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

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

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

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