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IIiiiiiii IiiiI I’iIIII; IiI . .1I ‘II1:i‘ 1 111111 :‘x... m I #333: 31:1”: f HES‘S Misfits: State University This is to certify that the thesis entitled COMPUTER SIMULATIONS OF GRAVITATIONAL ENCOUNTERS BETWEEN PAIRS OF BINARY STAR SYSTEMS presented by James Brian Hoffer has been accepted towards fulfillment of the requirements for Ph.D. degree in Physics Major professor Date W3 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES ”— RETURNING MATERIALS: PIace in book drop to remove this checkout from your record. FINES wil] be charged if book is returned after the date stamped below. s12”? wig: ’1 in '7? 3 II "-2. are «rpm. 5"? a“ COMPUTER SIMULATIONS or GRAVITATIONAL ENCOUNTERS BETWEEN SPAIRS or BINARY STAR SYSTEMS by James Brian Hoffer A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics and Astronomy 1983 IJV'XQE‘IU ABSTRACT COMPUTER SIMULATIONS OF GRAVITATIONAL ENCOUNTERS BETWEEN PAIRS OF BINARY STAR STSTEMS by James B. Hoffer Encounters (collisions) between pairs of binary stars were computer-simulated. The H1,564 collisions were divided into five mass families and all binaries initially had circular orbits. The exchanged energy cross-section for collisions between two binaries composed of identical mass stars was found to be roughly 2-3 times that for a single star colliding with a binary having components with masses equal to that of the single star. Other results cannot be stated so easily, but the energy released by hard binary collisions appears to be significant. A surprising.result is that roughly 40% of the binary- binary collisions in a globular cluster core precipitate a physical collision between two stars, possibly leading to their coalescence. To increase the speed of the integrator, a technique was developed whereby each tightly bound binary is treated as a James Brian Hoffer single star until it is intruded upon by another member of the system. Experiments have shown that this technique can decrease the required integration time by an order of magnitude without affecting the statistics of the collisions appreciably. Each collision was allowed a certain number of integration steps (50,000-100,000) to reach a final, stable configuration consisting of only single stars and binaries. If such a configuration could not be reached .within the prescribed limits, an attempt was made to find the (interme- diate) results and the collision was aborted. These results were not used in computing the statistics of that set of collisions. to my Florina ii ACKNOWLEDGMENTS I would first like to express my thanks to Dr. Jack G. Hills, my advisor throughout this project. He introduced me to the field of theoretical dynamics and suggested this dissertation project. He showed me that one of my favorite topics, orbital. motion, is still an area of active and interesting research. I am thankful for the support of Dr. Robert F. Stein in the form of computer time both for performing a large fraction of these collisions and for producing this disser- tation in the present physical form with the aid of his text processor. I am very appreciative of the support given to me by the Los Alamos National Laboratory through computer time. It was at LANL that I performed virtually all of the hard collisions and many of the soft collisions in this disserta- tion. The Departments of Physics, Astronomy, and Physics and' Astronomy graciously provided support in the form of graduate assistantships throughout my tenure at Michigan State University. I thank the Department of Physics for its support in the form of many hours of computer time also. iii I am appreciative of the support I have received from my family and friends at Michigan State University and elsewhere. Most importantly, for her enduring support throughout, I thank Florina, my fiancee, to whom this dissertation is lovingly dedicated. iv TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . vii LIST OF FIGURES . . .' . . . .' . . . . . . viii CHAPTER 1. INTRODUCTION . . . . . . . . . . . 1 1.1. Binary Systems in Star Clusters . 1 1. 2. Historical . . . . . . . . . . 3 1.3. Research Purpose . . . . . . . 4 2. COMPUTATIONAL TECHNIQUES . . . . . . . 7 2.1. Introduction . . . . . . . 7 2.2. The Regularization Technique . . . . 8 2.3. The Reduction of Close Binaries . . . 14 2. 4. Kepler' 3 Equation . . 17 2.5. Kinetic Energy at Infinite Separation . 21 3. THE PROGRAM . . . . . . . . . . . 24 3.1. Introduction . . . . . . 24 3.2. Organization of Collisions . . . . 24 3.3. Initializing a Collisions . . . . . 25 3.4. The Integration Routine . .. . . . 28 3.5. The Beginning and Ending of a Collision . . . . . . . . 3O 4. ANALYSIS OF RESULTS . . . . . . . . . 34 4.1. Introduction . . . . . . . . 34 4.2. The Exchange of Energy . . . . . . 35 4.2.1. Introduction . 35 4.2.2. The Exchange of Energy at Zero Impact Parameter . . . . 36 4.2.3. The Exchange of Energy Versus Impact Parameter . . . . . 45 2.4. The Exchanged Energy Cross- Section . . . . . . 48 4.3. The Final Configuration . . . . 52 4.4. The Average Eccentricity of the Surviving Binaries . . . . . . . 59 V 4.5. The Distance of Closest Approach CONCLUSIONS 5.1. Comparison of Present with Previous Results . 5.2. Future Investig Collisions LIST OF REFERENCES APPENDICES A. B. THE DATA - GLOSSARY OF TERMS vi ations of Binary-Binary 61 65 65 68 71 73 86 LIST OF TABLES TABLE Page 1. The results of least squares fits of the data to equation (4.6). . .~ . . . . . . . . 44 2. A summary of least squares fits of the data to equation (4.12). . . . . . . . . . . 47 3. A summary of least squares fits of the data in families A and B to equation (4.13). . . . . 48 4. Conversion factors from the present experiment to HF's experiment. . . . . . . . . . 66 5. A summary of the results of the collisions. . 74 vii FIGURE 1. 1o. 11. 12. 13. 111. 15. 16. LIST OF FIGURES A typical run-summary for a group of collisions. . . Two samples of Plots of family A. Plots of family B. Plots of family C. Plots of family D. Plots of family E. Plots of equation and 3. Probability 5 and g and E and g and 5 and collision results. .log(l€a1/21) versus logo log(lgc1/2I) versus loga 108(lfia1/21) versus logo log(l£a1/2|) versus loge log(l§a1/2l) versus loga log(|oEl) versus logu using (4.19) and the data from Tables 1 Probability of Probability Probability of no change versus loga. exchange versus logo. of dissociation versus logo. Average eccentricity versus logo. for for for for for of a single binary versus loga. Logarithm of the average distance of closest approach versus loga. The orientational orbital elements. The orbital plane. viii Page 26 27 37 38 39 4O 41 51 53 55 56 58 6O 62 88 89 CHAPTER 1 INTRODUCTION 1.1. Binary Systems in Star Clusters In 1972 Aarseth and Hills performed a computer experiment attempting to rectify the disagreement between theory and observation regarding open clusters. ‘Star- formation theory predicts that, if all the stars in a cluster are formed from the same gas cloud, there should be some clumpiness in their spatial distribution. This occurs because, as the cloud contracts under the action of its self-gravity, its density increases causing the gravita- tional Jeans length to decrease. The cloud then breaks up into subclouds. Each subcloud then contracts to the point where it breaks up into sub-subclouds. This division‘ process continues until a typical subcloud has dimensions appropriate for star-formation to occur. Thus the initial cloud forms a hierarchy of subclouds. According to this model, we should expect the density of a cluster to be non- uniform; a certain clumpiness in the spatial distribution of the stars should be observed. This clumpiness is not observed in mature open clusters, however, it is found in molecular-hydrogen proto-clusters observed by radio telescope (Larson 1981). 2 The Aarseth-Hills computer model began with a cluster of stars having a clumpy appearance. The stars were arranged so that a hierarchy of subclustering was present. The cluster was then allowed to develop according to the law of classical gravitation. :n: a single collapse time, it evolved a fairly homogeneous form, but almost more interesting was that the cluster was forming binary stars by three-star encounters. By the termination of the experiment at 4.2 collapse times, the cluster was composed of 6.2% binaries. After subtraction of the number of stars that had escaped from the cluster, this becomes 10.8%. While this is not an incredibly large percentage, these binaries had acquired more than 90% of the total binding energy of the cluster. Clearly the dynamical evolution of this small percentage of binary stars as mediated through collisions becomes important in determining the dynamical evolution of a cluster. As the number of binaries increases, the frequency of collisions involving them also increases. Another demonstration of the importance of binary stars in the dynamical evolution of a star system was given by Spitzer and Mathieu (1980) when they modeled the dynamics of some globular clusters.‘ In their models, they attempted to account for the effects of collisions between single and binary stars as well as between two binaries. The single- binary collisions were fairly well understood at the time, but binary-binary collisions were not. Their treatment of binary-binary collisions as successive single-binary colliSions seems less than adequate, but better than ignoring them altogether. Their globular cluster models initially contained 50% and 20% of the total mass in binaries. After 1600 Trh (Trh is the relaxation time of the stars in a sphere about the center of mass of the cluster and enclosing half of the mass), the central region of each cluster contained 90% and 80% of its mass in binaries. At such high concentrations of binaries, interactions involving them become extremely important. Clearly these interactions (collisions) must be understood if a correct model of the dynamical evolution of the core of a globular cluster is to be obtained. 1.2. Historical After the Aarseth-Hills investigation of the dynamical evolution of an open cluster, investigations were begun with the goal of obtaining an understanding of collisions involving binary systems. Probably because they are the simplest as well as the most common at low binary densities, collisions between single stars and binaries were investi- gated firstq This investigation was launched from 'two fronts. Heggie attacked the problem from a purely theoretical direction. His analytical treatment of the statistics of these collisions (Heggie 1975) is quite complete and gives a formalism into which experimental results can be cast. To verify the accuracy of several equations, he performed a 4 rather incomplete set of collisions on a computer. Hills took the computer-experimental approach (Hills 1975). With Fullerton (Hills and Fullerton 1980; Fullerton and Hills 1982), he has completed, analyzed, and published the results of some 65,096 computer collisions between single stars and binaries. While this treatment is complete as far as it goes, nearly all of these collisionSwere performed with the initial eccentricity of the binary being zero. Whether these statistics are representative of elliptical orbit statistics remains to be seen. Valtonen also has performed simulations of interactions between single and binary stars (Valtonen 1975). However, his interest was the decay of quasi-stable three-body systems and is not directly applicable here. All of the above work concerns the interaction of single stars with binaries. Presently only one experiment has been performed involving two binaries. Saslaw, Valtonen, and Aarseth (1974) have performed 200 simulations of the decay of quasi-Stable two-binary systems. No collisions between two binary systems have been performed. 1.3 Research Purpose The purpose of this investigation is to examine collisions between two binary star systems. These collisions are assumed to be completely Newtonian-gravita- tional in nature as well as independent of the structure and evolution of the stellar components. The gravitational 5 interaction among four point-masses is our only concern. 41,564 of these collisions were performed with the aid of several computers. The salient features of the computer program as 'well as the reduction of the data will be presented in this dissertation. The collisions are divided into five families, A-E, according to the masses of the components. Only the relative masses will be given since the equations of motion for the system are linear in the masses and, hence, can experience a mass scale change resulting only in a change of scale of the physical time. The families and their associated masses are: A (1-1)-(1-1) B (3-3)-(1-1) C (1-3)-(1-3) D (10-10)-(1-1) E (1-10)-(1-10). The notation (a-b)-(c-d) signifies that initially 21 star with mass a is part of a binary with another star of with mass b. This is similarly true for stars 0 and d. The two binaries are then caused to collide. Each family contains groups of collisions with usually 200 collisions in each group. Each group is specified by: the initial ratio of the kinetic energy of the binaries at infinite separation to the energy required for complete dissociation of the system (a), the ratio of the binding energies of the binaries (B), the 6 impact parameter (p) in units of the initial separation of binary (a-b), and the initial eccentricities of the binaries which are zero for all cases considered. The remaining quantities are randomly sampled by a Monte Carlo technique and will be discussed later in this dissertation. CHAPTER 2 COMPUTATIONAL TECHNIQUES 2.1. Introduction It is well known that the equations of motion for the gravitational three-body problem (TBP) have no analytic solution. Any analytic approximations break down nearly completely during very close approaches among the members of such a system during which the equations of motion become mathematically poorly-behaved. The gravitational four-body problem (FBP) suffers from the same difficulties as the TBP, but they are even more severe because of the increase in the likelihood of very close approaches. However, this difficulty can be significantly reduced through the process of regularization (simplification of the equations of motion by reparameterization) to be described in some detail later. Because the FBP is not analytically solvable, solutions to its equations of motion must be found by some approximation technique. In the present investigation, the solutions will be found numerically with the aid of a computer. In addition to integration and roundoff error, the length of time the computer requires to find the final, stable confi- guration of the entire system must be considered. In particular, the formation of quasi-stable configurations is 7 8 of interest. A common occurrence at low collision energies is the formation of a tightly bound binary as a component of a loosely bound binary, i.e. a quasi-stable trinary. While this configuration is not generally mathematically stable, an inordinately large quantity of computer time is usually needed to test its long-term stability unless some special technique is used to increase the integration speed. The two problems described above constitute the major problems encountered when integrating the equations of motion for the FBP. The techniques developed to reduce these difficulties introduced two fairly minor, additional problems. These techniques required that Kepler's Equation be solved more than 105 times. The large number of solutions needed requires that the method devised to find the solution be extremely reliable, sacrificing speed if need be. The other additional difficulty occurs at the end of a collision where the kinetic energies of all unbound bodies must be found as the separations become infinite. Since the FBP is not analytically solvable, approximations to the problem must be made if these quantities are to be found while the separations are still finite. 2.2. The Regularization Technique In integrating the equations of motion for the FBP, one finds that the separations between the masses can vary by as much as several orders of magnitude. In order to maintain accuracy, one would like to decrease the increment of the 9 independent variable (time) when the group is compact while increasing it when the group is dispersed. This can be accomplished either explicitly or by the more elegant technique of regularization which transforms the time coordinate so as to remove the singularity produced when two objects make an exceptionally close approach to each other. Such close approaches may cause a large error in the total energy of the system because of the large velocities involved and because of the finite precision afforded by computers. The regularization method employed for this calculation is the technique of multi-particle, quasi-regularization in time developed by Heggie (1972). The equations of motion are regularized by a replacement of the physical time with a regularized time. These are related by dT=h(xij, 1213-) dt (2.1) where dt is the increment of the physical time, dT is the increment of the regularized time, xij is the j—th (cartesian) component of the i-th mass, and h is called the regularizing function. The dot indicates the total physical time derivative of the quantity under it. The equations of motion can be written very simply as iij = aij (2.2) where aij is the j-th (cartesian) component of the accelera- tion (force per unit mass) of the i-th mass. Now, by the 10 chain rule of differential calculus, we have the operational relation d/dt : dT/dt d/dT (2.3) or, after applying equation (2.1), we can write d/dt = h(xij, iij) d/dT.. (2.4) If a primed quantity denotes the total regularized time derivative of the quantity, then we have from equation (2.4), iij = h(xij, iij)xij" (2.5) Total differentiation of equation (2.5) with respect to the physical time t gives iij = 5(xij, iij)xij' + h(xij, iij)iij'° (2.6) After applying equation (2.4) to equation (2.6), we obtain iij = h(xij, iij)h'(xij; iij)xij' + h2(xij, iij)xij"° (2.7) By solving equation (2.7) for Xij": we obtain a new set of differential equations in xij: xij" = h’2(xij, iij)aij - h'(xij, iij) xij'/h(xij, iij) (2.8) where aij has been written instead of in according to 11 equation (2.2). Even though equation (2.8) provides a recipe for finding xij, it is more convenient to rewrite it utilizing a regu- larizing function that contains the dynamical quantities and their regularized time derivatives rather than keeping their physical time derivatives. We define a new regularizing function g(xij, xij') such that g(xij. xij') = h(xij, iij). (2.9) This gives a new, consistent set of equations of motion. x.."- -2(x.. x..')a.. 13 - 8 13' 13 13 - g'(Xij, Xij')xij'/g(xij, Xij') (2.10) An appropriate g(xij, xij') will now be chosen so that equation (2.10) is well behaved as the separations of the component masses vanish. If equation (2.10) is to be convergent as the separations vanish, then the conditions lim lg'2(xij, xij')aijl < a (2.11) R+0 and lim Ig'(xij, Xij')xij'/g(xij, xij')| < a (2.12) R+0 must be satisfied. R is the minimum of the six separations of each body from each other body. Relation (2.11) can be 12 satisfied if g(xij, xij') diverges at leaSt as fast as I? vanishes since the dominating term in aij is proportional to 1/R2. A function satisfying this condition is the total potential energy of the system. For the FBP this is 4 i-1 U = - 2 2 Gmimk/rik (2.13) i=2 k=1 - where rik is the separation of the masses m1 and mk. To find whether relation (2.12) is satisfied, we must differentiate equation (2.13) with respect to the regularized time. s'(x1j. Xij') = dU/dT = - g. :1E1(Gmimk/rik3)hikoFik'. (2.14) i=2 k:1 We now see that the left side of relation (2.12) goes as x'2/R for very close approaches which, by equation (2.5), we can rewrite as isz. Conservation of energy tells us that this quantity converges as R vanishes (it varies linearly with R for the two-body case with zero angular momentum). The use of the potential energy for the regularizing function satisfies the necessary conditions and works well in practice, so we will use it. 13 In summary, the set of equations we wish to solve is Xii" = 8'2(Xij’ Xii'>aij - 8'(Xij, xij')xij'/g(xij, Xij') (2.15) with ' 4 1-1 » 8(xij1x ij) = U = - I Z Gmimk/rik (2.16) i=2 k:1 and u , 2 + aij : - kz1 Gmk/rik“ rik‘ (2.17) The primed summation indicates that the sum is over all k except for k=i. Notice that this set of equations contains only the dynamical variables x and x'; the physical time has been eliminated. When integrating the equations of motion, we need not concern ourselves with the physical time. Should it be required, we can compute it by integrating equation (2.1) in the form dt = dT/g(xij, Xij') (2.18) giving At : fT1 dT/S(Xij, xij') (2.19) T0 as the physical time interval corresponding to the regularized time interval T1-T0. 14 2.3. The Reduction of Close Binaries When a is less than three, the formation of quasi-stable configurations such as trinaries and two mutually bound binaries becomes common. Each of these may persist for a very long time until it evolves into a stable configuration composed of a combination of single stars and binaries. To aid the computer in solving these cases in a reasonable and affordable amount of time, we can often treat each tightly bound binary system as a single star. Consider a binary in the presence of a single star. The binary has components with masses ma and rub and orbital semi-major axis a. The single star has mass mi and is a distance R1 from the center of mass of the binary. The grav- itational force between the two members of the binary is Fb = Gmamb/aZ, ' (2.20) with the stars treated as point masses. The tidal force on the binary due to the field star is Ft, = 26(ma+mb)mia/Ri3. (2.21) It tends to dissociate the binary. We can define a measure of the total tidal force due to this and all other field stars in the system to be (2.22) 15 We will call the ratio of these two forces Q: Q : Fb/Ft‘ (2.23) When 0 is large, the disruptive efficiency of the field stars is small. When 0 equals or exceeds some value (105 was used in practice), the perturbative effect of the field stars is considered negligible and the binary is treated as a single star until 0 becomes small enough so that perturba- tive effects again become important. This replacement effectively eliminates a Ilarge term from the potential energy (the regularizing function), which increases the regularized time step and decreases the real computer time required to integrate the equations of motion. The process of replacing a tightly bound binary by a single star will be called "reduction" while the inverse process will be "resolution." The reduction of a binary is accomplished by a standard procedure: the coordinates of the binary components are first transformed into the binary center-of—mass coordinate system so the orbital elements can be found. The five classical orbital elements, the semi- major axis, the eccentricity, the inclination, the longitude of the ascending node, and the argument of periastron passage are then found. A sixth quantity, the true anomaly, giving the phase of the orbit relative to periastron (closest approach in orbit), is also calculated. (For definitions of these quantities, see Appendix B.) After saving these six quantities along with the physical time, we 16 replace the binary by a single star with a mass equal to the sum of the masses of the binary components and coordinates (both position and velocity) described by the motion of the binary center-of-mass. The integration routine then continues with one fewer star and the absence of a large, real time-consuming term in the regularizing function._ When a reduced binary system is perturbed strongly by another star or by a group of stars, it is resolved into its original components through a knowledge of its five orbital elements, the time, the true anomaly when reduction occurred, and the new time. Kepler's Equation is used to relate these last three quantities to the new true anomaly. It is worthwhile to note the magnitude of the contribu- tion to the random energy error intruduced by the reduction- resolution process. If a binary having components with masses ma and mb and semi-major axis a is perturbed by a star with mass m separated from the binary center of mass by a distance R, the random energy error introduced is of order (Hoffer 1982) (SE/E = 0.25 (a/R)2 = [mambnsmmambmfl/B, (2.21:) If ma=mb=m and 0:105, 6E/E=4.6x10'5. The maximum allowed relative energy error is 0T01 and this is well within that limit» In practice, the energy error was computed by actually resolving any reduced binaries and computing the energy associated with their components. The median relative energy error was approximately 10'”. This is 17 consistent with the median error obtained in previous similar experiments by Hills (1975). 2.4. Kepler's Equation When a reduced binary is resolved, the new true anomaly must be related to the new time, the old true anomaly, and the old time. The recipe to do this involves Kepler's Equation (Marion Chapter 8): 2n(t-to)/T : w — W0 - esinp + esinpo, (2.25) where t is the new physical time, to is the old physical time, T is the orbital period,'1p is the new eccentric anomaly, 1110 is the old eccentric anomaly, and e is the eccentricity of the orbital ellipse. Once p is obtained, the true anomaly a can be found through the relation tan(6/2) = [(1+e)/(1-e)11/2 tan(w/2). (2.26) (e and w are defined in Appendix B.) As long as e<1, no problems arise with equation (2.26). Notice that if the tangent function is defined to exist on the half-open interval [-n/2, n/2) radians, a unique 6 exists for each w in the interval [-n, n). Kepler's Equation must be solved numerically since all attempts at analytic solution have failed (Moulton p. 162f). Since we must extract a solution several thousand times in this experiment, the most important feature of the method developed must be its consistency. We would rather use a 18 fairly slow method which always converges on the solution than a fast one that works only 99% of the time. For the method to be usable, it should not fail more often than once in 10“l attempts. A higher failure rate than this would cause an unacceptable program failure rate. In this experiment, a hybrid method was used. This hybrid consists of two parts: the first part is used to find an approximate solution and the second part uses the approximate solution to find a solution to within an accuracy of one part in 10'10. The first part of the hybrid method is the technique of successive substitution. To utilize it, we must arrange Kepler's Equation so it has the form w = f(w). (2.27) We then substitute an approximate solution into the right side of equation (2.27) and compute a better approximation to the solution. We can write this process in iterative notation as vi+1 = fv MZHH n VHHUHmHzmoom I NHHUHMHZMUUM floumoqnumw.~ ~01mc~¢¢~o.¢ no NH 50 NA so mg no M“ nonmmmmmcu.m acumoq~¢NN.o fianmcumoam.¢ n zHZm u romWZN UZHGZHQ angummm.m m¢smom¢.ml mmm-a0.N mmmqum.¢1 mmmmmwm.~l ueomamo.¢l mmo¢oow.~ m~ow¢~o.~ I szm we Cg co - mo am we Nu calmmcfimum.~ ~o1um~a~o~.~ Noumommwwm.¢ golfinmmdmo.~ mnwowmm.~l mmNnoaq.ml mm~auca.n manaomw.al mmeueom.m Mmumm~¢.m mmcmofim.~l mwsauau.~l malnumqmm~.~ u wwmmzm UZHQZHQ I wommzm qunsz Noumwummmm.H denuncmmge.a .m mesmflm cc NH no Nfi co NH co m~ moomeum.o mfimc¢N¢.NI m-~¢mo.~n mmewmmo.ul mfifimmufi.o1 mmeoamm.~1 umownom.m m~om¢w¢.~ momma rummzm wummzm UHHNZHM m wummzm UHHMZHM N momma Mommzm mmmhm omum HAQZHm mgqum wcmmzm UHHmzHM ¢ a rmM<2HMH rummzm UHHWZHM a muqum szmom >m105; 3. Each subsystem is energetically unbound to all possible combinations of the remaining subsystems; 32 4. Each subsystem is moving away from all other subsystems. Determining whether a given star is a member of a binary is similar to deciding whether a binary is to be reduced to a single star. If the star has a negative energy with respect to another star, then the two stars form a binary. The binary is considered far enough from all other subsystems if Q for that binary exceeds 105. When all stars are single or are members of relatively isolated binaries, we have completed tests 1 and 2. We now move on to test 3. Testing the energy of a subsystem relative to all other subsystems is very simple. Consider the subsystem to be tested. Compute the total energy of it with respect to all possible combinations of the remaining systems. If all energies are positive, the system passes test 3. This leaves us with test 4. This test is also quite simple to carry out. Again consider the .subsystem to be tested. For all possible combinations of the remaining subsystems, compute the coordinates of the center of mass. If the two subsystems (the subsystem to be tested and the subsystem composed of combinations of the remaining subsystems) are moving apart, the inner product of the relative velocity and the relative position is positive and the subsystem to be tested passes test 4. When each subsystem passes each of the above tests, the collision is over and the various energies and other parameters are tabulated. It is difficult to ascertain 33 whether these tests will be passed only by stable systems. I suspect not, but I have not imagined a case these tests do not cover. CHAPTER 4 ANALYSIS OF RESULTS 4.1. Introduction In performing this computer experiment, we have simulated 41,564 collisions between pairs of binary stars, thereby producing prodigious amounts of data. Extracting a considerable portion of the possible generalizations and conclusions from this data could require years. We cannot do this here. Instead, we will reduce the data to a fairly concise form, a series of plots of the quantities given in Appendix A, and then draw some very general conclusions regarding this experiment. The set of quantities we will consider in this analysis will be divided into three categories: the exchange of energy; the final configuration (which stars are single and which are components of a binary) of the system; and other, miscellaneous quantities such as the average eccentricity of the surviving binaries and the average distance of closest approach. These quantities in IN) way exhaust the set of those that might be tabulated, however, if we understand the relationship of these quantities to the independent parameters, we probably have a good understanding of the dynamics of collisions between two binary systems. 34 35 4.2; The Exchange of Energy 4.2.1. Introduction When a binary star becomes more tightly bound because of a collision with another star or with a group of stars, it gives up its orbital energy to the other stars. The energy released or absorbed through a collision may be the most important quantity obtainable from this experiment since the dynamical structure of‘ a group of stars is extremely dependent upon the available kinetic energy. Since each of these collisions conserves energy, the kinetic energy added to the stellar system in which the two colliding binaries are imbedded divided by the initial total binding energy of the binaries, E, (hereafter relative energy exchanged) is equal to the relative binding energy increase of the surviving binaries as the separations of the unbound products become infinite. The total energy may be decomposed into two terms, Eext and Eint: the total macroscopic energy and the total internal energy of the binaries, respectively. From conservation of energy, the total energy before the collision equals the total energy after collision E : Eextf + Eintf. (4.2) 36 If we subtract equation (4.1) from equation (4.2) and divide by the initial total binding energy of the binaries, BEi, we obtain (Eextf‘Eexti)/BEi + (Eintf-Einti)/BEi = 0 (”-3) OI“ g = AEext/BEi = -AEint/BE1 . (4.4) where AEext = Eextr-Eexti and Mimi = Eintf'Einti° The binding energy is the energy required to dissociate 'both binaries. We can therefore write 5 = AEext/BEI : ABE/8E1 (4.5) where ABE : BEf-BEi. By examining the effect of a collision on the total binding energy of the four masses, we can find the kinetic energy released throUgh the collision. 4.2.2. The Exchange of Energy at Zero Impact Parameter In the upper graph in each of Figures 3-7, we show a plot of the exchanged energy (6) versus the common logarithm of a (loga) for one of the five families of collisions. (As a reminder, a is the initial kinetic energy of the binaries expressed in units of the minimum energy required to dissociate the binaries.) The impact parameter is zero for all cases. Each cross represents a data point and the vertical extent of the cross is the error associated with that datum. The curve drawn through the points as well as 37 logo 1/2 Figure 3. Plots of g and log(|ga family A. I) versus logo for 38 logo 1/2 'Figure 4. Plots of g and log(|§a family B. I) versus loga for 39 10g(|€al/2l)' logo 1/2 Figure 5. Plots of g and log(|ga |) versus logo for family 0. 4O logo 1/2 Figure 6. Plots of g and log(|§a family D. l) versus loga for 41 .3 1.0 - I N— 0.0 . \ 1-1 :5 w 210 O H -1.0 -2.0 1 ‘ -2 0 -1 0 0.0‘ 1 0 2 0 3 0 loge Figure 7. Plots of g and log(|gal/2|) versus logo for family E. 42 each lower graph will be explained below. Even though these plots are quite complicated, we can make some general statements about them. Each of the plots contains two regions, the region where g>0 (small a) and the region where §<0 (large a). We will call collisions with C>0 "hard" collisions and those with g<0, "soft." The point separating these two regions will be labeled (c0, 0) since 5:0 at this point. do is usually near unity. In the extremely large-a region (a>30), we might expect E to vary roughly linearly with the time the binaries are close enough to interact with each other. If this is true, then gal/2 should be roughly constant in this region. Taking this as a clue to possible later simplifications, we plot log(|€a1/2|) versus loga for each family. This results in the lower plots in Figures 3-7. As expected, gal/2 is approximately constant for a>30 for each of the plots with the remainder being surprisingly well-behaved. As a approaches zero, we expect 5 to approach some finite, non-zero value (Hills 1975) requiring that 5.11/2 vanish. In the region 00 causing €a1/2>0. Any function which is to pmedict gal/2 must increase from zero at a=0, peak, then decrease to negative values as a passes do. It must then become constant and negative as a passes 30. A function possessing these qualities is gel/2 = A11-exp<-ee)1[(eo/e)'/2-11 (4.6) 43 as we will now verify. AS a approaches zero, we have 5011/2 = Ab(01010)1/2 (4.7) so that E = AbaO1/2, (4.8) a cOnstant as expected. For extremely large a, we write Eel/2 = -A (4.9) or g = -A/e1/2, (4.10) again, as expected. Of course, if a:a0, then 5:0. By choosing the parameters appropriately, equation (4.6) can be made to fit the boundary conditions given. We have accomplished this fit by using the method of least squares. Since equation (4.6) can not be made linear in the parameters, we used a grid search of parameter space'to minimize XVZ as given by Bevington (Chapter 11). The results of these fits are given in Table 1. The curves in Figures 3-7 are those generated by graphing equation (4.6) with the parameters appropriate for each family as given in Table 1. Although “2 is quite large for most of the families, the curves do appear to roughly fit the data. In performing the least squares fits of the data to various curves, we minimized Xu2 in the form 44 Table 1. The results of least squares fits of the data to equation (4.6). Family A b do X02 A 3.048 0.286 0.959 3.34 B 1.083 1.338 0.461 0.86 c 4.676 0.137 0.928 8.41 D 0.170 329.524 0.156 1.99 E 14.381 0.009 0.030 31.31 N Xv2 e 1/(N-n) .21 [yi-y(xi)]2/012, (4.11) 1: where N is the number of data points, n is the number of parameters in the fitting function y(x), Yi is the i-th dependent data point, and xi is the corresponding independent data point. °i is the standard error of Y1 since each 3’1 is a mean obtained from roughly 200 data points. When defined in this manner, Xv is approximately the root-mean-square of the standard scores of the data. We consider an excellent fit to occur when sz‘l for the fit. This form of X02 is used throughout this dissertation. 45 4.2.3. The Exchange of Energy Versus Impact Parameter The relationship between the exchanged energy and the impact parameter appears to be fairly simple. The exchanged energy (E) is roughly gaussian in the impact parameter (p): a = 50 epr-(p/po>21, ' (4.12) where 50 is the energy exchanged at zero impact parameter and p0 is the width of the distribution. While the data do not follow this relationship exactly, they do follow closely enough for certain general features to be found. In allfairness, we must criticize the present use of equation (4.12). Examination of the data reveals that a gaussian is probably not representative of the relationship between E and p; the data have certain anomalies which preclude the validity of equation (4.12). For example, it is common to find a depression in the lgl versus p curve about p=0. This depression occurs most frequently with family A. Usually a curve with no depression can be fit to the data within the given error limits leaving one to wonder whether the depression is simply a statistical fluctuation. To reduce this uncertainty, 3 significant increase in the number of collisions contributing to these points must be obtained. The high cost of performing these collisions which are at small a prevents us from resolving this uncertainty at this time. Another difficulty with using a gaussian is that, for some values of a, the exchanged energy 46 often changes Sign as p becomes large. Even though they will not predict either of the above behaviors, we will use fits to a gaussian to provide us, in a systematic manner, information regarding the width and the depth of the distribution for the energy exchanged. Since the gaussian has only these two parameters, it is ideally suited to give us this information. Other reasonable curves. such as the sum of two gaussians have too many parameters for them to be fit to data consisting of only a few points. Preliminary investigations indicate that, when more data is available, the sum of two gaussians will be the curve used to fit the data at low kinetic energy. We performed least squares fits of equation (4.12) to ‘data of a given family with identical binding energy ratios (8) and identical kinetic energies (a). Families A and B (with 8:0.111) are the only ones with extensive data for non-zero impact parameters so they are the only ones for which these fits were performed. The results are summarized in Table 2. As expected, X02 is quite large for several of the fits, however, enough give values which are less than or roughly equal to unity to indicate that the gaussian function can provide a reasonable fit to the data. Since 50 is the exchanged energy at zero impact parameter, it should be calculable from equation (4.6) and the data of Table 2. Coupled with a relation giving p0) these allow us to find an expression for the cross-section for energy exchange at any a for a given family. 47 Table 2. A summary of least squares fits of the data to equation (4.12). a 50 P0 oE x92 °Eg FAMILY A: (1-1)-(1-1) 1.01 .300+-.035 1.881+-.076 -1.062+-.210 22.494 -1.462 3.00 -.615+-.018 1.599+-.044 -1.572+-.133 4.482 -1.586 5.00 .513+L.019 .1.397+-.047 —1 001+-.103 .902 -0.897 10.00 .554+-.015 1.164+-.039 -.750+-.071 1.168 -0.689 20.00 .527+-.017 .822+-.040 -.356+-.046 2.330 -0.359 30.00 .454+-.012 .802+-.031 -.292+-.030 .701 —0.337 100.00 .285+-.011 .651+-.032 -.121+-.017 2.191 -0.142 300.00 .139+-.007 .667+-.027 -.062+-.008 .617 -0.056 1000.00 .065+-.006 .652+-.030 -.028+-.005 .356 -0.028 FAMILY B: (3-3)-(1-1) 3.00 .337+-.028 1.384+-.065 -.645+-.113 3.890 -0.503 5.00 .260+-.027 1.272+-.067 -.421+-.088 2.829 -0.338 10.00 .212+-.026 1.037+-.097 -.228+-.070 -1.842 -0.191 20.00 .195+-.020 .985+-.077 -.189+-.049 2.820 -O.163 50.00 .115+-.016 .696+-.081 -.056+-.021 .613 -0.076 100.00 .101+-.007 .726+-.038 -.053+-.009 .758 -0.055 300.00 .035+-.005 .757+-.057 -.020+-.006 .865 -0.020 1000.00 .026+-.003 .582+-.047 -.009+-.003 .484 -0.008 48 We have found that p0 depends on a roughly as P0 = P» + k/log(a+1). (4.13) p, is the width of the E-p curve as a+w. Least squares fits of the data for families A and B give the parameters shown in Table 3. A summary of least squares fits of the data in families A and B to equation (4.13). Family pm k xvz A 0.5679 0.4584 0.035 B 0.4411 0.6033 0.006 Table 3. The Xv2 given in Table 3 were calculated with the errors in 01, 00,, set equal to unity since a is an input parameter and is taken to be exact. The fits appear to be quite good. 4.2.4. The-Exchanged Energy Cross-Section As a measure of the total effectiveness in releasing energy of a set of collisions from the same family with the same a and 8, we will define the exchanged energy cross- section: 49 total energy released (4 1”) energy incident per area‘ ' Of course, we would like to find a form for equation (4.14) that we can use for computational purposes. Consider- a single binary acted upon by a beam of binaries which interact with only the target binary. The total binding energy of each binary in the beam and the target binary is EL The kinetic energy added to the two binary system by the collision is AE and the beam has a cross-sectional surface number density of binaries, n. With these definitions, we can write 25 = 2wnf: AEbdb/nE (4.15) where b is the impact parameter. Since the units of b have not yet been specified, let us define a unitless impact parameter pzb/a (the impact parameter actually used in the experiment) and a corresponding unitless cross-section OE=ZE/fl82. We can then write OE : ZE/na2 : 2f: gpdp. (4.16) We can obtain an analytical eXpression for the cross- section by substituting for E from equation (4.12): CE = 280]: epr-(p/p0)2l pdp (4.17) which integrates to CE = $0002. (4.18) 50‘ In practice, we have computed the cross-section by two different techniques. We used equation (4.18) with the data in the third and fourth columns of Table 2 as well as plotting the data points (p, 5p) manually on graph paper, drawing a smooth curve through the points, and integrating by counting the squares under the cnuwe. Cross-sections computed by both of these techniques (9E and 058, respec- tively) are given in Table 2. We have now developed all the tools necessary to give an expression for the cross-section for any a for families A and B. By combining equations (4.6), (4.13), and (4.18), we can write 0E = A[1-exp(-ba)][(aO/a)1/2-1] xtp, + k/1og(a+1)]2/a1/2. (4.19) We can use this equation with the data in Tables 1 and 3 to obtain graphs of GE .versus a for families A and B. Since p, and k are similar for families A and B, we will assume they do not vary significantly for the other families and will use the average of the p... for families A and B for these families. With this assumption, we can include graphs of CE for families C, D, and E. All of the graphs are shown in Figure 8. 51 o.m meme on» use AmH.:v :OfiumSUo wean: emoH mammo> A_ o_vmoa ho muoam 0.m .m 0mm H mmHQmB EOLM dwoa 0.H 0.0 0.HI 0.NI I I r r A d l .L 0.m 0.: o.m .0 opswfim 52 4.3. The Final Configuration When a collision has been terminated, we would like to find quantities such as the relevant kinetic and binding energies and the eccentricities of the surviving binaries. Before we can find these quantities, we must ascertain which stars are binary components and which are single stars; we' must find the final configuraration of the system. By tabulating the final configurations of a group of collisions, we can find the probability of occurrence of processes such as an exchange collision, complete dissocia- tion of the system, and no change in the system. We will examine, in turn, each of the relevant quantities of the table in Appendix A. First we will examine the probability of no change (PNC) occurring. In this case the surviving binaries are the same as the initial binaries, only the kinetic energies and the orbital elements change. Plots of PNC versus loge are given in Figure 9. The plot for each family hovers near zero when loga<1. In this region, virtually all "memory" of the initial configuration is lost. Then, as a increases past unity, the plot for each family, except family E, increases, reaching at least 0.845 when loga=3. Beyond this we have not investigated. Another way of forming two surviving binaries is through an exchange collision whereby the two surviving binaries are not the initial ones, but are composed of a (different 53 PNC 1 ‘ .' I ' Family A I . O 8 Q . . c... l Family B O O O ‘F Family C O O 0 Q c c . Family D I) O 0 . . . o (o o l —a._i 0 Family E O o o - o c -2 -l 0 l 2 loge Figure 9. Probability of no change versus logo. 54 combination of stars. IR”: binary-binary colliSions, this implies that an exchange collision has taken place. When the masses of the stars are identical, the exchange collision is not important, but when each star has a distinct mass, it becomes interesting and potentially very important (Hills 1977). Plots of the probability of exchange (PE) versus logs for the five families are given in Figure 10. As expected, there is a much higher incidence of exchange collision for families with binaries having a large difference in their component masses (families C and E) than for families with binaries having identical mass components. However, the explanation of this effect is not the usual one. One might expect exchange collisions for families C and E to result in a binary containing the two most massive stars. This does not occur because in the region where exchange collisions become important, a is too large for the system to reach any degree of (dynamical equilibrium as required for the usual argument giving the ejection of the lightest stars. In fact, most of the exchange collisions (75% at 6:1.01 increasing monotonically with a for family C and 100% for all a for family E) were simply an exchange of the lighter components. This is not really surprising since it is much easier to change the course of’a less massive star than of'a more massive star. Quite often only a single binary will survive the 'collision. Figure 11 summarizes the probability that this occurs (PSB) for each of the families of collisions. These PE 1 Figure 10. 55 Family A P Family B . * 00-... C Family C t I» 0 ' - - -J Family D Family E t 1 1 l- O _ - -2 -l 0 l 2 3 loge Probability of exchange versus loga. PSB I Family A i_tf 1’ A H*+ 1* ‘P 0 Family B _‘_7 *It 0 Family C l 3 1 '1, 0 Family D . o_ 4—t_. y,, ‘Ttd q +1.. , ' 1' 0 Family E : -2 -l 0 l 2 3 logo Figure 11. Probability of a single binary versus loga. 57 plots are somewhat erratic making it difficult to draw many general conclusions from them, tun; we can describe their general forms. The plots for families C and E reach a local nunimum I“) the range a=10-100 after which they peak and then, at least for family C, asymptotically decrease to zero. Presumably the plot for family E would also exhibit this behavior were the_data extended to larger 8. The plots for families B and D are at a plateau until a exceeds approx- imately 50 where they both asymptotically approach zero. Family A, not surprisingly, appears to be a hybrid of the C- E and the B-D cases. The low-a plateau gives way to a lower plateau when a exceeds three. This plateau exists until a exceeds 50 where the plot asymptotically approaches zero. We will now examine the last possible final configura- tion of the system, that of complete dissociation (PD) or the formation of four single stars. The plots in Figure 12 summarize the dependence of PD on loga for the five families of collisions. -A feature we should note is that, for all families, PD=0 when a<1. This is reassuring since the total energy ceases to be positive in this region thereby precluding complete dissociation of the system. Upon comparing families A, C, and E, we find that PD increases with the difference in the component masses of the binaries, peaking at high values of a. It would be interesting to perform collisions between binaries with component mass ratios of 100 of 1000. Unfortunately, we have not done this here so we will not attempt to extrapolate these results to V t r 1 fl PD 1 Family A ‘I' T; ' «b T a. - a a ‘ .- Trt iv. t~ l~ ‘T1- 1 C Family B + 4' g j. 0 ;_, *1 *0r* it- Family 0 + 1 + I .§ '9 O a u .1 t’ ‘t—47 ‘1v~ If *1 Family D O o a O O . O . O c -1 #7 ‘T1L r. t t- l g - Family E + 4 1 i I. 6 0 a d —. 1 L + k -2 -1 0 l 2 3 logo Figure 12. Probability of dissociation versus loga. 59 their‘ asymptotic limits. We leave this to later experiments. When we compare families A, B, and D, we find that dissociation becomes extremely rare when the two most massive stars form a binary. We have already reached the asymptotic limit for these cases. 4.4. The Average Eccentricity of the Surviving Binaries It has been postulated (Hills 1977) that the eccentri- city of a binary can give a clue to its origin. One with a high eccentricity (e>0.5) was probably formed by an exchange collision between a binary and a single star. We will now examine the average eccentricity of the binaries surviving the collisions between two binaries. The _data regarding the average eccentricity of the surviving binaries when the collision impact parameter is zero are summarized by the plots of Figure 13. As one might expect, when a is small, the average eccentricity is near 2/3. This is the average eccentricity expected for a group of binaries in statistical equilibrium. The departure of family' D from this "rule" is not surprising since the massive binary should, on the average, be affected little by colliding with a binary of one-tenth its mass. For each family except family D, the peak eccentricity occurs for fairly large a. The error bars indicate these are real peaks, not just statistical fluctuations. By comparing the peaks for families A, C, and E, one finds that the height of the peak increases as well as shifting toward increasing t j: t ' I l 'l 09"..’... . l O 0 Family A L p‘t 4. t H l I 0 2 o O .. ‘ 0‘. o 0 Family t t 1 i a l 0 8 0 o '0 o . I O O l O . o 0 i_.i, l 09 O i -2 —l 0 l 2 3 logo Figure 13. Average eccentricity versus loge. 61 kinetic energy as the difference in the masses of the binary components increases. Thus, high-energy collisions can be very effective in disrupting one of the binaries. Figure 11 bears this out. As 0 increases for family E, the probability that only a Single binary will remain increases. We have been able to offer support to Hills's postulate, but we see TH) way of ascertaining whether a given binary was formed through a single-binary or a binary-binary collision. We can say only that if the eccentricity of a binary is less than approximately 0.2, the probability is high that the binary has not undergone a collision of some type. 4.5. The Distance of Closest Approach Although the collisions we have performed neglect the physical sizes of the stars, it is quite possible that a dynamical collision may precipitate a physical collision between stars in the system. If this occurs, then the assumptions implicit 131 this experiment 1K) longer apply. (This experiment does not allow us to begin with four stars and end with three!) We can, however, obtain the likelihood that a physical collision occurred by examining the distance of closest approach between any two stars in the system. Figure 14 summarizes the dependence of the logarithm of the average distance of closest approach on loga and on the masses of the binary components. The vertical axis is the logarithm of the minimum distance of closest approach in units of the initial semi-major axis of the binary 62 0 ' 7? m ‘\ c .H E L 3’, o " w .3-2 k 0 O -2 0 0 -2 t 0 Family D ' .. -2 .1 0 3 o 9 .. o 0 , o ‘2 l c I 41h: -2 -l 0 l 2 3 logo Figure 14. Logarithm of the average distance of closest approach versus logo. 63 containing m1 and m2. As expected, this distance decreases with decreasing a. The unexpected occurrence is with family E; the distance of closest approach is between 0.210 and 0.085 for all a considered. While we cannot predict actual physical collision rates without a knowledge of the stars composing the binaries or of the orbital elements of the binaries themselves, we can find the likelihood of physical collisions in a typical situation such as in dense, globular cluster cores. We will assume that the core contains equal mass binaries, each with components having mass m:0.4M, where M is the mass of the Sun. This gives a diameter for each star of d:0.5D, where D is the diameter of the Sun (Hills and Day 1976). Spitzer and Mathieu (1980) find from their compUter models of globular clusters that the maximum (x is about 0.03. A reasonable average a is then =0.1, which we will assume. We will also assume (V2>1/2=10 km/s as obtained by Hills and Day (1976). These values give a semi-major axis for the orbit of a binary of a=0.357 A.U. resulting in d/a=0.013. From our experiment, the median of Emin/a is 0.016 when a:0.1. Thus we may conclude that binary collisions can signifi- cantly increase the rate of physical collisions between stars in globular cluster cores. The data indicate that, for every 100 binary-binary collisions, 40-50 of them may involve physical collisions. It seems that globular cluster cores can no longer be treated as a group of point masses; the physical sizes of the stars must be included so that the 64 effects of physical collisions and the possible coalescence of stars as well as very close approaches resulting in significant tidal distortion can be included in the models. Otherwise, an accurate picture of the dynamics of globular clusters will not be obtained. CHAPTER 5 CONCLUSIONS 5.1. Comparison of Present with Previous Results As mentioned in the Introduction, computer experiments have been performed for collisions between binary and single stars (Hills 1975; Hills and Fullerton 1980; Fullerton and Hills 1982). Therefore, it would be most instructive to compare our results with these previous results. Rather than compare all of our results, we will compare only those which can be associated quantitatively: namely, the energy exchanged by a collision and the relevant cross-section. Before we can make comparisions between our data and the only comparable previously acquired data, that of Hills (1975) and Hills and Fullerton (1980), we must find conversion factors between their data and our data. (The experiments of Hills and Fullerton will be referred to col- lectively as HF.) HF'S energies were measured as multiples of the total initial binding energy as are our's, except they had only a single binary where we have two. We would like to convert our energies into energies measured in terms of the binding energy of only one of the binaries. The relation which accomplishes this is 65 66 5' : (1+f1/f2)5. (5.1) f2 is the fraction of the initial binding energy in the binary now considered to contain the reference energy and f1 is the fraction in the other binary. The quantity in parentheses is the factor by which we must multiply the energy given in column five of the table in Appendix A to convert it to HF'S units. These factors are given in Table Table 4. Conversion factors from the present experiment to HF's experiment. Present HF 1+f1/f2 (€'/5HF)a=0 (1-1)-(1-1) (1-1)-1 2 2.7 (3-3)-(1-1) (3-3)-1 1.11 4.7 (1-1)-3 10 3.7 (1-3)-(1-3) (193)-1 2 5.5 (1-3)-3 2 (10-10)-(1-1) (10-10)-1 1.01 253 (1-1)-10 101 748 (1-10)-(1-10) (1-10)-1 2 (1-10)-10 2 4. The first column gives the family in the present experiment in terms of the masses while the second gives the 67 masses of families in HF's experiments. The third column gives the conversion factors described by equation (5.1). Into the last column of Table 4, we have entered the ratio of our exchanged energy in HF's units to HF's when a=0. We used equation (4.8) to find the appropriate 5's for this experiment. a=0 was chosen because, if a¢0, a also must be transformed into HF's units. Direct comparison then becomes difficult because of the sparseness of data in both experiments. Blanks in Table 4 occur where data is not available for the single—binary collisions. While there appears to be no rule for finding 5' given 531:, we may conclude, not surprisingly, that a binary-binary collision releases several times the energy of a single-binary collision. The ratio (5'/5HF)a=O increases rapidly with the largest mass star partaking in the collision. alt is difficult to compare the cross-sections obtained in the present experiment with those obtained by HF for the reason mentioned above, namely, a is different for the two experiments. Rather than attempt a conversion of a, we note that each of the curves in Figure 8 reaches a peak in the region a>a0 as do the corresponding curves in Hills's (1975) Figure 8. As a comparison of the cross-section of the two experiments, we will compare the height of these peaks after applying equation (5.1) to the peak height of the present experiment. Hills's results will allow us to compare only the top. three entries in Table 4. We find that the (1-1)-(1-1) case has a cross-section which is approximately 68 2.0 times (flue (1-1)-1 case. (3-3)-(1-1) versus (3-3)-1 gives a factor of 1.8 while (3-3)-(1-1) versus (1-1)-3 gives 16, both larger in the present experiments. There are two reasons for the difference in the cross- sections of the binary-binary experiments compared with that of the single-binary, aside from simply a difference in the collision type. The binary-binary collisions have geometrical cross-sections that are a factor of four larger than the single-binary case. This is beacuse there are two finite-sized binaries rather than a binary and a point mass. The second is the reason for the application of equation (5.1) to adjust the units of GE. (M‘course, these two expla- nations do not account for all of the differences in the two experiments. They do, however, help one to live with the large differences in the released energy obtained above. 5.2. Future Investigations of Binary-Binary Collisions By performing this eXperiment and analyzing the subsequent data, we have begun to obtain an understanding of these collisions. However, the large majority of our collisions were soft collisions (6(0). While these collisions can be quite well understood after this experiment, hard collisions are not well understood. This is because hard collisions can cost 100 times more than soft collisions. ha addition, even with reduction, only about half of the hard collisions formed stable configurations within 105 integration steps. Because of -this, our 69 conclusions for these collisions may be unreliable. Therefore, we need to simulate more hard collisions, allowing perhaps 106 integration steps. This would require a significant amount of computer time. All collisions in this experiment were begun with the initial eccentricities of the binaries set to zero. While probably not a bad assumption if the binaries are undergoing their first collision, for subsequent collisions, the orbits are Ilikely 1x1 be eccentric. we should investigate the effect of eccentricity of the binaries on the collisions. We might, at some later time, wish to perform binary- binary collisions with differing values of 8. The B's used in this experiment were chosen so that the binaries had equal initial separations. While varying 8 might be interesting, when one binary is close, the collision results should be similar to those obtained in single-binary collisions. This should be verified, however. The five families investigated provide us with a reasonable mass spectrum. Families B and D appear to describe the case of a massive binary colliding with a light binary quite well. Familes C and E, however, which involve collisions between two binaries, each with components with quite different masses, do not appear to give a complete picture of this type of collision. We cannot state, with any degree of certainty, what will happen as the masses of the components become more discrepant. This also remains to be investigated. 70 The above "list of things to do" simply expresses the fact that there is a large parameter space to be investi- gated. While the completion of this investigation as described is important and necessary and should be carried out, a more expedient approach might be considered. Since one of the major uses of the results of this experiment was to be its application to clusters, we might consider modeling such clusters directly. The technique of reduction developed expressly for this experiment is ideally suited to such modeling. Not only would binary-binary collisions be considered, but higher order effects such as single-binary- binary and binary-binary-binary collisions would be automat- ically taken into account by the model. This might actually be the best next step, considering the expense of obtaining an understanding of the effects of binaries on real physical systems by the present technique. We have only begun to understand the effects of collisions between two binary systems on the evolution of a stellar system. LIST OF REFERENCES LIST OF REFERENCES Aarseth, S.J. and Hills, J.G. (1972) Astronomy and Astrophy- sics _2_1_, 255, The Dynamical Evolution of a Stellar Cluster with Initial Subclustering. Bate, R.R, Mueller, D.D., and White, J.E. (1971) "Fundamen- tals of Astrodynamics", (New York: Dover Publications, Inc.). Bevington, P.R. (1969) "Data Reduction and Error Analysis for the Physical Sciences", (New York: McGraw-Hill Book Company). Fullerton, L.W. and Hills, J.G. (1982) Astronomical Journal 87, 175, Computer Simulations of Close Encounters BEtween Binary and Single Stars: The Effect of the Impact Velocity and the Stellar Masses. Heggie, D.C. (1972) A Multi-Particle Regularisation Technique. "Gravitational N-Body Problem" pp. 148-152. Ed. M. Lecar. (Dordrecht-Holland: D. Reidel Publishing Company). Heggie, D.C. (1975) Monthly Notices of the Royal Astronomi- cal Society 173, 729, Binary Evolution in Stellar Dynamics. Hildebrand, F.B. (1974) "Introduction to Numerical Analysis", (New York: McGraw-Hill Book Company). Hills, J.G. (1975) Astronomical Journal 80, 809, Encounters Between Binary and Single Stars and Their Effect on the Dynamical Evolution of Stellar Systems. Hills, J.G. and Day, C.A. (1976) Astrophysical Letters 11, 87, Stellar Collisions in Globular Clusters. Hills, J.G. (1977) Astronomical Journal 82, 626, Exchange Collisions Between Binary and Single Sfars. Hills, J.G. and Fullerton, L.W. (1980) Astronomical Journal .85, 1281, Computer Simulations of Close Encounters Between Single Stars and Hard Binaries. 71 72 Hoffer, J.B. (1982) Celestial Mechanics, submitted, A Technique for the Computer Simulation of the Dynamics of Stellar Systems Containing Tightly-Bound Binary Star Systems. Larson, R.B. (1981) Monthly Notices of the Royal Astronomi- cal Society 194, 809, Turbulence and Star Formation in Molecular Clouds. Marion J.B. (1970) "Classical Dynamics of Particles and Systems" (Second Edition), (New York: Academic Press). Moulton P.R. (1914) "An Introduction to Celestial Mechanics" (Second Edition), (New York: The MacMillan Company). Saslaw, W.C., Valtonen, M.J., and Aarseth, S.J. (1974) As- trophysical Journal ‘HKL 253, The Gravitational Slingshot and the StrUEEure of Extragalactic Radio Sources. ' Spitzer, L. and Mathieu, R.D. (1980) Astrophysical Journal 241, 618, Random Gravitational Encounters and the Evolution of Spherical Systems. VIII. Clusters with an Initial Distribution of Binaries. Valtonen, M.J. (1975) Memoirs of the Royal Astronomical Society 89, 77, Statistics of Three-Body Experiments: Probability of Escape and Capture. APPENDICES APPENDIX A THE DATA The following table is a summary of all the data regarding collisions between two binary stars used in compiling the conclusions of this dissertation. All of the entries should be self-explanatory for anyone who has read the text of this dissertation. A quantity listed immediately after a +- is the absolute error of the preceeding entry. 73 74 00m mpo. 00—. 000. mow. :—0.1+mN0..PP0.I+0:m. PN0.1+N:F.I 00.N 00.F 00.m NP 00m mmo. 00m. m00. 0m0. mpo.l+mwm. mP0.1+0Nm. 0N0.1+0FN.I 0m.P 00.9 00.m 0P 00F PNP. 0P0. 0m0. N:N. 0P0.1+00P. 0F0.I+::0. 0N0.I+m::.1 00.P 00.— 00.m mp 00’ mwp. mww. moo. m:0. h00.1+mmp. 0—0.I+0m>. :m0.t+FP0.I 0m. 00.? 00.m :P 009 00m. >:>. mmo. moo. 000.I+00F. 0F0.I+00>. >m0.l+00:.l 00. 00.F 00.m mF 00m 000. 000. 000. 000.9 ~00.I+N00. 000.I+F90. 000.I+000. 00.5 00.F F0.P NP 00m 000. 000. 000. 000.F N00.1+mm0. Poo.l+0m0. 000.I+Poo. 00.0 00.F P0.P PP 00m 000. 000. 000. 000.? m00.I+F>0. m00.1+m00. P00.I+000. 00.m 00.P F0.P 0— 00m 000. 000. 000. 000.9 9P0.1+NFN. 000.1+:0m. 000.I+:00.I 00.: 00.P P0.F 0 :0, 000. Pup. :mo. pr. 0—0.1+0mm. :—0.1+mmm. hpo.t+000.l 00.m 00.F P0.P 0 0N9 000. 00:. >:0. mmm. mpo.1+mpm. mpo.l+mm0. 0N0.1+NNP.I 00.N 00.P 90.? b 0mP 000. 5mm. mpp. :00. >00.1+N00. 090.I+000. mm0.1+0mm.1 00.P 00.P 90.9 0 mp 000. 00». .:mp. 000. >00.1+m>0. mN0.I+m00. N>0.I+mmp.l 00. 00.P 90.9 m 0N9 000. :05. >0F. 0:0. :00.1+0m0. 0F0.1+:00. P:0.1+m0N. 00. 00.P 0m. : mm? 000. :Nm. wmr. 0P0. :00.1+mm0. 0—0.I+N00. >m0.1+0F>. 00. 00.? 0p. m Pm 000. m:0. 0mm. >00. 000.1+::0. 0N0.I+0N>. 00P.I+0m0. 00. 00.P P0. m 00 000. 000. 000. 000. 0N0.1+P::. Poo.1+moo. 000.1+F—0.1 00. F0. OF. P Apt—VIAPI—v “4 >4H2LMEEDm < .m mama—NH. 75 00m 000. 000. 000. 000.9 500.1+5m0. 000.1+FPP. 000.1+:P0.I 0m.N 00.P 00.0w 0m 00m 000. 000. 000. 000.P 050.I+P05. 000.|+05P. 050.1+:m0.1 00.m 00.5 00.0P 0m 00m moo. 0m0. 000. m:0. mF0.I+mm0. 0P0.I+:—m. 0po.l+m00.l 0m.F 00.F 00.09 50 00m 000. m0m. 000. m:0. mF0.I+5:m. :50.I+0Pm. 0N0.I+5NN.I 00.— 00._ 00.0P om 00m m:m. 05m. 000. m0P. 000.I+0Pm. 0P0.I+Pm0. :N0.I+00:.I 0m. 00.P 00.09 mm 00m 05:. 00m. 000. 000. 500.I+N0P. 0po.l+mm5. 5N0.1+0mm.1 mm. 00.5 00.09 :0 00m 0mm. m::. 000. mmo. 000.1+Nmp. NN0.1+:m5. 0m0.l+0:m.1 mp. 00.F 00.0w mm 00m 00m. 00m. moo. mmo. 000.I+Nm—. :N0.1+0m5. mm0.1+50:.1 00. 00.? 00.0w mm 00m 000. 000. 000. 000.F P00.1+050. P00.1+m90. 000.I+000. 00.0 00.— 00.m Pm 00m 000. 000. 000. 000.5 :00.1+0P0. N00.1+0m0. m00.1+~00. 00.: 00.? 00.m om 00m 000. 000. 000. 000.P m00.1+m00. m00.1+050. m00.|+000.| 0m.m 00.F 00.m 0m 00m 000. 000. 000. 000.P 500.1+::0. :00.1+:FP. m00.1+500.l 00.m 00.9 00.m 0N 00m 000. 000. 000. 000.? 0—0.I+505. 000.1+:5P. 000.1+5N0.1 0m.N 00.— 00.m 5N 00m 000. 050. 000. 0m0. N—0.I+005. 000.1+m0m. 0F0.I+F00.I 00.m 00.5 00.m 0N 00m 0m0. m5P. 000. m55. mP0.I+N0:. NF0.1+:P:. :N0.I+5mp.l 0m.~ 00.5 00.m mm 00m omm. mom. 000. m0:. P—0.I+0mm. 0F0.I+r0m. mmo.l+00m.l 00.F 00.F 00.m :N 00m 00m. 00:. moo. mmp. 000.I+m5p. 0P0.1+m00. Nm0.1+00:.| 0m. 00.5 00.m mm m0F 0::. 50:. moo. Pmo. 500.I+5mp. Fmo.l+0:5. Pmo.l+0mm.t mm. 00.P 00.m mm 00F 00:. 00:. 000. 090. 000.I+NmP. 0N0.1+:N5. m:0.1+50:.1 00. 00.5 00.m Fm 00m 000. 000. 000. 000.5 N00.I+5:0. ~00.I+mm0. 500.1+P00. 00.m 00.F 00.m 0N 00m 000. 000. 000. 000.9 m00.1+:00. m00.|+m50. N00.I+N00. 00.: 00.P 00.m 0— 00m 000. 000. 000. 000.P 000.I+N00. 000.1+P0P. 500.I+0F0.1 00.m 00.P 00.0 0? mega 00 000 mm 020 Am\cwsgv on w a 0 0 E32 .A.e.eeoev m magma 76 oom omo. mam. ooo. meo. ooo.1+mom. m_o.1+eea. mmo.1+mmm.1 om. oo.P oo.om _o oom mmm. mma. ooo. oom. ooo.1+omm. epo.-+amo. omo.1+zm=.1 oo. oo.P oo.om oo oom ooo. ooo. ooo. ooo.P Poo.-+=eo. ooo.1+eao. Poo.1+ooo.1 om.: oo., oo.om om oom ooo. ooo. ooo. ooo., moo.-+eoo. .oo.-+mmo. .oo.1+Poo.1 oo.: oo., oo.om om oom ooo. ooo. ooo. ooo.. moo.1+emo. .oo.1+omo. Poo.1+Poo.1 om.m oo.. oo.om em oom ooo. ooo. ooo. ooo.F moo.-+m=o. Poo.1+omo. moo.1+moo.1 oo.m oo., oo.om om oom ooo. ooo. ooo. ooo.? =oo.1+s.o. moo.1+emo. moo.-+moo.u om.m oo., oo.om mm oom ooo. ooo. ooo. ooo._ ooo.-+eso. moo.1+ooo. =oo.-+eoo.1 oo.m oo.F oo.om am oom ooo. ooo. ooo. ooo.? ooo.1+mes. moo.1+=oa. ooo.-+Pmo.1 oo.. oo.. oo.om mm oom mmo. omo. ooo. moo. epo.1+mo=. mao.-+omm. oao.1+oo..1 oo._ oo.F oo.om mm oom omp. mma. ooo. ma». mPo.1+=em. mPo.1+omz. mmo.-+moa.1 me. oo.F oo.om Pm mop _F_. mam. ooo. Pom. ooo.-+mom. apo.1+mmm. mmo.-+eom.1 om. oo._ oo.om om oom moo. ooa. ooo. mam. ooo.1+mam. SFo.1+o=o. omo.1+m==.1 mm. oo.P oo.om o: oom oom. mm=.. ooo. mo.. oao.1+oo_. omo.1+mmo. mmo.1+mez.1 mF. oo.. oo.om we was mom. ama. ooo. pop. oPo.1+eo_. =mo.1+omo. omo.1+mm=.1 oo. oo., oo.om a: oom moe. oom. ooo. mom. opo.1+mpm. omo.1+ooo. omo.1+:=:.1 oo. oo.? oo.om o: oom ooo. ooo. ooo. ooo.F ooo.1+m=o. moo.1+o.F. ooo.-+oPo.1 oo.m oo.. oo.om ma oom ooo. ooo. ooo. ooo.? Ppo.1+ome. eoo.1+=Pm. P_o.1+mmo.1 om.a oo.P oo.om a: oom meo. oma. ooo. moo. =_o.1+mm=. mpo.1+oo=. mmo.1+oma.1 oo., oo.? oo.om ma oom omP. oom. ooo. ooa. ooo.1+eam. o.o.1+Fom. omo.1+oam.1 om. oo._ oo.om m: eo_ mmm. mes. moo. mop. ooo.-+oo.. o.o.1+ooo. omo.-+oo=.1 mm. oo.P oo.om Pa oom ooo. mma. moo. ooo. oFo.1+ooP. mmo.1+oae. emo.1+eom.- oo. oo., oo.om o: mesa oa mma ma oza Ae\easev aev o a a 5:2 .A.o.eeoeo m eaoee 77 00m 000. 000. 000. 000.P 0P0.1+0F0. N00.1+500. 000.1+:00.1 00.F 00.? 00.000 :0 00m 000. 000. 000. 000.— Npo.l+5:5. N00.1+000. :00.1+000.1 0N.P 00.9 00.000 00 00m 000. 000. 000. 000.P 0—0.|+050. :00.I+000. 000.1+090.I 0P.— 00.P 00.000 m0 00m 000. moo. 000. m00. :P0.I+000. 000.I+:N—. 000.1+050.I 00.9 00., 00.000 .0 00m moo. mpo. 000. 000. :Fo.I+FNm. 000.I+mmp. 0—0.1+0N0.I 00. 00.F 00.000 00 00m 000. 000. 000. 050. 0P0.I+0::. P50.1+00p. NP0.I+P:0.1 00. 00.— 00.000 05 00m mwo. mro. 000. 050. Npo.l+000. 0P0.I+0PN. 0F041+000.I 05. 00.P 00.000 05 00m mpo. 000. 000. mm0. 090.1+N:0. 0P0.I+m0N. NP0.1+0:0.I 00. 00.F 00.000 55 00m 0N0. 000. .000. 0N0. 000.I+0P0. Npo.t+m0m. 0F0.I+N00.| 0m. 00.? 00.000 05 00m 050. omo. 000. 0:0. 000.1+m0m. mpo.I+—00. 5F0.I+NFP.I 0:. 00.P 00.000 m5 00m moo. m00. 000. 000. 000.1+00m. FP0.I+0:0. mP0.I+00F.I 00. 00.F 00.000 :5 00m 000. m00. 000. m00. 000.I+P5N. 000.I+P50. 0P0.I+:0—.I 0N. 00.5 00.000 05 00m 0N0. mop. 000. m50. 0~0.1+:0N. NP0.1+PN:. PN0.I+0mF.I 0F. 00.? 00.000 N5 00: N50. 0FP. 000. 000. 000.I+NN0. 000.I+0::. mF0.1+5mF.I 00. 00.P 00.000 F5 00m 000. 000. 000. 000.F 500.I+:m0. N00.I+000. :00.1+000.I 0m.F 00.— 00.00— 05 00m 000. mwo. 000. m00. :Fo.t+:0m. cpo.l+mpm. Ppo.l+0:0.l 00.P 00.P 00.00P 00 00m mpo. m:0. 000. 0:0. NF0.I+0F:. mpo.t+000. 5F0.1+050.I m5. 00.F 00.00F 00 00m mmo. omp. 000. mm0. 0P0.I+mp0. Npo.l+p50. 0—0.1+P0P.1 0m. 00.? 00.00P 50 00m 000. 00’. 000. 0P0. 000.I+00N. 0P0.1+0N:. 0N0.1+05P.1 0:. 00.P 00.00P 00 00m 000. 0:9. 000. 000. 000.I+P0m. NF0.1+55:. PN0.I+:0m.I 00. 00.9 00.00P m0 00m m50. mFm. 000. 0P5. 000.I+0:m. NP0.I+0:m. :N0.1+00N.I 0m. 00." 00.00P :0 00m 0mp. 00m. 000. 00m. 000.I+0:N. NP0.I+0P0. 5m0.1+0N0.1 op. 00.F 00.00F 00 00m m5p. 00m. 000. m0m. 0P0.1+50N. 0—0.I+NN0. 0N0.I+m0N.I 00. 00.9 00.005 N0 mczm 00 000 mm 020 Am\cHELv Amv 0 Q 0 5 8:: .A.U.ucoov m manmh 78 00m 000. 000. 000. 000.5 500.I+5P0. 500.1+5N0. Poo.1+—00.1 0m.P 00.P 00.000F :0F 00m 000. 000. 000. 000.P 000.1+050. ~00.I+500. P00.1+500.1 0:.P 00.9 00.000? 009 00m 000. 000. 000. 000.— PP0.1+:N0. 500.I+000. N00.I+F00.I 00.9 00.P 00.0005 mop 00m 000. 000. 000. 000.P 0P0.I+0m5. 500.1+0:0. N00.I+N00.1 0N.P 00.P 00.000F For 00m 000. 000. 000. 000.? :P0.1+000. N00.I+Nm0. 000.I+000.I 0F.P 00.P 00.0009 00F 00m 000. 000. 000. 000.P :—0.1+N00. 000.I+m00. 000.1+m00.1 00.P 00.5 00.0005 00 00m moo. 0P0. 000. m00. :50.1+0Nm. m00.1+000. m00.1+mpo.l 00. 00.9 00.000? 00 00m moo. 000. 000. m00. 0P0.1+Pm:. 000.1+Nop. 500.1+0F0.t 00. 00.5 00.000P 50 00m 000. moo. 000. m00. FF0.I+P00. 000.I+0NP. 000.I+5m0.L 05. 00.5 00.000F 00 00m 000. mpo. 000. m00. 050.I+m:0. 000.I+m:F. 000.I+m00.1 00. 00.9 00.000— m0 00m 000. opo. 000. 000. 000.1+0_0. 0~0.1+:0—. 000.1+000.| 0m. 00.9 00.000P :0 00m 000. 0N0. 000. 000. 000.I+05N. 000.I+00F. 0P0.1+000.I 0:. 00.P 00.000P 00 00m m00. 0:0. 000. mm0. 000.1+00N. 0F0.I+m0N. 050.1+0:O.I 00. 00.9 00.000F m0 00m 0—0. 0P0. 000. 000. 000.1+P0N. 000.I+0mm. mF0.I+mm0.1 0m. 00.? 00.000— —0 oom moo. 0:0. 000. mm0. FF0.1+00N. Ppo.l+05m. m50.l+050.l op. 00.5 00.000F 00 00m 0N0. mm0. 000. mm0. FF0.1+0F0. :F0.I+P90. 050.I+000.I 00. 00.P 00.000F 00 00m 000. 000. 000. 000.P 000.1+000. F00.I+000. N00.I+F00.1 o5.F 00.P 00.000 00 00m 000. 000. 000. 000.P :00.1+0P0. P00.I+::0. N00.1+F00.I 00.P 00.— 00.000 50 00m 000. 000. 000. 000.P 000.I+000. F00.1+0m0. N00.1+N00.I 0m.F 00.F 00.000 00 00m 000. 000. 000. 000.— 000.1+0m0. P00.I+0m0. 000.1+000.I 0:.P 00.P 00.000 m0 mcsm 00 000 mm 020 Am\CwELv on w a 0 5 E32 .A.o.eeoeo m eaoee 79 00m 000. 000. 000. 000.? :00.1+000. P00.I+0:0. P00.I+Poo.1 00.: PF. 00.0w 5P 00m 000. 000. 000. 000.5 500.I+500. N00.I+050. N00.I+N00.I 00.0 55. 00.0w 0' 00m 000. 000. 000. 000.5 N50.1+000. 000.1+50P. :00.I+050.1 00.m PF. 00.0p mp 00m 0P0. 05:. 000. 00m. m—o.l+m0:. N50.I+0m0. 050.I+N00.I 00.? PP. 00.05 :P oom mom. 0m5. 000. m:o. 090.I+:NN..mF0.I+50m. N00.I+00N.I 00. 55. 00.05 mp 00m 000. 000w 000. 000.— 000.1+mp0. N00.1+0m0. 500.1+500.1 00.: PP. 00.m NF 00m 000. 000. 000. 000.— 000.I+P00. :00.1+0op. N00.I+:00.I 00.0 PP. 00.m —P 00m 000. 000. 000. 0N0. m—0.I+0:5. 000.I+0—N. 000.I+:m0.1 00.m PP. 00.m op 00m m00. mmm. 000. 0F:. 050.I+N:0. 0'0.I+0P:. 0N0.I+000.I 00.5 PF. 00.m 0 00m m0m. m00. moo. mm0. 000.I+N0m. 550.I+050. 000.I+050.1 00. FF. 00.m 0 00m 000. 000. 000. 000.P 500.1+050. 000.I+050. 500.I+000.I 00.: P—. 00.0 5 00m 000. moo. 000. m00. p—0.I+000. 000.1+0:F. 000.I+000.I 00.0 PP. 00.0 0 00m 000. 00?. 000. 0N0. 5—0.I+P05. 000.I+0mN. 000.I+0:0.I 00.m PP. 00.0 m 00m m00. 000. 000. m00. m50.t+m0m. mpo.t+05:. 5N0.I+0NF.I 00.F PF. 00.0 : 00m mmm. 005. moo. 000. 000.t+05p. mP0.I+5N5. 0:0.1+NP:.I 00. PP. 00.0 0 :m 000. m50. 0:5. 500. NF0.1+00_. 0N0.1+500. F50.1+00N.1 00. PP. P0.P N NPP 000. 0:0. :m0. 000. 000.I+500. 0po.t+000. 0:0.I+P0:. 00. 5F. 05. P APIFVIA0100 um 54Hz