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

Kepler’s Laws of Planetary Motion

Second law of Kepler

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

Third law of Kepler

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

Newton’s Law of Gravity

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

Condition: $ r \leq r_E $

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

Lifting Work and Potential Energy

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

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

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