Standard gravitational parameter

In celestial mechanics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when one body is much larger than the other: For several objects in the Solar System, the value of μ is known to greater accuracy than either G or M. The SI units of the standard gravitational parameter are m3 s−2. However, units of km3 s−2 are frequently used in the scientific literature and in spacecraft navigation. The central body in an orbital system can be defined as the one whose mass (M) is much larger than the mass of the orbiting body (m), or M ≫ m. This approximation is standard for planets orbiting the Sun or most moons and greatly simplifies equations. Under Newton's law of universal gravitation, if the distance between the bodies is r, the force exerted on the smaller body is: Thus only the product of G and M is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product, μ, not G and M separately. The gravitational constant, G, is difficult to measure with high accuracy, while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision. For a circular orbit around a central body, where the centripetal force provided by gravity is F = mvr: where r is the orbit radius, v is the orbital speed, ω is the angular speed, and T is the orbital period. This can be generalized for elliptic orbits: where a is the semi-major axis, which is Kepler's third law. For parabolic trajectories rv2 is constant and equal to 2μ. For elliptic and hyperbolic orbits μ = 2a, where ε is the specific orbital energy. In the more general case where the bodies need not be a large one and a small one, e.g. a binary star system, 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) μ = Gm1 + Gm2 = μ1 + μ2, where m1 and m2 are the masses of the two bodies.