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EVOLUTION OF PREMAINSEQUENCE BINARY STAR SYSTEMS By JON KENNETH WEST A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979 ACKNOWLEDGMENTS There is no way I can give proper credit and acknow ledgment to all the people who have helped me in this en deavor. Time nor space allow me to fully express myself, but my heartfelt thanks goes particularly to: Dr. KwanYu Chen, my advisor and chairman of the dissertation committee. His genuine interest in my project and inspiring advice have been a real encouragement to me. Dr. F. B. Wood provided excellent literature for study and was instrumen tal in introducing me to astrophysics. I also thank Dr. John P. Oliver and Dr. T. D. Carr and Dr. R. T. Schneider for their efforts and time as members of my committee. Especially I would like to thank Dr. Mario Levio for his help in defining this problem. I would like to express my gratitude to Ken LeDuc, Butch Gould and Jim Madden for their continued support and help wherever needed. And I wish to thank the Advanced Degree Program of the General Electric Company Employee Continuing Educational Program for encouragement and funds needed to complete this degree. The funds for the computing time were made available by the Battery Business Department of General Electric ii Company. The computer was a Honeywell H6080 system oper ated by the Information Systems Business Department of General Electric Company. My appreciation also goes to Ginny McKann for her willingness to edit the various drafts and smooth out the rough edges of this dissertation. And a very special thanks to Carl George for bringing my attention to this area of study. Finally, I wish to thank my lovely wife, Donna, for her support through the many years of "after work" study. She graciously typed the many drafts and really made this work possible. Her love and understanding is a gift from God. iii TABLE OF CONTENTS ACKNOWLEDGMENTS . . ABSTRACT . . . CHAPTER I. CHAPTER II: CHAPTER III: INTRODUCTION . . PREMAINSEQUENCE BINARY MODEL A. KelvinHelmholtz Time Scale . B. PreMainSequence Mass Transfer . . C. Basic Assumptions for the Binary Model . . D. Equations of Stellar Structure . . E. The Hayashi Track . F. The Gravitational Contraction . G. Nuclear Energy Sources . H. Initial Conditions . I. Description of the Model Program . . RESULTS OF THE NUMBERICAL CALCULATIONS A. TV Cassiopeia . . B. IM Aurigae . . C. U Cephei . . D. AI Draconis . . E. 6 Librae . . Page i v CONTENTS continued . F. B Persei . .. 42 G. U Sagittae . 42 H. V505 Sagittarii .. .45 I. X Trianguli . .. 45 J. TX Ursae Majoris .. .48 K. W Ursae Minoris . 48 CHAPTER IV: SUPARY AND CONCLUSIONS .. 52 APPENDIX A: PROGRAM LISTING . .. .58 APPENDIX B: TEST MODEL . .. 66 APPENDIX C: BINARY SYSTEM MODELS. .. 79 REFERENCES . .. 210 BIOGRAPHICAL SKETCH . .. .212 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EVOLUTION OF PREMAINSEQUENCE BINARY STAR SYSTEMS By Jon Kenneth West June 1979 Chairman: KwanYu Chen Major Department: Astronomy This paper develops a quasistatic polytropic model. It represents the premainsequence contraction phase. The model was applied to both components of close binary star systems. The initial conditions for the models were determined from the Roche Lobe. The period and masses for the systems determine the maximum stable radii for both stars. Both stars were allowed to contract from the ini tial models. In the early stages, it may be necessary for a secondary to evolve along the Hayshi Track. The time for each contraction had to be calculated by an iterative procedure. The time steps were corrected for nuclear en ergy sources between iterations when needed. Models were calculated for eleven observed semi detached close binaries published in the literature. The vi combined mass of each system's components is between 2.5 and 6 solar masses. First, the results of the model were compared with the observed systems, then a determination was made concerning their premainsequence nature and age. The names and ages of the three that are likely to be premainsequence are: U Cephei (6.1 X 105 years), X Trianguli (4.6 X 106 years), and W Ursae Minoris (1.87 X 106 years). The following four systems were found not to fit the premainsequence model in this investigation: TV Cassiopeia, AI Draconis, B Persei and TX Ursae Majoris. Finally, the next four systems could be premainsequence with modifications to the model: IM Aurigae (3.5 X 105 years), 6 Librae (8.2 X 105 years), U Sagittae (3.78 X 105 years), and V505 Sagittarii (1.89 X 106 years). vii CHAPTER I INTRODUCTION The problem of the formation of a close binary star system has not been solved. No detailed understanding exists for the physical process involved. The processes for the formation of a single protostar are still in complete. At least three factors make the formation of a selfgravitating protostar very difficult: conservation of angular momentum, magnetic fields, and interstellar gas dispersion. Conventional studies of binary evolution usually do not consider any mechanism of formation. Usually model calculations, e. g. Pacznski (1971), are started with both stars on the main sequence. The star that is originally the more massive is called the primary. In most cases only the evolution of this component is followed in detail. Mullen (1974), however, followed the detailed evolution of both stars in the case of a W Ursae Majoris type binary model. Nevertheless initial conditions were that both stars were on the main sequence. The evolution of binary stars from the main sequence is identical to that of single stars until the more mas sive star expands to fill its Roche Lobe. At this point, 1 2 Crawford (1955) proposed that a large fraction of the mass is transferred to the secondary. The roles of the primary and secondary may actually reverse. Sahada (1962) asked whether such a star, originally a certain spectral type, after acquiring mass from the other star could become a more massive object that nevertheless would show a "normal" spectrum of an earlier type. The mass transfer, during the reversal, goes through a rapid thermal phase. Then the star has a tendency to expand slowly on a nuclear time scale because of hydrogen burning in the core. The nuclear transfer phase is the most likely to be observed. Assuming that a binary can be formed, Thomas (1977) has stated that the prototype of close binary evolution with mass transfer has confronted theoreticians with an increasing number of problems. He considers the post main sequence evolution to be understood in principle. The problems arise as soon as we investigate individual systems. Recently, three investigations indicate the need to study individual systems of the premainsequence type. Some observed systems are shown to be premainsequence (Field; 1969), some observed systems violate the require ment that a zeroage contact system cannot exist (Leung andWilson; 1976), and some show mass transfer rates that are on a thermal time scale (Hall and Neff; 1976). These cases are reviewed below. 3 Field (1969) found that some close binary systems appeared to be premainsequence. Field used data for the contraction of single premainsequence stars published by Iben (1965) to find the age of the stars. Both com ponents of the binary were assumed to be formed simulta neously as single stars on their respective Hayashi Tracks (see Chapter II). Four systems were determined to be pre mainsequence. They are KO Aquilae, TV Cassiopeia, WW Cygni, and Z Herculis. Their total orbital angular momen tum are similar to that predicted by Roxburgh (1967) for stellar fission. Leung and Wilson (1976) have presented evidence that zeroage contact binary systems exist. Results of their photometric investigation show both components of V701 Scorpii and those of V1010 Ophinchi are in contact. Their locations on the HertzsprungRussell diagram and their radii both suggest that the systems are zeroage. However, this result presents a problem. This is in contradiction to the postmainsequence model for the origin of contact systems presented by Lucy (1968). This model requires the components of a zeroage contact system with a common en velope should have equal masses. However, many detached binaries with a mass ratio of unity are not observed. Con tact binaries have been observed to be on the zeroage mainsequence, but none have a mass ratio of unit. In con nection with this, Whelan (1970) showed the possibility that a mass ratio of unity was unstable for premainsequence binaries. An alternative possibility is that the adiabatic constants of the stellar envelopes are not equal. The later possibility implies short lifetimes for the individual sys tems. Bierman and Thomas (1972) allowed for different adia batic constants by requiring a luminosity exchange to main tain contact of the system. Their model did not solve the problem of the light curves for W Ursae Majoris systems, nor did it propose a mechanism for the required amount of luminosity transfer. Therefore, their origin remains a theoretical problem. Hall and Neff (1976) evaluated the period changes of 24 observed semidetached binary systems. Their model as sumes mass transfer from the cooler to the primary compo nent. The mass loss results in a period change because of conservation of angular momentum. Using the observed period changes they calculated the mass transfer rate for each of the 24 systems. All systems showed mass transfer on a thermal time scale. This time scale differs from that of the conventional postmainsequence models for the forma tion of a contact or a semidetached binary system. For in stance, Mullen (1974) found that each component of a post mainsequence binary expands away from the mainsequence on a nuclear time scale. Thus, the problem of the pre or postmainsequence nature of a binary has not been solved. Each system must be evaluated on an individual basis. The problem that the writer wishes to investigate is the evaluation of individual systems with a premainsequence model. The main purpose of this investigation is to find 5 possible premainsequence binary systems. This will be accomplished by comparing the positions of the observed binary on the HertzsprungRussell diagram with the evolu tionary tracks generated by the premainsequence model. If both components appear to have the same age and fall near the tracks of the model, then the system is likely to be premainsequence. The investigation will include the evaluation of eleven binaries each with a total mass between 2.5 and 6 solar masses. In all cases the secondary star is overluminous. These systems are classified as semidetached. The model being considered is independent of the ori gin of a binary star system. The origin may be either con densation of both components as described by Wood (1962) or fission of a rapidly rotating premainsequence core as described by Whelan (1970). Yamasaki (1971) studied the case immediately after fission where mass transfer may occur. As the stars contract, they must eventually detach themselves from their Roche Lobes. Yamasaki ends his in vestigation when mass transfer stops or a single star is formed because of excessive mass loss. The writer assumes the first case as the initial configuration of the pre mainsequence model. At this stage the individual stars fill their Roche Lobes. Each star is allowed to contract gravitationally and independently toward the mainsequence. As a comment on the study of origins, I would like to quote Whitcomb and Morris (1961) page 213. After all, any real knowledge of origins or of earth history anticedent to human histor ical records can only be obtained through divine revelation. Since historical geology [or stellar origin], unlike other sciences, cannot deal with currently observable and reproductible events, it is manifestly impossible ever really to prove, by the scientific methods, any hypothesis relat ing to prehuman history. This idea applies directly to the problem of binary star evolution and origin. Experiments cannot be set up and conducted that will prove or disprove any case of binary origin. Only models of specific sets of conditions can be made and compared with observed binary stems. The "best" model which can be found has the fewest number of adhoc assumptions and predicts the greatest number of observed phenomena. CHAPTER II PREMAINSEQUENCE BINARY MODEL The masses and periods for eleven observed close binary systems were selected from data tabulated by Gian none and Giannuzzi (1974). The stellar structure is rep resented by a polytrope of index n=3 :for "he radiative solution. A polytcope of index n,: 5 ijs used for the Hayashi solution. The Hayashi solution is 'eprc:r:ntec by tL'e line AB on the HertzsprungRussell diagram shown in Figure 1. The radiative solution is line BC. Both solutions are calculated for each model. The Hayashi solution is selected as long as it is above the radiative solution. After the solution is determined, the time step is calculated by an iterative procedure described in a later section. The mod el calculations are stopped when the primary reached the mainsequence. The variables and symbols used throughout this chapter are listed below: T KelvinHolmholtz contraction time (years) AL Time step between Inolels AR Radial contraction between mod'is R Stellar radius M Stellar mass r Radial distance from Stellar C:ntLer p Density P Pressure T Temperature a Radiation constant / /FOCRBIDDEN REGION C\ B MAIN SEQUENCE < LOG Teff FIGURE 1. Schematic diagram of the evlr,' ona;r trSrks on the HertzsprungRussell :iiaT'.anm or a ciLV" mass. Hayashi solution line / 8, 'he rdi, tive solution line B C. A 0 J 0 _j _ I__~~ c Velocity of light T(ff Effective temperature y Ratio of specific heats n Polytropic index 5 Polytropic radius l Polytropic radius at surface of star K Constant in polytropic relationship K Opacity a StefanBoltzman Constant H Mass of hydrogen atom L Total Stellar Luminosity Ln Luminosity due to nuclear sources La Luminosity due to gravi national contraction L Luninosity due to the Hayashi Track A Separation between the centers of the binary components G Gravitational Constant C Total energy gener !;.io1 ial C En Energy generation rate J',r nuclear sources g9 Energy generation rr'te foyr gravitational contrac tion R Gas constant A. KelviniHelmholtz Tijm Scale The KelvinHelmholtz contraction is known as the total premainsequence phase of stellar evolution. The source of energy is derived totally from the gravitational potential. The time for this phase can be estimated by the following relationship: S= ro (M/M0) Where To = 2.75 X 107 years and No is the solar mass, Lo is the solar luminosity. A fraction of this contr;ii oin tojim,e w is used as the firstl estimate for time step r .hPe modr calculations. The first estimated time step was found from the following: At T (AR) 3.17 X 107 seconds. R B. PreMainSeauence Mass Transfer The study of premainsequence mass transfer shows that a binary will evolve rapidly from an overcontact configuration to a stage where both stars will fill their Roche Lobes. Mass transfer during condensation has been proposed, e. g. Wood (1962). He describe; formation mechanism for a close binary in which nuc '.i are poor in hydrogen compared to the mixtur from which they are con densing. Whenever two condensing nuclei of comparable size are present, the formation of a double star, rather than a single star or planets may occur. The difference between the formation of a double star and a single star is that the double star is constrained by the inner critical equipotential surface known as the Roche Lobe. Wood argues that the more massive component would continue to attract the most gas. Hayashi and Nakano (1965) and Nakano, Okyama and Hayashi (1968) proposed a method for the formation of a single star that could lead to the formation of a close binary. Theyhey propose tht the .Frefall ola [ n opaque proLostIr is stopped by ine bounce 2n ,ntr;l core. For a protostar of one solar mass this takes ap proximately ton years. The bounce causes a shock wave. 11 The subsequent propagation of the shock wave toward the stellar surface causes a sudden flareup. The star finally settles down to a quasihydrostatic equilibrium state and rapidly moves toward the Hayashi Track. If the protostar condenses with sufficient rotational angular momentum, the above theory will predict the formation of a rapidly rotat ing core. When the bounce shock occurs it could be non spherical. The result may be the formation of a promain sequence binary system by fission. Yamasaki (1971) studied the case of a binary system immediately after fission of a collapsing protostar, He suggested that a substantial fraction of the matter would be crowded out of the critical lobes and would drift around the components. The total orbital angular momentum of the confined mass must be above the critical value to cause rotational instability. He assumed that the balance of the mass outside of the Roche Lobes would be lost from the outer Lagrangean point. The loss rate is a parameter of the problem. If only one component overfilled its Roche Lobe, then the whole excess mass was assumed to fall on the other component. It does not escape from the system provided that the other does not fill its Roche Lobe. The model calculations were based on a polytropic star with n = 1.5. This represents fully convectiv LP stars )ontrac t ing on the Hayashi Track. The initial mass for all cil culations was held constant at 13.5 solar masses. In all cases of stable mass transfer rates and mass loss rates, 12 the systems detached within several hundred years. For a critical value of mass exchange rate the system would not recover from the instability and form a single star. The final configurations of these models still had separa tions from 2 to 4 A. U. Although these separations are a minimum, they do not represent observable close binary systems. The writer evaluated the case of premainsequence mass transfer from the secondary to the primary. Table 1 shows the initial and final configuration after mass transfer. TABLE . INITIAL AND FINAL CONFIGURATrION OF 1A3SS TRANSFER MODEL INITIAL FINAL MI/Mo 3.0 3.128 M2/M0 1.5 1.372 Period (days) 3.0 3.459 Separation (A. U.) 0.067 0.074 The mass transfer rate is chosen to be 1 X 107 solar mass per year. This is representative of observed rates in semidetached binary systems, (e. g., Hall and Neff; 1976). Table 2 shows the calculated models of the secondary during mass transfer. Column 1 is the model number. Column 2 is the age of the model in second!. Colu in 3 is th logarithm of the luminosity in solar units. Column 4 is the logarithm of the effective temperature. Column 5 is the ratio of the radius to the Roche Lobe. Finally, column 6 is the radius in solar units. The mass transfer stops 13 when the secondary is smaller than its Roche Lobe. The initial radius of the secondary is arbitrarily chosen to be 20% greater than the Roche Lobe. Conservation of angular momentum and of mass requires the binary system to increase in separation during mass exchange. Therefore, the Roche Lobes of each star increase as the mass ratio increases. The mass transfer stops within 28 years for this case. This is similar to the results found by Yamaski (1971). The initial configuration chosen by the writer is immediately after the binary components form a minimum contact binary. Both components will fill their Roche Lobes initially. TABLE 2 MODELS OF THE SECONDARY DURING MASS MODEL TIME LOG(L/LSUN) LOG(TEFF) 1 2.506E+07 4.8421 3.2855 3 6.452E+07 4.7893 3.2888 5 1.044E+08 4.7364 3.2920 7 1.446E+08 4.6834 3.2953 9 1.855E+08 4.6304 3.2985 11 2.310E+08 4.5772 3.3018 13 2.830E+08 4.5239 3.3051 15 3.423E+08 4.4705 3.3083 17 4.101E+08 4.4170 3.3116 19 4.876E+08 4.3634 3.3149 21 5.761E+08 4.3097 3.3181 23 6.773E+08 4.2558 3.3214 25 7.930E+08 4.2019 3.3247 27 9.254E+08 4.1478 3.3279 TRANSFER ROCHE ER 1.19 1.17 1.16 1.14 1.12 1.11 1.09 1.08 1.06 1.04 1.03 1.01 1.00 '' Q R/RSUN 5.50 5.45 5.39 5.34 5.29 5.23 5.18 5.13 5.08 5.03 4.98 4.93 4.88 4,83 C. Basic Assumptions for the Binary Model The proposed model represents a preamainsequence binary star system. The initial configuration introduced is constrained by the Roche Lobes. The basic assumptions which must be made are as follows: 1. The first law of thermodynamics requires mass and energy to be conserved both for the in dividual stars and the binary system as a whole. 2. The second law of thermodynamics requires that the entropy for any closed system be greater than or equal to zero. This means that there is no external source (or sink) of energy available to do work. The system dynamics are determined from the initial conditions. 3. The binary system obeys conservation of angu lar momentum. Circular, Keplarian orbits were assumed. 4. The stars are spherically symmetric. Thus, we are assuming that the stars are not affected by rotation during the model calculations. 5. The stars are in quasihydrostatic equilibrium. This assumption is required to calculate each model. A contracting star is not, by defini tion, in hydrostatic equilibrium. Yamasaki (1971) showed each contraction occurs over a time step which is long compared to the dyna mical time scale of readjustment. Thus, the approximation of hydrostatic equilibrium is good. D. Equations of Stellar Structure There are four basic equations of stell; dP = GM(r) hvdrosta dr r2 dM(r) dr = 4Tpr2 continue dL(r) 2 dr) = 4per2 thermal dr and either dT 3 KP L(r) for radii dr 4ac 3 4crr eauilibr or 1 dT Y 1 1 dP for conv T dr y P dr equilibr M(r) and L(r) are the mass and luminos ar structure: tic equilibrium ty of mass equilibrium active ium active ium ity a t radial distance r. Along with these four differential equations have the following relations: P = P(p, T, chemical composition) < = (r, T, chemical :omcosi on) c = E(p, T, chemical comrositLon) we must the equa tion of state I ;e Op' & c i.ty the energy Generation rate Der unit mass Where 16 The boundary conditions for this problem are as fol lows: At r = O, M(r) = O L(r) = 0 At r = R, T = Teff P = 0 M(r) = M A polytropic star is one which obeys an equation of the following form: 11 P = Kp + n Since the pressure is an explicit function of density only, polytropic stars are determined by the equation of hydro static equilibrium and the equation of continuity of mass. By combining these two equations and using dimensionless variables, one differential equation defines the structure of a polytropic star: 1 d ( 2 d0) n S2 dE dE This is known as the LaneEmden equation. The vari ables are defined as follows with the subscript c indicat ing the core valves. On = p Pc 0 = T Tc S_ r r Where rn is known as the Emden unit of length: rn [(n + 1) P  4lTGpc2 In addition, we have R = 1 r The polytropic model provides a fair approximation to the structure of certain types of real stars. This idealized model is often useful in qualitative and in rough quantitative discussions and aid considerable in gaining an overall] in sight into the structure of stars,(e. g, Cox and Guili; 1968) E. The Hayashi Track Hayashi (1961) proposed a phase of gravitational con traction of a convective star known as the Havashi Track. This is shown in Figure 1 as the nearly vertical line, AB, on the HertzsprungRussell diagram. The convection may result from the large opacities caused by ionizing hydro gen. The adiabatic relation between the pressure and temperature is a good approximation for most of the mass of completely convective stars. P = KT2.5 Here, the ratio of specific boats, y, is constant and equal to 5/3. Even including the photosphere, this adiabatic relationship i a good approximation. 18 For stars in convective equilibrium, the adiabatic temperaturepressure relation can be rewritten in terms of Schwarzschild dimensionless variables as p = Et2.5 where p = {(47r/G) R4/M2) P and t = {(R/G)R/uLM}T and E = 47K(p/R)2.5 G1.5 M0.5 R1.5 If the convective envelope extends all the way to the cen ter of the star, then a limiting .alue of i foi comnolete]y convective stars can be found to be: E = 45.48 Hayashi described the "Forbidden Region" for a star of a given mass to be to the right of the line AB. Line AB represents the surface solution for the luminosity based on the limiting value of E. A star finding itself in this forbidden region would not be in hydrostatic equilibrium and would have to adjust itself to another structure on roughly a "freefall" time scale. The luminosity for fully convective gravitationally contracting stars, can be found by solving for radiative transfer in the photosphere. Cox and Giuli (1968) found a solution for the luminosity for stars /:. h L > L to have a strong dependence on the effective temperature in the following form: L cBlpB2MB3 B4 LIP M Teff By assuming complete ionization of hydrogen at the base of the photosphere, the equation for the luminosity has been shown to be: L a C9 0.9M1.56 T16.9 eff Schatzman (1963) found the solution for the luminosity in terms of solar type population I chemical composition and mass to be: Lh = 1087 M Teff15.3 Equation 1) Where M, is the stellar mass in solar units. F. The Gravitational Contraction For a premain sequence star, the only significant sources of energy come from the gravitational potential. The relationship between the gravitational contraction and luminosity can be determined from the equation of thermal equilibrium: dL d 4Tapr2 By using the ideal gas law, allowing the r.:.io of specific heats, y, to be equal to 5/3, and defini:9 c= n + Cg, the equation for thermal equilibrium becomes dL d 4,r2p(n 3 2/3 d (P By using the equation for hydrostatic equilibrium the fol lowing relationship holds: 3 2/3 d P 3 P L dR a dt ( 5/3) 2 P P at Substituting, en = 0 into the equation of thermal equilib rium we get the following: dL. P dR S 6r2 P dr R dt Now this relation is rearranged in the form of polytropic variables: dL 6rrn2 p n+l 1 dR d = 6rn P c6 R dt Finally, the dR being rewritten in different form and by dt noting that R = 51 Ar,, the relation becomes dL = 6TrrnPc E28n+l AR 51 At Where AR is the radial contraction and At is the time step for the contraction, the total contributio to luminosity from a gravitational contraction oL a pcl 'upe is found by integration: L = 6iPcrn2 Arn f r2Gn+1 da (Equation 2) At 0 where El = 3.65375 for n = 1.5 and E = 6.89685 for n = 3.0. This relation for the luminosity has the term Arn. The At Arn represents the polytropic radial contraction while At is the corresponding time step. Arn is selected simply to optimize computing time. The time step, on the other hand, must be determined from the Virial Theorem and conservation of energy. The Virial Theorem simply states that half the energy made available by a contraction causes an increase in the stellar temperature, while the other half is radiated away. The potential energy 2 is given by: R p = + f VdMr For a polytrope the potential energy is given by: = ( 3) GM2 5n R If only half the potential energy is available for the luminosity, then the time step, At, is determined by ^t  where ponding time step. Therefore, we have At = ti+1 ti = ( 3 ) GM2 (1 1 5n (Equation 3) and where the index i corresponds to the ith model. The problem then is to determine, by an iterative procedure, selfconsistent values for the new value of the luminosity Li+l and the time step ti+1 t. An initial guess for the time step was obtained from the Kelvin Helmholtz time scale for contraction onto .he mainosequence. The estimate was obtained by proportioning the time scale for the radial contraction step. An estimate for the lumin osity, L', is calculated for the polytrope. An estimate for the time step, At', is calculated from L'. Then using At' a new value of the luminosity, L", is calculated and compared to L'. If L' is greater than L", then a new in creased time step, At", is estimated. Likewise, if L' is less than L", then the new estimate for the time step is de creased. This can be written as follows: At = At" +jAt' At" l where + was used if L" The primed values are the old estimates and the double prime indicates the new estimate. After L" and L' approach each 23 other to the desired accuracy then L" is set equal to Li+I. The time step, At", is stored for that model as At and be comes the first estimate for the time step for the next model. G. Nuclear Energy Sources If the assumption is that no contraction takes place then the equation for thermal equilibrium becomes dLn dr = 4Er2pn By using the appropriate polytropic var:iabLes and the nuclear energy generation rate cn for the protonproton reaction, the luminosity is Ln = 4Trrn3 P2Tc4 oX2 f 22n+4d (Equation 4) where Eo 10 X 1030 ergs/sec/gm and X is the hydrogen fraction of stellar material. The nuclear luminosity is calculated and compared to the luminosity of the "zeroage" mainsequence star of corresponding mass. The model calculations ended when Ln approached the mainsequence value. The time that a particular componernt of the binary remained on the rmainsequence is considered to evaluate the systems. Generally the primary should reach the main sequence before the secondary. It is of interest whether 24 the primary has enough time to evolve off of the mainse quence before the secondary can reach the mainsequence. A logarithmic regression is performed on data published by Iben (1965), and Strothers (1963) (1964) (1966). The fol lowing Table 3 represents mean published values for the time on the mainsequence tIg, Novotny (1973). TABLE 3 TIME ON THE MAINSEQUENCE M/Mo log M/Mo log tMS 60 1.7782 6.4914 45 1.6532 6.5563 30 1.4771 6.6721 15 1.1761 7.0043 9 .9542 7.3242 5 .6990 7.8088 3 .4771 8.3443 2.25 .3522 8.7259 1.5 .1761 9.2966 1.25 .0969 9.6053 1.0 0 10.0086 0.5 .3010 10.4771 0.25 .6021 10.8541 The following relationship can be used to represent the above logarithmic values: (The maximum error is less than 5% over the entire range of masses.) log tMS = (2.00998) log (M ) + 9.62409 SM In all cases evaluated, the secondary has sufficient time to contract onto the mainsequence before the primary evolves away. H. Initial Conditions With the masses of each component and their period being free parameters, the Roche Lobe for each star was determined by the program. The following relation, given by Paczynski (1971) for the Roche radius, Rr, was used: Rr = A (0.38 + 0.2 log (M1/2) ) The Roche radius was determined for each ftar and was taken to be the initial stellar radius for the first model of each star. The program then calculated the sequence of contraction models. The chemical composition is chosen to be of popula tion I type and assumed to be the same for each star. The hydrogen fraction, X, is selected to be 0.750, the helium fraction, Y, selected to be 0.224, and the heavy metal fraction, Z, is selected to be 0.026. This selection is arbitrary but is chosen to allow comparison with the work of Iben (1965). The total mass for each model is between 2.5 and 6 solar masse. Eleven close bi ry systemO are chosen from Giannone and Giannuzzi (1974). The parameters are listed in Table 4. Column 1 is the name of the System. Column 2 is the period in days Column 3 is the mass in solar units 26 for both the primary and the secondary. Column 4 is the logarithm of their luminosities in solar units. Finally, column 5 is the logarithm of their effective temperatures. TABLE 4 OBSERVED SEMIDETACHED BINARY SYSTEMS SYSTEM PERIOD M/Mo LOG (L/Lo) LOG (TEFF) TV Gas 1.243 3.10 2.070 4.029 1.39 1.028 3.787 IN Aur 1.245 2.97 1.999 4.029 0.89 0.203 3.679 U Cep 2.493 3.19 2.066 4.079 1.53 0.875 3.678 AI Dra 1.199 2.18 1.467 3.982 1.03 0.650 3.756 6 Lib 2.327 2.96 1.976 3.982 1.31 0.770 3.675 8 Per 2.867 3.15 2.082 4.079 0.74 0.797 3.696 U Sge 3.381 4.27 2.593 4.134 1.60 0.573 3.567 V505 Sag 1.183 2.22 1.501 3.958 1.18 0.414 3.693 X Tri 0.972 1.72 1.044 3.947 1.00 0.000 3.642 TX U Ma 3.063 3.13 1.997 4.079 0.90 1.088 3.746 W U Mi 1.701 2.68 1.824 3.947 1.19 0.541 3.681 I. Description of the Model Program The program calculates both stellar model sequences simultaneously. This allows comparison of time steps and structure of each component when possible. The initial mass of each component and the period of the binary are parameters of each model sequence. Each observable system can be evaluated in turn. 27 Figure 2 represents a flow chart for the premainse quence model program. By using the KelvinHelmholtz con traction time, an estimate for the time step, At, is made for the initial radial contraction AR. The initial value of the luminosity, L', is arbitrarily chosen from the mass luminosity relation for mainsequence stars: log (L ) = 4.1 log (L) 0.1 Lo Mo The radial contraction, Arn, wps chosen in all cases to be (Arp)i1 = 0.005 Ri A polytropic model is calculated for both n = 1.5 and n = 3. By using Equation 2, the first value for Lg is calculated. The effective temperature is calculated by using the follow ing relation: 4 L Teff 4' R2 By using Equation 1, the Hayashi Track luminosity, Lh, is found and compared to the radiative luminosity, Lg. If Lg the time step can be calculated fjrm th(? Viral Theorem using Equation 3. If the old falue fo r the urnmi nosity L' is equal to L" within sufficient accuracy, then the pro 'dure is stopped; if not, the time step is adjusted ESTIMATE FOR INITIAL TIME STEP AND LUMINOSITY CONSERVATION OF ANGULAR MOMENTUM N L FIGURE 2. Flow chart of the program for the premainsequence model 29 as described previously and the procedure starts over again. After the error between L" and L' is less than 1 part in 105, the values for R, Teff, L, and At are stored for that model. If the program limits have been met, Ln is large enough; the program stops; if not a radial con traction is made and the procedure is restarted. Appendix A lists the main program, all subroutines and polytropic constants. The language is Honeywell's Mark III version of General Electric Company's BASIC. A test model is compared with the evolutionary tracks published by Iben (1965). The primary for the test model is 3 solar masses, the secondary is 1.5 solar masses, an6 the period is 3 days. Initially both stars are filling their Roche Lobes and are on the Hayashi Track. Figure 3 shows the plots of the test model and Iben's premain sequence tracks for single stars on the HertzsprungRussell diagram. The initial point for Iben's models is labeled "O". The points 1 through 5 can be used for age compari sons with the test model. Table 5 shows the ages at these points labeled on the evolutionary tracks in Figure 3 for each star. The same numbering is used for the test model. The test model tabulation for both the primary and the secondary can be found in Appendix B. Each table has 6 columns. Column 1 is the model number after the initial model. Column 2 is the accumulated time ir seconds after the initial model. 2.5 o 0 2.0 \. 1.5 6 \ / 1 3 A \ /o S1.0 5  0o< 1o o0. 3 0 0.5  1.0  4.2 4.0 3.8 3.6 3.4 LOG Teff FIGURE 3, Evolutionary tracks of a premainsequence test model for a binary (S = 3Mo, 0 1.5 Mo) and the evolutionary tracks for single stars published by Iben (1965). TABLE 5 COMPARISON OF AGE (106 YEARS) FOR THE TEST MODEL PRIMARY = 3 Mo SECONDARY = 1.5 Mo AGE POINT IBEN TEST MODEL IBEN TEST MODEL 1 0.034  0.23 0.205 2 0.208 0.06 2.36 1.391 3 0.763 0.747 5.80 3.483 4 1.135 1.202 7.58 4.962 5 1.250 1.351 8.62 5.949 6 1.465 1.432 10.43  7 1.741  13.39 8 2.514  18.21 Column 3 is the logarithm of the luminosity in solar units. Column 4 is the logarithm of the effective iemperatiue. Column 5 is the ratio of the model radius to that of its Roche Lobe. Finally, column 6 is the radius of the model in solar units. These ages and evolutionary tracks of the test model show reasonable agreement with Iben's results. Up to point 5, the age of a component of the test model compares with the age of Iben's models. Suggestions for future im provements and changes are discussed in Chapter IV. The major differences in ages occur near the mainsequence. These differences are caused by the limitations of a poly tropic model to adjust for the core's changing chemical composition. In particular, Iben followed 'he depletion of C12 in detail. Normally, C12 is maintEined in equilib rium for the CN reaction chain. However, during the pre mainsequence contraction the CN reaction has not started. 32 Only very near the mainsequence is there sufficient ther mal energy in the core to initiate this reaction. C12 must be depleted to form the equilibrium concentrations of the "secondary" elements in the CN reaction chain. After equilibrium is reached, the concentration of C12 remains constant. This resulted in slowing down the con traction near the mainsequence as C12 was depleted in the core. The test model is polytropic and cannot follow detail changes in stellar structure. It is intended as a survey tool to determine whether or not a binary system is premainsequence. CHAPTER III RESULTS OF THE NUMERICAL CALCULATIONS The age computed by the writer's model is the time after both components filled their Roche Lobes. It may not be the same as would be calculated for a single star contracting onto the mainsequence. Appendix C lists the numerical values for each of the premainsequence models calculated as described in the previous chapter. The Appendix C contains 22 tables. Each table is identified by the name of the binary system. Each table has 6 columns. Column 1 is the model number after the initial model. Column 2 is the accumulated time in seconds after the initial model. Column 3 is the logarithm of the luminosity in solar units. Column 4 is the logarithm of the effective temperature. Column 5 is the ratio of the model radius to that of its Roche Lobe. Finally, column 6 is the radius of the model in solar units. The computed binary model and an observed binary sys tem are compared by age and position on the Hertzsprung Russell diagram. The ages of each component aust match and the observed binary must fall near tne evolutionary tracks 34 of the binary model for the systems to be premainsequence. Individual systems are discussed in the following sections. A. TV Cassioneia It appears unlikely that TV Cassiopeia is a premain sequence binary. The secondary is either overluminous or it has too great an effective temperature. This is in con tradiction to the results published by Field (1969) for TV Cassiopeia. Field determined the premainsequence nature of the system by estimating the radii and age of each star. The estimate is made by interpolation of pub lished models of a single star. Figure 4 shows the plotted contraction of both com ponents of a premainsequence model for TV Cassiopeia on the HertzsprungRussell diagram. The observed luminosity and effective temperature for both components are plotted also. The problem of the overluminosity of the secondary is clearly shown. If the age of the primary can be determined from the model, it is approximately 6.3 X 105 years since the in itial model. To estimate the age of the secondary, either the radiative solution or the Hayashi solution must assume to be correct. Comparing the luminosity with that of the model secondary on the Hayashi Track yields an age of 1.5 X 105 years. If the radiative solution is assumed, the age is 3.1 X 105 years. This premainsequence model does not fit the observed system for either case. F M % 2.5 2.0 1.5 `\O0o' I f 6.3 x 105 4.0 YEARS 3.8 LOG Teff FIGURE 4. TV CASSIOPEIA (A = 3.1 Mo, A = 1.39 Mo). The evolutionary tracks of the premainsequence binary model is represented by the circles (primary 9, secondary o). 0 8 o \o9. 0 0 0 O 0 1.0 0.5 0.0 0.5 .O 4.2 3.6 3.4 __ B. IM Aurigae Figure 5 shows the plotted contraction of a pre mainsequence model for IM Aurigae on the Hertzsprung Russell diagram. The observed stellar luminosity and effective temperature for both components also are plotted. The observed system falls close to the contraction paths of the premainsequence model. However, the "ages" do not match. The age of the primary seems to be 3.1 X 105 years. The age of the secondary appears to be between 9 X 105 and 3.1 X 106 years according to this mo:l. C. U Cephei Figure 6 shows the plotted contraction of a premain sequence model of U Cephei on the HertzsprungRussell dia gram. The observed luminosity and effective temperature for both components also are plotted. This system appears to be premainsequence. First of all, the observed system falls close to the contraction path of the model. This is true for both com ponents. Secondly, the ages match within 20% at 3.15 X 105 years if the secondary is on the Hayashi Track and has a slightly different chemical composition. Iben (1965) \ ,CO 2.5 2.0 1.5 0 0 3.5x105 YEARS 1. 01 4.2 4.0 3.8 LOG Teff 3.6 3.4 FIGURE 5. IM AURIGAE (A = 2.97 A =z 0.9 Mo). The evolutionary tracks of the premainsequence binary model is represented by the circles (primary a, secondary o). 0 0 1.0 0.5 0.0  0.5 I 99 000 A \ 000 0 N 0 6.1 x 105 YEARS I 4.0 LOG Teff FIGURE 6. U CEPHEI (A = 3.19 Mo, A = 1.53 Mo). The evolu tionary tracks of the premainsequence binary model is represented by the circles (primary *, secondary o). 2.5 2.0 1.5 S0 0.5 0.0 0.5 1.0 4.2 3.6 3.4  I 39 showed a slight decrease in the metals composition would move the Hayashi Track toward higher effective temperatures. This would match ages and positions on the Hertzsprung Russell diagram. This model predicts that U Cephei is a premainsequence binary star system with population type I chemical composition for both components. D. AI Draconis Figure 7 shows both the path of the premainsequence contracting model and the observed binary system AI Draconis plotted on the Hert.spr ngRussell diagram. The system does not appear to be a premainrsequence object according to this model. This problem seems to be the same as TV Cassiopeia. The ages do not match. The secon dary component of the observable system is overluminous; or the effective temperature is too high for it to be con sidered a premainsequence star for this particular mass. E. 6 Librae Figure 8 shows the premainsequence contraction model for 6 Librae plotted on the HertzsprungRussell diagram. The observed binary system may possibly be premain sequence if it is assumed that the two stars have different chemical compositions. This would follow the description by Wood (1962) of the formation of a binary with different compositions. As stated before, Ibn (1965) has shown that such a change in composition could increase the effective 2.5 o 0 \ o 2.0 \o 0 I .5 \ \ o SI .0 1 o o 0.5  J 0 0.0 / 0.5 \ 2.6 x 106 YEARS 1.0 \ 4.2 4.0 3.8 3.6 3.4 LOG Teff FIGURE 7. AI DRACONIS (A = 2.1.8 Mo, A = 1.03 Mo). The evolutionary tracks of the premainsequence binary model is represented by the circles (primary *, secondary o). 0 0 0 2.5 2.0 1.5 8.2 x 105 YEARS 4.2 4.0 3.8 3.6 3.4 LOG Teff FIGURE 8. 6 LIBRAE (A = 2.96 IMN, A = 1.31 Mo) he evolu tionary tracks of the premai asequence binary model is represented by the circles (primary o, secondary o). 0 0 0 0 0 O oo 0 1.0 0.5 0.0 0.5 1.0 '~OleCshq as~ 42 temperature of a star on the Hayashi Track. With this in mind, the ages of both components are in agreement. The age after both components filled their Roche Lobes is approximately 8.2 X 105 years. F. 8 Persei Figure 9 shows the observed binary 8 Persei plotted along with the path of a premainisequence contracting model. This system does not fit the model for a premain sequence binary star system. It suffers LLom the same problems as TV Cassiopeia and AI Draconis. The secondary is not represented by this preLmainsequence model. The age is also a problem. If the chemical compositions are different enough to fit the model, the age for the secondary would be younger than the primary by a large factor. This is caused by the decrease in contraction time as the lumin osity increases. The luminosity must be much higher for a given effective temperature for the secondary to fall on the Hayashi Track. G. U Sagittae Figure 10 shows the plot of the binary system U Sagittae on the HertzsprungRussell diagram. Comparing the evolutionary tracks of the model with tile obser ved system, the binary could be premainequence. The ages of both stars appear to match at approximately 3.78 X 105 ,0 \ o 1.14xl 6 YEARS  I I I 4.0 3.6 LOG Teff FIGURE 9. 3 PERSET (A = 3.15 Mo, A = u. Mo) The evolu tionary tracks of the premara 4 quence binary model is represented by the circles (primary e, secondary o). 2.5 2.0 1.5 1.0  0.5 0.0 0.5  1.0 4.2 3.4 C$99, 2.5 s A 0 \ ** o 2.0 \ o o \ o 1.5 \ \ o a \ o oo 0 0 0. 5 \ _J 0.0  0.5 3.75x 05 YEARS 1.0  I I I I 4.2 4.0 3.8 3.6 3.4 LOG Teff FIGURE 10. U SAGITTAE (A = 4.27 1M, A = 1.60 I1)). The evolutionary tracks of tIhe premainsequence binary model is represented by the circles (primary o, secondary o). 45 years. An increase in metals content of the secondary would move the Hayashi Track toward lower effective tem perature. U Sagittae is probably a premainsequence ob ject. Therefore, another alternative is that U Sagittae may not have a fully convective premainsequence secondary. The secondary falls in the forbidden region described by Hayashi. If this is actually the case, then the star is not in hydrostatic equilibrium, and its structure would be changing rapidly. Larson (1969) described premain seauence contractions that may not be on the Hayashi Track. These evolutionary tracks had rpiidly varying luminosities as a function of effective temperature, caused by different depths of the convective envelope. H. V505 Sagittarii Figure 11 shows the paths of a contracting premain sequence model plotted on the HertzsprungRussell diagram. Both components of the observable binary system V505 Sagit tarii are plotted along with the model. By using the same argument that the components could have slightly different chemical compositions, the system could be premainsequence. The age of the system would be approximately 1.89 X 106 years. I. X Trianguli Figure 12 shows the paths of the contracting binary model and both components of the binary X Trianguli plotted 0 0 0 0 ** 0 0 0 0 00 1.89 x 106 YEARS I I 0O 4.2 4.0 3.8 3.6 3.4 LOG Teff FIGURE 11. V505 SAGITTARII (A = 2.22 t,, A = 1.18 M,). The evolutionary tracks of the prenrain sequence binary model is represented by the circles (primary e, secondary 0o) 2.5 2.0 I.5 1.0 0.5 0.0 0.5 4.6 x 106 4.0 e o 0 0 0 YEARS 3.8 3.6 3.4 LOG Teff FIGURE 12. X TRIANGUILI (A = 1.72 Mo, A = ].00 M). The evoltuionary tracks of the premainsequence binary model is represented by the circles (primary a, secondary o). 2.5 2.0 5 1.0 0.5 0.0 1 .0 4.2 \A 0 48 on the HertzsprungRussell diagram. The observed binary system falls close to the premainsequence path of the model. The ages, however, do not quite match. The dif ference in the ages is a little less than a factor of two. This could possibly be caused by a difference in chemical composition. If we allow for a slight increase in effec tive temperature, caused by a decrease in heavy metals, then the time of contraction would decrease. This is re quired by conservation of energy and the Vrial Theorem. With these assumptions the binary X Trianguli could pos sibly be a premainsequence object. J. TX Ursae Majoris Figure 13 shows the observed binary TX Ursae Majoris plotted on the HertzsprungRussell diagram. The contract ing premainsequence model is plotted also. The system does not appear to fit the model. The age would have to be _9.5 X 105 years as determined by the primary. This yields the same problems that were found with 8 Persei and others. The secondary is overluminous for a premain sequence star with this particular mass. K. W Ursae Minoris Figure 14 shows the observed contact binary system W Ursae Minoris plotted on the HertzsprungRussell diagram. The path of the contracting premainsequence model also $ eve \^A * as9 9 e* 9.5x 105 YEARS o \ o _ _ 4.2 4.0 3.8 LOG Teff 3.6 3.4 FIGURE 13. TX URSAE MAJORS (A = 3.13 Mo, A = 0.90 Mo) The evolutionary tracks of the premain sequence binary model is represented by the circles (primary o, secondary o). 2.5 2.0 1.5 0 0 0 o 1.0 0.5 0.0 0.5 .0 L I_ 0 0 0 o \ o  \o 2.5 2.0 1.5 4.0 3.8 3.6 3.4 LOG Teff FIGURE 14. W, URSAE MINOEIS (A = 2.68 Mo, A = 1.19 M0)  The evolutionary tracks of the premain sequence binary model is represented by the circles (primary e, secondary o). \\o A o 0O \ 0 !.87 x 06 YEARS 1.0 0.5 0.0 0.5 1.0 4.2 51 is plotted. Both components of the observable system are relatively close to the path of the model. The exciting part of this analysis is that the ages for the premain sequence model match at about 1.87 X 106 years for a sys tem slightly different from the observed system. Iben (1965) showed that an increase in metals could account for the increased luminosity of the primary on the radiative path. He also showed that a decrease in metals could in crease the effective temperature for the secondary on the Hayashi Track. The results of this model cannot be dis counted concerning the possible premainseque'ce nature of W Ursae Minoris. CHAPTER IV CONCLUSION AND DISCUSSION This binary model allows discovery of premain sequence systems. Initially both stars must be forced into their inner equipotential surfaces rapidly. Evolu tionary tracks calculated from his initial configuraLion define a new means for evaluating actual binary systems. The major conclusions cf tnis evaluation are as follows: 1. Premainsequence close binary star systems can be found by comparing observed binaries with this model. Some observed closed binary systems are found to fit this premainsequence model. 2. The chemical composition of the two components of the premain sequence binaries appeared to be slightly different. Even though the actual dif ferences in composition were not quantified, both stars could still be considered population type I. 3. There does not seem to be a correlation between the orbital angular momentum, ages, or mass ratios for the premainsequence binary systems. 52 Table 6 summarizes the evaluation of the eleven binaries. The distribution falls fairly evenly into three groups, those binaries that are likely to be premain sequence, those that are possibly, and those that are un likely to be. Improvements in the model could allow better determination of the premainsequence nature of a binary system. A more realistic model, instead of assuming a polytropic structure would guarantee more accurate struc ture and age determination. It would allow for changing the chemical composition of both stars. The close binary systems under consideration most likely are in synchronous rotation. The periods are approximately a few days. This rotation has been neglected in the calculations, but it surely would slow the gravitational contraction to some extent. Roxburgh (1967) evaluated the effects of rotation and magnetic fields on premainsequence evolution of binary systems. Rotation of an individual star could be uniform or differential. Uniform rotation requires a large vis cosity of the stellar material to couple the core to the envelope. If the star is in convective equilibrium or has a sufficiently large magnetic field (>1 Gauss), then the viscosity of the stellar material should he great enough to llow uniform rotation. In this case, Roxburgh predicts loss of material at the stellar equator. Thus, rotating premainsequence stars in convective equilibrium will fol low a more vertical Havashi Track. o 1 W i I n W L mo CO r Lo 4 4moNN 44N in L* N t ) 11 IN 01 o CO mm ":iT In CN ma OcNiIn r In M "IT IN nr>N "rr q Cl x X X X X x x x in I Ln rI4 N m f, Lnr Lo )o c D r4co o ooocsoroiO td tM : r; 10 to 3 a o * U <; w C r ( trin n > I Hi Ln Ei H D < a > X rl x3 X U ~U ~ tI z En HE H DD Z 2 00 X En cc2 MOH 0 HI PI I Ci (./i o: 55 Stars in radiative equilibrium do not have the "con vective viscosity" mechanism. They are more likely to approach the critical angular momentum valve and fission may result. He concludes that contact binary systems could be formed by fission of a single premainsequence star on the radiative track. For binary systems that are formed before they reach the radiative track, the rotational angular momentum should not reach the critical value. Both components should be in synchronous rotation and most of the angular momentum would be orbital. Bierman and Hall (1976) and Oliver (1978) evaluated the possibility that RS Canum Venaticorum systems could be premainsequence. Their arguments were based on space densities and probabilities assigned by evaluating ages. Because of the large number of these systems, they concluded that they must be postmainsequence. This conclusion is valid as long as the universe is the accepted cosmological age. However, there is no direct observational evidence that these systems must be postmainsequence. The writer would like to apply an appropriate premainsequence model to RS Canum Venaticorum type systems. Bierman and Hall concluded that RS Canum Venaticorum system resulted from the fission of a postmainsequence star. It would be in teresting to model a thermal contraction f hydrogen de pleted secondary. This inay represent a bi.ary immediately after fission. In studies of premainsequence and postmainsequence evolution, the origin of a forming system is assumed. The writer is assuming the origin of the binary system. Clayton (1978) and Larson (1978) and others have presented theories for stellar formation. They admit that unresolved problems remain. These include gas dispersion, viscosities, and turbulences within a collapsing gas cloud. The prob lems are more severe for the formation of a binary star system. Many scientists have assumed the origin when mathe matical modeling beyond a particular point is difficult or impossible. Whitcomb and Morris (1961.) imve proposed special creation beyond the known laws of physical science. Special creation states that God created the universe (Genesis 1:1). He created each object with a specific purpose ( I Corinthians 15:40, 41) There are also celestial bodies and bodies terrestrial: but the glory of the celestial is one, and the glory of the terrestrial is another. There is one glory of the sun, and another glory of the moon, and another glory of the stars: for one star differeth from another star in glory. APPENDIX A PREMAINSEQUENCE BINARY MODEL PROGRAM LISTING 17n "T" c(?,50) 1 o7 "T T ( 2io nr) T (?IT'(( 23 n0T)' "(T ) 21.4 n TU 'T(2) 2r0 fTP' ,(2) 5.4 nr r P., ( r) 27n ~)" C(2) 260 I) T 'I1(2,2) 3r)n nT'T P(2) 31 0 nTI T(2.,95 ) 120 0T'. V(2.25') 330 nT7' 4(?.2T) 14n nTi" 1(2,250) S42 D ,NTIP .I 0f M A' ?" 3.42 T!PUT AP 350 PP TPT'l"i.,. r T M )r T T'T MS T " 3j60 TIPTlJr '1I 39O THPUT "( 1 .(2) 17. D0 OP T r '"f',': ( r PQ.' f rV 40 q r A, 7'3\) s02 DRT.JTI(' Qon P T T ',r) DRACTT)'7T (01 ,02 ?" C4 TTMP!IT 01 ,0T "0n ppT'"TTVIDP Trrr Tr "'~vc= 302 Tr'P'IT PI 304 pD.7fiTln Aq<(: TPA.rygcp ,ATC (,n/VCAD) ?" 30o/ TI'PUT TI 4 ; r)7:1, p (r)nv`irQOA TT)'l c'rT'AqTVT Tq t 405 n rfI n , AIn C01". 00 420 r (1 )=''( (1 .=ooT 01 43)0 S(2)'=i(2 '1 .o c1 +4 440 0( ) =A :! 1 .5 +1 1+fL tn n n o3f ('((1i ) /*(( )) /2.3 r250) 450 O(3)=^ I1 .F+ + ( 1.' 0 n.o "nl r o( ('(3)/'( 1 )) /2.30250o) 455 ?PF IM!TNT TAT pT r rnCrlTIn' ' 460 (i ,1)=(0 )+ 1 47n P(2,1)0()102 rn I) rp7f CAr C '(EAn? "rIT CfIT A 'T T. 51 Y r 75r 3? ") A 0' i i .)'(?"'* ^ v / '7 ? 5 ) Dr?) '. f*2)p)'T V 'T n T TL 'nF Y 57 'r?3  0on rlf)Qntc 5V1)n A *2n cop T=I T) 5 ,4Fn "JPYT I aSrSo C0!0 T1 r" 1 ,QLn crop T=1 T' 9 .62 l=n (T. ') S^4 r)nl=n(T,2) A,5 no cp 'fn~in 477 I TJ? TIT:! 70n so0n r on;i 7A ," 700 'Th;!V9 22)2 7no oqin ,? ) 710n r)qjI 2^q 751 ) niD '=1 Tr) In 730 T r(,T.p) P T'4U 75 740 T ^ A^ P (( 2_( T. ) I/"( I,')) ". n"o l TIrl'l T qon 75n 0r) ll;[ i Q 754 TF 02>(T,.2) T'"~ 70; 756 n( T.2)= ( T. 2)+( T A' ( ( T.2 1n ) 4) r T2. * 75) fl'Trn 770 7,V n( T )=n(T. )(h ( n(T,? 1"2 )) *'* . 77P C( T79)  700 NIMyT v 8on pcr, CmUiTT'l!irU ql n T I,1 )nli 9q ( I. 1 ),)lI qAin irn;[qlq Airn q5 T Tl . ". A q59, T1=T1." 