Physics

Gravity and orbits

Newton's gravity, Kepler, surface g, and circular orbits.

Basics

Newton's gravity

Two point masses attract with F = G M m / r². The direction is along the line of centres; the size falls as distance squared. Outside a uniform sphere the mass acts as if at the centre. Gravity only attracts — there is no repulsive counterpart as with charge. Laboratory G comes from Cavendish-style measurements and is less precise than most other constants.

Kepler's laws

1: Orbits are ellipses with the centre at one focus. 2: Equal areas in equal times — faster at periapsis. 3: Period squared scales with semi-major axis cubed. The three rules follow from Newtonian gravity plus angular-momentum conservation. In the solar system the Sun dominates, so T² ∝ a³ is nearly exact.

Surface g

Near Earth’s surface, weight mg approximates G M m / R². The standard g = 9.80665 m/s² is conventional; local g shifts with latitude, altitude, and underground density. The equator is a little lower than the poles because of spin and the equatorial bulge. Free-fall acceleration is g when air resistance is neglected.

Circular orbit

In a circular orbit gravity is the centripetal force. v = √(G M / r), period T = 2π r / v. Low circles are fast and short; farther out they slow. A geostationary orbit (equator, one-day period) is the communications-satellite slot. Escape speed from that radius is √2 times the circular speed.

Formulas

Newton’s gravity

F = G M m / r²

Attraction of two point masses. The vector points toward the other body.

Symbols

G gravitational constant
r distance between centres

Gravitational field

g(r) = G M / r²

Force per unit mass. At a surface, g = G M / R².

Symbols

R body radius

Circular-orbit speed

v = √(G M / r)

From gravity = m v² / r. Lower orbits are faster.

Symbols

r orbital radius

Kepler III

T² = 4π² a³ / (G M)

When the central mass M dominates. a is the semi-major axis.

Symbols

T period
a semi-major axis

Key table

Standard g	9.80665 m/s² (defined conventional value)
Low Earth circular orbit	period ~90 min at ~400 km altitude
Geostationary radius	about 4.2×10⁴ km from Earth’s centre