href="#fb3_img_img_bccd90c2-bb42-5fe8-a9bc-6f6e06e2723c.png" alt="images"/> or in Fig. 2.5), and is a measure of the ellipticity (or oblateness) of the body. The higher order term, indicates the pear‐shaped or triangular harmonic, whereas and are the measures of square and pentagonal shaped harmonics, respectively. For a reasonably large body, it is seldom necessary to include more than the first four Jeffery's constants. For example, Earth's spherical harmonics are given by , and .
The acceleration due to gravity of an axisymmetric body is obtained by taking the gradient of the gravitational potential, Eq. (2.118), with respect to the position vector, , and can be resolved in the radial, , and the north polar, , directions (Fig. 2.5) as follows:
where the following identities have been employed:
The acceleration can be alternatively resolved in two mutually perpendicular directions, and (see Fig. 2.5). The unit vectors and denote the radial and southward directions, as shown in Fig. 2.5, while the unit vector signifies the direction of the increasing longitude; that is, the eastward direction. These unit vectors constitute a moving coordinate frame, , attached to the test mass as shown in Fig. 2.5. Such a frame is called a local‐horizon frame (see Chapter 5). The acceleration given by Eq. (2.120) is resolved in and by substituting , resulting in
(2.121)
where
(2.122)
and
(2.123)
Due to a non‐zero transverse gravity component, , the direction of differs from the radial direction, while its radial component, , is smaller in magnitude when compared to that predicted by a spherical gravity model; that is, the model derived by assuming a perfectly spherical mass distribution of a body of radius .
2.7.4 Spherical Body with Radially Symmetric Mass Distribution
A spherical body of constant radius with a radially symmetrical mass distribution has the density , varying only with the radius, . A body consisting of concentric spherical shells, each of which has a different (either constant or radially varying) density, has a radially symmetrical mass distribution. Such a distribution results in the vanishing of all the integrals of and in Eqs. (2.109)–(2.111), which produces , thereby implying a potential which is independent of the co‐latitude and longitude. Hence the gravitational potential of a body with a spherically symmetric mass distribution of radius, , is given by
(2.124)
Clearly, a spherical body with a radially symmetric mass distribution behaves exactly as if all its mass were concentrated at its centre. The acceleration due to gravity is thus given by
(2.125)
Exercises
1 A particle, , is launched vertically upward with velocity, , from the surface of a spherical planet of radius , rotating about the polar axis, , from west to east at the rate . The launch takes place at the latitude, , and longitude, , measured from a fixed meridian, , as shown in Fig. 2.6. Derive the expressions for the net velocity and acceleration of the particle, and resolve the vector components in a reference frame, , with its origin, , at the centre of the planet, and with one axis along the polar axis, , another along the meridian, , and the third axis, , normal to both and .Figure 2.6 Geometry for Exercise 1.
href="#fb3_img_img_bccd90c2-bb42-5fe8-a9bc-6f6e06e2723c.png" alt="images"/> or 