In a one-dimensional system it is always possible to define a potential energy corresponding to any given f(x); let
/ x
|
(1) U(x) = - | dx' f(x') ,
|
/ x_0
where x_0 is an arbitrary position at which U = 0.
Different choices of x_0 produce potential energies differing
by an additive constant; this constant has no influence on the
dynamics of the system.
In a space of n > 1 dimensions the analogous path integral,
/ x
|
(2) U(x) = - | dx' . f(x') ,
|
/ x_0
may depend on the exact route taken from point x_0 to
x; if it does, a unique potential energy cannot be defined.
One condition for this integral to be path-independent is that the
integral of the force f(x) around all closed paths vanishes.
An equivalent condition is that there is some function U(x)
such that
(3) f(x) = - grad U .Force fields obeying these conditions are conservative. The gravitational field of a stationary point mass is the simplest example of a conservative field; the energy released in moving from radius r_1 to radius r_2 < r_1 is exactly equal to that consumed in moving back from r_2 to r_1.
In astrophysical applications it's natural to work with the path integral of the acceleration rather than the force; this integral is the potential energy per unit mass or gravitational potential, Phi(x), and the potential energy of a test mass m is just U = m Phi(x). For an arbitrary mass density rho(x), the potential is
/
| 3 rho(x')
(4) Phi(x) = - G | d x' -------- ,
| |x - x'|
/
where G=6.672x10-8cm3g-1sec-2
is the gravitational constant and the integral is
taken over all space.
Poisson's equation provides another way to express the relationship between density and potential:
(5) div grad Phi = 4 pi G rho .Note that this relationship is linear; if rho_1 generates Phi_1 and rho_2 generates Phi_2 then rho_1 + rho_2 generates Phi_1 + Phi_2.
Gauss's theorem relates the mass within some volume V to the gradient of the field on its surface:
/ /
| 3 |
(6) 4 pi G | d x rho(x) = | dS • grad Phi ,
| |
/ V / S
where the infinitesimal vector dS is an element of
surface area with an outward-pointing normal vector.
Consider a spherical shell of mass M; Newton's first and second theorems imply
/ r / r
| | M(x)
(7) Phi(r) = - | dx a(x) = G | dx ---- ,
| | x^2
/ r_0 / r_0
where the enclosed mass is
/ r
| 2
(8) M(r) = 4 pi | dx x rho(x) .
|
/ 0
A point of mass M:
M
(9) Phi(r) = - G - .
r
This is known as a Keplerian potential since orbits in this
potential obey Kepler's three laws. The velocity of a circular orbit
at radius r is v_c(r) = sqrt(M/r).
A uniform sphere of mass M and radius a:
{ -2 pi G rho (a^2 - r^2 / 3) , r < a
(10) Phi(r) = {
{ - G M/r , r > a
where rho = M / (4 pi a^3 / 3) is the mass density. Outside
the sphere the potential is Keplerian, while inside it has the form of
a parabola; both the potential and its derivative are continuous at
the surface of the sphere.