Figure 2.4 An elemental mass, , of a body with centre of mass O, and a test mass, , located away from the body.

For all the particles constituting the mass, M, of the attracting body, let the limit of an infinitesimal elemental mass, , be taken as , whereby the summation in Eq. (2.71) is replaced by the following integral:

      (2.73)

      which results in the following expression for the acceleration of the test mass:

      (2.74)

where s is the distance of the test mass, , from the elemental mass, , as depicted in Fig. 2.4, and can be expressed as follows:

      (2.75)

      with and being the position vectors of the test mass, , and the elemental mass, , respectively, from the centre of mass of the attracting body, and , being the angle between , and as shown in Fig. 2.4.

      From Fig. 2.4 it follows that

      (2.76)

      and is a constant, because the attracting body is assumed to be a rigid body. When the position vectors and are resolved in the Cartesian coordinates, we have

      (2.77)

the gravitational potential of the mass distribution is given by

(2.78)

      and the gravitational acceleration at from the centre of mass of the attracting body is the following:

      (2.79)

       2.7.1 Legendre Polynomials

To carry out the integration in Eq. (2.78), it is assumed that the body is entirely contained within the radius measured from its centre of mass; that is, for all points on the body. It is then convenient to expand the integrand in the following series:

(2.80)

Equation (2.80) is an infinite series expansion in polynomials of , and is commonly expressed as follows:

(2.81)

      where is the Legendre polynomial of degree k, defined by

      (2.82)

      with denoting the largest integer value of given by

      (2.83)

The first few Legendre polynomials are the following:

      (2.84)

Clearly the Legendre polynomials satisfy the condition , which implies that the series in Eq. (2.81) is convergent. Therefore, one can approximate the integrand of Eq. (2.78) by retaining only a finite number of terms in the series.

      By writing and , the general expression for the Legendre polynomials is given in terms of the following generating function, :

      (2.85)

      The generating function can be used to establish some of the basic properties of the Legendre polynomials, such as the following: