Gravitation

Gravitational Force

$F = \dfrac{G \cdot m_1 \cdot m_2}{r^2}$

G = Gravitational Constant = $6.673 \cdot 10^{-11} Nm^{2}{Kg}^{-2}$

r = Distance from the centers of mass (m)

Gravitational Field Strength

$g = \dfrac{F}{m} = \dfrac{GM}{r^2}$

Gravitational force per unit mass

Gravitational Potential Energy

$E_p = -\dfrac{GMm}{r}$

Ep = Gravitational potential energy (J)

m = Mass experiencing the potential (kg)

Note that the gravitational potential energy at $r = \infty$ is $-\infty$

Gravitational Potential

$\phi = -\dfrac{GM}{r}$

ϕ = Gravitational potential (J/kg)

Work done to bring unit mass from infinity to a point in the field

It is the gravitational potential energy per unit mass

Escape Velocity

$v = \sqrt{\dfrac{2GM}{r}}$

v = Escape velocity (m/s)

The minimum speed required to escape a gravitational field without further propulsion

Orbital Velocity

$v = \sqrt{\dfrac{GM}{r}}$

v = Orbital speed for a stable circular orbit (m/s)

Total Orbital Energy

$E = -\dfrac{GMm}{2r}$

Total energy = Kinetic + Potential in a bound circular orbit

Potential Gradient

$\text{Potential Gradient} = -\dfrac{\Delta\phi}{\Delta r} = g$

The negative gradient of gravitational potential is the gravitational field strength

Kepler’s Third Law

$(\frac{T_1}{T_2})^2 = (\frac{r_1}{r_2})^3$

or

$T^2 \propto r^3$

Where T is the orbital period and r is the radius of orbit