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The standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of that body. For two bodies, the parameter may be expressed as G ( m 1 + m 2 ) , or as GM when one body is much larger than the other: μ = G ( M + m ) ≈ G M . {\displaystyle \mu =G(M+m)\approx GM.}
After reducing the problem to the relative motion of the bodies in the plane, he defines the constant of the motion c 3 by the equation ẋ 2 + ẏ 2 = 2k 2 M/r + c 3, where M is the total mass of the two bodies and k 2 is Moulton's notation for the gravitational constant. He defines c 1, c 2, and c 4 to be other constants of the
The product GM is the standard gravitational parameter and is often known to higher precision than G or M separately. The potential has units of energy per mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero.
The quantity is often termed the standard gravitational parameter, which has a different value for every planet or moon in the Solar System. Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by 2 {\displaystyle {\sqrt {2}}} :
= (+) is the standard gravitational parameter of the bodies; h {\displaystyle h\,\!} is the specific relative angular momentum of the same body referenced [ 2 ] to the center of mass. In other context h is used in the sense of a total for two bodies expressed as relative angular momentum of the system divided by the reduced mass, giving the ...
where μ is the standard gravitational parameter, a constant for any particular gravitational system. If the mean motion is given in units of radians per unit of time, we can combine it into the above definition of the Kepler's 3rd law, = (),
μ = G(M + m), a gravitational parameter, [note 2] where G is Newton's gravitational constant, M is the mass of the primary body (i.e., the Sun), m is the mass of the secondary body (i.e., a planet), and; p is the semi-parameter (the semi-latus rectum) of the body's orbit. Note that every variable in the above equations is a constant for two ...
The mean anomaly changes linearly with time, scaled by the mean motion, [2] =. where μ is the standard gravitational parameter. Hence if at any instant t 0 the orbital parameters are (e 0, a 0, i 0, Ω 0, ω 0, M 0), then the elements at time t = t 0 + δt is given by (e 0, a 0, i 0, Ω 0, ω 0, M 0 + n δt).