8
Stellar-Dynamical Chapter Systems
A wide range of self-gravitating systems may be idealized as congurations of point masses interacting through gravity. But in galaxies, the effects of interactions between individual stars accumulate so gradually that they can be neglected even over timescales vastly longer than the age of the universe. This permits a simpler description, the collisionless Boltzmann equation (BT87, Ch. 4).
8.1 N-body Equations of Motion
Any system in which physical collisions are rare may be idealized as a collection of N point-sized bodies, each with mass mi , position ri , and velocity vi . The hamiltonian for such a system is H ri vi
2 mi v i 2
1
i
iji
Gmi m j r j ri
(8.1)
where H depends on all body positions and velocities, the rst sum runs over all N bodies, the second runs over all pairs of bodies, and G is the gravitational constant. Then the equations of motion are dri dt dvi dt
vi
j
Gm j r j ri r j ri 3 i
(8.2)
where the sum runs over all bodies except body i. N-body systems obey several basic conservation laws. In BT87 (Appendix 1.D.2) these laws are derived by manipulating (8.2). However, they may also be recognized directly from the form of the hamiltonian; Noethers theorem states that each symmetry of H gives rise to a conservation law. A symmetry is a transformation which leaves the physical system unchanged. For example, translation in time, t t t, is a symmetry of (8.1) because H is not an explicit function of time; consequently the total system energy E T U H is conserved. Likewise, symmetry with respect to translation in space, r r r, implies conservation of total linear momentum, and symmetry with respect to rotation gives rise to conservation of total angular momentum. 59
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CHAPTER 8. STELLAR-DYNAMICAL SYSTEMS
8.2 Virial Parameters
Another general result shown by manipulating (8.2) is the scalar virial theorem (BT87, Chapter 8.1.1), which states that for a system in equilibrium, 2T
U
0
(8.3)
where T and U are the total kinetic and potential energy, respectively, and the angle-brackets indicate time-averages. Since E T U, the time-averaged kinetic and potential energies are related to the conserved total energy by T
E
U
2E
(8.4)
The total mass M and total energy E of an N-body system thus dene characteristic velocity and length scales M2 M2 T E 2 RV G G M M U 2E These are sometimes known as the virial velocity and radius, respectively.
2 VV
2
(8.5)
The quantity tc RV VV is an estimate of the time a typical body takes to cross the system. This timescale may be expressed in several different ways; for example, in terms of the total mass M and energy E, it is tc G
M5 8 E
3
(8.6)
Note that M and E are conserved, so tc is a constant even for systems which are far from dynamical equilibrium. In such cases tc approximates the time-scale over which the system evolves toward equilibrium.
2 Another expression for tc follows from the substitution VV equilibrium:
GM RV valid for systems near (8.7)
tc
GM
R3 1 V
2
Here the quantity M R3 , which has units of density, appears. In systems with galaxy-like density V proles, the virial radius is approximately proportional to the half-mass radius: Rh 0 4RV . Using this relationship, it follows that tc 1 36Gh1
2
(8.8)
where h is the mean density within Rh . Since the crossing time is just supposed to indicate a typical time-scale for orbital motion, it is usual to drop the numerical constant, and dene tc
Gh
1
2
(8.9)
8.3 Relaxation Time
Consider an encounter with impact parameter b and velocity v between two stars of mass m, shown in Fig. 8.1. Using the impulse approximation, the transverse velocity acquired is
vt
2Gm bv
(8.10)
The impulse approximation is well-justied in systems with large N because large-angle deections are very rare; the impact parameter leading to a large deection is bmin Gm v2 RV N (8.11)
8.3. RELAXATION TIME
61
b
vt
Figure 8.1: An impulsive passage; as a result of the gravitational pull of the passing star at top, the test stars acquires velocity vt . where the second equality follows by assuming that v VV and using the virial theorem. This is much smaller than the typical distance between particles, which is of order RV N 1 3 . In a typical stellar-dynamical system there is roughly one close encounter per crossing time, regardless of N. A key assumption made in considering the effects of interactions between individual stars is that encounters are not correlated with one another; thus collective effects are neglected. This assumption works well in many cases, though examples of collective relaxation will come up later in this course. If each encounter is uncorrelated with the last, the cumulative effect of many encounters is a random walk in velocity; perturbations add in quadrature. During a single passage through the system, a typical star has roughly N 2bdb (8.12) dn R2 V encounters with impact parameters between b and b db. Here the rst factor is just the surface density of stars, and the second factor is the area of an annulus with radius b and width db. Adding velocity perturbations in quadrature, the deection due to these dn encounters is dv2
v2 dn t
8N
Gm RV v
2
db b
(8.13)
and the total velocity perturbation acquired in one crossing time is v2
RV bmin
dv2
8N
Gm RV v
2
ln
RV bmin
(8.14)
Here the logarithmic factor arises from the integration over impact parameter from bmin to RV ; each decade between bmin and RV contributes equally to the total deection. Thus, even though a single wide encounter, with b bmin , scarcely perturbs the star, the cumulative effect of such encounters typically dominates the evolution of the system! Finally, estimating the encounter velocity v from the virial velocity v2 VV 8 ln N 2 VV N
GNm RV gives (8.15)
for the total change in a typical stars velocity per crossing time tc . The relaxation time is the time over which the cumulative effect of stellar encounters becomes comparable to a stars initial velocity. From (8.15) this is tr
2 VV N tc tc v2 8 ln N
(8.16)
In stellar systems with large N this time is much longer than the crossing time; the evolution of such systems proceeds on two widely-separated timescales. Relaxation due to stellar encounters plays an
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CHAPTER 8. STELLAR-DYNAMICAL SYSTEMS
important role in the evolution of star clusters. But a typical galaxy has 1011 stars but is less than 100 crossing times old, so the cumulative effects of encounters between stars are pretty insignicant. This justies the next step, which is to idealize a galaxy as a continuous mass distribution, effectively . taking the limit tr
8.4 Collisionless Dynamics
In the continuum limit, each star moves in the smooth gravitational eld r t of the galaxy. Thus instead of thinking about motion in a phase space of 6N dimensions, we can think about motion in a phase space of just 6 dimensions. This is a vast simplication!
8.4.1 Distribution function
Rather than keeping track of individual stars, a galaxy may be described by the one-body distribution function; let f r v td 3 rd 3 v (8.17) be the mass of stars in the phase-space volume d 3 rd 3 v at r v and time t. This provides a statistically complete description if stars are uncorrelated, as assumed above.
8.4.2 Collisionless Boltzmann equation
The motion of matter in phase space is governed by the phase-ow, r v
v
(8.18)
How does this affect the total amount of mass in the phase space volume d 3 rd 3 v? Consider the 2-D example shown in Fig. 8.2, where a cell of volume 2 r 2v is located at r0 v0 . The mass within the cell is 4 r v f r0 v0 t (8.19)
Matter ows in to the cell through the left and top sides, and out through the right and bottom sides; the rate of change of the mass within the cell is 2 v f r0 r v0 r r0 r v0 f r0 r v0 r r0 r v0
2 r f r0 v0 v r0 v0 v f r0 v0 vv r0 v0 v v
(8.20)
4rv
f r f v r v
r0 v0
Here the rst line is the difference between the mass owing in at the left and out at the right, the second line is the difference between the mass owing in at the top and out at the bottom, and the third line follows from recognizing these differences as derivatives. Equating the rate of change to the time derivative of the mass within the cell yields
f t
which is the equation of continuity.
f r f v r v
0
(8.21)
8.4. COLLISIONLESS DYNAMICS
63
v
v0
2v 2 r
r0
r
Figure 8.2: Phase space cell of volume 4 r v located at r0 v0 . The arrows represent the phaser space ow eld v v , which transports matter in to and out of the cell. The analogous continuity equation for a 6-D phase-space is
f t
f r f v r v
0
(8.22)
Using (8.18) for the phase-ow yields the collisionless Boltzmann equation:
f t
v
f f r v
0
(8.23)
The collisionless Boltzmann equation or CBE describes the evolution of the distribution function f r v t. It contains Newtons F ma [via (8.18] as well as conservation of matter, and therefore serves as the fundamental equation of galactic dynamics. In a galaxy we often deal with several distinct kinds of collisionless matter; for example, stars and dark matter1 . We can dene separate distribution functions fstars and fdark to describe these ...