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3.1 Preliminaries

First, let's write down formulas for quantities that we will need lateron (Heiskanen and Moritz, 1967). First the relationship between Earth flattening parameter $ f$, semi-major and semi-minor axes $ a$ and $ b$:

$\displaystyle f$ $\displaystyle =\frac{a-b}{a}$   or (1)
$\displaystyle b$ $\displaystyle =a(1-f)$   (2)
$\displaystyle E$ $\displaystyle =\sqrt{a^{2}-b^{2}}$ (linear eccentricity) (3)
$\displaystyle e$ $\displaystyle =\frac{E}{a}$ (first numerical eccentricity) (4)
$\displaystyle e'$ $\displaystyle =\frac{E}{b}$ (second numerical eccentricity) (5)
$\displaystyle m$ $\displaystyle =\frac{\omega^{2}a^{2}b}{GM}$ (an abbreviation) (6)
$\displaystyle \gamma_{a}$ $\displaystyle =\frac{GM}{ab}\left(1-\frac{3}{2}m-\frac{3}{14}e'm\right)$ (gravity acc. at equator) (7)
$\displaystyle \gamma_{b}$ $\displaystyle =\frac{GM}{a^{2}}\left(1+m+\frac{3}{7}e'm\right)$ (gravity acc. at poles) (8)
$\displaystyle x$ $\displaystyle =\frac{a\cos\varphi\cos\lambda}{\sqrt{1-e^{2}\sin^{2}\varphi}}$ (Cartesian coordinates (9)
$\displaystyle y$ $\displaystyle =\frac{a\cos\varphi\sin\lambda}{\sqrt{1-e^{2}\sin^{2}\varphi}}$ relative to center (10)
$\displaystyle z$ $\displaystyle =\frac{a(1-e^{2})\sin\varphi}{\sqrt{1-e^{2}\sin^{2}\varphi}}$ of Earth) (11)


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