Foundations of Space Dynamics. Ashish Tewari

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       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:

      (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)

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