Foundations of Space Dynamics. Ashish Tewari
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(2.95)
2.7.2 Spherical Coordinates
To evaluate the gravitational potential given by Eq. (2.94), it is necessary to introduce the spherical coordinates for the mass distribution of the body, as well as the location of the test mass. Let the right‐handed triad,
(2.96)
where
(2.97)
where
The coordinate transformation between the spherical and Cartesian coordinates for the elemental mass is the following:
(2.98)
differentiating which produces
(2.99)
or the following in the matrix form:
An inversion of the square matrix on the right‐hand side (called the Jacobian of the coordinate transformation) yields the following result:
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