Collisionless Stellar-Dynamical Systems

Astronomy 626: Spring 1995

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).


The N-body Equations of Motion

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 G is the gravitational constant. Then the equations of motion are
                                    --
            dx_i            dv_i    \       x_ji
(2)         ---- = v_i ,    ---- = G | m_j ------- ,
             dt              dt     /            3
                                    -- j   |x_ji|
where x_ji = x_j - x_i and the sum runs over all bodies except body i.

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.

Characteristic Time-Scales

Crossing time

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 virial velocity and radius, respectively. Their ratio is an estimate of the time a typical star takes to cross the system.

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)    ,
            V
where rho_h is the mean density within R_h. Since the crossing time t_c is just supposed to indicate a typical time-scale for orbital motion, it is usual to drop the numerical constant, and define
                           -1/2
(6)         t_c = (G rho_h)    .

Relaxation time

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 v
This 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 N.

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 encounters
Here the logarithmic factor arises by converting the sum to an integral over impact parameter b from b_min to R. In effect, each decade between b_min and R contributes equally to the total deflection.

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
                   v
This is known as the relaxation time.

Collisionless Dynamics

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.

Distribution function

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 dv
be the mass of stars in the phase space volume dx dv centered on (x,v) at time t. This is a complete description provided that stars are uncorrelated, as above.

Collisionless Boltzmann Equation

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 dx dv? The rate of change of the mass is just the inflow minus the outflow; that is, the flow obeys a continuity equation in 6 dimensions:
            df   d     .    d     .
(13)        -- + -- (f x) + -- (f v) = 0 ,
            dt   dx         dv
where the derivatives with respect to t, x, and v are understood to be partial derivatives. Using the expression for the phase-flow yields the collisionless Boltzmann equation:
            df       df   d        df
(14)        -- + v . -- - -- Phi . -- = 0 .
            dt       dr   dx       dv

Gravity

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

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

Conservation of Phase Space Density

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       dv
In other words, phase-space density is conserved along any orbit.

Homework

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