun-like stars in the sense that they have comparable mass and outer convective envelopes. Therefore asymptotic theory is a hopeful tool with which asteroseismology can probe the internal structure of MPSDs.  35  3  METAL-POOR SUBDWARFS  The Milky Way Galaxy is currently believed to have rapidly collapsed from a protoGalactic Nebula where the spherical halo formed much earlier than the disk. The old halo stars have maintained their nearly radial trajectories resulting in high velocities through the disk along their highly elliptical orbits. The earliest catalogue that contained proper motions of high-velocity stars was that of Miczaika (1940). Roman (1955) selected from Miczaika's catalogue all stars with proper motion velocities higher than 100 km/s to incorporate in her Catalogue of HighVelocity Stars, which contained 600 stars. From this sample, Roman was the first to note: An indication of the weakness of the metallic lines in the F- and Gtype spectra can be found from a comparison of the B - V and U - B colours for each star. Ultraviolet excess, ranging from 0.0 to more than 0.2 mag, which is found in the stars with the highest velocity, is well correlated with the weakness of the lines. Moreover, both anomalies are correlated with velocity, in the sense that the stars with the weakest lines also have the largest ultraviolet excesses and the largest space velocities.  Stars such as these appeared to have absolute magnitudes intermediate between main sequence dwarfs and white dwarfs. Chamberlain and Aller (1951) dubbed them "intermediate white dwarfs" or "subdwarfs".  3.1 Observational Properties of MPSDs The metal-poor subdwarfs (MPSDs), as they are now known, arefieldhalo stars passing through the solar neigbourhood. A "reverse" line blanketing effect can explain the UV  36  excess correlation with weak metal lines noted by Roman (see Figure 3.1) causing MPSDs to appear sub-luminous relative to the main sequence. Line blanketing is the redistribution of flux caused by the dense metallic absorption lines in a stellar spectrum of a "normal" dwarf star, which increases the flux in the red. This causes the effective temperature to appear lower than the blackbody equivalent. The opposite effect of line blanketing occurs in MPSDs due to their weak metallic lines. The net result causes a blueward shift in the observed colour of the star or a shift towards higher effective temperature for its luminosity. Thus, on an H-R Diagram, an MPSD seems subluminous, falling below the standard main sequence. Figure 3.2 contains two H-R diagrams, each showing an obvious subdwarf main sequence, which is clearly shifted relative to the standard fiducial Hyades main sequence. MPSDs are Population II stars of spectral type from late A to early K, which represent an old generation of stars in the Galaxy. Late-type MPSDs have three characteristics in common: (1) high space velocities of about 100 km/s on average (Fuchs et al. 1999), (2) highly eccentric Galactic orbits, and (3) low metallicity relative to the Sun. Spectroscopically, it is quite clear that MPSDs are deficient in the heavy elements. Figure 3.3 illustrates the pronounced metal deficiency of typical MPSDs through observations of weak-metallic spectral lines. The metal abundance of a star can be expressed by its [Fe/H] value defined by  [Fe/H] = logfFe. / Fe ) - log(H* / H ) @  @  (3.1)  37  where Fe is the iron abundance (comprising most of the heavy element abundance Z, by mass) and H is the hydrogen abundance (also denoted by X). Both are mass fractions such that X + Y + Z = 1 where Y is the helium abundance. The observed range of [Fe/H] among MPSDs is typically between -2.5 and -0.3. There are metal-poor stars which are not old halo stars such as X Bootis stars whose atmospheres are overabundant in the lighter elements (C, N, O) leaving them metal-deficient in heavier elements like Mg, Al, Si, S, Mn, Fe and Ni (Baschek and Slettebak 1988). This is thought to be a consequence of heavy-element diffusion: radiative acceleration of some elements and gravitational settling of others. (Michaud, 1970). Diffusion cannot explain MPSDs, though, since most MPSD show an overabundance of a-elements (O, Mg, Si, Ca, and Ti) relative to the Fe abundance (which is an indicator of the star's enrichment history. If diffusion was the cause for the low [Fe/H] of MPSDs these a-elements would have surely undergone gravitational settling since their cross-sections for radiative acceleration are relatively low. The largest survey to date of subdwarf metallicities and kinematics was compiled by Carney et al. (1994). Prior studies were carried out by Schuster and Nissen (1989) and by Ryan and Norris (1991). The primary source for stars in the Carney et al. survey was the Lowell Proper Motion Catalogue (Giclas et al. 1971, 1978) in which the blue magnitude limit was B < 16 and the proper motion limit was > 0.26"/year (tangential velocity > 80 km/s). For fainter stars, down to B = 20, Carney et al. included stars taken from the New Luyten Two-Tenths Catalogue (Luyten 1979, 1980). The limit on the proper motion was set at 0.18"/year. The Carney et al. catalogue contains spectroscopic  38  -3.0  -2.5  I  I  •  Dwarfs  o  Dwarfs with poor T values  1  I  B  +• Subgionts  -2.0  x  Close binaries  +•  X  -1.5  -1.0  -0.5  + 0.5  -0.5  0  0.5  .10  .15  wlm  .20  6(U-B)  .25  dm  . 30  .35  .AO  0 6  Figure 3.1: Adopted stellar metallicites, [Fe/H], versus the normalized ultraviolet excess, 8(U - B). The dots are single stars, plus signs are subgiants, and crosses are binaries. The circles are singles stars with uncertain temperatures. Large dots have higher uncertainties than the small dots. (Figure taken from Carney 1979.)  data for 1447 non-degenerate stars, ranging in spectral type from late A to early K, for which temperature, metallicity, distance, space motion, and Galactic orbital parameters are reported. Of these, 25% have [Fe/H] < -1.2 and 27% of the stars in the sample have -1.2 < [Fe/H] < -0.5. The complete range of metallicity in Carney et al.'s catalogue is -2.0 < [Fe/H] < -0.3 (1994). Table 3.1 is a representative (but not complete) list of bright late-type MPSDs selected over a range of metallicities down to V = 9.  39  •a 3  OX  Figure 3.3: Spectra of high-velocity subdwarfs, except for HD 6755, a very weakline dwarf. BD +17° 31 is the most metal-poor subdwarf of this sample. (Figure courtesy of H. Richer.)  41  Table 3.1  A Sample of Bright Metal-Poor Subdwarfs (B-V) [Fe/H] Log(g)* Spectral T ff * v„ (cm/s ) (K) Type (mas) H D 59374 F8V 8.48 0.52 -1.02 20.00 4.30 5851 H D 64090 sdG2 8.27 0.61 -1.60 35.29 3.54 5362 H D 64606* G8V 7.43 0.73 -0.93 52.01 4.0 5040 HD 73898* G4III 7.89 0.90 -0.49 13.83 3.03 5030 HD 74011 F8 7.42 0.58 -0.57 21.51 4.15 5741 H D 76813* G9III 5.20 0.93 -0.82 10.19 4.2 6072 H D 76932** F7 5.82 0.52 -1.10 46.90 3.5 5860 H D 78558 G3V 7.34 0.57 -0.40 27.27 4.28 5767 HD 78747* GO 7.74 0.56 -0.74 25.16 4.41 5860 HD 79452* G6III 5.97 0.86 -0.85 7.17 2.20 4165 HD 82210* G4III 4.60 0.73 -0.38 30.89 3.58 5305 HD 83212 G8III 8.33 1.01 -1.51 1.96 1.2 4460 H D 84937 sdF5 8.33 0.40 -2.07 12.44 4.5 5929 HD 88261 G3V 8.07 0.59 -0.66 21.97 3.98 5550 H D 89125* F8V 5.81 0.50 -0.38 44.01 4.51 6000 H D 96434* F0 5.52 0.97 -0.50 5.76 2.75 4755 HD 101227 GO 8.45 0.69 -0.40 26.74 4.68 5504 HD 103095 G8V 6.42 0.75 -1.22 109.22 4.5 4500 HD 108076 GOV 8.03 0.57 -0.75 26.94 4.20 5733 HD 110897* GOV 5.95 0.56 -0.51 57.57 4.15 5793 H D 114762 F9V 7.30 0.52 -0.67 24.65 4.0 5860 H D 126512 F9V 7.27 0.58 -0.56 21.32 4.20 5753 HD 128279* GO 7.97 0.63 -2.05 5.96 3.5 5478 HD 130551* F5V 7.18 0.44 • -0.62 20.94 4.25 6237 HD 132475 F5 8.57 0.53 -1.04 10.85 3.58 5860 H D 134169* G1V 7.67 0.56 -0.72 16.80 3.8 5793 H D 140283* sdF3 7.20 0.48 -2.38 17.44 4.80 6300 H D 148816 F8V 7.27 0.54 -0.68 24.34 4.0 5538 H D 157089 F9V 6.95 0.57 -0.51 25.88 4.0 5663 HD 157214* GOV 5.4 0.61* -0.36* 69.48 4.27 5600 H D 159482 GOV 8.37 0.58 -1.08 20.90 3.92 5500 H D 165401* GOV 6.80 0.61 -0.44 41.00 4.37 5758 H D 165908 F7V 5.05 0.53 -0.54 63.88 4.2 5727 H D 224930* G5V 5.74 0.67 -0.67 80.63 3.35 5305 Data was taken from Table 1 of Reid (1998) and references within. * Data taken from Cayrel de Stroble et al.'s (1996) catalogue of late-type field stars. Parallaxes are from the Hipparcos Catalogue (Perryman et al. 1997). High-resolution spectra were taken of these stars in May 2000 at the Canada-France-Hawaii Telescope by Shkolnik and Matthews (in preparation). MOST prime target. Bold-faced stars are located in MOST's Continuous Viewing Zone. Name  0  c  2  §  i  42  3.2 Importance of MPSDs The chemical evolution of our Universe is governed by the enrichment of the gas by nucleosynthesis in the cores of stars and its redistribution by mass loss through supernova explosions. It is widely accepted that Type II (single massive star) supernovae released a-elements into the interstellar medium early in Galactic history and that iron was freed from Type I (slowly evolving binary system) supernovae. Thus the oldest stars should be more metal-deficient than more recent generations. The most obvious populations of very old stars are the globular clusters (GCs). GCs are composed about 10 to 10 metal-poor stars of varying masses that all formed at 5  6  essentially the same time from the same interstellar gases, and are therefore expected to have the same chemical composition. Since the MPSDs and GCs are believed to have formed in the same epoch, subsets of subdwarfs with comparable metallicities are used to calibrate zero-age main sequence (ZAMS) isochrones to fit observed GC colourmagnitude diagrams (CMDs); e.g., Pont et al. (1998). Well-constrained photometric sequences can be compared with theoretical isochrones to determine a relative age for the cluster. However, deriving absolute ages to better than 20% has been a challenge with this method (cf. Chaboyer et al. 1998). Like MPSDs, the lower the heavy element abundance of the GCs, the older they probably are. However, there is general consensus that almost all GCs that have [Fe/H] < -2 were formed at the same time regardless of location in the Galaxy (Stetson et al. 1996; Harris et al. 1997). Accurate absolute ages rely on accurate distances to GCs, which in turn depend critically on the reddening corrections. Because of this, fitting globular cluster isochrones to a ZAMS defined by nearby subdwarfs of known distance (and hence  43  luminosity) is a leading method to determine absolute ages for GCs (e.g., Gratton et al. 1997). The uncertainties of this technique are attributed to the poor definition of the subdwarf luminosity scale. More precise parallaxes provided by the Hipparcos astrometric satellite (ESA 1997) for nearby subdwarfs now permits a better calibration thanks to improved main-sequence loci on the observational ((B - V) , My) plane (Pont 0  et al. 1998). However, precise metal abundance determinations of the MPSDs are as important as accurate distances since this method works only if the GC has similar metallicity to the MPSD sequence. Pont et al. (1998) derived new metallicities for their work using the Coravel radial velocity spectrometer (Baranne et al 1979) which correlates the observed star's spectrum with a physical template of more than a thousand weak metallic lines. Mayor (1980) demonstrated that this method of cross-correlation works particularly well for late-type dwarfs (F-and G- dwarfs) which is appropriate for MPSDs.  M92 is a good example of a metal-poor GC ([Fe/H] = -2.03) believed to be very old (Stetson and Harris 1988). Figure 3.4 shows Pont et al.'s (1998)fitto the position of the turn-off and the subgiant branch of the halo subdwarfs. The fiducial sequence of M92 matches the subdwarf sequence remarkably well based on isochrones by VandenBerg et al. (1997). Using only 17 subdwarfs with [Fe/H] < -1.8, the distance modulus determined by Pont et al. is /A, = (m - M ) = 14.67 ± 0.08 mag with (<7 I n) < v  n  15% (i.e., distance d = 8.59 ± 0.32 kpc) and an age > 14 Gyr. Reid (1998) obtains a maximum distance modulus of jUo « 14.9 mag for M92 corresponding to a minimum age of 11 Gyr using MPSDs with -1.8 < [Fe/H] < -0.7. Fusi Pecci et al. (1998) derive an intermediate distance modulus. Pont et al. (1998) suggest a few possible reasons for  44  these differences in distance moduli: number of calibrators, metallicity scale, treatment of biases, magnitude corrections for detected and undetected binaries in the subdwarf sample, and errors in the colour and reddening values used. The fit to M92 sets a lower limit to the age of the Universe at 11 - 14 Gyr. This limit corresponds to a Hubble constant, H < 48 km/s/Mpc if Q = 1 and A = 0. If H is 0  0  larger, then one or more of the above assumptions is incorrect. However, most recent estimates of H are higher than -50; e.g., H = 71 ± 4 (random) ± 7 (systematic), from the 0  0  Hubble Space Telescope Key Project (Sakai et al. 2000). The higher values of H imply 0  a younger age of the Universe. Table 3.2 presents a sample of lower limits to the age of the Universe deduced from various methods. There remains a need to independently confirm the true ages and/or luminosities of MPSDs, for which asteroseismology should be an excellent tool.  45  I  I  I  I  I  0.4 (B-V)  0  0.6 [mag]  I  I  I  — I  0.8  Figure 3.4: The fiducial sequence of M92 fitted to the -2.6 < [Fe/H] < -1.8 subdwarfs. The M92 sequence is plotted with jUo = 14.61 mag (solid line) and 14.68 mag (dotted line). Suspected or detected binaries are shown as open circles. The diagonal line is the Hyades main sequence. (Figure taken from Pont et al. 1998.)  46  O O oo ^ 5 J3 o Q oo U P3  CD  _> '3 D  T3 CD  CD  CN  3 «  H  J3  O CD  x  00  <  60  2  (N  Cn  u  £ 2 CD 5 5  I O _e  T 3  o CD  .E  2  « •  2  cd  * 2 £2  00  B  7 3  —i  rj  00  CD J3  CD  Si E "2  ~  S  o  u  cd  g  O  c  I  CD  c  H PC  CD  6  5 >  o « c O  .s  -a .s 00  cn C  IS  q  \<  o  j u 2  g  = O  CD  .5 a. 3 q cd cd o  CD  o  «J  a S  UJ  O  s2 cd  '2 2 O CD  1/1  ©  IO CD  w  CD u  CD  M  cd  § a  00 3  o 1-5 fl o CD I S E S a  F  o  +1  CN  o — Trr-.fN+i--i2 Al  A l Al -  Al -  Al Al  3.3 MPSDs as Targets for MOST The primary science targets for MOST include bright solar-type stars (e.g., Procyon, n Bootis), roAp stars (e.g., HR 1217, HD 176232), Wolf-Rayet stars (e.g., WR 113, WR 128), and bright Sun-like stars with known extra-solar planets (e.g., T BOO). An updated list of the MOST targets can be found at The MOST research team also hopes to make thefirstdetections of p-mode oscillations in MPSDs. These findings will then be used to recalibrate independently the ages of distant Globular Clusters (GC) and refine the lower limit to the age of the Universe (MOST Phase A report, Matthews 1997).  The two priority MPSD targets identified are HD 224930 and HD 76932. Both stars of late spectral type, are in the MOST's continuous viewing zone and are relatively bright. (Their observational data is listed in Table 3.1.) Both have low surface helium abundances, suggesting that they are old enough to have allowed significant gravitational settling of He. HD 224930 is known to have low chromospheric activity and is not too close to the end of its hydrogen-burning lifetime. HD 76932 is near its main-sequence cut-off and has already been estimated to have an age of 11.7 ± 1.2 Gyr by Nissen et al. (1999) through observations of Li isotope abundances.  3.4 History of MPSD Models Modeling of MPSDs has been driven mainly by the desire to understand the properties of GCs, the dynamical and chemical evolution of the Galaxy, and the age of the Galactic halo. Modeling of low-mass stars began in earnest in the 1970's (Grossman 1970; Hoxie  1970; Copeland et al. 1970; cf. VandenBerg et al. 1983). Early models lacked the detailed molecular opacities for cool atmospheres that are available today. The first such opacity table (Alexander 1975) revolutionized stellar model accuracy for low-mass, lowZ models. VandenBerg (1983) generated an extensive grid of models which were evolved through the main sequence to the giant branch. The masses ranged from 0.6 to 1.3 M  @  and the heavy element abundances from Z = 0.0001 to 0.0169 (or [Fe/H] = -2.27 to -0.09). The grid was calculated assuming two values of helium abundance, Y = 0.2 and 0.3, various mixing-length parameters, 1.5 < a < 1.7, (see §4.1.2) and using what were the latest opacities from the Los Alamos Astrophysical Opacity Library (Huebner et al. 1977). The models neglected "complications" such as diffusion, convective overshooting, mass loss, rotation, and any effects due to magnetic fields. VandenBerg obtained his best fits to GCs with a = 1.6 ± 0.1 and a common initial helium abundance of Y = 0.2 ± 0.04. VandenBerg's (1985) next generation of low-mass, low-Z stellar evolution models included updated Los Alamos opacities and the Alexander et al. (1983) low-temperature opacities that incorporate the essential atomic and molecular absorbers. He also adopted improved model atmospheres of Kurucz (1979), Gustafsson et al. (1975) and Eriksson et al. (1981) for all regions of the H-R diagram. Charbonnel et al. (1996) extended an existing grid of stellar models of Schaller et al. (1992) by modeling lower masses (0.8 to 1.7 M ) for two metallicities, Z = 0.02 and @  0.001. These models included the OPAL radiative opacities from Iglesias and Rogers (1993) with the Kurucz opacities (1991) for low temperatures. The mixing-length parameter was assumed to be a = 1.6 and Y = 0.24 + (dY/dZ) Z where 0.24 is the  49  primordial He abundance and dY/dZ is the average relative ratio of helium to metal enrichment, usually taken to be about 3 (Peimbert 1985). (See §4.1.1.) VandenBerg et al. (2000) has published the most recent grid of MPSD models. The grid was computed for 17 [Fe/H] values from -2.31 to -0.30, assuming Y = 0.235 + 2Z, where the primordial helium abundance of 0.235 was taken from Olive, Steigman, & Skillman (1997). This relation was chosen because it produces a value close to the solar ZAMS (Y, Z) = (0.274, 0.0188). VandenBerg et al. adopted a = 1.5, determined by GC fitting to an earlier grid of subdwarfs (VandenBerg 1983). Again, diffusion, convective overshooting, mass loss, rotation, and magnetic fields were ignored. Masses range from 0.5 to 1.0 M in steps of 0.1 M . These models vary the ct-element abundances such that @  @  each mass is calculated for [ct/Fe] = 0.0, 0.3, and 0.6. This investigation of a-element enhancement improved VandenBerg's (1992) previous stellar models, revised the bolometric correction scale to better determine stellar ages, and refined the conversion from effective temperature to (B - V) colour indices. This study improves on previous work in two ways. Firstly, the models presented here include gravitational settling of both helium and heavier elements as well as the most recent opacity and equation-of-state tables and updated nuclear reaction rates. (The solar calibration of a adopted here is a different, but not necessarily better, way of setting this parameter.) Secondly, the MPSD models were "pulsed" to obtain the asteroseismic observables A v and Sv. All models prior to these have focused only on observables such as absolute magnitude and colour indices. Within the next two to three years A v and Sv will no longer be potential observables but will become new windows into the fundamental properties of nearby stars.  50  4 EVOLUTIONARY MODELS OF MPSDs  Prior to analyzing the nonradial oscillations of MPSDs, structural stellar models throughout various stages of their main-sequence evolution are required. Schwarzchild (1958) and Clayton (1968) derive the equations of stellar structure in their classic texts. The time-independent equations are:  Hydrostatic equilibrium  — = -G  (4.1)  dr  r  dM, Mass conservation  ->  - = 4m• p  (4.2)  = 4m- pe  (4.3)  dr Energy conservation  2  dr ~  •  Radiative transport  ^  Convective transport  — = dr  dT  3. Kp L. :—; r4ac T 4nr  dT  (,  .. (4.4)  2  \\um  GM  — = - 1 - — y-^^v rj k r H  r  2  . ,. . . .  (adiabatic)  (4.5)  B  where K is the average opacity, c is the speed of light, and a is the radiation constant.  4.1 Modeling Procedure All stellar models presented in the four-dimensional (Y, Z, mass, age) grid were calculated using the Yale Stellar Evolution Code with Rotation (YREC; Prather 1976; Pinsonneault 1988; Guenther et al. 1992) which solves the conservation and transport 51  equations of stellar structure given above. The Henyey relaxation method (cf. Clayton 1968, p. 451) for solving boundary value problems is applied to the equations. Assuming an initial model structure for the star, this method corrects the physical variables at a specific spatial position and time making the differential equations of stellar structure more nearly satisfied. The process is iterative until the solutions to the equations are within user-specified tolerances such that the corrections become insignificant. The interior (inner 95% of the radius; 99.98% of total mass), the envelope (outer 5% of the radius; 0.02% of total mass) and the atmosphere are each modeled with approximately 600 shells. The atmosphere begins at the base of the photosphere where optical depth x = 2/3 and is used to constrain the outer boundary conditions and is modeled using the Krishna Swamy (1966) T-x relation. This resolution of shells along the stellar radius is sufficient to produce p-mode frequencies within a numerical accuracy of ± 0.5 /Mz with the nonadiabatic, nonradial pulsation code developed by Guenther and Sarajedini (1988). This resolution is comparable to the frequency resolution expected of observations obtained by MOST. (See discussion in §5.) The nuclear reaction network (energy generation and nucleosynthesis) is solved separately for each shell in the interior of the model. The YREC code was run in its non-rotating configuration. Most MPSDs have a relatively low vsim and are chromospherically inactive implying that they are not rapid rotators. Slow rotation ( V (equator) « 10 km/s) does not distort the equipotential structure of the star so rotation can be safely ignored in the structural models. However, as discussed in Chaboyer et al. (1995), rotation rates greater than 20 km/s tends to restrain the effects of diffusion by a factor of approximately 2.  52  The OPAL opacities and their associated equation-of-state (EOS) tables were adopted for the structural models from the Lawrence Livermore National Laboratory (LLNL; Rogers 1986; Rogers, Swenson and Iglesias 1996). The most recent OPAL opacity and EOS tables were computed for a range of composition encompassing the Sun and MPSDs (0.00 < Z < 0.04) through an interpolation program. This becomes important when heavy-element diffusion is taken into effect and Z varies from shell to shell over time. In each model, the EOS table was calculated for the ZZAMS- Due to the relatively small effects on the equation-of-state from changes in surface Z over time, this ZAMS EOS table was used throughout the model's evolution. The most recent OPAL opacity tables (OPAL98; Iglesias and Rogers 1996) were used for the higher temperatures in the interior (logT > 4.12). For the cooler envelope and atmosphere (logT < 4.0), Alexander and Ferguson's (1994) opacity tables were used. In the intermediate temperature range, a ramped average between the two opacity tables was implemented. Guenther, Kim and Demarque (1996) tested other EOS tables for effectiveness in the Sun. They compare predicted solar p-modes calculated with the OPAL EOS and a popular alternative, the MHD EOS tables (Mihals, Dappen and Hummer 1988) which use full internal partition functions for C, N, O, their ionic subspecies, Fe and Fe . However, +  for the highly ionized subspecies, only ground states are accounted for whereas the OPAL EOS tables include all stages of ionization and excitation. The MHD tables are based on a single composition corresponding to the solar Ross-Aller mixture (Ross and Aller 1976). This clearly is not an accurate description of MPSD compositions. The complete grid ranges in mass from 0.7 to 1.0 M , in Y from 0.235 to 0.255, @  and in Z from 0.0002 to 0.01.  53  4.1.1 Initial Helium Abundance The average helium abundance Y is a free parameter, but there are limits on this value. The initial Y for MPSDs must be less than that of the Sun (Y , ZAMS = 0.274) and but no Q  less than the cosmic primordial helium abundance, Y . Table 4.1 lists published values p  of Y selected over the past decade. The highest estimates are Y < 0.255; the smallest, p  p  Y > 0.221 and the average is 0.24 ± 0.0091. One method of determining Y is by p  p  measuring the Y and Z of different objects in the Galaxy and comparing them to the Sun's values. By extrapolating the best fit line of the (Y, Z)-plot back to Z = 0, a value for Y is determined. The slope of the line, dY/dZ = (Y*-Y )/(Z* - Z ), is used as a p  @  @  measure of the Galactic chemical evolution. A recently adopted value for this slope is 2.3 ± 1.0 (Thuan and Izotov 1998). The spread in dY/dZ was used to confine the MPSD grid values for Y and Z. Only the two extreme corners of the grid go beyond the range 1.3 < dY/dZ < 3.3; i.e., (Y, Z) = (0.235, 0.01) and (0.255, 0.0002). The grid of models spans a range in Y of 0.235 to 0.255 in steps of 0.005. The heavy element abundance range for the grid was guided by observed [Fe/H] values for MPSDs (e.g., Reid 1998). The models were calculated for Z values of 0.0002, 0.0004, 0.0008, 0.001, 0.002, 0.004, 0.008, and 0.01. This range corresponds to -2.0 < [Fe/H] < -0.3. The resolution in Z was chosen such that the resulting p-mode spectra would have large enough differences such that they would not be hidden in the numerical uncertainty.  54  o  ON  E 00 ON ON  > o o  <^ C  OO ON OS  C8  811  <« -5  E ^ O £  c >  C/5 u  o CQ « > O  o — ea u c o (U  l o g  £i  P2 PQ H  §5 J3.  « £1 N c  2  m ON ON  5  O K  CO  x .ca 00  cd  -a c  E '= c o  3  'ob  <  X)  | CW  J3  •i  "5  "s •5 i~  S  =  c  K  H  c  CO  <u o c  e  o  DC  .aX  u a c« eo <4oo  2 '5b  O  HH  .2 £ CC X rCN K <+-. 2 c c£n <2* o £ S -° 2 '5b b .2 oo a*O CO q> ca 00 U  T3 3  E  "3  00  *"  ^ g. 8 E  3  ca J D oo o S3  C  1 ^ •o -8 8ti o 2 O 5b O (U ' £  3  OJ J O  0 o o .a xs — o. a CD a2 S 'ca o E e S V "> E " 3 SP o c B E P3 o ? >N « JJ a. C/3 o ca t / 3 O X _) CQ  i.£  1  W  CN O O  CN O  o © ©  o  © +1  cJ  CN  CN  O  O  1  +1  CN O  CN  CN  CN  ©'  ©'  ©  V  CN CN  V ©  + l |  m CN  ©1  4.1.2 Diffusion Michaud (1970) first proposed that diffusion of helium occurs in stars that possess thin surface convection zones and later, pointed to its importance even in stars like the Sun with thicker convective envelopes (Michaud 1986). YREC includes diffusion of He as well as the effects of gravitational settling of heavy elements. Dappen & Gough (1986) and Antia & Basu (1994) showed that the inclusion of He diffusion in stellar models produces significantly better agreement between the observed and theoretical solar pulsation spectrum. Guenther and Demarque (1997) have determined both He and heavy element diffusion are required in the Standard Solar Model in order to achieve simultaneous close agreement with the observed p-mode spectrum and the observed (Z/X) ratio. Both Y- and Z- diffusion were included in the MPSD models. 0  4.1.3 Treatment of Convection No widely accepted theory exists which rigorously describes the physical process of convection and mixing within a star. The most widely used approximation in stellar evolutionary models is the Mixing Length Theory (MLT), a parameterization first introduced by Bohm-Vitense (1958). In this approximation, it is assumed that convective energy is carried upwards a fixed distance, the mixing length I, by a gaseous "bubble" of a fixed size that instantly dissipates into the surrounding gas through radiative diffusion. The distance that this bubble travels is inversely proportional to the pressure scale height and is parameterized by a ( = / /H ), which is known as the mixing length parameter (see p  Equation (1.12)). The choice of a is constrained by matching models with observations of luminosity and effective temperatures of well-studied stars. Most modelers use the  56  observations of the Sun to fix a, usually adopting values between 1.5 and 2 (e.g., VandenBerg 1985; Charbonnel et al. 1996). The value adopted for a within this range is determined mainly by the preferred opacities used in a stellar model. For other theoretical treatments of stellar convection, see Kim et al. (1995, 1996), Canuto & Dubovikov (1998) and Kupka (1999). To determine a for the MPSD models, a solar model (i.e., M = 1 M and Z = Z @  @  = 0.02) was evolved from the ZAMS to the Sun's observed luminosity and effective temperature. This solar calibration model adjusts a to best fit the current solar observations for L and T . The fundamental characteristics of the solar calibration @  model are summarized in Table 4.2. The optimum value, a = 1.78, was adopted for the MPSD grid since the MLT can be applied to any stellar model with a convective envelope regardless of composition.  57  Table 4.2 Calibrated Solar Model Mass 1.0 M log(L/L ) 0.0 log(T ) 3.7619 Age 4.53 Gyr 0  0  eff  X  0.7057  i n i l  Zinit X Z  0.02 0.7374 0.018 4.79  s u r f  s u r f  M , log(T ) log(P )  7.1956 17.3716  log(Pc) X Z  2.1846 0.3442 0.0200  b0  a  c  c  c o r e  c o r e  M  0.02438 M  b c e  Rc.e. [Z/X] 3 r 4 5  a  G y  0.71174 R 0.02441° 1.78  g  0  Central conditions Mass and radius of convective envelope °This value agrees with the published value of 0.244 ± 0.001 by Gerevesse, Noels, & Sauval 3  b  (1996).  4.2 The Age and Mass Range of the Grid Beginning with a chemically homogeneous Z A M S model of solar mass and composition, each model in the grid is rescaled to the specified mass and composition. The rescaling steps consisted of altering one or more of the initial model parameters ( M , XZAMS, ZZAMS) and calculating a new Z A M S model. For models whose parameters are most different from the calibrated solar models, up to ten rescaling steps were required to generate a stable model.  58  The mass range spanned 0.7 to 1.0 M . Any stars with masses below 0.7 M are @  @  too faint to be accurately observed by the small-aperture space telescopes currently being built (see §1.6). Whereas a 1.0 M may be bright, but evolves so quickly that any @  observed MPSD with such a mass has by now exhausted all its core hydrogen. The rescaled model was then evolved in equal time steps determined by the code. For most models, this time-step was on the order of 10 years. The eigenfrequency 8  behavior becomes more complicated after core hydrogen exhaustion. (See discussion and Figure 5.2 in §5.2.) The information on stellar fundamental parameters derived from irregular A vand 5v cannot be accurately described by asymptotic theory and is therefore quite limited. Models of Procyon, an evolved star, more massive than the Sun, have shown this irregularity in A v (Chaboyer, Demarque and Guenther 1999). The exhaustion of hydrogen in the core and a mean molecular weight discontinuity leaves an obvious signature on the p-mode spectrum caused by mode bumping. This phenomenon is induced by g-modes from the interior overlapping (or bumping) the p-modes causing an observable distortion in the p-mode spectrum. A detection of this disturbance can provide direct information on the stellar core as well as determine whether the star is in its subgiant phase of evolution. The models that reached core hydrogen exhaustion early on, were still evolved to ages of cosmological interest. Table 3.2 presents a sample of lower limits to the age of the Universe. Based on these criteria, models were evolved from the ZAMS to the sub-giant branch, or to a maximum age of 16 Gyr, whichever occurs first.  59  4.3 Sensitivity of the Models to Input Physics and Parameters Figure 4.1 shows the evolutionary tracks on an H-R diagram for models with masses of 0.7, 0.8, 0.9 and 1.0 M for the median composition of the grid, (Y, Z) = (0.245, 0.001), @  and with two values of a (discussed below). The higher the mass of the model, the higher the temperature and luminosity causing the model to evolve faster. The 1.0-M  @  model reaches turn-off at 4.9 Gyr (half the main-sequence lifetime of the Sun) whereas the 0.7-M model does not reach turn-off even after 16 Gyr. The lower mass models @  have cooler central temperatures and burn their nuclear fuel at a much slower rate allowing them to remain on the main sequence for much longer. Similarly, the Sun remains on the main sequence longer than an equal-mass lower-Z model due to the Sun's higher central opacity which inhibits radiative transfer while slowing down nuclear burning. The physical characteristics of the four models shown in Figure 4.1, at oldest ages calculates, are presented in Table 4.3.  Table 4.3 Model Characteristics for Reference Composition: (Y, Z) = (0.245, 0.001) 1.0 0.9 0.7 0.8 Mass (M ) 0.7807 0.4846 0.6239 -0.0671 log(L/L j Q  0  lOg(Teff)  3.7771  3.7810  3.8150  3.8395  Final Age (Gyr)  16.0  12.4  8.3  5.7  Xjnit  0.7540  0.7540  0.7540  0.7540  Xsurf  0.8400  0.8673  0.7899  0.7540  Zjnit  0.0010  0.0010  0.0010  0.0010  Zsurf  0.00067  0.00049  0.00087  0.0010  M i  4.96  3.58  3.23  2.84  log(T )  7.244  7.407  7.365  7.376  log(Pc)  17.912  18.991  18.709  18.676  'Og(Pc)  2.819  3.698  3.485  3.448  ^core 7 ^ core M ,.(M )  0.0534  0.000121  0.0000  0.0000  0.00118  0.00114  0.00101  0.00100  0.0080  0.0048  0.0000  0.0000  Rc.e(R»)  0.7492  0.8103  1.0000  1.0000  b o  c  C  0  60  s-  o «  oo o -4—»  T-H  CD  ^  S NO" '5 © o ^ -*-»  cd  »•  CD NO  O  00  >  CD  cd  CD  SO  T3  S3  GO -g  <z CD »  co  3 3  cd  ^ CD 0 0 S3  £ on  oo CO  o o  CD  CN CO CO  cu  .3  o  in  00 CL> 7 5 00  S3  ^  oo OO  i  II  o  -—•,  CO  N  o  B  oc  ts  N" 00  CN  CO CD  o  00  co  00  M H  CO  o  CD b  CM o  No  CN o I  ( 1/l)6o| e  d  CD O  CO  CD  c OH  & CJ S3  oo co  IH  © -a  - a oo"  CD  a  o CM  O  g CD  00  00 CD  00  CD S3  o  S3  3  00  _o -3 _3 O >  a> 00  I  W .. -tt >> 1 3 o "5b .> CD  3  DC  CD CD  Q-  00 CD IH  To test sensitivity to the mixing length parameter a, models were calculated for a = 1.78 and 1.88. Figure 4.1 shows the slight rise in effective temperature due to increasing a. The maximum effective temperature change is 40 K for the models of mass 0.8 M . In MLT, a bubble of gas dissipates after travelling a mixing length / and 0  releases its excess heat to its surroundings. The pressure of the bubble is assumed to be equal to the pressure of its surroundings at all times. Therefore, when a is increased, the bubble effectively travels a larger distance originating from a deeper, hotter layer in the convection zone. This causes more heat to be transported to the upper envelope increasing T ff. There is a considerably smaller shift in luminosity than in temperature. e  However, both increase with the age of the model. VandenBerg (1983) modeled metalpoor stars for a large range in Z ( = 0.0001 to 0.0169) with 1.5 < a < 1.7. He showed that for higher values of Z ( > 0.01), the deviations between models with difference a increase significantly. This is so because models of higher Z are cooler and more opaque and therefore have larger convection zones which are more affected by changed in a. Since MPSD as a group have relatively low Z, the grid of models presented in this study is relatively insensitive to small adjustments in a. Figures 4.2 and 4.3 show the effects of varying Y and Z, respectively on the evolutionary tracks and the ages of models for afixedmass. Two very different physical processes occur in these two cases which have opposite effects on the H-R diagram. When Y is increased, the density and mean molecular weight p is also increased causing a higher central temperature (Hansen and Kawaler 1994), T « '" " pk P f  .  n  (4.1)  B  62  Here, kg is the Boltzmann's constant and mn is the hydrogen mass. The helium-richer models thus evolve at a slightly faster rate. For example, main-sequence age of a 0.8 M  @  model with Y = 0.235 is 11.1 Gyr and with Y = 0.255 is 10.0 Gyr, for the same Z of 0.001. The mean molecular weight p for a completely ionized gas (as it is in the center of stars) can be written as  n  ^ \  where z is the proton number and A is the atomic number of an atom for all /' atoms considered. Since 1 + z, = z, for heavy elements and Aj = 2 z,, the weighted average over all elements heavier than helium becomes (4.3)  Equation (4.2) then reduces to 1  3  Z  — &2X + -Y + - . Mi  4  (4.4)  2  The mean molecular weight /J. is far more sensitive to the helium abundance than to the heavy-element abundance since Z « Y. Any increase in ju due to a higher Z has no significant effect on the central temperature. However, an increase in Z greatly affects the opacity of the star inhibiting radiative transfer and thus decreasing the bolometric luminosity and increasing its lifetime.  63  The effects of Y and Z diffusion are shown in Figure 4.4. Settling of He increases /J. in the center of the model (increasing T ) while decreasing /u in the outer c  regions of the star (decreasing T ff). Adding Z diffusion to the models has little effect on e  p. but increases the central opacity. Again, T increases and T fr decreases. With age, the c  e  influence of diffusion increases however the 1.0 M model evolves too quickly for the 0  effects of diffusion to be apparent on the H-R diagram. The turn-off ages decrease slightly with the inclusion of diffusion. For example, for the 0.8 M model, the turn-off 0  ages are 11.5, 11.0, and 10.8 Gyr for the no-diffusion, Y diffusion, and Y and Z diffusion models, respectively. In reality, the composition of an observed star will not be known exactly. The grid resolution in Y is small enough that a 2% error in the inferred helium abundance of star will correspond to by another existing model in the grid. The resolution in Z is not as fine. Tests were calculated for an anticipated error of 10% in an observed heavy element abundance (equivalent to a 3% uncertainty in [Fe/H]). The evolutionary tracks showing small variations in Z are presented in Figure 4.5. The maximum shift in temperature is ± 34 K with a ± 10% change in Z. To summarize the relative sensitivity to a, Z, Y, and diffusion, Figure 4.6 compared various tracks of test models to the reference model for afixedmass of 0.8 M . The dashed box shows the combined uncertainty in effective temperatures of 0  evolved models, not counting the effects of neglecting diffusion (which helioseismology indicates should be included). The error box spans 48 K in T ff and 0.0183 in log(L/L ). e  0  66  — r  0  o  -a  > a  o  11  h-  co  S3 & -2  g CD  Jg -13  CO  N  *  c  o o ^  0  CO  u  co  CO CO  CNJ CO  OH  cu  o  II 55 I 0*  S-— l 1  fiBO  ?  *—'  " "I  CO  N  <u  w  co  jg  2 £  CO  6 CD CO  CO  co co CO  o  CD O  CNJ  CNJ  o  o  ( in)6o 0  n  CO  o  o  I  2 + §->  s «  21 32 — O  CD  CO  b  ©  S c  .2  ' 1  o > H  d  .2  !-  i  3 M l •— —  O 3  OS NO  £  Jj  "So 00  M  o ~o  CO  °* §  co  CU  »  "3 fc | H J .s  m co hco  r3 co  og cu  ( u  00  CU  IS  —  CD  oo oo  O  O cu  co  •a -s 3 3  > 5S  in cn  co co co  _  1>  O O  X) M  ^ o o o , cu  o ©  N w  0 0  — F CU  © 11  uo o  -O  CO CO  o s  II  o cu s  cu Ci  C*H  CU  —  CO CO  oo C3  13 -o o S  cu - c3u -3 3 00  o  in T —  CO  co  rm)6oi  C3  i->  S3 i0 0  S3  e i .2 I 3 CU -3 O ^ > 3  W •= oo  . . cu  a "* .5 NO OJ  S3  a  c  S  <D  on o  5 NONRADIAL OSCILLATIONS OF MPSDs  We focus on the low-^ modes since they are the only ones visible in unresolved light. They have the deepest inner turning points and so are maximally sensitive to the internal structure of the star relative to moderate and high-^ modes. In the Sun, very low-n modes are not excited to high amplitudes, presumably due to the power spectrum of convective turbulence driving the oscillations (see §1.5). The solar eigenspectrum in integrated light (Figure 2.2) shows the expected asymptotic behavior for n » £. We assume the same conditions apply for MPSDs.  5.1 The Code For the structural models described in Chapter 4, synthetic pulsation eigenspectra were generated with a nonadiabatic pulsation code developed by Guenther (1994). The code operates in five basic steps: 1) It solves the turning-point equation (1.13),  c(>,) r t  =  o~  V^l)'  70  using the run of sound speed c(r ) in the structural model with a specified £ for the t  turning point r,. The inner turning point of the acoustic mode cavity is determined by the Lamb frequency, I? = a . 2  e  2) It then solves the adiabatic equations of stellar oscillation (without the outer boundary conditions) using the Henyey relaxation method. The equations are (Unno et al. 1989, §14)  L^  r  r dr  1_±L  c.  p dr  2  J  V  (5.1)  2 2  PCs  pc  a r  (5.2)  dr  s  and \_d_ r dr 2  d® dr  v  y 2  cD = AnGp  p'  N  2  \  (5.3) j  where O is the gravitational potential and the primes denote derivatives with respect to r. The solution is an initial guess of the eigenfrequencies for the adiabatic eigenfunctions. 3) Using the solution from Step (2), the code solves the complete linearized adiabatic oscillation equations, again with the relaxation method. 4) The nonadiabatic eigenfunctions, without the outer mechanical boundary condition, are then calculated with the approximate frequency from Step (3) by calculating the work integral, W(r) (see § 1.4), and the kinetic energy, Wk'.  71  W(r) = -An r lm((5p*)% ), 2  (5.4)  2  r  k = ^o-  w  2  ^'\Sr\ dm 2  d  .  (5.5)  The work integral sums up the work done through each shell of the model and is positive during the excitation of a mode, and negative when a mode is dampened. 5) The eigenfunctions from Step (4) are then used to solve the complete eigenfunctions for the nonadiabatic frequencies. The Henyey method is again used where (after approximately eight iterations) a solution is usually converged to an accuracy of one part in 10 . 5  The pulsation code takes into account nonadiabatic effects that allow heat transport only due to radiation (not due to convection) where the opacity is high. The effects of nonadiabaticity on the oscillations increase with frequency as the eigenfunctions with high frequencies have higher concentrations of nodes near the stellar surface. The effects of nonadiabaticity are illustrated in Figure 5.1.  72  -0.0045 1000  1500  2000  3000  2500  3500  Non-adiabatic v (nHz)  ra -0.01  111  112  113  114  115  116  117  Non-adiabatic Av (//Hz)  Figure 5.1: Plots demonstrating the increasing relative difference between the adiabatic and nonadiabatic frequencies (top) and large frequency spacings (bottom) for a model with (Y, Z) = (0.245, 0.001) and M = 0.8 M . @  73  5.2  The Pulsation Models  Nonradial, nonadiabatic oscillation frequencies for each model in the evolutionary grid (described in §4) were calculated for modes of low £ ( = 0, 1,2, and 3) and up to high n (= 0 to 40), spanning a frequency range of 250 to 4550 //Hz (corresponding to a period range of ~ 3 to 70 minutes). Guenther and Demarque (1997) have demonstrated with their seismic tests of the Sun that p-mode sensitivity to the structure of the core decreases with increasing £. A similar decrease in sensitivity occurs with n>25. For £ < 5 and n < 25, Guenther et al. (1993) find there is "little resolvable sensitivity" to the deep interior in the small frequency spacings. A typical model eigenspectrum is shown in Figure 2.1 in which the large (A v) and small (Sv) spacings are obvious. For each model, A v and 8v^i were calculated from averages over 10 < n < 25. This range corresponds to eigenfrequencies from 1300 to 3400 //Hz. The large spacing A v is plotted against n in Figure 5.2 for models of different ages and masses. There is obvious irregularity in A v for modes with n < 10 (where the asymptotic approximation begins to break down) and n > 26. At approximately n « 32 for the M = 0.8 M model and at n » 26 for the M = 0.9 M model, there is a brief 0  e  increase in A v. This may be a mode counting problem where the code misidentifies overtones. However, it could be physical. As the nodes of the eigenfunction move from low n to high n, there could be a radial node at one of a few possible discontinuities in physical variables in the upper atmosphere. The depth at which this discontinuity  74  appears can, in principle, be determined by observations of such a glitch in the otherwise constant A v (Guenther 1994). Some of the irregularity in A v for low n is due to mode mixing or mode bumping (see §4.2). All stars, regardless of mass, experience mode bumping. However, it is a more prominent effect in evolved stars whose core densities have increased causing the g-mode frequency spectrum to increase further and "bump" into more of the /?-mode spectrum (Guenther 1991). Figure 5.3 identifies the bumped modes in two models at different evolutionary stages.  75  250 M = 0.8, Age = 8 Gyr  A  o M = 0.8, Age = 10 Gyr  200  N  x <  150  100  . A 5 A A A  ooo  ° °  s e  0  ^AAAAAAAAAAAAAAAAAAAAA^  §  0 e e  * |  ooooeeoooooooooo@g§eog  0 0 0  A  A  A  A  ooo  o  °6  50  A A  O  A M = 0.8, Age = 8 Gyr o M = 0.9, Age = 8 Gyr  200  A A  ^150  <  A*  A A A A  AA,  A. A.A.A.AAAAAAAA ^ A  A A A A A A A  A  *!A,AAJ  100 AA o o o O ° 8 g e o 8 g ° ° o O § o o o o o g o o o o a  B  OOOOOOOOOOo 8  O  Bo  50 o  o  o  o  0 i 10  20  30  50  40  Radial Order, n Figure 5.2: The large frequency spacings, Av(n, 1) are plotted again the radial order, n. for the p-modes of the indicated models. The composition for each model is (Y, Z) = (0.245, 0.001). The top panel shows the effects of evolution while the bottom panel shows the effects of a differing mass at the same age. The M = 0.9 M model reached core hydrogen exhaustion. @  76  fl  H  fl A ci O  fl 03  >"  fl S  CD ^ _  CD  LO CM  -EH  ^  CD o O  t/3  .2 £  c/3 CX s — oi ^ U /—s CX * J  II  II CD O)  o CNJ  < <  o  oo" CD" ci o  o o  II  o  o  ,6  Q.  c  ;  crt  CD  i  .2  C/3  CD  CD  O CD  fl 03  ^  *—H  1  [So  o o o  03 CX  CD  •- O CD O  o o  GO o 1-1  fl CD  o  O O It  CD  C/3  £-ir >N^fl  O O I O OO I  CD  fl „  l b  O OOO I  O O  o  CD fl  o o  I  S  COD « M  ^| | in  o o  O  1  '  o  O O  | fy O fl  o 1  "o  c/i  «>  CD fl ^>  00  II  A  CX  ©P >  C3  >>  II CD  U  - °  .£ £ A  — 1  •I  3  <u X>  CD C/5  X CD  m  fl  cd CD  OO I I  O OOOOOO I I I O OO O  o o  Cd c/i  OOO <D O I  I  I  I  I  I I  I  I  I  -G—I—I—I—I—I—I—I—I—I  in  CM  o  in  in  CM  u  oj  11 > fl  ^ .2 £ "5 fl -° OS J S 2A ££ CD  w  H  CD fl •• O  I 1*1  O CD  In  CX  "3 g o g  £a  V) i CU ftO• t / i La CD {JO o S si  wo,*to o  ™-l  fl  12 o  5.3 Diagnostics from Eigenspectra The average value of A v over n= 10 to 25 for a representative model with Av = 120 /Mz has a standard deviation of 1%. (A v)„ = 10..25 was also averaged over £ = 0 and £ = \ while Sv = Svoj. = M),«+i - ^2,« only. Figure 5.4 a) is an echelle diagram where a set of model eigenfrequencies has been folded at the large frequency spacing A v (rather like a phase diagram of light curve folded at the pulsation period). The £ = 0, 2 pair of lines on the two panels of Figure 5.4 represents the Svo 2 and the £ = 1, 3 pair of lines displays the t  larger value of 8v\j. (The ratio of SVQ^I  8V\$  is relatively constant.) Figure 5.4 b) is the  same stellar model at a later age. Both A vand Jvhave decreased but the change in Sv'is more pronounced. The physical dependencies of A von mass (if radius is known) and Svon age are apparent in the form of a (A v, Sv) diagram, popularized by Christensen-Dalsgaard (1984), (see also Christensen-Dalsgaard 1999). One so-called "asteroseismic H-R diagram" for the median composition in the grid (the reference model with (Y, Z) = (0.245, 0.001), is shown in Figure 5.5. The solid lines connect models of the same mass. The dashed lines are isochrones (constant age) starting at 5 Gyr with a step-size of 1 Gyr. The O.7-M model reaches the maximum age of 16 Gyr before turn-off. The 0.8 M 0  0  model reaches turn-off soon after 11 Gyr and the 0.9 M model, after 7 Gyr. The 1 M 0  0  model reached turn-off before 5 Gyr and is not included in this plot. Since the large spacings become more irregular at late stages of evolution and are no longer as easy to interpret, as shown in Figure 5.2, the (Av, <5v)-plots are truncated just before the model reaches hydrogen exhaustion in the core. Figure 5.6 shows asteroseismic diagrams for  78  the four extreme compositions of the grid: (Y, Z) = (0.235, 0.0002), (0.235, 0.01), (0.255, 0.0002) and (0.255, 0.01).  20  40  60  80  100  120  140  [Remainder of (v/A v)] x A v (JJ Hz)  Figure 5.4: Echelle diagrams for the eigenfrequencies of a M = 0.8 M models with (Y, Z) = (0.245, 0.001). The model at 7 Gyr (Av= 151.49 //Hz) is in panel (a) and panel (b) shows the model at 10 Gyr with A v= 114.32 //Hz. @  79  5.4 Sensitivity to Input Physics and Uncertainties Figure 5.7 shows the effects on A vand Svdue to changes in Y and Z. For both cases, the median value is compared with the grid's maximum value: Y = 0.245 and 0.255, Z = 0.001 and 0.01. For the model with increased Y, the mean molecular weight is higher causing a decrease in sound speed. Since Svx A v which grows with c , the frequency s  spacings are shifted to smaller values. In Figure 5.7 (bottom), the largest shift in Au occurs for the lowest mass models (0.7 M ) and has an average shift of 9.2 //Hz. For Sv, 0  the average shift is 0.6 //Hz. Both are relative shifts of approximately 5%. The same relative shift occurs because the global increase in Y effects both the outer regions, where Avis most sensitive, and in the interior, where Svis most sensitive. A different effect is observed when comparing asteroseismic diagrams of models with different Z values. Figure 5.5 (top) shows (A v, Sv) diagrams with Z = 0.001 and 0.01 for Y = 0.245. Models with larger Z cause a significant increase in opacity, especially in the partial ionization zones forcing the temperatures to increase. Since the sound speed increases with temperature, the frequency spacings shift to larger values by ~ 30% with a 10x increase in Z. The inclusion of He diffusion increases the mean molecular weight in the interior of the model. As discussed above, the sound speed is then reduced implying smaller frequency spacings (by about 5%). When Z diffusion is also taken into account, the decrease in the outer opacity reduces the frequency spacings. The relative shift is ~ 10% for both A vand Sv. These effects are illustrated in Figure 5.8.  20 18 16 14  Iii 2  I  310  &Q  8 6 4  (Y,Z) = (0.245, 0.001)  2  (Y,Z) = (0.245, 0.01)  18 16 14  i  12 10 8 c 0 4  (Y,Z) = (0.245, 0.001) (Y,Z) = (0.255, 0.001)  2 0 70  90  110  130  150  170  190  210  230  250  A v (//Hz) Figure 5.7: (Av, Sv) diagrams illustrating the shift in frequency spacings due to a shift in Z (top) and in Y (bottom).  83  CD  Si  E—  1  CO  00  G 'o  o  OH  co CJ  C3  <u  c o >> w c 3  i  N ro  =3  CT CD  O  g  CD  Si  i W  -4—»  co c  3  "9 o  >- >- c  O CM CM  'co  •-3  5 5 $  I+ +  N  O  N X  o fe  CD CD  <  Si -*-»  00  g  13 S o  T3  CM  ^ CD CJ  C  00  > H . O  M-H  co CD C3  n i-i cd CD CD  1 / 3  si . O  co  C  CD  o o CM  CO  CD  CO  CM  (ZH r*)  9  A  CD  CM  i  B  tn  CD  C3  •-3 ^ ^  CD  50  60 C  < -5  W  ..  CD  00  en S. 00 k  u. g  An increase in a from 1.78 to 1.88 results in a systematic increase of 40 K in T fr as e  shown in Figure 4.1. The effects of this small increase in T ff can be seen in the (A v, Sv) e  diagram shown in Figure 5.9. Larger T ff increases the local sound speed and hence, Av e  has increased for each model. Changing a from 1.78 to 1.88 increases Av by ~1.7 //Hz. This 1% shift is only about 1/5 of the average shift in A v due to an increase in age of a model by 1 Gyr (9.8 //Hz). Therefore, uncertainties in a do not play a significant role in the accuracy of age determination. There is virtually no shift in £vdue to the increase of a, since Sv is primarily sensitive to the interior of the model, whereas a describes convection in the outer envelope. To test sensitivity of A vand Svdue to a 10% error in Z, models were generated for Z = 0.001 ± 10 %. The (A v, Sv) diagram is shown in Figure 5.10 for Z = 0.0009, 0.001, and 0.0011. The shifts in the frequency spacings increase with evolution reaching a maximum of 1.8 //Hz for A vand 0.07 //Hz for Sv. This is equivalent to a 1.2% shift in both. Again this is small when compared to the shift in A vdue to an age increase of 1 Gyr. Accumulating the possible errors incurred due to the uncertainties described above, the total error (added in quadrature) for A vis 12.5% or 0.0125 M and for Sv, the 0  cumulative error is comparable at 12% or 0.12 Gyr.  85  o in  CM  <J>o T—  o o o d  II N  t  O o o II N  T— 7  o p d  II N  in CM CM  + • O  o  CM  m  N  o in  m CM  o o  in  CD  oo  r-  (zH *) *9  o CM  00  CO  O CM  O  (zH>0 *f  CO  CD  CM  in  X 3  £ 1  6.  MATCHING OBSERVATIONS TO THE GRID: IS THERE A UNIQUE SOLUTION?  This grid of MPSD models was generated as a tool to predict the masses and ages of target stars for which A vand Jvhave been observed. But the grid of models allows for various Y and Z. How uniquely can the observations be fit to a specific model? Figure 6.1 shows schematically the complete 5x8 grid in Y and Z of (Av, Sv) diagrams. Each model is defined by six physical quantities: initial helium abundance Y, initial heavy element abundance Z, mass M, age A, luminosity L and effective temperature T ff. The problem is essentially to solve for the first four parameters, Y, Z, e  M, and A with four observables: A v b , Sv b (from the eigenspectrum), log(L/L ) b , and 0  s  0  S  Q  0  s  log(T ff)obs (from the parallax, apparent magnitude and spectrum or colours). However, a e  datum point (A v b , Sv \,s) on the (A v, Sv) diagram will yield a different age and mass for 0  S  0  each combination of Y and Z. A simple "hare and hounds" exercise (a blind test of model identification) is presented for illustration. David Guenther provided a set of observations (A v b , Sv t, ) = 0  S  0  s  (165, 11.5) /Mz from an evolutionary model (unknown to the author) calculated with input parameters that fell within the ranges of the existing grid. This datum point falls in a 2 x 2 subset of the grid shown in Figure 6.2, with four models: (Y, Z) = (0.25, 0.001), (0.25, 0.002), (0.255, 0.001) and (0.255, 0.002).  88  0.235  0.0002  0.24  0.245  , /  -  0.0008  0.255 /  // •  0.0004  0.25  . / / • . / / - .///  •  •  •  0.001 0.002  .//  0.004 0.008 0.01  Figure 6.1: Schematic diagram representative of the entire pulsation grid with Y = 0.235 to 0.255 and Z = 0.0002 to 0.01. The dashed box outlines the four models used in the example described in the text and are enlarged in Figure 6.2. The black dots represent the complete set of plots whose icons are not shown here.  89  Since each observed point on the (A v, Sv) diagram corresponds to a specific M, A, log(L/L ) and log(T ff), there are 40 (= 5 x 8) "points" in the six-parameter space: (Y, Z, @  e  M, A, log(L/L ), log(T ff)),= i.. 40- We propose a series of interpolation steps adapted @  e  from Guenther (2000) to isolate a unique solution:  1) Table 6.1 lists the model characteristics that each panel in Figure 6.2 produces for (A v bs, <5vbs) = (165, 11.5) //Hz. Since most observed points will not directly 0  0  coincide with a theoretical point on the plots, interpolation gives an intermediate value for M, A, log(L/L ) and log(T ff). @  e  Table 6.1 Interpolated model characteristics for the test point, for (A v , <Sv ) = (165, 11.5) /Mz from the (Av, <5v)-diagrams in Figure 6.2. obs  Z 0.001 0.002 0.001 0.002  Y 0.25 0.25 0.255 0.255  obs  Age (Gyr) 7.0 6.8 7.0 6.8  M(M^) 0.7714 0.7840 0.7692 0.7790  log(L/L^) -0.088 -0.108 -0.067 -0.089  log(Teff) 3.7897 3.7838 3.7900 3.7841  2) There is a relationship between Z and L, and between Z and T ff for each value of Y e  as shown in the H-R diagram in Figure 4.3. The eight points of (log(L/L ), Z) and @  (log(Teff), Z) for each Yj = 1.5 value givesfivepairs of plots. Over small ranges, a straight line is sufficient from which a Z-value can be determined from the log(L/L ) bs and one from the log(T ff) bs- (If the range in Z is large, a higher-degree @  0  e  0  91  polynomial fit may be necessary to start. This can produce more than one solution but those that are not physically realistic can be eliminated.) For the test case, Figure 6.3 shows plots of log(T ff) vs. Z and log(L/L ) versus Z for Y = 0.25 and 0.255. e  @  3) A value for Z is determined for each Y using by the observed log(L/L ) and log(T ff). @  e  Plotting these two data sets with a best-fit line through each identifies a point of intersection, which isolates the unique model-determined combination of Y and Z for the observed star. 4) If the range of the grid is large in Y and Z , then the accuracy of the interpolations decreases. Therefore, Steps 2) and 3) can be repeated to reduce the Y and Z range (which when small enough can be fit with straight lines, as shown in the example.) 5) Returning to the theoretical grid, the model closest to the unique (Y, Z) is the best choice to describe the target star.  The composition determined by the interpolation method for the test case is (Y, Z)derived = (0.2509,  0.0018). These values for Y and Z are within 5% of the "true"  composition of the test model. The model in the grid closest to this composition and consistent with the masses and ages in Table 6.1 has (Y, Z, M, A, log(L/L ), log(T ff)) = @  e  (0.25, 0.002, 0.8, 7, -0.053, 3.786). At this stage, one could choose to increase the resolution of the grid in the region surrounding the above parameters or calculate a new model with the obtained values.  92  3.791 • Y = 0.25  3.79  • Y = 0.255  3.789 3.788 — 3.787 O) o 3.786 3.785 3.784 3.783  (a) T  1  0.0007  1  1  0.0011  1  1  ,  i  0.0015  1  1  10.0019  z  0.003  0.0025  0.002  N 0.0015  0.001  0.0005  0.249  0.25  Figure 6.3: Theoretical values for the effective temperature (a) and luminosity (b) plotted against Z for Z = 0.001 and 0.002. The dashed horizontal lines represent the test's "observed" values of log(L/L ) = -0.1 and T ff= 3.785. The vertical dashed lines identify the corresponding Z for each Y (c). These values of Z(L bs) and Z(T b ) for each value of Y tested (0.25 and 0.255). The point of intersection, (Y,Z) = (0.2509, 0.0018), is the optimal value derived from the interpolation method. Q  e  0  0  S  93  7 SUMMARY AND CONCLUSIONS  A large grid of evolutionary models for MPSDs was constructed whose parameters spanned the following values:  Z: Y: Mass: Age:  0.0002, 0.0004, 0.0008, 0.001, 0.002, 0.004, 0.008, 0.01 0.235, 0.24, 0.245, 0.25, 0.255 0.7,0.8,0.9, 1.0 M 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 Gyr (or until turn-off). 0  This grid is thefirstfor metal-poor stars to extend beyond evolutionary models and traditional stellar parameters (luminosity and effective temperature) into the asteroseismic observables (v, Av, Sv). The evolutionary models presented here are also thefirstto incorporate up-to-date physics such as the OPAL equation-of-state tables, OPAL high-temperature opacity tables, and the Alexander low-temperature opacities. Both helium and heavy-element diffusion were included in the models as another new feature. The effects of rotation, magnetism and mass loss were, however, excluded. Each model was evolved from its ZAMS through to either the subgiant branch or to a maximum age of 16 Gyr. From a solar calibration model, the mixing length parameter a was set to 1.78. Eigenfrequency spectra were calculated with Guenther's (1994) nonadiabatic pulsation code from which the large and small characteristic frequency spacings were derived. For each (Y, Z) combination, a (A v, Sv) diagram was constructed. This type of  asteroseismic H-R diagram demonstrates the effects of varying mass and age on the spacings.  7.1 Sensitivities and Uncertainties We tested how systematic or random errors in the input quantities could affect the results derived from this grid: 1. The mixing length parameter was changed from a = 1.78 to 1.88 (a difference of 6%). This produced a relatively small shift in the frequency spacings, 1% in A vand no shift in Sv. The shift in A v is only on average about 1/5 of the change caused by an increase in age of 1 Gyr (~ 10 //Hz). 2. Others (e.g., Guenther and Demarque 1997) already showed that the match between models and the observed solar eigenspectrum is greatly improved by including Y and Z diffusion. Diffusion is an effect that accumulates with a model's evolution, particularly for very long-lived models of lower mass. For example, in a model with (M, A, Y, Z) = (0.7, 11, 0.245, 0.001), the effects on the frequency spacings (by neglecting diffusion and including Y diffusion only) are listed below:  A Vno-diff. - AVy&Z-diff.  = 11.8//Hz (6.1%)  A Vy-diff. - A Vy&Z-diff. =  5.30 //Hz (2.9%)  ^Vno-diff. - ^VV&Z-diff.  = 1.05 //Hz (10.3%)  ^Vy-diff. - ^Vy&z-diff.  = 0.56 //Hz (5.5%).  95  3. The effects of nonadiabaticity calculated for radiation effects near the surface on the eigenspectrum were calculated for each model. The difference between the adiabatic and nonadiabatic frequencies increases with frequency up to a relative difference of 0.45% for v= 3200 /Mz. This translates into a change in A v of less than 1.5%. 4. It is reasonable to assume that Z will be known for a target star to better than 10%> (or an uncertainty in [Fe/H] of 3%). Testing this uncertainty in Z in the pulsation models translates as a maximum error in A vand Svof < 1%.  In both mass and age, sensitivity is lost with increasing evolutionary stages for the lower mass models. Models of 1 M models with Z < 0.004 (for all Y values) evolve off s  the main sequence within 6 Gyr and are not represented in the (A v, Sv) diagrams. Since models with higher initial Z evolve much more slowly than lower-Z models, there is still reliable sensitivity to the 1.0-M models with Z > 0.004. 0  By monitoring stars for up to 7 weeks, MOST will achieve eigenfrequency resolution of ~ 0.1 /Mz. This implies an uncertainty in Svof ± 0.2 /Mz. This source of uncertainty is included in Table 7.1 with reasonable expected errors in the input parameters.  96  Table 7.1 Uncertainties in Frequency Spacings Attributed to Errors in Model Input Parameters Source of Error  Relative error in A v  Relative error in Sv  a ± 6%  1%  0%  Y±4%  5%  5%  Z±10%  1.2%  1.2%  Observational Resolution  0.2%  2.2%  Adding the errors in quadrature from the four quantities produces a global uncertainty in A v of 5.2% and in Sv of 5.6%. With the expected observational resolution, the grid could be used to specify the mass of a target star to ± 0.0052 M and its age to ± 0.056 @  Gyr (for the median composition in the grid and assuming the input physics are correct).  7.2 The Next Steps Observed values of A vand c5vcan match various physically feasible models. Since the suggested interpolation technique described in §6 can be quite cumbersome and timeconsuming, there are plans to automate the procedure and incorporate it as an interactive web-tool. Astronomers who will have asteroseismic data will be able to access this program through the MOST website. More extensive tests are necessary to check the uniqueness of the solutions from this method. Two further physical effects on the stellar models should also be investigated: (1) rotation and (2) the nonadiabatic effects due to convection on the eigenfrequency spectrum.  In May 2000, Shkolnik and Matthews (in preparation) used the Coude spectrograph at the Canada-France-Hawaii Telescope to obtain high-resolution, high-S/N observations of eight MPSDs. (They are included in Table 3.1.) The spectroscopic data will provide better model input parameters, (e.g., effective temperatures, surface gravities, and atmospheric compositions) for these stars that, in turn, can be used to generate more accurate evolutionary and pulsational models. Asteroseismology of MPSDs has the potential to reveal information in many diverse areas including setting an independent lower limit to the age of the Universe, improving the correlations between stellar ages and kinematics and representing the evolutionary sequences of globular clusters on colour-magnitude diagrams. With high-quality asteroseismic observations expected to arrive within the next couple of years, efforts such as this pulsation grid will equip astronomers with the diagnostic tools they require to interpret the data and answers old questions with new techniques.  98  REFERENCES  Alexander, D.R., 1975, ApJS., 29, 363. Alexander, D.R., and Ferguson, J.W., 1994, ApJ, 437, 879. Ando, H., and Osaki, Y., 1975, Publ. Astron. Soc. Japan, 27, 221. Antia, H.M., and Basu, S., 1994, ApJ, 426, 801. Balbes, M.J., Boyd, R.N., and Mathews, G.J., 1993, ApJ, 418, 229. Ballantyne, D.R., Ferland, G.J., and Martin, P.G., 2000, ApJ, 536, 773. 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