A wide range of self-gravitating systems may be idealized as configurations of point masses interacting through gravity. But in galaxies, the timescale over which the cumulative effects of interactions between individual stars cause significant deflections is so long that galaxies can be accurately approximated as continuous systems. This produces a simpler description, the collisionless Boltzmann equation (BT87, Chapter 4).

Any system in which stellar collisions are rare may be idealized as
a collection of `N` point-sized *bodies*, each with mass
`m_i`, position `r_i`, and velocity `v_i`. The
hamiltonian for such a system is

-- -- 1 \ 2 G \ m_i m_j (1) H(r,v) = - | m_i v_i + - | ----------- , ~ ~ 2 / 2 / |r_i - r_j| -- -- i i, j#iwhere

-- d(r_i) d(v_i) \ m_j (r_j - r_i) (2) ------ = v_i , ------ = G | --------------- , dt dt / |r_j - r_i|^3 -- j#iwhere the sum runs over all bodies except body

N-body systems obey several basic conservation laws. In BT87
(Appendix 1.D.2) these laws are derived by manipulating Eq. 2.
However, they may also be recognized directly from the form of the
hamiltonian; Noether's theorem states that each *symmetry* of
`H` gives rise to a corresponding conservation law. A symmetry
of the hamiltonian is a transformation which leaves the physical
system unchanged. For example, translation in time, `t -> t +
Delta t`, is a symmetry of Eq. 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 + Delta r`, implies
conservation of total linear momentum, and symmetry with respect to
rotation gives rise to conservation of total angular momentum.

Another general result shown by manipulating Eq. 2 is the *scalar
virial theorem* (BT87, Chapter 8.1.1), which states that for a
system in equilibrium,

(3) 2<T> + <U> = 0 ,where

(4) <T> = - E , <U> = 2 E .

The total mass `M` and total energy `E` of an N-body
system thus define characteristic velocity and length scales

2 2 2 <T> |E| M M (5) V = 2 --- = 2 --- , R = - G --- = G ---- . M M <U> 2|E|These are sometimes known as the

The quantity `t_c = R/V` is an estimate of the time a typical
star 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

1/2 (6) t_c = G (M^5 / 8 |E|^3) .Note that

Another expression for `t_c` follows from the substitution
`V^2 = G M/R` valid for systems near equilibrium:

-1/2 (7) t_c = (G M / R^3) .Here the quantity

-1/2 (8) t_c = ~1.36 (G rho_h) ,where

-1/2 (9) t_c = (G rho_h) .

Consider an encounter with impact parameter `b` and velocity
`v` between two stars of mass `m`. Using the impulse
approximation, the transverse velocity acquired is

2 G m (10) v_t = ----- . b vThis approximation is well-justified in systems with large

2 (11) b_min = G m / v = ~ R / N ,where the second equality follows from the virial theorem. In the entire system, approximately one close encounter occurs per crossing time, regardless of

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 may 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

(12) dn = (N / pi R^2) 2 pi b dbencounters with impact parameters between

2 2 2 (13) d v = v_t dn = 8 N (G m / R v) db/b ,and the total velocity perturbation acquired in one crossing time is

/ R 2 | 2 2 (14) Delta v = | d v = 8 N (G m / R v) ln(R/b_min) . | / b_minHere the logarithmic factor arises from the integration over impact parameter from

2 8 ln N 2 (15) Delta v = ------ V Nfor the total change in a typical star's velocity per crossing time

The **relaxation time** is the time over which the cumulative
effect of stellar encounters becomes comparable to a star's initial
velocity. From Eq. 15 this is

V^2 N (16) t_r = --------- t_c = ------ t_c . Delta v^2 8 ln NIn stellar systems with reasonably large

A typical galaxy has `10^11` stars but is only `100`
crossing times old, so the cumulative effects of encounters between
stars are pretty insignificant. This justifies the next step, which
is to idealize a galaxy as a continuous mass distribution, effectively
taking the limit `t_r -> infinity`.

In this limit, each star moves in the smooth gravitational field
`Phi(x,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
simplification!

Rather than keeping track of individual stars, a galaxy may be described by the one-body distribution function; let

(17) f(r,v,t) dr dvbe the mass of stars in the phase-space volume

The motion of matter in phase space is governed by the phase-flow,

. . (18) (r, v) = (v, - grad Phi) .How does this affect the total amount of mass in the phase space volume

df d . d . (19) -- + -- (f r) + -- (f v) = 0 , dt dr dvwhere the derivatives with respect to

df df df (20) -- + v . -- - grad Phi . -- = 0 . dt dr dvThe collisionless Boltzmann equation or CBE describes the evolution of the distribution function

The gravitational field `Phi(x,t)` is given by Poisson's
equation,

/ | (21) div grad Phi = 4 pi G | dv f(r,v,t) . | /Eqs. 20 & 21 may be viewed as a pair of coupled PDEs which together completely describe the evolution of a galaxy.

Let `(r,v) = (r(t),v(t))` be the orbit of a star. What is
the rate of change of `f(r,v,t)` along the star's orbit? The
answer is

Df df . df . df (22) -- = -- + r . -- + v . -- Dt dt dr dv df df df = -- + v . -- - grad Phi . -- = 0 , dt dr dvwhere the first equality is just the definition of the convective derivative in phase-space, the second equality follows on substituting the phase-flow (Eq. 18), and the last equality follows from the CBE (Eq. 20). Thus,

This fundamental and completely general result shows that the CBE
has a much greater level of symmetry than the N-body equations of
motion; whereas the latter conserves a fairly small set of parameters,
the CBE conserves `f(r,v,t)` along an *infinite* number of
stellar orbits. We can take advantage of this infinite array of
conservation laws to obtain some important results even when we can't
explicitly solve the CBE.

Due date: 2/18/97

8. Assuming an encounter velocity `v = 300 km/s`, compute
the large-angle impact parameter `b_min` for a solar-mass star,
and compare it to the radius of the Sun. Some useful numbers are
given in BT87, Appendix 1.A.

9. A physically meaningful phase-space distribution function must
never be negative. Prove that if `f(r,v,t_0)` is not negative
at time `t_0` then it is not negative at all later times.

Joshua E. Barnes (barnes@galileo.ifa.hawaii.edu)

Last modified: February 12, 1997