Orbits of planets and satellites (A-level only): Synchronous orbits.

The idea

A synchronous orbit has an orbital period equal to the rotation period of the body it orbits, so the satellite returns to the same position in the sky after each rotation. For a satellite of mass m in a circular orbit of radius r around a planet of mass M, gravity supplies the centripetal force: GMm/r² = mω²r. Substituting ω = 2π/T and cancelling m gives r³ = GMT²/(4π²), so the orbital radius is fixed once T is chosen. A geosynchronous orbit above Earth's equator has T = 24 hours (about 8.64 × 10⁴ s), giving r ≈ 4.22 × 10⁷ m from Earth's centre, an altitude of about 3.6 × 10⁷ m. Geostationary orbits are a special case: equatorial, circular and in the same sense as Earth's rotation, so the satellite appears fixed above one point.

What earns the marks

• Defines a synchronous orbit as one whose orbital period equals the rotation period of the central body, so the satellite repeats its position relative to the surface.
• Equates gravitational force GMm/r² to centripetal force mω²r or mv²/r for a circular orbit and cancels the satellite mass m.
• Uses ω = 2π/T to obtain r³ = GMT²/(4π²) or an equivalent relationship linking orbital radius and period.
• Calculates the orbital radius or altitude for a given period, for example about 4.2 × 10⁷ m from Earth's centre for a 24-hour period.
• Distinguishes geostationary from geosynchronous: geostationary must be equatorial, circular and in the same rotational sense as the planet.
• Explains that only one orbital radius satisfies a chosen synchronous period for a given central mass, so synchronous orbits form a single shell.

Common mistakes and how to put them right

• Confusing orbital radius with altitude above the surface.

  Put it right by calculating r from r³ = GMT²/(4π²) and then subtracting the planet's radius to find altitude.

• Assuming any 24-hour orbit is geostationary.

  Put it right by stating that a geostationary orbit must also be equatorial, circular and in the same sense as the planet's rotation.

• Using the satellite's mass in the force equation and failing to cancel it.

  Put it right by cancelling m from both sides, showing the orbital radius depends only on M and T.

Exam tips

• Convert the period to seconds before substituting into r³ = GMT²/(4π²), since G is in SI base units.
• Show the force equation and the substitution ω = 2π/T so the derivation is clear and the relationship between r and T is visible.
• When asked about geostationary satellites, state all three conditions: equatorial plane, circular orbit and same rotational sense as the planet.