The gravitational acceleration (g) experienced by a spacecraft orbiting a planet of mass M at a distance of r from the planet's center is
where G is the universal gravitational constant. The gravitational acceleration is the integral of the gravitational potential (U), therefore:
Since all of the planet's mass lies inside the surface, the gravitational potential exterior to the planet satisfies Laplace' s equation:
The general solution to Laplace's equation in spherical coordinates is (Hubbard, 1984):
U(r, h, k) = &mrl + bimr-(l+1)\Yim (h; k). (3 .4)
This equation gives the gravitational potential at a particular spot on the planet, where r=distance from center, h is the colatitude (=90°-latitude), and k is the longitude; alm and fjlm are constants. In planetary physics applications, alm is set to zero so that the potential vanishes at infinity. The index, l, is the degree of the harmonic and indicates the rate at which the gravitational potential varies in latitude.
Zonal Sectoral Tesseral
Zonal Sectoral Tesseral
Figure 3.3 The expansion of the gravity potential produces three types of harmonics. Zonal harmonics provide information on the mass distribution parallel to the equator while sectoral harmonics provide the longitudinal distributions of mass. Tesseral harmonics subdivide the planet into small blocks to determine the localized distributions of mass.
The index, m, is the order and indicates how rapidly U varies in longitude. Yim are spherical harmonics, defined by
The Legendre polynomials, Pjn, are defined by
(-1)m(1 - x2)m/2 dl+m(x2 - 1)1 Pl (x) = 2ll! dxi+m . (3:6J
The coefficients fjlm are associated with the mass distribution inside the planet. Since the potential is measured outside of the planet, the Legendre polynomials are related to the spherical harmonics, and Eq. (3.4) is typically rewritten as
U = (çM) 11 + è X (7) Pm(cos h)[Clm cos(mk) + Sm sin(mk)^. (3.7)
The radius of the reference sphere or ellipsoid is R and n is the limiting degree of the expansion. Here Cl0 are the zonal harmonics, which provide information on mass distributions parallel to the equator (Figure 3.3). The sectoral harmonics, Sl0, define the mass distributions perpendicular to the equator. Tesseral harmonics (Clm and Slm, m > 0) further subdivide the planet into smaller blocks, allowing localized variations in mass to be discerned.
Geophysicists often use J instead of the Cl0 nomenclature for the zonal harmonics:
If the center of the reference figure coincides with the center of mass of the body, J1 = C11=S11=0. Unfortunately for Mars, the COM-COF offset makes these coefficients non-zero (Smith etal., 2001a). J2 is the zonal harmonic representing the largest deviation from a perfect sphere, which results from the planet's equatorial bulge. The principal moments of inertia (A, B, and C) are related to the lower order and degree harmonics by
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