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) ,
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) .
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.
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 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
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 dv
In 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