Idealizing a galaxy as a collection of point masses enables us to write down the N-body equations of motion. An estimate for the time-scale for interactions between stars to lead to significant deflections shows that the mass distribution in a galaxy can be approximated as continuous. This leads to a simpler description, the collisionless Boltzmann equation (BT87, Chapter 4).

As physical collisions between stars are extremely rare, a galaxy
may be idealized as a collection of `N` point-sized
*bodies*, each with mass `m_i`, position `x_i`, and
velocity `v_i`, governed by the two-body interaction potential

(1) u(r) = - G / r ,where

-- dx_i dv_i \ x_ji (2) ---- = v_i , ---- = G | m_j ------- , dt dt / 3 -- j |x_ji|where

By manipulating the above equations, it is possible to prove several important and useful facts (BT87, Appendix 1.D.2):

**1**. Conservation of total energy:

-- -- 1 \ 2 \ (3) E = T + U = - | m_i v_i - G | m_i m_j / |x_ji| . 2 / / -- i -- i, j < i

**2**. Conservation of total linear momentum.

**3**. Conservation of total angular momentum.

**4**. The scalar virial theorem (BT87, Chapter 8.1.1): `<
2T + U > = 0`, where the angle brackets indicate a time-average.

Given the total mass `M` and total energy `E` of an
N-body system, one may define a characteristic velocity and length

2 2 |E| M (4) V = 2 --- , R = G ---- . M 2|E|These are sometimes known as the

In systems with galaxy-like density profiles, the virial radius is
approximately proportional to the half-mass radius: `R = ~2.5
R_h`. Using this relationship, it follows that

R -1/2 (5) - = ~1.36 (G rho_h) , Vwhere

-1/2 (6) t_c = (G rho_h) .

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

2 G m (7) v_t = ----- . b vThis approximation is justified because large-angle deflections are rare; the impact parameter leading to a large deflection is

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

A key assumption made in considering the effect of many stellar
encounters is that distinct encounters are *uncorrelated* with
each other; collective effects are neglected. This assumption seems
well-justified in practice. Because each encounter is uncorrelated
with the last, the cumulative effect of many encounters is equivalent
to a *random walk*; velocity perturbations add in quadrature.
Thus the total transverse velocity acquired in one crossing time is

-- 2 \ 2 2 (9) V_t = | v_t = ~ 8 N (G m / R v) ln(R / b_min) . / -- all encountersHere the logarithmic factor arises by converting the sum to an integral over impact parameter

The characteristic time-scale over which the cumulative effect of stellar encounters becomes comparable to a star's initial velocity is

2 V_t N (10) t_r = --- t_c = ~ ------ t_c . 2 8 ln N vThis is known as the

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 the
motion of one point in a phase space of `6N` dimensions, we can
think about the motion of `N` points in a phase space of just
`6` dimensions.

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

(11) f(x,v;t) dx dvbe the mass of stars in the phase space volume

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

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

df d . d . (13) -- + -- (f x) + -- (f v) = 0 , dt dx dvwhere the derivatives with respect to

df df d df (14) -- + v . -- - -- Phi . -- = 0 . dt dr dx dv

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

/ | (15) div grad Phi = 4 pi G | dv f(r,v;t) . | /

Let `(x,v) = (x(t),v(t))` be the orbit of a star. What is
the rate of change of `f(x,v;t)` along the star's trajectory?
Using Eq. 14, the answer is

Df df . df . df (16) -- = -- + x . -- + v . -- Dt dt dx dv df df d dv = -- + v . -- - -- Phi . -- = 0 . dt dx dx dvIn other words,

Due date: 2/2/95

9. 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.

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

Last modified: January 26, 1995