Foundations of Space Dynamics. Ashish Tewari
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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,
(2.106)
which is the addition theorem for the Legendre polynomials of the second degree,
The substitution of the addition theorem into Eq. (2.94) results in the following expansion of the gravitational potential:
where
with
(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,
(2.113)
whose substitution into the triple integrals in Eqs. (2.110) and (2.111) leads to the integration in the longitude,
(2.114)
This implies that
(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
(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,