Foundations of Space Dynamics. Ashish Tewari

      In terms of the associated Legendre functions and the Legendre polynomials of the first degree, Eq. 2.104 becomes

      (2.105)

      which is referred to as the addition theorem for the Legendre polynomials of the first degree, . In terms of the Legendre polynomials of the second degree, , we have

      (2.106)

      which is the addition theorem for the Legendre polynomials of the second degree, . Extending this procedure leads to the following addition theorem for the Legendre polynomials of degree , :

(2.107)

The substitution of the addition theorem into Eq. (2.94) results in the following expansion of the gravitational potential:

(2.108)

      where

(2.109)

(2.110)

(2.111)

      with denoting the maximum radial extent of the body. The mass of the body is evaluated by

      (2.112)

Equation (2.108) is a general expansion of the gravitational potential which can be applied to a body of an arbitrary shape and an arbitrary mass distribution. However, the evaluation of the series coefficients by Eqs. (2.109)–(2.111) is often a difficult exercise for a body of a complicated shape, and requires experimental determination (such as the acceleration measurements by a low‐orbiting satellite).

       2.7.3 Axisymmetric Body

A body whose mass is symmetrically distributed about the polar axis, , has its density varying only with the radius and the latitude; that is, . Upon neglecting the longitudinal variations in the mass distribution, the additional theorem for the Legendre polynomial of degree , Eq. (2.107), becomes the following:

      (2.113)

whose substitution into the triple integrals in Eqs. (2.110) and (2.111) leads to the integration in the longitude, , being carried out independently of and as follows:

      (2.114)

      This implies that and

      (2.115)

      These simplifications allow the gravitational potential of an axisymmetric body to be expressed as follows:

      (2.116)

      where

      (2.117)

      A more useful expression for the gravitational potential can be obtained as follows in terms of the non‐dimensional distance, , where is the equatorial radius of the axisymmetric body:

(2.118)

      where and

      (2.119)

      are called Jeffery's constants, and are unique for a body of a given mass distribution. Jeffery's constants represent the spherical harmonics of the mass distribution, and diminish in magnitude as the order, k, increases. The largest of these constants, , denotes a non‐dimensional difference between the moments of inertia about the polar axis, , and an axis in the equatorial plane (

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