arXiv:physics/0603066v1 [physics.class-ph] 9 Mar 2006 a rtitoue nselrssesb Subrahmanyan by systems stellar Chandrasekhar. in introduced first was ne rgfrewe oigi sa fmc lighter much of “sea” a particles in moving star when force drag a ences n isptv systems. theorem dissipative work-energy and the the fric- between discussing All for connection background important mechanics. a on are problems lectures related in tion alike students and ers exist, courses introductory a rcin namely: friction, cal astronomy ee mn h ufcsi contact. in- in surfaces matter, the complicated molecular among and a atomic level the is at origin interactions state complex to its volving needs and invariably one friction end, be- that the its At the and mass where studied. solved, a havior medium, is of viscous equation motion a differential the inside corresponding is spring force a friction-like to of example attached a classical between of A distinction effect made. a the is and friction surface, kinetic and a static on block material a books systems. oreo icsinadeape ntahn.Ap- mechan- cellular teaching. of modeling in processes the examples re- ical from and intensive range discussion of plications of area source an a foundations be its in to both search, continues sciences, ical ena nitouigtesbeto friction. of by subject mentioned the as introducing kind”; on real Feynman some must day, the present of about the approximations for discourse least involve of at system, system that approxima- a then involve have world, we already if thing and single tion, a on that forces experiments, laboratory “the in particularly and courses, eea o-yia xmlso ehnclfito for friction mechanical of examples non-typical Several h ups fti ae st rn neapefrom example an bring to is paper this of purpose The hspeoeo suulyitoue ntext- in introduced usually is phenomenon This lsia ehnc,prasteods ftephys- the of oldest the perhaps mechanics, Classical ndsrbn auesuet er rmterfirst their from learn students nature describing In 10,11,12,13 autdd inis nvria ainldlCnr ela de Centro del Nacional Universidad Ciencias, de Facultad 6,7,8 25,26,27 iuain eea rbesadpoet r ugse to suggested are projects and problems Several simulation. oini oeraitccs sdn.Teerslsaecom are results numeri These A done. is case motion. realistic of syste more a type gravitational in oscillator homogeneous motion harmonic an two- in underdamped hyperbolic particle an using massive way dynamica elaborate a experi of more of features a particles General in star and fri momentum. lighter elementary and This much energy of of presented. “sea” change is a mechanics, through in moving class undergraduate an 5 m nitoutr xoiino hnrska’ gravitatio Chandrasekhar’s of exposition introductory An oslrsse dynamics system solar to .INTRODUCTION I. n etrsb osdrn h ld of slide the considering by lectures and 28,29 ∗ nttt eAtoo´a NM praoPsa 7,Ensen 877, Postal Apartado Astronom´ıa, UNAM. de Instituto yecagn nryadmomentum. and energy exchanging by lsl eae osadr mechani- standard to related closely nbif asv particle massive a brief, In yaia rcini tla ytm:a introduction an Systems: Stellar in Friction Dynamical yaia friction dynamical 22,23,24 18,19,20,21 1,2 n applications, and hthl ohteach- both help that 14,15,16,17 hsprocess This . 3 n galactic and m 9 Dtd uy2,2013) 26, July (Dated: experi- 3,4 a´ aColosimo Mar´ıa ´ co Aceves H´ector and tutr fdr ao urudn galaxies, surrounding halos dark of structure clusters globular thsbe rpsdt xli h omto fbina- of galaxies. formation of the Kuiper-belt centers explain the to the in proposed in ries been holes has black It of motion the e f o xml:teft fglx satellites galaxy of fate the example: for of, ies presenta- for suitable courses. introductory hence in basic and tion some mechanics, only from requires ideas understanding elementary An aspaesi te oa ytm rmteotrparts outer ( the formed from presumably systems they solar where other in planets mass ia itne ( distances bital tee a encniee ntemto fcosmic of motion the in considered been strings. has even It ruprudrrdaecaso mechanics on class upper-undergraduate or srnmcluisadsadr nt sdi gravita- in used some simulations. units contains computer standard tional appendix and An study units IV. further astronomical of Section projects in well and provided as problems are comments for ideas Final some afterwards. as computer done equa- a with is the Comparison experiment of presented. integration is motion numerical of exam- the tion astronomical requires realistic more that is a oscillator ple IV harmonic Section damped In homogeneous a found. ideal solved; an is in system particle for stellar massive problem a analytical of simple motion a the III is Section In collisions Keplerian considered. approxi- hyperbolic Chandrasekhar’s used two-body Secondly, using are mation arguments them. elementary friction elucidate dynamical Firstly, to of theory presented. the of are elements basic II tion behavior and systems. research fate stellar astronomical the of of of understanding area the an for students of important glimpse Furthermore, a obtain mechanics. another will to classical them of expose and example nature describing in physics physics, putational yaia rcini motn natooia stud- astronomical in important is friction Dynamical h rsnaino hstpct tdns nalower a in students, to topic this of presentation The h raiaino hsppri sflos nSec- In follows. as is paper this of organization The ∗ rvni eBeo ie.Tni,Argentina Tandil, Aires. Buenos de Provincia † 40 tdnsfrfrhrstudy. further for students nal ssle nltcly htlasto leads that analytically, solved is m ae otoeo an of those to pared to eut hnamsieparticle massive a when results ction ne eadn oc ut nex- an to du force retarding a ences oyitrcin.Teobtldecay orbital The interactions. body yaia friction dynamical rcinaepeetd ohi an in both presented, are friction l a nerto fteeuto of equation the of integration cal d,BC 20,M´exico. 22800, B.C. ada, ∼ < 33 0 43 riigterhs aais h sub- the galaxies, host their orbiting . U twihte r observed. are they which at AU) 1 ilehneterapeito of appreciation their enhance will 38 n h irto fJupiter- of migration the and , N prpit for appropriate , bd computer -body ∼ > U otesalor- small the to AU) 1 41,42 34,35,36 30,31,32 rcom- or and or 39 37 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). From the symmetry of the problem, is better m Z m Zbmin 3 α µ In a homogeneous sea of stellar masses all perpendicular V0 α deflections cancel by symmetry. However, the parallel ve- locity changes are added and the mass m will experience b b θ r min a deceleration. ∆ The calculation of the total drag force due to a set V of particles m∗ is as follows. Let f(v∗) be the number density of stars. The rate at which particle m encounters ∆V V0 stars with impact parameter between b and b + db, and velocities between v∗ and v∗ + dv∗, is FIG. 2: Dispersion of a “reduced” mass particle µ in the 3 potential of a static body of mass (m + m∗) moving with 2πb db V0 f(v∗) d v∗ , (18) relative speed V0. The scattering angle is θ. · · 3 where d v∗ is the volume element in velocity space. The total change in velocity of m is found by adding all the to decompose ∆V in terms of perpendicular and parallel contributions of ∆v due to particles with impact pa- | m||| components: rameters from 0 to a bmax and then summing over all velocities of stars. At a particular v∗ the change is ∆V = ∆V|| + ∆V⊥ , (9) v bmax d m 3 with = V0 f(v∗) d v∗ ∆v 2πb db . (19) v m|| dt ∗ · 0 | | Z ∆V|| = V0 cos θ and ∆V⊥ = V0 sin θ , (10) | | | | The required integral is where θ is the angle of dispersion and V0 the initial speed −1 at infinity; this being the same after the encounter since bmax b2V 4 = 1+ 0 b db only kinetic energy changes are considered; see Figure 2. I G2(m + m )2 Z0 ∗ From geometry, the angle α in Figure 2 is related to the max max b b db 1 s ds orbit’s eccentricity e by46 = 2 = , 0 1+ ab 2a 1 s 1 θ Z Z cos α = cot = e2 1 , (11) 4 2 2 2 e → 2 − where a = V0 /G (m+m∗) and s = 1+ab , with smax = 2 p 1+ abmax. Evaluating the integral yields where θ +2α = π. Physically e is given by 2 2 1 G (m + m∗) 2 2EL2 = 4 ln 1+Λ , e = 1+ , (12) I 2 V0 s µ3κ2 where where E = µV 2/2 in the kinetic energy and L = µbV 0 0 b V 2 b the angular momentum magnitude. Since Λ max 0 = max . (20) ≡ G(m + m∗) bmin θ 2 θ 2 tan 2 1 tan 2 sin θ = 2 θ and cos θ = − 2 θ , (13) Putting these results together in equation (19): 1 + tan 2 1 + tan 2 v d m 2 2 after some algebra it is found that = 2πG ln(1 + Λ )m∗(m + m∗) dt v∗ −1 3 2 4 v∗ vm 2bV0 b V0 f(v ) d3v − . (21) ∆V⊥ = 1+ , (14) ∗ ∗ v v 3 | | G(m + m ) G2(m + m )2 × ∗ m ∗ ∗ | − | The quantity ln Λ is called the Coulomb logarithm in b2V 4 −1 analogy to an equivalent logarithm found in the theory of ∆V =2V 1+ 0 . (15) | ||| 0 G2(m + m )2 plasma. The factor ln Λ reflects the fact that the cumu- ∗ lative effect of small deflections is more important than Using equation (8) the perpendicular and parallel mag- strong or close encounters. This may be seen geometri- nitudes of the components of ∆vm follow: cally from Figure 2, were the stronger the deflection the smaller is the parallel component contributing to the slow 3 2 4 −1 v 2bmV0 b V0 down of m. ∆ m⊥ = 2 1+ 2 2 , (16) | | G(m + m∗) G (m + m∗) The determination of the limits bmin and bmax is not an easy matter and depends on the problem at hand. In 2 this approximation bmin satisfies V0 = Gm/bmin, where 2 4 −1 2mV0 b V V depends on the relative velocity of m and m . If the ∆v = 1+ 0 . (17) 0 ∗ | m||| (m + m ) G2(m + m )2 motion of m is relatively slow in comparison to that of ∗ ∗ 4 the stars, V0 can be approximated for example by the root-mean-square value velocity of stars Vrms. The outer limit b is in principle the radius at which stars no max m longer can exchange momentum with m. If m is close r F to the center of a stellar system b can be taken as a df max v particular scale-radius of the system; for example, where Fg the star density falls to half of its central value. In typical astronomical applications Λ 1. For ex- ample, consider the motion of a massive≫ black hole of 5 M mass m 10 M⊙ near the center of a dwarf galaxy. These galaxies≈ have V V 30 km s−1, character- rms ≈ 0 ≈ R istic radii bmax 3 kpc and stars of masses m∗ 1M⊙. Using these values≈ we obtain Λ 6.3 103. This≈ allows to use the approximation ln(1 +≈ Λ2) ×2 ln Λ. Note that ≈ FIG. 3: Forces acting on a massive particle m moving with ln Λ shows a weak dependence on V0 that is usually ne- velocity v inside a stellar system: Fg is the gravitational force < < glected. Values of 2 ln Λ 20 are typically found in and Fdf the dynamical friction force. The latter acting oppo- astronomical literature.∼ ∼ site to the direction of motion of m. Now, the integration of equation (21) over the velocity space of stars is required. Writing equation (21) as 2 2 2 where y = v /2σ and X vm/(√2σ). Integrating by dv ∞ ρ(v )(v v ) ∗ ≡ m = G ∗ ∗ − m d3v , (22) parts results in dt v v 3 ∗ Z0 ∗ m | − | n 2X 2 v v = 0 Erf(X) e−X , ρ( ∗) 4πG(m + m∗)m∗ ln Λf( ∗) , Im 4π − √π ≡ 2 it is noticed that represents the equivalent problem of where Erf(x) = (2/√π) x e−y dy is the error function. finding the gravitational field (acceleration) at the “spa- 0 v v If ρ0 = n0m∗, the density of the background of stars, and tial” point m generated by the “mass density” ρ( ∗). assume that m m , theR deceleration of m inside an From gravitational potential theory,6,47 the acceleration ≫ ∗ r homogeneous stellar system with isotropic velocity dis- at a particular spatial point is given by tribution is: ∞ r′ r′ r 3r′ r r ρ( )( ) d G ′ 3 ′ v a(r)= G − = ρ(r ) d r . d m v r′ r 3 − r3 = Γdf (25) Z0 | − | Z0 dt − 2 2 This is the known result that only matter inside a par- 4πG ln Λρ0m 2X −X Γdf 3 Erf(X) e . ticular radius contributes to the force. In analogy to the ≡ − vm − √π gravitational case, the acceleration is given by the total “mass” inside v∗ < vm, is III. AN ANALYTICAL EXAMPLE dv Gv vm m = m ρ(v ) d3v . dt − v3 ∗ ∗ m Z0 A simple application of Chandrasekhar’s formula (25) For an isotropic velocity distribution: for an homogeneous spherically symmetric stellar system, although not infinite, is presented. The problem consists dvm = v , (23) in determining the motion of a massive particle m subject dt −Cdf m vm to gravitational and dynamical friction forces. The stellar 16π2G2m (m + m )lnΛ f(v )v2 dv . system has a radius R and total mass M; see Figure 3. Cdf ≡ ∗ ∗ ∗ ∗ ∗ Z0 The equation of motion for m is This is called Chandrasekhar dynamical friction formula. d2r It shows that only stars moving slower than vm con- m = Fg + Fdf = m ϕ(r)+ m adf , (26) tribute to the drag force on the massive particle. dt2 − ∇ If stars have a Maxwellian velocity distribution func- where ϕ(r) is the gravitational potential, and adf is given tion, by equation (25). 2 2 n0 −v∗/(2σ ) Equation (26) is not in general tractable by analytical f(v∗)= e , (24) (2πσ2)3/2 methods, so some approximations are required. Zhao35 has found an approximation to the term associated with the integral in (23) in done by an elementary method. In the velocity distribution in equation (25), namely: dimensionless form it is
X 2 n 2 1 2X −X 1 = 0 e−y y2 dy , χ(X) Erf(X) e , (27) Im π3/2 ≡ X3 − √π ≈ 4 + X3 Z0 3 5
2 2 2 that works to within 10 percent for 0 X < . When where 2β = γ/m and ξ = β ω0, with ω0 = k/m. m moves slow in comparison to the velocity≤ dispersion∞ of The behavior of m is dictated− by the relative values of β stars, v σ, χ(0) 3/4, and when v = σ/2 χ(1) 3/7. and ω. The values of ln Λ andp σ are first to be estimated. ≪ ≈ ≈ 2 Note that in the case of a very fast relative motion of m Take bmax R and bmin = Gm/V0 . An estimate of V0 dynamical friction is negligible; a situation analogous to may be obtained≈ from the virial theorem,10 that relates when a block of material slides fast. Using the previous the kinetic T and potential energy W of the system by: approximation, and considering χ = 3/4, the frictional force F in equation (26) becomes: 2T = W . (35) df − 2 2 3πG ln Λρ0m For an homogeneous system of size R the potential energy Fdf = madf v = γ v , (28) ≈− (√2σ)3 − is R 2 2 2 3 3 GM where γ =3πG ln Λρ0m /(√2σ) . W = 4πG ρ Mr dr = , (36) F − 0 −5 R To determine g recall that the potential is related to Z0 the density through Poisson equation, 2 and the kinetic energy is taken as T = MV0 /2. This 2ϕ(r)=4πGρ(r) , (29) leads to ∇ whose solution for a spherically symmetrical system of 2 3 GM 2 V0 3σ ; (37) radius R is ≈ 5 R ≈
1 r R where the last term provides an estimate of the one- ϕ(r)= 4πG ρ(r)r2 dr + ρ(r)r dr . (30) dimensional velocity dispersion under the assumption of − r " Z0 Zr # isotropy in the velocity distribution of stars. Using equa- tion (37) β and ω are: In a constant density ρ0 system the potential is 2 2 1 2 45 5 G mM ln Λ GM ϕ(r)= 2πGρ0 (R r ) , (31) β = 3 3/2 , ω0 = 2 . (38) − − 3 16r2 R (GM/R) r R and the gravitational force on m is The resulting Coulomb logarithm is ln Λ = ln[3M/(5m)]. 4 To compare the numerical values of β and ω0 is better Fg = m ϕ(r)= πGρ0m r = k r , (32) to use another system of units than a physical one. Let − ∇ −3 − G = M = R = 1, that is a common choice in N-body with k =4πGρ0m/3. This is the well known result from simulations in astronomy; to return to physical units one introductory mechanics that a particle inside an homoge- can use Newton’s law and set G to the appropriate value neous gravitational system performs a harmonic motion. (see Appendix). In these units, relations (38) become Combining equations (28) and (32) the resulting equa- tion of motion is 45 5 3 β = m ln , ω0 =1 . (39) 2r 16 2 5m d r v r m 2 + k + γ =0 . (33) dt If m =1/100 then lnΛ 4 and β 0.2 <ω . Hence an ≈ ≈ 0 This is the same equation, for example, as that of a underdamped harmonic motion for the massive particle results. If m = 1/10 then ln Λ 2 and β 0.8 ω , mass attached to a spring with stiffness constant k in- ≈ ≈ ≈ 0 side a medium of viscosity γ; i.e., a damped harmonic so the motion of m will be strongly damped. Note that oscillator.10,11,12,13 The solution of equation (33) in a an upper limit to m is set when m = 3/5, leading to plane, under arbitrary initial conditions ln Λ = 0; i.e., no dynamical friction results. For larger m a negative β is obtained. Clearly, the model fails and the x(0) = x0, x˙(0) = u0 ; y(0) = y0, y˙(0) = v0 , behavior of the dynamics is unrealistic. For cases of interest, where m M, it follows that 48,49 ≪ where the dot indicates a time derivative, is β<ω0 and the resulting motion (34), after some algebra, −(β+ξ)t is e 2ξt x(t) = (e 1)u0+ 2R − u0 + βx0 −βt x(t) = x0 cos ωt + sin ωt e , (e2ξt 1)β + (e2ξt + 1)ξ x , ω − 0 v0 + βy0 −βt y(t) = y0 cos ωt + sin ωt e , (40) −(β+ξ)t ω e 2ξt y(t) = (e 1)v0+ 2ξ − 2 2 2 where ω = ω0 β . Note that a time-scale when the (e2ξt 1)β + (e2ξt + 1)ξ y , (34) orbit decays 1/e− is given by τ =1/β. − 0 df 6
stellar system is considered, both using a semi-analytical method and N-body simulation, to illustrate further the application of dynamical friction.
A. Semi-analytic treatment
A simple representation of a stellar system, such as a globular cluster or an elliptical galaxy, is provided by the Plummer model. Its potential and stellar density are, respectively:6,7
GM 3Ma2/4π ϕ(r)= , ρ(r)= , (41) −(r2 + a2)1/2 (r2 + a2)5/2
where M is the total mass, and a the scale-radius of the system. In a spherical system with isotropic velocity distribution the equation of “hydrostatic” equilibrium62 is satisfied:
2 1 d(ρσ ) dϕ 2 ϕ(r) FIG. 4: Orbital decay of a massive particle m in a homoge- = σ (r)= . (42) ρ dr dr 6 neous stellar system due to dynamical friction. Initial con- − → − ditions are (x0,y0) = (0, 0.59) and (u0, v0) = (0.44, 0). Left The last result follows from noticing that ρ ϕ5, and panels are for m = 0.01 and right ones for m = 0.02. Top 2∝ panels show the orbit (solid line) in the xy-plane and bottom imposing boundary conditions that both ρσ and ϕ go ones the time evolution of the distance r from the center. The to zero at infinity. dynamical friction time scale τdf = 1/β is indicated by an ar- Equations (41) and (42) will be used in equation (25) row. Dashed lines are orbits without considering dynamical to compute the orbital motion of a massive particle m. It friction. rests to determine bmin and bmax. The former is evaluated 2 at local values, bmin = Gm/[3σ (r)], and the latter is set fix to bmax =a. Take as example m = 0.01. If the initial position of The equation of motion (26) for m can now be inte- m is at (x, y)=(0.59, 0.0) and its velocity (0.0, 0.44) the grated numerically using standard methods,50,51 or using resulting orbit is that shown in Figure 4-left with solid the one discussed by Feynman9 for planetary orbits ( 9). line. Doubling m results in the orbit shown in Figure 4- Here a fourth-order Runge-Kutta algorithm with adap-§ right. The dashed lines in correspond to the orbit of m tive time-step was used. The initial conditions for m are without dynamical friction. The effect of increasing the the same as those used in the analytical case. mass of m on the orbit, and on the decay time τdf , is In Figure 5 the resulting orbit from the numerical in- clearly appreciated. tegration is shown as a dashed line. Also, the behavior of This example shows the basic features of, for example, the x and y coordinates, and of the distance r of m to the the orbital decay of a satellite galaxy toward the cen- center, as a function of time are shown. The typical de- ter of its host larger galaxy. It may be applied also to cay of the orbit is evident. In the same figure results from the motion of a massive black hole near the center of a an N-body simulation are displayed, that are described galaxy or star cluster, where to some approximation the next. gravitational potential can be taken as harmonic. More realistic situations require however the numerical integra- tion of the orbit and/or an N-body computer simulation. B. N-body simulation A particular case of these are treated next. The use of N-body simulations allows to study more realistically the different dynamical phenomena that oc- IV. A MORE REALISTIC EXAMPLE cur in stellar systems.8,52 Several N-body codes with dif- ferent degrees of sophistication have been developed for Chandrasekhar’s formula (25) although derived as- astronomical problems in mind.53,54,55 Some low-N sim- suming an infinite homogeneous system may be ap- ulations can be run nowadays using a personal computer plied, to some degree, when stellar systems are non- with publicly available N-body codes.63 homogeneous.6 In this case, local values for the density Barnes’ tree-code in Fortran, and some of his pub- ρ(r) and the velocity dispersion σ(r) are used. Here the lic subroutines are used to simulate the motion of m motion of a massive particle m inside a non-homogeneous inside a Plummer model. A numerical realization of 7
Both approximations overestimate the decay rate of m in comparison to the N-body simulation. Taking bmax = a/5 leads to a somewhat better agreement, but does not reproduce the N-body result. Rather surprisingly, the analytical result does a fair job in reproducing the overall orbital decay in this case.
V. FINAL COMMENTS
The approximations in deriving Chandrasekhar for- mula limits, obviously, its application to more complex stellar systems than the one considered here. However, it is remarkable that equation (21) leads to reasonably well results when used with values under a local approx- imation. In similar vain to the study of the friction between surfaces,15,16,17 dynamical friction is a complex subject. Elaborate calculations based on Brownian motion,56 lin- ear response theory, resonances, and the fluctuation- 30,57,58 FIG. 5: N-body simulation of the orbital decay (solid line) dissipation theorem exist. These that are steps for- of a massive particle m inside a Plummer stellar model. The ward toward a more complete physical theory for this semi-analytical (dashed line) and the analytical calculation process. (dotted line) of Figure 4 are drawn for comparison. These Instead of listing explicitly some of the shortcomings overestimate the effect of dynamical friction in comparison to of Chandrasekhar dynamical friction formula6 when ap- the numerical simulation. plied to gravitational systems, the student is encourage to think on some of them and possible improvements on such formula. this model with N = 105 particles is used with indi- From the point of view of an introductory or interme- vidual “star” masses of m =1/N. The massive particle ∗ diate class on mechanics the exposure of students to non- m = 1/100 with initial conditions (y , x˙ ) = (0.59, 0.44) 0 0 typical problems, as the one presented here contributes is set “by hand” inside the numerical Plummer model. to further their understanding and appreciation of the In N-body units the scale radius is a = 3π/16 = 0.59 subject. (see Appendix). Some ideas that may lead to problems and/or projects The circular period at radius r is τ =2πr/V (r), where c for students are: the circular velocity and integrated mass for a Plummer model are given, respectively, by: 1. How would the analytical solution considered here would be changed if the Plummer model is used? GM(r) M (r/a) What type of approximations would be required to Vc(r)= ,M(r)= 2 3/2 . r r [1 + (r/a) ] make? How does ln Λ change? From this, an orbital period of τa = 4.8 time units at 2. If σ2 is a measure of the kinetic energy per unit r = a results. The simulation was run for t = 10 2τa tree-code≈ mass of stars, what is an estimate for its mean in- time units. The parameters for running the crease due to the energy lost by the massive particle in serial were those provided by Barnes at his Internet during its decay? site for an isolated Plummer evolution. The quadrupole moment in the gravitational potential is activated. The 3. How would the orbital decay time be changed for simulation took about 5.2 cpu hours on a PC with an different types of initial eccentricities of the massive Athlon 2.2GHz processor, and 512 KB of cache size. En- particle? ergy conservation was 0.04 percent, that is considered ≤ 6 very good. 4. Consider a star cluster (m = 10 M⊙) in circular Figure 5 shows the orbital evolution of the massive orbit at a distance of r = 5kpc from the center of 11 particle m in the N-body system as a solid line. The our galaxy (M 6 10 M⊙, R 150 kpc). Would dashed line corresponds to the semi-analytical calculation it be expected≈ to fall× to the center≈ within the age of Section IV A. This follows closely the orbit of m in the of the universe, say t = 1010 yr? Typical velocities N-body simulation for about τa time units. Afterwards, for stars and dark matter particles at that distance it deviates from the N-body result. In the r-t panel, the are about 200 km/s, and the scale-radius may be analytical solution (40) is shown as a dotted line; that is, around 5 kpc. What if instead of a star cluster assuming the total system was homogeneous. we have a galaxy satellite, such as the Magellanic 8
TABLE I: Astronomical units TABLE II: From N-body units to astronomical
Unit Equivalence Stellar system ul um uv ut a × 11 Astronomical unit AU = 1.496 10 m M⊙ km/s Myr 5 Parsec pc= 2.063 × 10 AU Globular cluster 50 pc 106 9.3 5.3 = 3.261 light-years Galaxy 10 kpc 1011 207.4 47.2 3 Kiloparsec kpc=10 pc Cluster of galaxies 5 Mpc 1015 927.4 5271.4 30 Solar mass M⊙ = 1.989 × 10 kg Year yr = 3.156 × 107 s
aMean sun-earth distance The Gravitational constant can be expressed in terms of typical astronomical values, for example, as:
10 2 3 Clouds, with m 10 M⊙ and at a distance of −3 km pc −3 pc ≈ G =4.3007 10 2 =4.4984 10 . 100 kpc? × s M⊙ × Myr M⊙ 5. How do results change if instead of a Plummer The transformation of G using length units such as kpc 6 model a more pronounced density profile is used, or Mpc (10 pc) is direct. Choosing ul and um the unit 59 such as the Hernquist model? How does the num- of velocity and of time ut, under an appropriate G value, ber of particles N in a simulation affect the decay are rate? 3 Gum u 6. As the massive particle moves through the stellar u = , u = l . v u t Gu system it induces a density wake behind it. Can r l s m this be detected in an N-body simulation on a home In this way the transformation from N-body units, computer? How about looking for this wake in the where G = M = R = 1, to physical ones can be made. phase-space diagram (e.g. a plot ofx ˙–x) of stars Table II lists some values for different choices of u and near the the massive particle? l um, and the resulting units of uv and ut. The entries 7. How good do the local approximation works if in- correspond to using the approximate size and mass of a stead of a massive particle one has an extended globular cluster, a disk of a spiral galaxy, and of a cluster object, small in comparison to its host galaxy? of galaxies, respectively, as units ul and um. In the standardized gravitational N-body units8,60 the Textbook problems are designed in general to yield one total energy of a system is E = 1/4. This follows from − correct answer, the above ideas for problems are rather the virial theorem (2T + W = 0), where vague but this is on purpose. The reason is twofold. On 1 GM W GM one hand, to promote in students a spirit of research by W = E = = . setting an approximate physical model and to look for the −2 R → 2 4R required data and “tools” to solve it; some of them can Here R is strictly what is called the virial radius of the be found in the references. On the other hand, no single system; that does not necessarily coincides with the total definite answer can be given. A feature proper of the extent of the stellar system, but is a very good approxi- way physics evolves toward describing and understanding mation. The potential energy of a Plummer model is nature. 1 ∞ 3π GM 2 W = ρ(r)ϕ(r)4πr2 dr = . 2 − 32 a APPENDIX A: ASTRONOMICAL AND N-BODY Z0 UNITS Thus the total energy is E = (3πGM 2)/(64a). In N- body units this leads to a value− of the Plummer scale- Several quantities in astronomy are so large in compar- radius of a =3π/16. ison to common “terrestial” values, that special units are used. Table I lists some of these and their equivalences in physical units. In the mks system of units the Gravitational constant is G = 6.67 10−11 m3 kg−1 s−2. A natural system of units for gravitational× interactions is that where the gravitational constant is set to G = 1; in the same way as for quantum systems Planck’s constant is usually set 2 toh ¯ = 1. On dimensional grounds [G]= uvul/um; where um, ul, and uv correspond, respectively, to units of mass, length and velocity. 9
∗ Electronic address: [email protected] Astrophys. J. 515, 50 (1999). † Electronic address: [email protected] 35 H. Zhao, Mon. Not. R. Astron. Soc. 351, 891 (2004). 1 J. E. Marsden and T. S. Ratiu, Introduction to Mechanics 36 J. S. Bullock and K. V. Johnston, Astrophys. J. 635, 931 and Symmetry (Springer, NY, 1999). (2005). 2 D. Hestenes, New Foundations for Classical Mechanics 37 S. S. Kim, D. F. Figer, and M. Morris, Astrophys. J. Lett. (Kluwer, Dordrecht, 1999). 607, L123 (2004). 3 C. D. Murray and S. F. Dermot, Solar System Dynamics 38 P. Goldreich, Y. Lithwick, and R. Sari, Nature (London) (Cambridge University Press, NY, 1999). 420, 643 (2002). 4 F. Diacu and P. Holmes, Celestial encounters: the ori- 39 A. Del Popolo, S. Yesilyurt, and N. Ercan, Mon. Not. R. gins of chaos and stability (Princeton University Press, NJ, Astron. Soc. 339, 556 (2003). 1996). 40 P. P. Avelino and E. P. S. Shellard, Phys. Rev. D 51, 5946 5 D. Boal, Mechanics of the Cell (Cambridge University (1995). Press, Cambridge, 2002). 41 J. R. Taylor, Classical Mechanics (University Science 6 J. Binney and S. Tremaine, Galactic Dynamics (Princeton Books, CA, 2005). University Press, NJ, 1987). 42 T. W. B. Kibble and F. H. Berkshire, Classical Mechanics 7 W. C. Saslaw, Gravitational Physics of Stellar and Galactic (Imperial College Press, London, 2004). Systems (Cambridge University Press, Cambridge, 2003). 43 R. L. Spencer, Am. J. Phys. 73, 151 (2005). 8 S. J. Aarseth, Gravitational N-body Simulations (Cam- 44 W. A. Mulder, Astron. Astrophys. 117, 9 (1983). bridge University Press, Cambridge, 2003). 45 R. T. Coffman, Mathematics Magazine 36, 271 (1963). 9 R. P. Feynman, R. B. Leighton, and M. Sands, Feyn- 46 J. W. Adolph, A. Leon Garcia, W. G. Harter, R. R. Shiff- man lectures on physics, Volume 1 (Addison-Wesley, MA, man, and V. G. Surkus, Am. J. Phys. 40, 1852 (1972). 1963). 47 G. W. Collins II, The Foundations of Ce- 10 M. Alonso and E. Finn, Physics (Addison-Wesley, MA, lestial Mechanics (Parchart, AZ, 1989), URL 1992). http://ads.harvard.edu/books/1989fcm..book. 11 D. Halliday, R. Resnick, and K. S. Krane, Physics, Vol- 48 R. Weinstock, Am. J. Phys. 29, 830 (1961). ume 1 (Wiley, NY, 2002). 49 R. S. Luthar, The Two-Year College Mathematics Journal 12 C. Kittel, W. Knight, and M. A. Ruderman, Mechanics. 10, 200 (1979). Berkeley Physics Course, Volume 1 (McGraw-Hill, NY, 50 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and F. B. 1973). P., Numerical Recipes: The Art of Scientific Computing 13 A. P. French, Newtonian Mechanics. MIT Introductory (Cambridge University Press, NY, 1992). Physics Series, Volume 1 (Norton, NY, 1971). 51 A. L. Garcia, Numerical Methods for Physics (Prentice 14 F. Palmer, Am. J. Phys. 17, 336 (1949). Hall, NJ, 2000). 15 E. Rabinowicz, Am. J. Phys. 31, 897 (1963). 52 R. W. Hockney and J. W. Eastwood, Computer simulation 16 J. Krim, Am. J. Phys. 70, 890 (2002). using particles (Hilger, Bristol, 1988). 17 J. Ringlein and M. O. Robbins, Am. J. Phys. 72, 884 53 J. Barnes and P. Hut, Nature (London) 324, 446 (1986). (2004). 54 V. Springel, Mon. Not. R. Astron. Soc. 364, 1105 (2005). 18 D. G. Parkyn, Am. J. Phys. 26, 436 (1958). 55 W. Dehnen, Astrophys. J. Lett. 536, L39 (2000). 19 I. R. Lapidus, Am. J. Phys. 38, 1360 (1970). 56 S. Chandrasekhar, Rev. Mod. Phys. 21, 383 (1949). 20 M. I. Molina, Phys. Teach. 42, 485 (2004). 57 J. D. Bekenstein and E. Maoz, Astrophys. J. 390, 79 21 J. C. Simbach and J. Priest, Am. J. Phys. 73, 1079 (2005). (1992). 22 B. Sherwood, Am. J. Phys. 51, 597 (1983). 58 R. W. Nelson and S. Tremaine, Mon. Not. R. Astron. Soc. 23 A. J. Mallinckrodt and H. S. Leff, Am. J. Phys. 60, 356 306, 1 (1999). (1992). 59 L. Hernquist, Astrophys. J. 356, 359 (1990). 24 A. B. Arons, Am. J. Phys. 67, 1063 (1999). 60 D. Heggie and R. Mathieu, Standardised Units and Time 25 C. Sagan, Cosmos (Random House, NY, 1980). Scales. in: The Use of Supercomputers in Stellar Dynam- 26 T. T. Arny, Explorations: an introduction to astronomy ics. S. L. W. McMillan and P. Hut, eds. (Springer Verlag, (Mosby-Year Book, MO, 1994). Berlin, 1985). 27 61 2 F. H. Shu, Physical Universe (University Science Books, Contributions from (∆v⊥) are linked to the concept of relax- CA, 1982). ation time in stellar systems.6,7 28 S. Chandrasekhar, Astrophys. J. 97, 255 (1943). 62 In mechanical equilibrium a change in pressure dP is balanced 29 S. Chandrasekhar, Principles of Stellar Dynamics (Dover, by the gravitational “force” −ρ(r)∇ϕ(r) dr. The pressure is here 2 NY, 1960). P = ρσ , similar to that of an ideal gas where P = kT ρ/m. 30 M. D. Weinberg, Mon. Not. R. Astron. Soc. 239, 549 The equation used is a particular case of that called in stellar dynamics Jeans equation. (1989). 63 tree-code 31 The reader may obtain, for example, Barnes’ at H. Velazquez and S. D. M. White, Mon. Not. R. Astron. http://www.ifa.hawaii.edu/faculty/barnes/software.html. The Soc. 304, 254 (1999). site contains also programs, both in C and Fortran, to generate 32 M. Fujii, Y. Funato, and J. Makino, Publ. Astron. Soc. some stellar systems and initial conditions. The Gadget code Jap. (2005), astro-ph/0511651. of Springel is at http://www.mpa-garching.mpg.de/gadget/. 33 S. L. W. McMillan and S. F. Portegies Zwart, Astrophys. Dehnen’s tree-code is included in the Nemo package under gyr- J. 596, 314 (2003). falcON at http://bima.astro.umd.edu/nemo/. 34 F. C. van den Bosch, G. F. Lewis, G. Lake, and J. Stadel,