The shape of the Earth and the other terrestrial planets is well approximated by spheres and thus the gravitational acceleration all over the surface points almost exactly towards the center of the planet. However, as we have seen, the outer planets spin very rapidly and, since the interiors of these planets are essentially fluid, the planets bulge out at the equator in response to the centrifugal force. This non-sphericity means that the shape of the gravitational field is modified from the spherical form with which we are more accustomed.
In general the gravitational acceleration at some point in a gravitational field is given by the gradient of the gravitational potential energy function U, g = -VU. (2.17)
and outside a planet U satisfies Laplace's equation,
In spherical polar coordinates, Laplace's equation has a number of well-known solutions. If we limit ourselves to solutions which are axially symmetric (i.e., those that do not depend on the azimuth angle) then we find
where 0 is the zenith angle; and Pn are Legendre polynomials. If we further limit ourselves to solutions that possess north/south symmetry (which to a very good approximation all the giant planets have) then we only need consider even powers of n. Additionally we may put An = 0 for all n, since the potential must tend to zero as r tends to infinity. Adding in centripetal effects we then obtain (Lindal et al., 1985)
where 0 is the planetocentric latitude, which is (90 — 0). Taking the gradient of Equation (2.20) we obtain an expression for the gravitational acceleration as a function of radial distance r and planetocentric latitude 0 in the radial direction
9r(r,<t>) = —— (1— E(2n + 1)J2^-J ^2n(sin 0)1 + fQ2r[1 — P2(sin 0)] (2.21)
and in the latitudinal direction g,(r,0) = —CM (p^^—^r^ (2.22)
and the total gravitational acceleration is g = ^J(g2 + g0). For the approximately spherical terrestrial planets the so-called "J-coefficients" are negligible and thus gr ^ g</>. However, the oblateness of the giant planets means that J-coefficients are substantial and thus the "surface" gravitational field does not point directly towards the center of the planet, but is instead displaced by a small angle where -0 = arctan(g^/gr), which varies with 0. This offset leads to a second definition of latitude and longitude, the planetographic system. The planetographic latitude is defined as the inclination of the local normal to the equatorial plane. This is equal to 0g = 0 + -0 and is shown in Figure 2.8. The curve described by the limb of a planet is, to a first approximation, an ellipse and thus the planetocentric and planetographic latitudes are more simply related by
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