364
The Galactic System of Stars
[ch. XIV
kinetic theory of gases with the collisions left out and a gravitational field
thrown in. Just as, in the kinetic theory, the gas may be imagined divided up
into a system of showers of parallel-moving molecules, so in stellar dynamics
the stars may be imagined divided up into a system of parallel-moving clusters.
But there is the essential difference that in stellar dynamics these clusters
retain their identity through long periods of time, whereas in gas-theory they
do not.
Confining our attention to a small region dxdydz of space, let us suppose
that the number of stars within this region, whose velocity-components u,v,w
lie within prescribed limits dudvdw, is
f(u, v, w, x, y, z, t) dudvdw dxdydz (333'1).
If V is the gravitational potential of the whole system of stars, the motion
of each of these stars will be determined by the equations
du^dV dv = dV dw_dV
dt dx ’ dt dy ’ dt dz
After a time dt the parallel motion of these stars will take them to
a position x + udt, y + vdt, w + zdt, while their gravitational accelerations
dV
will have increased their velocity components to w + r- dt, etc. Hence
ox
expression (333’1) must be equal to
/ dV dV dV \
f\ +1 bx V d~ ^ w + dt, x + udt, y + vdt, z + wdt, t + dtj ,
since the stars specified in both groups are identical. We must accordingly
have
df dVdfdVdf dV df df df df „
+ a + t~ f- + uf-+vf + wf = 0... (333-2).
dt ox ou dy ov dz ow ox oy oz 7
This is the differential equation which must be satisfied by the distri
bution-function f throughout any motion whatever of a system of stars. It
will be seen to be identical with the corresponding equation in the kinetic
theory of gases, except that the terms arising from collisions are left out.
334. If the stars are in a state of steady motion, f does not change with
the time, so that the first term d// dt in equation (333‘2) must be omitted.
To solve the resulting equation for f Lagrange’s rule directs us to find all
possible integrals E x = cons., E 2 = cons., etc., of the system of equations
du dv dw dx dy dz
sV = dV = dv=T = 'J = w (3m >-
dx dy dz
The solution of the equation is then simply
/=<¿>0^, E 2} ...) (334-2)
where </> is any arbitrary function.