Satellite Orbits and Kepler’s Laws

A satellite in a circular orbit of radius r around a mass M moves at speed v = √(GM ÷ r) with period T = 2π√(r³ ÷ GM), and Kepler’s laws describe all planetary orbits: orbits are ellipses with the Sun at one focus, a planet sweeps out equal areas in equal times, and T² is proportional to a³. In brief: gravity provides the centripetal force: GMm ÷ r² = mv² ÷ r. Higher orbits are slower and have longer periods. A geostationary satellite has a period of one sidereal day and an orbital radius of 42,164 km, about 35,800 km above the equator, so it appears fixed in the sky.

The International Space Station, GPS satellites and the planets obey the same laws. This guide derives the orbital speed and period, states Kepler’s three laws, discusses geostationary and low Earth orbits, compares orbits in a table, works through numerical examples and corrects common mistakes.

How are circular orbits derived?

For a satellite of mass m in a circular orbit of radius r (measured from the center of the central body), gravity is the only force and acts as the centripetal force: GMm ÷ r² = mv² ÷ r. The mass m cancels, so

v = √(GM ÷ r), and the period T = 2πr ÷ v = 2π√(r³ ÷ GM), or T² = (4π² ÷ GM) r³.

A larger orbit means a smaller speed and a longer period. The orbital speed does not depend on the mass of the satellite. The total energy of the satellite is E = −GMm ÷ 2r, which is half of the (negative) potential energy; the kinetic energy is +GMm ÷ 2r. To move to a higher orbit, the satellite gains energy (E becomes less negative), but its speed decreases. A satellite is “weightless” because it is in free fall, continually falling toward the Earth but missing it because of its sideways speed.

What are Kepler’s laws?

1. First law (ellipses): the orbit of each planet is an ellipse with the Sun at one focus. (Circles are a special case.)
2. Second law (equal areas): the line from the Sun to a planet sweeps out equal areas in equal times. A planet moves faster when it is closer to the Sun (at perihelion) and slower when it is farther (at aphelion). This expresses the conservation of angular momentum.
3. Third law (harmonic law): the square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ a³. In units of years and astronomical units (AU), T² = a³ for bodies orbiting the Sun. Newton’s derivation shows that the constant is 4π² ÷ GM.

The laws apply to any orbiting system: planets, moons, satellites, and binary stars (using the total mass for the latter).

What types of satellite orbits are there?

• Low Earth orbit (LEO), 200 to 2,000 km altitude: periods of 90 to 130 minutes; used for the ISS, Earth observation and many communication constellations.
• Medium Earth orbit (MEO), such as the 20,200 km altitude of GPS satellites, with a period of about 12 hours.
• Geostationary orbit (GEO), at 35,786 km above the equator, with a period of 23 h 56 min (one sidereal day): the satellite stays above one point on the equator, which is used for communications and weather satellites.
• Polar and sun-synchronous orbits: pass over the poles, scanning the entire Earth as it rotates beneath, used for mapping and monitoring.

Orbits compared

Worked example: orbital speed, period and Kepler’s third law

1. ISS. r = 6.371 × 10⁶ + 4.0 × 10⁵ = 6.771 × 10⁶ m. v = √(3.986 × 10¹⁴ ÷ 6.771 × 10⁶) = √(5.887 × 10⁷) = 7,673 m/s = 7.67 km/s. T = 2πr ÷ v = 2π × 6.771 × 10⁶ ÷ 7,673 = 5,545 s = 92.4 minutes, about 15.6 orbits a day.
2. Geostationary orbit. The period T = 86,164 s. r³ = GMT² ÷ 4π² = 3.986 × 10¹⁴ × (86,164)² ÷ 39.48 = 7.495 × 10²², so r = 4.216 × 10⁷ m. The altitude is 4.216 × 10⁷ − 6.371 × 10⁶ = 3.58 × 10⁷ m = 35,800 km. The speed = 2π × 4.216 × 10⁷ ÷ 86,164 = 3,075 m/s.
3. Kepler’s third law. Mars is at a = 1.524 AU. T = a^3/2 = 1.524^1.5 = 1.88 years. Jupiter (a = 5.20 AU): T = 5.20^1.5 = 11.9 years.
4. Moon. r = 3.844 × 10⁸ m: T = 2π√(r³ ÷ GM) = 2π√(5.68 × 10²⁵ ÷ 3.986 × 10¹⁴) = 2π × 1.194 × 10⁵ = 2.37 × 10⁶ s = 27.5 days (the sidereal month is 27.3 days; the difference is from the Moon’s mass and the Sun’s influence).
5. Changing orbit. A satellite moves from a radius r to 2r: the speed falls by a factor √2, and the period increases by a factor 2^3/2 = 2.83.

How were the laws discovered?

Tycho Brahe made precise observations of Mars in the late 1500s. Johannes Kepler analyzed them and published the first two laws in 1609 (Astronomia Nova) and the third in 1619 (Harmonices Mundi). Newton showed in 1687 that Kepler’s laws follow from the inverse-square law of gravitation. Sputnik 1, the first artificial satellite, was launched on 4 October 1957, and Arthur C. Clarke proposed geostationary communication satellites in 1945. The first geostationary satellite, Syncom 3, was launched in 1964.

Which mistakes are common?

Mistake 1: Using the altitude instead of the orbital radius

r is measured from the center of the Earth: r = R + h.

Mistake 2: Saying that a heavier satellite orbits faster

The mass cancels, so the speed depends only on the orbital radius.

Mistake 3: Thinking that a satellite has to speed up to move to a higher orbit and stay there faster

It gains energy in the transfer, but the final circular orbit has a lower speed.

Mistake 4: Believing that astronauts float because there is no gravity

Gravity at the ISS is about 90 percent of that at the surface. They are in free fall.