Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 34

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:

(2.86)

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)