An Introduction to Modern Astrophysics · Problem 24.26

The Early Universe — Carroll & Ostlie

Physics — Astrophysics · gravitational-gauss-law

Problem

The

r^{-2}

dependence of Coulomb's electrostatic force law allows the construction of Gauss's law for electric fields, which has the form

\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\rm in}}{\epsilon_0},

where the integral is taken over a closed surface that bounds the enclosed charge,

Q_{\rm in}

. Because Newton's gravitational force law also varies as

r^{-2}

, it is possible to derive a gravitational ``Gauss's law.'' The form of this gravitational version is

\oint \mathbf{g} \cdot d\mathbf{A} = -4\pi G M_{\rm in}, \tag{24.56}

where the integral is over a closed surface that bounds the mass

M_{\rm in}

, and

\mathbf{g}

is the local acceleration of gravity at the position of

d\mathbf{A}

. The differential area vector (

d\mathbf{A}

) is assumed to be normal to the surface everywhere and is directed outward, away from the enclosed volume. Show that if a spherical gravitational Gaussian surface is employed that is centered on and surrounds a spherically symmetric mass distribution, Eq. (24.56) can be used to solve for

\mathbf{g}

. The result is the usual gravitational acceleration vector around a spherically symmetric mass.

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