where is a surface harmonic of degree 2. It is assumed that .

The net gravitational acceleration in the vicinity of the planet takes the form

(C.9) |

where is the gravitational potential. According to standard gravitational theory,

(C.10) |

where is the mass density distribution. Thus, we can write

where

(C.12) |

and

(C.13) |

[See Equations (E.2) and (E.3).] Here, is a Dirac delta function (Riley 1974b). The physical boundary conditions are

(C.14) | ||||||

and | (C.15) |

as . As is well known, the solutions to

and

(C.18) | ||

(C.19) |

where

(C.20) |

is the gravitational acceleration at the planet's surface. Note that is a solid harmonic of degree 2 inside the planet (i.e., ).