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where the prime stands for the derivative with respect to the argument, . The generating function, , is also used to generate a Legendre polynomial from those of lower degrees with the help of recurrence formulae, such as
(2.87)
The reciprocal of the generating function, , can be regarded as the radical portion, , of the real root, , to the following quadratic equation:
(2.88)
where the positive sign is taken to correspond to the smaller of the two roots. The choice and yields
(2.89)
or
(2.90)
Since , the following series expansion (called Lagrange's expansion theorem) can be applied to Eq. (2.90) (Abramowitz and Stegun 1974):
(2.91)
The differentiation of Eq. (2.91) with results in
(2.92)
where corresponds to the root . Equation (2.92) yields the following expression for the Legendre polynomials, called Rodrigues' formula:
(2.93)
The gravitational potential is expressed as follows in terms of the Legendre polynomials by substituting Eq. (2.81) into Eq. (2.78):
(2.94)
It is possible to further simplify the gravitational potential before carrying out the complete integration. The integral arising out of in Eq. (2.94) yields the mass, M, of the planet, thereby resulting in
(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, , and , represent the axes, , , and , respectively, of the inertial frame, (OXYZ), with the origin, , at the centre of the body. The location of the test mass, , is resolved in the spherical coordinates, , as follows (Fig. 2.5):
(2.96)
where is the co‐latitude and is the longitude. Similarly, let an elemental mass, , on the body be located by using the spherical coordinates as follows (Fig. 2.5):
(2.97)
where and are the co‐latitude and longitude, respectively, of the elemental mass.
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:
(2.100)
An
