Lecture 3
A simple description of the mass distribution
and gravitational potential of a galaxy is to represent the stars as point
masses. This is a good approximation for elliptical galaxies, since these
usually contain very little gas. It is an adequate approximation for spiral/disk
galaxies, because they usually contain significantly more stars than gas.

Following Newtonian gravitational theory, we can represent the
potential F at a distance R from a point mass
by

F(R) = -GM/R.

and the force field F(r) around the point mass as

F(r) = -GMr/R3

where r is the vector from the point mass to the point
at which we evaluate the force. This may be more familiar in the form

F(r) =
-GMer/R2

where er  is the unit vector directed
along r.

In a galaxy, we have a large number of point masses, N, which
we denote by the subscript i, where i =1,2,...,N.

The gravitational potential is additive, so that the total
potential at some position in space denoted by the vector r, is given
by

F(r) = - S
GMi / |r-r'|

where we denote the particle masses by Mi

In galaxies, there are typically 109 to
1011 stars (or point masses) so that the sum above contains
109 to 1011 terms, clearly an impractical number to do by
hand.

There are two ways around this. Firstly, the stars can be
represented by point masses in a computer simulation. Secondly, the stars can be
considered to have a smooth distribution in space, so that we may apply
Poisson's Equation in order to recover the potential from the matter
distribution. Poisson's equation is derived in BT 2.1

The following results are from
Binney and Tremaine "Galactic Dynamics"

Princeton University Press 1987
--- (BT) see section 2.1

Poisson's Equation : div grad F = 4pGr.

The force F(r) generated at a
position r is given by

F(r) = -
grad F(r)

These two equations can be summarised in words as follows : from the
distribution of the stars in space r we can determine
the total potential F, and from the gradient of the
potential we can determine the force at any given point.

Newton derived two useful theorems concerning spherical shells
of matter.

1) A body inside a spherical shell of matter eperiences no net
gravitatinal force from the shell.

2) The gravitational force on a body lying outside a closed
spherical shell experiences a gravitational force which is the same as if all
the matter in the shell were concentrated at a point at its center.

These two theorems allow us to deduce that the gravitational
attraction of a spherically symmetric density distribution r(r') on a test mass at radius r is
determined by the mass interior to r. The mass exterior to r has no
effect.

A spherical mass distribution can therefore be characterised in
terms of a circular speed vcirc(r), which is the speed
a test particle would have in a circular orbit at radius r.

F(r) =  v2circ(r) / r

but since

F(r) = - grad
F(r) = -dF/dr

we have

v2circ(r) = r dF/dr = GM(r)/r

where M(r) is the mass interior to r.

The kinetic energy of a particle with mass m and velocity v is
defined as KE = mv2/2. Gravitational fields are conservative,
which means that the energy used to move between two points within the field is
independent of the path taken.  The total energy of the system is the sum
of the kinetic energy and gravitational potential energy:

E = KE + F.

In a conservative field the potential
can be defined to within a fixed constant with no effect on the dynamics of the
system. Consequently, it is quite common to define the total energy of the
system to be identically zero

KE + F = 0.

If a test particle of unit mass at infinity falls into this
gravitational potential field, then its KE at radius r is

KE = -F(r)

i.e. v2 = -2F(r).

Note that F is
always negative.

The escape speed vesc is defined by

v2esc =  2 |F(r)|

This is just the kinetic energy a particle needs at radius r to
move infinitely far from the center of the potential field.

Some basic examples
A point of mass M:

F(r) = -
GM/r

This is called a Keplerian potential, since it is that
obeyed by the planets in the solar system and Kepler's three laws.

The velocity of a circular orbit at radius r is

vcirc(r) = sqrt(GM/r).

A uniform sphere of mass M and radius a:

{ -2 p  G r (a2 - r2 /
3)        r < a
F(r)  =

{ -
GM/r
r > a

note that the density of the sphere is given by r = M / (4 p
a3 / 3). Inside the sphere the potential grows
parabolically to the surface, while outside the sphere it falls off as in the
Keplarian case (as we would expect).

An isothermal sphere has a density profile given
by

r(r) = r0 (r/r0)-2

and the potential is

F(r) = 4pG
r0 r20
ln(r/r0 )

The circular velocity is

vcirc =  sqrt ( 4 p G
r0 r20).

This is an interesting potential because the rotation speed is
constant (unchanging with radius). This makes it a useful potential when trying
to understand flat rotation curves (see end of  lecture 2).