Gravity

$t$	$\mathrm{s}$	time
$T$	$\mathrm{s}$	period of circulation
$a$	$\mathrm{m}$	half-axis of a planet
$A$	$\mathrm{m^2}$	swept area
$F_G$	$\mathrm{N}$	weight force
$m$	$\mathrm{kg}$	mass
$m_g$	$\mathrm{kg}$	mass of the earth
$f$	$\mathrm{m^3/kg\, s^2}$	gravitational constant
$r$	$\mathrm{m}$	radius, distance between two pointmasses
$e$		unit vector
$g$	$\mathrm{m/s^2}$	gravitational acceleration
$\phi$	$\mathrm{m^2/s^2}$	gravitational potential
$W$	$\mathrm{J}$	work
$E$	$\mathrm{J}$	energy

Second law of Kepler

$$ \frac{\Delta A}{\Delta t} = \mathrm{const.} $$

Third law of Kepler

$$ \frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3} $$

$$ \vec{F}_G = – f \, \frac{m_1 \, m_2}{r^2} \, \vec{e}_r \qquad
f = 6.673 \cdot 10^{-11} \,\frac{N\,m^2}{kg^2} $$

Condition:

$ r \leq r_E $

$$ g = – f \, \frac{ m_E }{ r^2 } \, \vec{e}_r $$

$$ W = – f \, m_E \, m_K \, \left( \frac{1}{r_2} – \frac{1}{r_1} \right) \qquad
W_M = – f \, \frac{m_E}{r} \, m $$

$$ W = \phi \, m \qquad
\phi = – f \, \frac{m_E}{r} \qquad
W = m \, ( \phi_2 – \phi_2 ) $$

$$ \phi = – \int\limits_\infty^r \! \vec{g}(r) \;\mathrm{d}\vec{r} \qquad
\nabla \phi = – \vec{E} $$