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1983Apj. . .274. . .53W the Astrophysical Journal .53W . .274. The Astrophysical Journal, 274:53-61, 1983 November 1 . © 1983. The American Astronomical Society. All rights reserved. Printed in U.S.A. 1983ApJ. SIMULATIONS OF SINKING SATELLITES Simon D. M. White Department of Astronomy and Space Sciences Laboratory, University of California, Berkeley Received 1983 February 11; accepted 1983 April 6 ABSTRACT The orbital decay of satellite galaxies can be investigated using simulation methods which follow the dynamical process in a variety of different approximations. A comparison of the results shows that the decay rate depends on the global response of the parent galaxy. It is artificially enhanced by fixing the center of the parent, and it is suppressed by neglecting the self-gravity of the response. Chandrasekhar’s local dynamical friction formula cannot, therefore, be a complete description of the underlying physics, although it predicts decay rates which are approximately correct. A simulation method which calculates the galactic gravitational field to second order in a multipole expansion gives results in agreement with full iV-body experiments, but it runs several hundred times faster for N = 5000; it is only a factor of 2 or 3 slower than methods based on a generalization of the restricted three-body problem. Subject headings : galaxies : clusters of — numerical methods — stars : stellar dynamics I. BACKGROUND Satellite galaxies orbit deep within the sphere of influence of their giant neighbors. Although usually too puny to be more than an occasional annoyance to their allies, they can suffer considerable damage as a result of this unequal association. Their peripheral regions may be stripped by the giant’s force field, and their orbits gradually decay, causing an ever more disruptive relationship between giant and satellite which may culminate in the disintegration and total absorption of the latter. This orbital decay is at root a manifestation of the general tendency of closed systems to eliminate diversity. The detailed mechanism involved has been described in two complementary ways. On a “microscopic” level encounters lead, on average, to a transfer of kinetic energy from the satellite to individual stars in the halo of the giant; the transfer rate can be modeled by an effective force known as dynamical friction which provides a purely local description of the interaction (Chandrasekhar 1942). An apparently more rigorous “macroscopic” description considers the effect on its orbit of the global response the satellite stimulates in its companion (e.g., Kalnajs 1970). Unfortunately while a straightforward application of Chandrasekhar’s formula predicts a significant decay rate, calculations of the global response usually find it to be in phase with the satellite and so suggest negligible orbital decay. The resolution of these conflicting predictions may lie in a proper treatment of resonant stars, but a full understanding still eludes us (Tremaine 1981). It is certainly of interest to know how fast satellites are dragged into their parent galaxies, and direct numerical simulation offers a useful complement to pure thought. An early full A-body simulation of mine seemed to agree with a simple model based on dynamical friction (White 1978). Nevertheless, although a single experiment with a small number of particles can greatly strengthen prejudices, it can never be conclusive. Lin and Tremaine (1983) recently tackled the problem in a far more systematic way using a novel technique to speed up their simulations and to suppress certain spurious effects. Their method is an extension of the restricted 3-body approach. Two massive particles represent the satellite and the core of the parent galaxy. In their combined field move a large number of particles representing the outer halo of the parent. These particles exert forces on the satellite but not on each other; in this way the halo response can influence the satellite, but the time needed to compute particle accelerations is much less than in a full iV-body calculation. Lin and Tremaine term this method a “ semirestricted A-body ” approach. The decay rates in their experiments agreed with the predictions of Chandrasekhar’s local formula in size, in dependence on satellite mass and diameter, and in their insensitivity to previous orbital history. Much impressed by these results, I decided to write a semirestricted A-body code of my own. While on a visit to the Institute for Advanced Study, I duly set out the equations for a core-halo-satellite system, eliminated all forces between the halo particles, and wrote a program to follow motions in the resulting force field. An obvious first test was a repeat of one of Lin and Tremaine’s experiments, so I constructed a galaxy-satellite system, loaded it into the computer, and waited eagerly for the results. First runs rarely come out well, and I was disappointed to find that my satellite’s orbit did not decay as fast as expected; indeed, it appeared to have little wish to reach the center of its parent at all. Some checks showed that my code, although unable to produce friction, could conserve energy and angular momentum very well. I retired baffled to read Lin and Tremaine’s preprint 53 © American Astronomical Society • Provided by the NASA Astrophysics Data System .53W . .274. 54 WHITE Vol. 274 . more carefully. An examination of the text between Abstract and Conclusions showed that I had missed a key ingredient of their method; in order to eliminate all particle-particle relaxation, they had elected to nail down 1983ApJ. the core of their galaxy. The offending equation of motion was easily removed from my code, and a new run produced a much higher decay rate consistent with their results. Apparently the decay rate depended strongly on whether the core of the giant galaxy could pursue its own little orbit about the barycenter of the pair. By chance Tremaine himself was visiting, and so I showed him my results. After a few moments of introspection he declared the reason for my discrepant results to be self-evident : after all, if the center of the Earth were nailed down, what would happen to the tides? This insight did, indeed, offer a plausible explanation—but why then did my full AT-body simulation agree with dynamical friction theory and with Lin and Tremaine’s experiments, while it disagreed with the new experiment in which the center was free? Clearly something remained to be understood. Let us look at the situation in more detail. II. THE PROBLEM The equations of motion which I originally chose to represent a core-halo-satellite system were the following: = mÄxs - xc)Psc~3 + m '£(x - x )p -3 ; (1) h i j i c ic Mxc Xs)Pcs "I” ^ (Xi Xs)Pis ? (2) i 3 3 Xi = mc(xc - Xi)pci- + ms{xs - xjpr ; i = 1, Nh . (3) In these equations the subscripts c, s, and i (=1, Nh) denote properties of the core, satellite, and halo particles 2 2 2 respectively, the “softened” lengths are defined by psc = \xs — xc\ + e and analogous formulae, and Newton’s constant has been set equal to one. They are derived from the total potential energy, 1 1 1 W = msmcpsc- + mhY,(rncPci~ + mspsi- ), (4) i and as a result a system which obeys them conserves energy as it evolves. In addition because each contribution to the force on a particle is balanced by an equal and opposite force on another particle, the system conserves both linear and angular momentum. I set up an equilibrium galaxy for my experiment as follows. Around a core particle with mc = 10, I put Nh = 5000 halo particles. Their total mass was Nh mh = 10, and they had a spherical distribution with density profile p ccr~2 truncated at r = 20. I then gave each particle a randomly directed velocity equal to the circular velocity at its distance from the core (taking into account the imposed softening e2 = 10). This procedure produced an initial system which was in virial equilibrium but was not properly phase mixed. I therefore let it evolve for 200 time units (about 10 crossing times) with the satellite mass set to zero. At this time it had reached a steady state, had retained its initial density profile within r = 14, and had spread out at the edges to blur the sharp initial cut-off. This system has a higher central concentration than Lin and Tremaine’s standard “galaxy” and is a closer approximation to what one expects of a real system. I set the satellite on a circular orbit around this “galaxy” with ms = 2, r = 30; it thus had one-tenth the total mass of its companion, and its orbit enclosed 95% of the halo mass. The large softening I chose allowed me to integrate the equations of motion rapidly with a simple fourth order Runge-Kutta routine. In addition it implied an effective size for the satellite which agreed roughly with that expected for a galaxy in orbit about a companion of 10 times its mass. The above specifications resulted in the slow orbital decay which so surprised me. After eight full orbits the satellite had spiraled in only as far as r = 11, and it took 22 orbits to get to r = 7; the dynamical friction formula predicts that one or two orbits should be sufficient to reach the center from r= 11. Nailing down the core to mimic Lin and Tremaine’s experiments was quite easy. I just removed equation (1) from my code and set xc = 0. The remaining equations of motion still conserve energy since they are derived from equation (4). However, they do not conserve linear momentum because the forces exerted by the core are no longer reflected in its acceleration. Angular momentum about the core is conserved because the core force is purely radial.
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