Kepler's constant

The Kepler constant is a parameter resulting from Kepler's 3rd law . It is the quotient of the square of the period of revolution of a celestial body and the third power of the semi-major axis of its orbit : ${\displaystyle C}$

${\displaystyle C={\frac {T^{2}}{a^{3}}}}$

This quotient is constant for a central object . With the sun as the central star (i.e. for the planets orbiting it, etc.) the following value applies , which is often given in formulas :

${\displaystyle C_{\mathrm {s} }=2{,}97\cdot 10^{-19}\,{\frac {\mathrm {s} ^{2}}{\mathrm {m} ^{3}}}}$

With the help of this Kepler constant, the orbital time or the major semiaxis of the orbit of a planet can be calculated if the other value is known. Often planetary orbits are simplified as circular orbits and the major semi-axis is equated with the radius .

Alternative calculation

The Kepler constant can also be determined without knowing the semiaxis and the period of revolution of a planet. From Kepler's third law, with the aid of the law of gravitation, the following results :

${\displaystyle C={\frac {4\,\pi ^{2}}{G\,(M+m)}}}$

in which

• G is the gravitational constant
• M is the mass of the central body
• m is the mass of the orbiting body (planet).

This shows that the Kepler “constant” basically depends on the planet under consideration. But since is as a rule , the planetary mass m can  usually be neglected: ${\displaystyle M\gg m}$

${\displaystyle C\approx {\frac {4\pi ^{2}}{G\,M}}}$

Earth as a central body

For artificial satellites orbiting the earth , one would have to use the semi-major axis and the orbital time of the moon as a first approximation when calculating the Kepler constant ; this value would then also apply to the satellite, since it also orbits the central body of the earth.

The Kepler formula is based on the idealized assumption that the mass of the celestial body is negligibly small compared to that of the central body. In fact, however, the moon - and the earth - revolve around their common center of gravity , which results from the relevant mass ratio. The earth-satellite situation, however, corresponds to the model assumption, i. H. the mass of the satellite is actually negligible compared to that of the earth. Therefore, the calculation via the Earth's moon for artificial satellites does not lead to the goal. For more details see under satellite orbit element .

Individual evidence

1. ^ Rudolf Pitka: Physics - the basic course. Harri Deutsch Verlag, 2009, ISBN 978-3817118526 , p. 127 ( online  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. ).