What Is The Semi-major Axis

Kepler's Third Law

The squares of the sidereal periods of the planets are proportional to the cubes of the semi-major axes of their orbits.

The semi-major axis of a planet is equal to the hateful altitude of the planet, and then one tin can also say that the cube of the mean distance of a planet is proportional to the square of its sidereal menses.

[NMSU, N. Vogt]

If a ₁ and T ₁ refer to the semi-major axis and sidereal menses of a planet P _ane moving about the Sun,

a ₁ ⁱⁱⁱ / T _one ² = constant,

the constant existence the same for whatsoever of the planetary orbits. If a _two, a ₃,..., and T ₂, T ₃, ..., refer to the semi-major axes and sidereal periods of the other planets P ₂, P ₃, ..., moving most the Sun, then

a _i ⁱⁱⁱ / T _i ² = a ₂ ³ / T ₂ ^two = a ₃ ^three / T ₃ ² = constant.

The most convenient class for the abiding is obtained past taking the planet to be the Earth and expressing the distance in astronomical units and the time in years. Hence, for the Globe, a = one and T = 1, so the constant becomes unity. For any other planet, consequently,

a ³ = T ²,

showing that if nosotros measure the sidereal period, T, of a planet in Earth-years, we tin can obtain its mean distance from the Sun, a, in AU. Kepler showed that the 4 moons of Jupiter discovered past Galileo besides obeyed his 3rd law, though with a different value of the constant, confirming its wide applicability. The and then-called abiding in Kepler'south tertiary law is really dependent on the masses of the two angelic bodies in orbit about each other. Nevertheless, when one of the bodies is significantly more massive that the other, such as the Sun compared to the planets, or Jupiter compared to its satellites, Kepler's third law is a very close approximation to the truth. Only when the outer-most retrograde satellites in the solar organisation are considered, or shut satellites of a non-spherical planet, do Kepler'southward laws fail to describe the beliefs of such bodies with pregnant precision.

This is simply a specific case of the general relation between semi-major axis a and orbital period P for a point mass in orbit around a mass M, governed by the gravitational force.