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standard gravitational parameter


In astrodynamics, the standard gravitational parameter \mu \ of a celestial body is the product of the gravitational constant G and the mass M:

     \mu=GM \

The units of the standard gravitational parameter are km3s-2

Small body orbiting a central body


Under standard assumptions in astrodynamics we have:
     m << M \
where:
m \ is the mass of the orbiting body,
M \ 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:
     \mu = rv^2 = r^3\omega^2 = 4\pi^2r^3/T^2 \
where:
r \ is the orbit radius,
v \ is the orbital speed,
\omega \ is the angular speed,
T \ is the orbital period.


The last equality has a very simple generalization to elliptic orbits:
     \mu=4\pi^2a^3/T^2 \
where:
a \ is the semi-major axis.
See Kepler's third law.


For all parabolic trajectories r v^2 \ is constant and equal to 2 \mu \ ;.

For elliptic and hyperbolic orbits \mu \ 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 \mathbf{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)
\mu={G}(m_1 + m_2) \ (the sum of the two \mu \ values)
where:
m_1 \ and m_2 \ are the masses of the two bodies.

Then:
for circular orbits rv^2 = r^3 \omega^2 = 4 \pi^2 r^3/T^2 = \mu\!\,
for elliptic orbits: 4 \pi^2 a^3/T^2 = \mu \ (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get a^3/T^2 = M)
for parabolic trajectories r v^2 \ is constant and equal to 2 \mu \
for elliptic and hyperbolic orbits \mu \ 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 and equals 1.32712440018 m3s-2.

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