Semimajor axis via Kepler’s constant and total energy

The semi-major axis of an orbiting planet depends on the Kepler’s constant of the star—planet system and the total energy of the planet per unit of its mass.

Notes:

1. Works for both elliptical (\(\varepsilon < 0\)) and hyperbolical (\(\varepsilon > 0\)) orbits.

Links:

1. Sivukhin D.V. (1979), Obshchiy kurs fiziki [General course of Physics], vol. 1, p. 317, (58.2).

semimajor_axis

The semimajor_axis of the planet’s orbit.

Symbol:

a

Latex:

\(a\)

Dimension:

length

kepler_constant

The kepler_constant, whose value is determined by the mass of the orbited star.

Symbol:

K

Latex:

\(\mathfrak{K}\)

Dimension:

length**3/time**2

specific_energy

specific_energy of the planet. Can be negative or positive depending on the sign of the planet’s energy.

Symbol:

epsilon

Latex:

\(\varepsilon\)

Dimension:

energy/mass

law

a = 2 * pi^2 * K / Abs(epsilon)

Latex:

\[a = \frac{2 \pi^{2} \mathfrak{K}}{\left|{\varepsilon}\right|}\]