Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 35

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 :

(2.102)

If the mass density at the location of the elemental mass is given by , then the elemental mass is the following:

(2.103)

The angle between , and (Fig. 2.4) is related to the spherical coordinates by the following cosine law of the scalar product of two vectors:

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 and order (Abramowitz and Stegun 1974):

(2.104)

where is the Legendre polynomial of degree . Some of the commonly used associated Legendre functions are

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, . In terms of the Legendre polynomials of the second degree, , we have

(2.106)

which is the addition theorem for the Legendre polynomials of the second degree, . Extending this procedure leads to the following addition theorem for the Legendre polynomials of degree , :

(2.107)

The substitution of the addition theorem into Eq. (2.94) results in the following expansion of the gravitational potential:

(2.108)

where

(2.109)

(2.110)

(2.111)

with denoting the maximum radial extent of the body. The mass of the body is evaluated by

(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).