Standard gravitational parameter

Body	μ μ {displaystyle mu } (m)3s-2)
Sun	1.327 124 400 18(9)×1020 [chuckles] ]
Mercury	2.2032(9)×1013
Venus	3.248 59(9)×1014
Earth	3.986 004 418(9)×1014 [chuckles] ]
Mars	4.282 837(2)×1013
Jupiter	1.266 865 34(9)×1017
Saturn	3.793 118 7(9)×1016
Uranus	5.793 939(9)×1015
Neptune	6.836 529(9)×1015
Pluto	8.71(9)×1011

In astrodynamics, the Standard gravitational parameter (μ μ {displaystyle mu \,}) of a celestial body is the product of universal gravitation constant (G{displaystyle G,}) and its mass M{displaystyle M,}:

μ μ =G⋅ ⋅ M{displaystyle mu =Gcdot M!,}

The units of the standard gravitational parameter in the International System are m³s^-2 although it is frequently expressed in km³s^-2

Small body that orbits a central body

Under the standard astrodynamic hypotheses we have:

<math alttext="{displaystyle m_{1}<m1;m2{displaystyle m_{1}{2}{2},}<img alt="{displaystyle m_{1}<

where:

• m1{displaystyle m_{1}! is the mass of the orbiting body,
• m2{displaystyle m_{2}! is the mass of the central body,

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

For all circular orbits:

μ μ =rv2=r3ω ω 2=4π π 2r3T2{displaystyle mu =rv^{2}=r^{3}omega ^{2}={dfrac {4pi ^{2}r^{3}{T^{2}}}}}}}}}

where:

• r{displaystyle r,} It's orbital radio,
• v{displaystyle v,} It's orbital speed,
• ω ω {displaystyle omega \,} It's angular speed,
• T{displaystyle T,} It's the orbital period.

The last equation has a very simple generalization for elliptical orbits:

μ μ =4π π 2a3T2{displaystyle mu ={dfrac {4pi ^{2}a^{3}{T^{2}}}{2}}}}}

where:

• a{displaystyle a,} It's the ellipse's biggest semieje. This is Kepler's third law

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

Two bodies orbiting each other

In the most general case where the bodies are not necessarily one large and one small, they are defined:

• the vector r is the position of one body in relation to the other
• r, v, and in the case of an elliptical orbit, the major semieje a, are defined respectively (and r is distance)
• μ μ =G(m1+m2){displaystyle mu ={G}(m_{1}+m_{2}),} (the sum of the two μ values)

where:

• m1{displaystyle m_{1}! and m2{displaystyle m_{2}! are the mass of the two bodies

Then:

• For circular orbits rv2=r3ω ω 2=4π π 2r3T2=μ μ {displaystyle rv^{2}=r^{3}omega ^{2}={dfrac {4pi ^{2}r^{3}}{T^{2}}}}}}{mu }
• For elliptical orbits: 4π π 2a3T2=μ μ {displaystyle {dfrac {4pi ^{2}a^{3}{T^{2}}}}{mu }
• For parabolic trajectories rv2{displaystyle rv^{2}} is constant and equal to 2μ μ {displaystyle 2mu }
• For elliptical and hyperbolic orbits μ μ {displaystyle mu } is twice 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 precision

The value of the Earth is called geocentric gravitational constant and is equal to 398 600.441 8 ± 0.000 8 km³s^-2. So the precision is 1/500,000,000, much more precise than the precisions of G and M separately (1/7000 each).

The value of the Sun is called heliocentric gravitational constant and whose value is 1.32712440018×10²⁰ m³s^-2.