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Let us adopt the right-handed Cartesian coordinate system $ x_1$ , $ x_2$ , $ x_3$ . Consider a homogeneous ellipsoidal body whose outer boundary satisfies

$\displaystyle \frac{x_1^{\,2}}{a_1^{\,2}}+ \frac{x_2^{\,2}}{a_2^{\,2}}+\frac{x_3^{\,2}}{a_3^{\,2}}=1,$ (D.1)

where $ a_1$ , $ a_2$ , and $ a_3$ are the principal radii along the $ x_1$ -, $ x_2$ -, and $ x_3$ -axes, respectively. Let us calculate the gravitational potential (i.e., the potential energy of a unit test mass) at some point $ P\equiv (x_1,\,x_2,\,x_3)$ lying within this body. More information on ellipsoidal potential theory can be found in Chandrasekhar 1969.

Richard Fitzpatrick 2016-03-31