General Expressions for the Gravitational Potential due to An Arbitrary MassGeorge H. BornMay 7, 2001Figure 1. Potential of an arbitrary shaped bodyUniversity of Colorado at Boulder George H. BornColorado Center for Astrodynamics Research 5/7/2001General Expressions for the Gravitational Potential due to An Arbitrary MassGeorge H. BornMay 7, 2001Figure 1. Potential of an arbitrary shaped bodyThe geometry of the problem is shown in figure 1. It is desired to derive an expression for the gravitational potential, which will exist at 2P due to a mass Mof arbitrary shape and density distribution. The orthogonal coordinate system ZYX ,, is located at an arbitrary point in Mand is inertial, i.e. it is undergoing neither acceleration nor rotation. The point2Pis defined by222,, ZYX or 222,,r. Consider an element of mass dM, which is located at the point 1P. The point 2P is an exterior point to Mand it is understood that for any point of 12, rrM  i.e., the potential function derived here is only valid outside the arbitrary mass, M. The mass density of M, an arbitrary function of position, may be designated byGravitational Potential112r1rdY X Z),,(1111ZYXP),,(2222ZYXPo2r2211dMMUniversity of Colorado at Boulder George H. BornColorado Center for Astrodynamics Research 5/7/2001 111,,r(1)also,111121cosdddrrdvdM (2)The gravitational potential at 2Pdue to mass dM is, 12rGdMd  massunitP 2(3)where G is the universal gravitational constant.Hence,VrdvG12 .cos11112121dddrrrG(4)By the law of cosines of plane triangles2121222112cos211rrrrr 21221212cos211rrrrr. (5)The quantity 2122121cos21rrrr is the generating function for an infinite series of zonalsolid harmonics (Hobson, page 105, 1965); hence eq.(5) may be written as  021212cos11lllPrrrr(6)where  coslP is the lth degree Legendre polynomial of the 1st kind. By the law of cosines for spherical triangles,Gravitational Potential2University of Colorado at Boulder George H. BornColorado Center for Astrodynamics Research 5/7/2001 .coscoscossinsincos122121(7)By the use of the addition theorem for Legendre polynomials (Whittaker and Watson, page 395, 1965) given by             1212121cossinsin!!2sinsincosmPPmmPPPmmmllllllll, (8)eq. (6) may be expressed in terms of the polar angles and  as             0121212121212cossinsin!!2sinsin11lllllllllmPPmmPPrrrrmmm         lllllllllll121210210212sinsin!!2sinsin1mmmPPrrmmPPrrr  )sinsincoscos(212mmmm. (9)Here  1sinmPl is the associated Legendre polynomial of the 1st kind, and of the lth degree and mth order, and where by definition of these polynomials,  0sin mPl for lm. Substituting this expansion for the distance between 1P and 2P into eq.(4) the potential becomes    111111222sincos1sindddrPrMRPrRrGMl2llll0l    112212coscos!!2sin2lllll0lllrmmmMRPrRmmGravitational Potential3University of Colorado at Boulder George H. BornColorado Center for Astrodynamics Research 5/7/2001    1111111211111sinsincossinsincosdddrPmrmdddrPmmm l2ll     22212222sincossinsinmSmCPrRPrRArGMmmmmlllll0lll0ll (10)where    111111111sincos,,1dddrPrrMRAl2lll (11)     1111111111sincoscos,,!!2dddrPmrrmmMRCmm l2lllll(12)     .sincossin,,!!21111111111dddrPmrrmmMRSmm l2lllll(13)Note that in eq.(10) the total mass M has been introduced and the summations, along with quantities not participating in the integration, have been taken outside the integral. The quantityR has been introduced; it is a dimensional parameter which is characteristic of the body of massM and which defines the ratio,Rr2, to be the distance of 2P from the origin as measured in units of R. (R is generally assumed to be the mean equatorial radius.)The coefficients mCAll, and mSl are functions of the size, shape and density distributions of the body of mass M, are a set of constant characteristics of that body. If the shape and density distributions are known, the integrations involved in these coefficients may be carried out resulting in a set of theoretical values for these coefficients. When such information is lacking, however, a theoretical determination of the coefficients of the potential function is impossible. In the case of the Earth for instance the values of these coefficients have been estimated by astronomical measurements and more recently by methods of satellite geodesy. For convenience the coefficients mJl and the phase angles ml, may be defined byGravitational Potential4University of Colorado at Boulder George H. BornColorado Center for Astrodynamics Research 5/7/2001mmmmJClllcos mmmmJSlllsin(14) .llJA Equation (15) give the alternate expression for the potential     mmmmmPrRJPrRJrlllll0lll0llcossinsin1(15)whereGMand the subscript 2 which is no longer necessary has been omitted. Equation (15) may be simplified since 1sin0Pgiving, from eq.(11), for l = 0.11cos1111110