Planetary motion (5.4.3)

Kepler’s first law of planetary motion is related to the shape of orbits.

Orbiting objects (such as planets, moons, or artificial satellites) move in elliptical orbits with the object they are orbiting (such as a star or planet) at one of the two foci.

It is useful to know that all the planets in our solar system (including Earth) orbit in elliptical paths with low eccentricity, meaning that the orbits are approximately circular.

Kepler’s second law of planetary motion is related to the movement of planets.

A line joining an orbiting object and the object that it is orbiting sweeps out equal areas in equal times

This implies that orbiting objects move faster when closer to the sun (perihelion) and slower when farther from the Sun (aphelion).

Kepler’s second law results from the conservation of angular momentum, where the orbiting body’s speed adjusts to maintain a constant motion around its host.

Kepler’s third law of planetary motion relates orbital time to radius.

The square of the orbital time period is directly proportional to the cube of the orbital radius:

Here is the semi-major axis of the elliptical orbit, which is the greatest distance between the planet and the centre of the elliptical orbit.

The third law mathematically demonstrates how the time for one orbit increases with distance from the body being orbited.

Kepler’s laws provide an excellent approximation of planetary motion using Newtonian physics. However, they do not fully account for the effects of general relativity.

An example of this is Mercury’s orbit of the Sun. It deviates slightly from Keplerian predictions due to the curvature of spacetime caused by the Sun’s immense gravitational field. This discrepancy, known as the precession of Mercury’s perihelion, was accurately explained by Einstein’s theory of general relativity.

The centripetal force acting on a planet is the gravitational force of the sun on the planet.

The direction of the centripetal force is towards the sun and perpendicular to the direction of motion of the planet.

It is important to note that for simplified models typically found at A level, the orbital path of most planets can be approximated as circular.

Kepler’s third law applies to any satellite or moon orbiting a planet, provided the dominant mass is much greater than the orbiting object:

It is useful to note that if is at least 100 times larger than , the error in using Kepler’s Third Law without modification is generally very small (less than 1%).

Examples of systems where Kepler’s law can be applied, other than our solar system:

• Artificial satellites orbiting Earth
• Moons orbiting a planet
• Exoplanets orbiting other stars

A geostationary orbit is a circular orbit around Earth in which a satellite remains fixed above the same point on the equator. The satellite’s orbital period is exactly 24 hours, matching Earth’s rotational period.

This is a special case of a geosynchronous orbit, where the orbital period is 24 hours, but the satellite does not necessarily stay above the same point.

Conditions for a geostationary orbit:

• The orbital period equals 24 hours; the satellite must complete one orbit at the same time that Earth completes one full rotation.
• The satellite moves from west to east (in the same direction as the Earth’s spin).
• For an equatorial orbit, the satellite must orbit directly above the equator (zero inclination), which means that it remains in the same position above the Earth’s surface.
• The orbit must be perfectly circular, not elliptical.

The table below lists some of the applications of geostationary satellites.

The table below lists the advantages and disadvantages of the use of geostationary satellites.