The laws of planetary motion

1.7.3 The laws of planetary motion

From the invaluable database of planetary positions provided by Tycho, Kepler was able to draw up his three empirical laws of planetary motion. The word ‘empirical’ indicated that these laws were not based on any deeper theory, but accurately described the observed motion of the planets. The fi rst two were pub- lished in 1609 and the third in 1618.

The fi rst law states that:

Planets move in elliptical orbits around the Sun, with the Sun positioned at one focus of the ellipse.

Figure 1.13 shows a planet in an elliptical orbit around the Sun and defi nes some of the terms associated with the orbit. The second law states that:

The radius vector – that is, the imaginary line joining the centre of the planet to the centre of the Sun – sweeps out equal areas in equal times.

Figure 1.14 shows Kepler’s Second Law graphically.

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Figure 1.13 The parameters of an elliptical orbit.

Figure 1.14 Kepler’s Second Law.

It is worth pointing out the fundamental physics that lies behind the second law.

A planet possesses both potential and kinetic energy. The potential energy relates to its distance from the Sun, reducing as it nears the Sun. The kinetic energy relates to its speed. As the planet is moving in space, there is no mechanism for it to lose energy, so the sum of the potential and kinetic energy must remain constant. In an elliptical orbit, the planet varies its distance from the Sun and so, for its total energy to be conserved, when its potential energy reduces as it nears the Sun its kinetic energy must increase. It must thus move faster along its orbit – exactly as implied by the second law.

22 Introduction to Astronomy and Cosmology

The third law relates the period of the planet’s orbit, T, with a, the semi-major axis of its orbit and states that:

The square of the planet’s period, T, is proportional to the cube of the semi-major axis of its orbit, a.

One point should be made here: if an orbit is circular, then the semi-major axis is simply the radius of the circle and thus the distance of the planet from the Sun. When the orbit is not far from circular, then the semi-major axis is very close to the mean distance of the planet from the Sun. The third law is often stated in the form:

The square of the planet’s period is proportional to the cube of its mean distance from the Sun. However, this is not strictly accurate. It should also be noted that Kepler’s Third Law as stated above is only applicable

when one of the bodies is signifi cantly more massive than the other – as is always the case for the planets of our Solar System.

Writing the third law mathematically:

T 2 αa 3

Thus:

T 2 ⫽k⫻a 3

where k is a constant of proportionality. The value of k, which will be the same for all objects orbiting the Sun, depends on the units chosen. It is conventional to measure the period, T, in units of Earth years and the semi-major axis, a, in units of the Earth’s semi-major axis, which is termed an Astronomical Unit (AU), in which case k ⫽ 1.

The semi-major axis of the asteroid Ceres, which orbits the Sun every 4.60 years can thus be found using T 2 ⫽k⫻a 3 . With k ⫽ 1, this becomes a 3 ⫽T 2 , and thus:

a ⫽T 2/3 giving a ⫽ 2.77AU.

Kepler’s Third Law can, of course, be applied to any system of planets or sat- ellites orbiting a body. Only the value of the constant of proportionality will be different.

An example

Satellite television signals are broadcast from what are termed ‘Geostationary’

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satellite will orbit the Earth once per day and so remain in the same position in the sky as seen from a location on the Earth’s surface, thus allowing a fi xed reception antenna. How high above the surface of the Earth at the equator would such an orbit be?

The radius of the Moon’s orbit is 384 400 km and its orbital period is 27.32 days. (You may be worried about this value for the period and might think that it should be 29.53 days. This latter value is the period between two New Moons, and is called the synodic lunar month. It is obviously related to the position of the Moon related to the Sun and so depends both on the Moon’s motion about

the Earth and the orbital motion of the Earth–Moon system around the Sun. The

27.32 day sidereal lunar month, the value that we need to use, is determined by the time it takes for the Moon to return to the same place on the celestial sphere after one orbit of the Earth. Incidentally, Richard Feynman in his famous Feynman Lectures on Physics made this mistake and used 29.5 days as the Moon’s orbital period – but he was a world-famous physicist, not an astronomer, so perhaps he can be forgiven!)

Using these values, we can calculate the constant of proportionality that applies to satellites around the Earth:

k ⫽ (27.32) 2 /(384 400) 3 ⫽ 1.314 ⫻ 10 ⫺14 .

For our geostationary satellite, T is 1, so we derive “‘a’ from: 1 ⫽k⫻a 3

a ⫽ (1/k) 1/3 ⫽ 42 377 km.

The surface of the Earth is 6400 km from the centre, so geostationary satellites are ∼36 000 km above the surface of the Earth.