Planets and Satellites in Circular Orbits: Mechanics Guide

The Moon circles Earth, Earth circles the Sun, and thousands of satellites circle Earth — all following one rule. In a circular orbit, gravity supplies exactly the centripetal force needed to bend the body's path into a circle. No engines, no ropes: just falling sideways fast enough to keep missing the planet.

This matters because it is the master key to the whole solar system and every satellite mission. One equation predicts orbital speeds, periods and distances — from the 90-minute space-station lap to the Moon's 27-day journey to Neptune's 165-year year. Kepler's laws, discovered from observations, all drop out of this single force balance.

Key Concepts

• Force balance: gravitational pull = required centripetal force, so GMm/r² = mv²/r. The satellite's mass cancels — orbits don't care how heavy you are.
• Orbital speed v = √(GM/r): set by the central mass and radius only. Farther means slower.
• Orbital period T = 2π√(r³/GM): the time for one lap. Squaring gives T² ∝ r³, Kepler's third law.
• Angular momentum L = mvr is conserved, which is why orbits stay stable without any steering.

Formulas

• mv²/r = GMm/r² (force balance) → v = √(GM/r).
• T = 2π√(r³/GM); equivalently T² = (4π²/GM) r³.
• Geosynchronous check: T = 86,164 s around Earth gives r ≈ 42,164 km.
• Units: v in m/s, T in s, r in m; G = 6.674 × 10⁻¹¹ N·m²/kg².

Worked Example

Use the period formula to predict the Moon's orbital period from its distance (r = 3.84 × 10⁸ m).

1. r³ = (3.84 × 10⁸)³ = 5.662 × 10²⁵ m³.
2. r³/GM = 5.662 × 10²⁵/3.986 × 10¹⁴ = 1.4205 × 10¹¹ s².
3. √(r³/GM) = √1.4205 × 10¹¹ ≈ 376,906 s.
4. T = 2π × 376,906 ≈ 2.368 × 10⁶ s = 2.368 × 10⁶/86,400 ≈ 27.4 days.

The formula nails the Moon's observed 27.3-day sidereal period from nothing but its distance and Earth's mass — the same calculation works for Mars's moons, Jupiter's moons and exoplanets.

Common Misconceptions

• "Something must keep pushing the satellite forward." Nothing does — with no air in space there is no drag, so the satellite coasts forever by Newton's first law while gravity only bends its path.
• "Kepler's laws and Newton's law are rivals." Kepler's laws are consequences of Newton's gravity: T² ∝ r³ falls straight out of mv²/r = GMm/r².
• "All orbits are circles." Real orbits are ellipses; the circle is the special equal-energy case. Planets' ellipses are nearly circular, but comets show the full range.

Real-World Applications

• Satellite constellations: Starlink, GPS and weather satellites all sit at radii chosen through T = 2π√(r³/GM) for their required periods.
• Exoplanet discovery: a star's wobble period reveals an unseen planet's orbital radius via Kepler's third law.
• Weighing the cosmos: timing any moon's orbit gives the planet's mass — how we "weigh" Jupiter, Saturn and distant stars.

Circular orbits run on one balance — mv²/r = GMm/r² — yielding v = √(GM/r) and Kepler's T² ∝ r³. It predicts the Moon's month from its distance alone, sizes every satellite constellation, and weighs planets across the galaxy. Gravity bends the path; inertia does the rest.