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The Gaussian gravitational constant (symbol ''k'') is an astronomical constant first proposed by German polymath Carl Friedrich Gauss in his 1809 work ''Theoria motus corporum coelestium in sectionibus conicis solem ambientum'' ("Theory of Motion of the Celestial Bodies Moving in Conic Sections around the Sun"), although he had already used the concept to great success in predicting the orbit of Ceres in 1801. It is equal to the square root of ''G'' where ''G'' is the Newtonian gravitational constant and is the solar mass and roughly equal to the mean angular velocity of the Earth in orbit around the Sun. The Gaussian gravitational constant is related to an expression which is the same for all bodies orbiting the Sun. A different constant is needed for the objects in orbit about another body. The value that was calculated by Gauss,〔 Reprint of 1857 translation of Theoria Motus〕 in the astronomical system of units, was still used until recently. It formed the basis of the definition of the international second from 1956 to 1967, and was a defining constant in the astronomical system of units from 1952 to 2012. == Derivation == In ''Theoria Motus'', Gauss gave an expression for all bodies orbiting the Sun that had a constant value. To derive this expression we need the specific relative angular momentum, ''h'', which is related to the areal velocity and a constant of the motion of a planet and the formulae for free orbits. We note that ''h'' is equal to the area, Δ''A'', swept out by the radius divided by the time, Δ''t'', and also related to the parameter, ''p'' = ''h''2/μ, so, : Where m is the mass of the body divided by the mass of the sun. On dividing by the variable quantities on the right associated with the orbiting body we get, : For a 1 AU circular orbit ''p'' = 1 AU, the area bounded by the orbit is Δ''A'' = π AU2 and Gauss sets Δ''t'' = 365.2563835, the sidereal period, and the mass of the Earth, ''m'', equal to which yields ''k'' = 0.01720209895. Gauss used relative values for his measurements so his value for ''k'' is unitless and measured in radians. If we treat mass and distance as relative measurements and use the day as the unit of time then the units for ''k'' are radians per day. The Gaussian gravitational constant is now an IAU defining constant used to define the astronomical unit. Gauss' constant can be used as the constant of proportionality in the formula for the mean daily motion, ''n'' (in radians per day), for bodies in elliptical orbits. The mean motion is a function the semi-major axis, ''a'', in AU. : In general relativity this formula is sometimes written as ω2''a''3 = ''M''. In the case of nearly circular planetary orbits about the Sun one can show in general relativity that the equation for the orbit is approximately the same as the classical orbit with the exception that the plane of the orbit precesses slowly about the Sun resulting in an advance in perihelion. To first approximation we still have the parameter ''p'' = ''h''2/μ. So the derivation of the constant function above is also valid in general relativity to the order of the approximation but we have to use the precessing orbital plane and its slightly decreased mean motion to determine the perihelion period. The term "gravitational constant" comes from the fact that ''k''2 is related to the standard gravitational parameter expressed in a system of measurement where masses are measured in , time is measured in days and distance is measured in semi-major axes of the Earth's orbit. By transforming the system of measurement, Gauss had been able to greatly simplify the calculation of planetary orbits. This basic system (slightly modified in the definitions of the base units) is still used today as the astronomical system of units. 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「Gaussian gravitational constant」の詳細全文を読む スポンサード リンク
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