Geocentric gravitational constant - Definition and Overview

In astrodynamics, the standard gravitational parameter (<math>\mu\!\,<math>) of a celestial body is the product of the gravitational constant (<math>G\!\,<math>) and the mass <math>M\!\,<math>:

<math>\mu=G*M\!\,<math>

The units of the standard gravitational parameter are km³s^-2

Small body orbiting a central body

Under standard assumptions in astrodynamics we have:

<math>m_1 << m_2\!\,<math>

where:

• <math>m_1\!\,<math> is the mass of the orbiting body,
• <math>m_2\!\,<math> is the mass of the central body,

and the relevant standard gravitational parameter is that of the larger body.

For all circular orbits around a given central body:

<math>\mu = rv^2 = r^3\omega^2 = 4\pi^2r^3/T^2\!\,<math>

where:

• <math>r\!\,<math> is the orbit radius,
• <math>v\!\,<math> is the orbital speed,
• <math>\omega\!\,<math>is the angular speed,
• <math>T\!\,<math> is the orbital period.

The last equality has a very simple generalization to elliptic orbits:

<math>\mu=4\pi^2a^3/T^2\!\,<math>

where:

• <math>a\!\,<math> is the semi-major axis.

For all parabolic trajectories rv² is constant and equal to 2μ.

For elliptic and hyperbolic orbits μ is twice the semi-major axis times the absolute value of the specific orbital energy.

Two bodies orbiting each other

In the more general case where the bodies need not be a large one and a small one, we define:

• the vector r is the position of one body relative to the other
• r, v, and in the case of an elliptic orbit, the semi-major axis a, are defined accordingly (hence r is the distance)
• <math>\mu={G}(m_1+m_2)\!\,<math> (the sum of the two μ-values)

where:

• <math>m_1\!\,<math> and <math>m_2\!\,<math> are the masses of the two bodies.

Then:

• for circular orbits <math>rv^2 = r^3 \omega^2 = 4 \pi^2 r^3/T^2 = \mu\!\,<math>
• for elliptic orbits: <math>4 \pi^2 a^3/T^2 = \mu\!\,<math>
• for parabolic trajectories <math>r v^2\!\,<math> is constant and equal to <math>2 \mu\!\,<math>
• for elliptic and hyperbolic orbits <math>\mu<math> is twice the semi-major axis times the absolute value of the specific orbital energy, where the latter is defined as the total energy of the system divided by the reduced mass.

Terminology and accuracy

The value for the Earth is called geocentric gravitational constant and equal to 398,600.441,8 ± 0.000,8 km³s^-2. Thus the uncertainty is 1 to 500 000 000, much smaller than the uncertainties in G and M separately (1 to 7000 each).

The value for the Sun is called heliocentric gravitational constant.