## Gravitational Perturbations Mass Distribution

Consider the gravitational attraction between two point masses, M1 and M2, separated by a distance r. The gravitational potential between these two bodies, U, can be expressed as

The gravitational force on M2 resulting from the interaction with Mi can be derived from the potential as the gradient of U; that is,

where r is the position vector of M2 with respect to M1 given by r = xux + yuy + zuz

Figure 2.3.4: Defiiit ion of position vectors and differential mass for a body with arbitrary mass distribution.

Figure 2.3.4: Defiiit ion of position vectors and differential mass for a body with arbitrary mass distribution.

dx dy dz

If a body of total mass M has an arbitrary distribution of mass, it can be modeled as the collection of a large number of point masses. The gravitational potential sensed by a point mass, m', at an external location is

where 7 is the mass density associated with an element of mass dm, dx dy dz is the differential volume, and p is the distance between the differential mass dm and the external mass m' (see Fig. 2.3.4). For convenience, the external mass is usually taken to be unity (m' = 1), and the integral expression is written in a more compact notation

Jm p where the integral notation represents integration over the entire mass of the body. Equation (2.3.10) can be directly integrated for some bodies, such as a sphere of constant density 7. This integration (see Danby, 1988) shows that the constant density sphere is gravitationally equivalent to a point mass.

The position vector of m' with respect to the coordinate system (x, y, z) is r. The location of the origin, O, and the orientation of (x, y, z) is arbitrary; however, for convenience, the (x, y, z) system is assumed to be body-fixed. Hence, the (x, y, z) axes are fix ed in the body and rotate with it. For the Earth, as an example, the (x, y) plane would coincide with the equator, with the x axis also coinciding with the Greenwich meridian.

Equation (2.3.10) can be expanded into the infinite series

where — is the distance between the origin O and the differential mass dm, and Pi is the Legendre polynomial of degree I, with argument equal to the cosine of the angle between the two vectors, R and r. Using the decomposition formula (Heiskanen and Moritz, 1967, p. 33), the Legendre polynomial can be expanded into spherical harmonics and terms that are dependent on the mass distribution collected into coefficients to yield

+1 0