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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.}
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, = (),
universal gravitational constant: newton meter squared per kilogram squared (N⋅m 2 /kg 2) shear modulus: pascal (Pa) or newton per square meter (N/m 2) acceleration due to gravity: meters per second squared (m/s 2), or equivalently, newtons per kilogram (N/kg) magnetic field strength: ampere per meter (A/m)
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
For an object of mass the energy required to escape the Earth's gravitational field is GMm / r, a function of the object's mass (where r is radius of the Earth, nominally 6,371 kilometres (3,959 mi), G is the gravitational constant, and M is the mass of the Earth, M = 5.9736 × 10 24 kg).
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).
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}}} :