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Dynamical Friction in Stellar Systems: an Introduction

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Dynamical Friction in Stellar Systems: an Introduction Dynamical Friction in Stellar Systems: an introduction H´ector Aceves∗ Instituto de Astronom´ıa, UNAM. Apartado Postal 877, Ensenada, B.C. 22800, M´exico. Mar´ıa Colosimo† Facultad de Ciencias, Universidad Nacional del Centro de la Provincia de Buenos Aires. Tandil, Argentina (Dated: July 26, 2013) An introductory exposition of Chandrasekhar’s gravitational dynamical friction, appropriate for an undergraduate class in mechanics, is presented. This friction results when a massive particle moving through a “sea” of much lighter star particles experiences a retarding force du to an ex- change of energy and momentum. General features of dynamical friction are presented, both in an elementary and in a more elaborate way using hyperbolic two-body interactions. The orbital decay of a massive particle in an homogeneous gravitational system is solved analytically, that leads to an underdamped harmonic oscillator type of motion. A numerical integration of the equation of motion in a more realistic case is done. These results are compared to those of an N-body computer simulation. Several problems and projects are suggested to students for further study. I. INTRODUCTION An elementary understanding requires only some basic ideas from mechanics, and hence suitable for presenta- Classical mechanics, perhaps the oldest of the phys- tion in introductory courses. ical sciences, continues to be an area of intensive re- Dynamical friction is important in astronomical stud- 1,2 3,4 search, both in its foundations and applications, and ies of, for example: the fate of galaxy satellites30,31,32 or a source of discussion and examples in teaching. Ap- globular clusters33 orbiting their host galaxies, the sub- plications range from the modeling of cellular mechan- structure of dark halos surrounding galaxies,34,35,36 and ical processes5 to solar system dynamics3 and galactic the motion of black holes in the centers of galaxies.37 6,7,8 systems. It has been proposed to explain the formation of bina- In describing nature students learn from their first ries in the Kuiper-belt38, and the migration of Jupiter- courses, and particularly in laboratory experiments, that mass planets in other solar systems from the outer parts “the forces on a single thing already involve approxima- where they presumably formed (> 1 AU) to the small or- tion, and if we have a system of discourse about the real bital distances (< 0.1 AU) at which∼ they are observed.39 world, then that system, at least for the present day, must It even has been∼ considered in the motion of cosmic involve approximations of some kind”; as mentioned by strings.40 Feynman on introducing the subject of friction.9 The presentation of this topic to students, in a lower This phenomenon is usually introduced in text- 41,42 10,11,12,13 or upper-undergraduate class on mechanics or com- books and lectures by considering the slide of 43 a material block on a surface, and a distinction between putational physics, will enhance their appreciation of static and kinetic friction is made. A classical example of physics in describing nature and expose them to another the effect of a friction-like force is the motion of a mass example of classical mechanics. Furthermore, students attached to a spring inside a viscous medium, where the will obtain a glimpse of an area of astronomical research corresponding differential equation is solved, and its be- important for the understanding of the fate and behavior havior studied. At the end, one invariably needs to state of stellar systems. that friction and its origin is a complicated matter, in- The organization of this paper is as follows. In Sec- volving complex interactions at the atomic and molecular tion II basic elements of the theory of dynamical friction level among the surfaces in contact.14,15,16,17 are presented. Firstly, elementary arguments are used Several non-typical examples of mechanical friction for to elucidate them. Secondly, Chandrasekhar’s approxi- introductory courses exist,18,19,20,21 that help both teach- mation using two-body hyperbolic Keplerian collisions is arXiv:physics/0603066v1 [physics.class-ph] 9 Mar 2006 ers and students alike in lectures on mechanics. All fric- considered. In Section III a simple analytical problem for tion related problems are a background for discussing the the motion of a massive particle in an ideal homogeneous important connection between the work-energy theorem stellar system is solved; a damped harmonic oscillator is and dissipative systems.22,23,24 found. In Section IV a more realistic astronomical exam- The purpose of this paper is to bring an example from ple that requires the numerical integration of the equa- astronomy25,26,27 closely related to standard mechani- tion of motion is presented. Comparison with a computer cal friction, namely: dynamical friction. This process experiment is done afterwards. Final comments as well was first introduced in stellar systems by Subrahmanyan as some ideas for problems and projects of further study Chandrasekhar.28,29 In brief, a massive particle m experi- are provided in Section IV. An appendix contains some ences a drag force when moving in a “sea” of much lighter astronomical units and standard units used in gravita- star particles m∗ by exchanging energy and momentum. tional computer simulations. 2 ∆v ∆ m where we set ρ0 = n0m∗, the background density, and b * b and b are a minimum and maximum impact pa- ∆ min max b v rameter, respectively. Letting ln Λ be the resulting inte- m gral, the deceleration of m due to its interaction with an vm homogenous background of particles stars is ∆v 2 dvm πG ρ0m v ∆ t ∆v 2 ln Λ . (4) m dt ≈ vm FIG. 1: Illustration of the deceleration a heavy particle m ex- The velocity impulse on m∗ has a perpendicular ∆v⊥ periences when moving in an homogeneous and infinite “sea” and parallel ∆v|| component; see Figure 1. It is not diffi- of much lighter particles m∗. cult to see that a mean vector sum of all the ∆v⊥ contri- butions vanishes in this case. This is not true however for 61 the mean square of ∆v⊥. Thus the dynamical friction II. DYNAMICAL FRICTION force is along the line of motion of m. Several key features of dynamical friction are observed Two equivalent approaches to compute the dynami- from equation (4) in this elementary calculation, that cal friction a massive particle m experiences as it moves appear also in more elaborate treatments. (1) The de- through a stellar system of much lighter stars m∗ are the celeration of the massive particle is proportional to its following.6 (1) Particle m produces a region of star over- mass m, so the frictional force it experiences is directly density behind it, much like the wake behind the motion proportional to m2. (2) The deceleration is inversely pro- of a ship, that in turn exerts a gravitational pull on m portional to the square of its velocity vm. leading to its deceleration.44 (2) Particle m moves in the “sea” of lighter particles m∗ and an energy exchange oc- curs, increasing that of the lighter ones at the expense of B. Chandrasekhar formula the heavy one leading to a breaking force for m. In the latter picture the basic features of dynamical friction are A further step in calculating the effect of dynamical easier to compute and understand by elementary meth- friction is to consider hyperbolic Keplerian two-body en- ods. Here the latter picture is taken. counters. Such analysis was done by Chandrasekhar.28,29 The resulting formula is provided in textbooks on stel- lar dynamics.6 For completeness such calculations is pro- A. Elementary estimate vided here, following Binney & Tremaine. Use of well known results from the Kepler problem for 10,11,12,13,45 Consider a particle m moving with velocity vm in an two bodies in hyperbolic encounters are used. homogeneous background of stationary lighter particles The two-body problem can be reduced to that of the of equal mass m∗; see Figure 1. Assume only changes motion of a particle of reduced mass µ = mm∗/(m+m∗) in kinetic energy. As m moves through, a particle m∗ about a fixed center of force: incoming with impact parameter b will be given a velocity κ impulse of about the acceleration a times the duration of µ¨r = ˆr , (5) −r2 the encounter ∆t. This can be approximated as where κ = Gmm∗, r = r∗ rm is the relative vector Gm b − position of particles m and m∗, and ˆr its unit vector; see ∆v∗ 2 . (1) ≈ b × vm Figure 2. The relative velocity is then V = v∗ vm, and a change in it is − The kinetic energy gain of m∗ is therefore V v v 2 ∆ = ∆ ∗ ∆ m . (6) 1 2 1 Gm − ∆E∗ m∗(∆v∗) m∗ . (2) ≈ 2 ≈ 2 bv The velocity of the center-of-mass of m and m∗ does not change, hence The total change in velocity of the massive particle is given by accounting for all the encounters it suffers with m∗ ∆v∗ + m ∆vm =0 . (7) particles m∗. The number of encounters with impact pa- rameter between b and b+∆b is ∆N n (v ∆t) ∆(πb2); From equations (6) and (7) the change in velocity of m ≈ 0 m where n0 is the number density of background stars. The is total change in velocity of m at the expense of the energy v m∗ V lost by stars is then ∆ m = ( ) ∆ . (8) − m + m∗ 2 bmax dvm 1 dE∗ πG ρ0m db Once ∆V is determined, ∆v can be found from equa- dN , (3) m dt ≈ mv dt ≈ v2 b tion (8).
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