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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 body. : For several objects in the Solar System, the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''.〔This is mostly because ''μ'' can be measured by observational astronomy alone, as it has been for centuries. Decoupling it into ''G'' and ''M'' must be done by measuring the force of gravity in sensitive laboratory conditions, as first done in the Cavendish experiment.〕 The SI units of the standard gravitational parameter are . == Small body orbiting a central body == 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 . 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,〔. A lengthy, detailed review.〕 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 ''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 ''rv''2 is constant and equal to 2''μ''. For elliptic and hyperbolic orbits , where ''ε'' is 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 (the two-body problem), 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'' = ''Gm''1 + ''Gm''2 = ''μ''1 + ''μ''2, where ''m''1 and ''m''2 are the masses of the two bodies. Then: * for circular orbits, ''rv''2 = ''r''3''ω''2 = 4π2''r''3/''T''2 = ''μ'' * for elliptic orbits, (with ''a'' expressed in AU; ''T'' in seconds and ''M'' the total mass relative to that of the Sun, we get ) * for parabolic trajectories, ''rv''2 is constant and equal to 2''μ'' * for elliptic and hyperbolic orbits, ''μ'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「standard gravitational parameter」の詳細全文を読む スポンサード リンク
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