Ashish Tewari - Foundations of Space Dynamics

(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)

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 inversion of the square matrix on the right‐hand side (called the Jacobian of the coordinate transformation) yields the following result:

(2.101)

Since the determinant of the matrix on the right‐hand side of Eq. (2.100)equals , and that of the inverse matrix in Eq. (2.101)is , the two sets of coordinates are related by the following expression for the elemental volume at :