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Foundations of Space Dynamics. Ashish Tewari
Читать онлайн.Название Foundations of Space Dynamics
Год выпуска 0
isbn 9781119455325
Автор произведения Ashish Tewari
Издательство John Wiley & Sons Limited
Since the determinant of the matrix on the right‐hand side of Eq. (2.100) equals
(2.102)
If the mass density at the location of the elemental mass is given by
(2.103)
The angle
Figure 2.5 Spherical coordinates for the gravitational potential of a body.
To derive the gravitational potential given by Eq. (2.94) in spherical coordinates, it is necessary to expand the cosine law (Eq. 2.104) in terms of the Legendre polynomials. To do so, consider the following associated Legendre functions of the first kind, degree
where
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,