Gravitation and orbital mechanics

Newton's law of gravitationKepler's third laworbital velocity formulaescape velocity formulasatellite equations

Newton's inverse-square law and what falls out of it in orbit — field strength, orbital speed and period, Kepler's third law, and escape velocity.

Newton's Law of Universal Gravitation

F = \frac{G m_1 m_2}{r^2}

Attractive force between two masses, with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2018).

Gravitational Field Strength

g = \frac{GM}{r^{2}}

Gravitational acceleration produced by a mass M at distance r from its center.

Weight (W = mg)

W = m g

Weight of a mass at Earth's surface, using standard gravity g = 9.80665 m/s² (exact by definition).

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Speed of a body in a circular orbit of radius r around a central mass M.

Orbital Period

T = 2\pi \sqrt{\frac{r^{3}}{GM}}

Time for one circular orbit of radius r around a central mass M — Kepler's third law in Newtonian form.

Kepler's Third Law (Ratio Form)

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

For two bodies orbiting the same central mass, the squares of their periods are in the same ratio as the cubes of their orbital sizes.

Gravitational Potential Energy (Orbital)

U = -\frac{GMm}{r}

Gravitational potential energy of a mass m at distance r from a central mass M, taking zero at infinite separation.

Escape Velocity

v = \sqrt{\frac{2GM}{r}}

Minimum launch speed needed to escape the gravity of a mass M starting from distance r, with no further propulsion.

How they fit together

Newton's law of universal gravitation is the parent of everything else here. Set the gravitational force equal to the centripetal force needed for a circular orbit and orbital velocity drops out; solve that for time around and you get orbital period; square the period and you have Kepler's third law, which Kepler had found empirically from Tycho Brahe's observations some seventy years before anyone could explain it. Escape velocity comes from a different balance — kinetic energy against the gravitational potential well.

Choosing is mostly about which mass and which radius. Orbital formulas need the mass of the central body only; the satellite's own mass cancels, which is why a bolt and a space station in the same orbit travel at the same speed. The radius is measured from the centre of the planet, not from its surface — forgetting to add the Earth's 6371 km to an altitude is the single most common wrong answer in the topic. Use g = GM/r² for a field strength at a distance and W = mg only near a surface where g is effectively constant. And escape velocity is independent of launch direction and of the escaping object's mass, but it assumes one unpowered impulse: a rocket that keeps thrusting can leave far slower.