In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field (the geopotential). The Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. However, a spherical harmonics series expansion captures the actual field with increasing fidelity. If Earth's shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), it could be integrated numerically (when combined with a reciprocal distance kernel) to find an accurate model for Earth's gravitational field. However, the situation is in fact the opposite: by observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately. The best estimate of Earth's mass is obtained by dividing the product GM as determined from the analysis of spacecraft orbit with a value for the gravitational constant G, determined to a lower relative accuracy using other physical methods.
Background
From the defining equations (1) and (2) it is clear (taking the partial derivatives of the integrand) that outside the body in empty space the following differential equations are valid for the field caused by the body:
Functions of the form ϕ = R ( r ) Θ ( θ ) Φ ( φ ) {\displaystyle \phi =R(r)\,\Theta (\theta )\,\Phi (\varphi )} where (r, θ, φ) are the spherical coordinates which satisfy the partial differential equation (6) (the Laplace equation) are called spherical harmonic functions. They take the forms:
where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference:
also P0n are the Legendre polynomials and Pmn for 1 ≤ m ≤ n are the associated Legendre functions. The first spherical harmonics with n = 0, 1, 2, 3 are presented in the table below. [Note that the sign convention differs from the one in the page about the associated Legendre polynomials, here P 2 1 ( x ) = 3 x 1 − x 2 {\displaystyle P_{2}^{1}(x)=3x{\sqrt {1-x^{2}}}} whereas there P 2 1 ( x ) = − 3 x 1 − x 2 {\displaystyle P_{2}^{1}(x)=-3x{\sqrt {1-x^{2}}}} .]
Formulation The model for Earth's gravitational potential is a sum
where μ = G M {\displaystyle \mu =GM} and the coordinates (8) are relative to the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with z-axis in the direction of the polar axis. The zonal terms refer to terms of the form:
P n 0 ( sin θ ) r n + 1 n = 0 , 1 , 2 , … {\displaystyle {\frac {P_{n}^{0}(\sin \theta )}{r^{n+1}}}\quad n=0,1,2,\dots }
and the tesseral terms terms refer to terms of the form:
P n m ( sin θ ) cos m φ r n + 1 , 1 ≤ m ≤ n n = 1 , 2 , … {\displaystyle {\frac {P_{n}^{m}(\sin \theta )\cos m\varphi }{r^{n+1}}}\,,\quad 1\leq m\leq n\quad n=1,2,\dots }
P n m ( sin θ ) sin m φ r n + 1 {\displaystyle {\frac {P_{n}^{m}(\sin \theta )\sin m\varphi }{r^{n+1}}}}
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