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 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.
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) , Vwhere 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) .
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 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 encountersHere 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 vThis is known as the relaxation time.
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 dx dv centered on (x,v) at time t. This is a complete description provided that stars are uncorrelated, as above.
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 dvwhere 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
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, phase-space density is conserved along any orbit.
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.
Last modified: January 26, 1995