The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

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The Poisson Equation [I.D.1] is derived straightforwardly from Isaac Newton's inverse-square law of gravitation. In presenting this derivation, we follow closely the presentation in Chapter 2, § 1 of Binney & Tremaine (1987).1

According to Isaac Newton's inverse-square law of gravitation, the acceleration a(x) felt at any point in space x due to the gravitational attraction of a distribution of mass r(x') is obtained by integrating over the accelerations exerted by each small mass element, r(x') d³x', as follows:

First, if we adopt the following

and notice that the gradient of the function | x' - x |^-1 with respect to x is

we find that we may write the gravitational acceleration as

Next, realize that the divergence of the gravitational acceleration takes the form,

Now

When (x' - x) ¹ 0 we may cancel the factor | x' - x |² from top and bottom of the last term in this equation to conclude that

Therefore, any contribution to the integral (on the right-hand-side of Eq. [I.H.5]) must come from the point x' = x, and we may restrict the volume of integration to a small sphere ¼ centered on this point. Since, for a sufficiently small sphere, the density will be almost constant through this volume, we can take r(x') = r(x) out of the integral. Via the divergence theorem (see BT87 for details), the remaining volume integral may be converted into a surface integral over the small volume centered on the point x' = x and, in turn, this surface integral may be written in terms of an integral over the solid angle d2W to give:

Finally, by combining expressions [I.H.4] and [I.H.8], we derive the

which serves as one of the principal governing equations in our examination of the structure, stability, and dynamics of self-gravitating systems.

¹Text in green is taken verbatim from Chapter 2, § 1 of Binney & Tremaine (1987).

² Note that throughout his book entitled, "Ellipsoidal Figures of Equilibrium," Chandrasekhar adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention used herein.

3 Note that there is a typographical error in Eq. (2-7) of BT87. As printed, the first term on the right-hand-side of the equation is [ - 3 | x' - x |-1 ] whereas it should be [ - 3 | x' - x |-3 ] as written here in [I.H.6].