3.6 Reference ellipsoid

An important part of the computation is the subtraction of the reference ellipsoid, defined by the semi-major axis of the Earth $a$ and the flattening parameter $f$. To do so, the zonal coefficients of the spherical harmonic gravity model $\bar{C}_{2n,0}$, $n=1,2,3,4,5$ are corrected by

$$\bar{C}_{2n,0} \longrightarrow \bar{C}_{2n,0}+ \frac{GM}{GM_{\mathrm{g}}}\left(\frac{a}{a_{\mathrm{g}}}\right)^{n} \cdot\frac{J_{2n}}{\sqrt{4n+1}}.$$ (20)

The factor $(GM/GM_{\mathrm{g}})\cdot(a/a_{\mathrm{g}})^{n}$ accounts for the fact that neither the gravity mass constants nor the semi-major axes (just a scaling factor for the geopotential model) have to be the same for geopotential model and reference ellipsoid. In fact, very often they are not the same.³ In general, it is sufficient to correct the zonal coefficients $\bar{C}_{2,0}$ through $\bar{C}_{10,0}$. The coefficients $J_{2n}$ are determined from the parameters $a$, $f$, and the angular velocity $\omega$ of the Earth by an expansion (Heiskanen and Moritz, 1967, p.73, eq.2-92):

$$J_{2n} = (-1)^{n+1}\frac{3(E/a)^{2n}}{(2n+1)(2n+3)} \left(1-n+5n\frac{C-A}{ME^{2}}\right).$$ (21)

This equation involves new parameters: the moments of inertia with respect to any axis in the equatorial plane $A$ and with respect to the axis of rotation $C$ (not to be confused with the spherical harmonic coefficients $\bar{C}_{n,m}$), the total mass $M$, and the linear eccentricity $E=\sqrt{a^{2}-b^{2}}$. They can be computed by (Heiskanen and Moritz, 1967)

$$\frac{C-A}{ME^{2}} =\frac{1}{3}\left[1- \frac{2}{15}\left(\frac{me'}{q_{0}}\right)\right]$$ (22)

$$m =\frac{\omega^{2}a^{2}b}{GM}$$ (23)

$$q_{0} =\frac{1}{2}\left[\left(1 +\frac{3}{e'^{2}}\right)\arctan{e'} -\frac{3}{e'}\right].$$ (24)

$e'=E/b$ is again the second numerical eccentricity. Subtracting the reference ellipsoid is a very sensitive operation, because two large numbers are subtracted from each other and this is always prone to errors. So if your coefficients $J_{2n}$ are only slightly wrong, this will have a big effect on your solution. If you are lucky, the precomputed values of the coefficients $J_{2n}$, which are in principle defined by $a$, $f$, $\omega$, and $GM$, are provided with the reference ellipsoid.