Gaussian gravitational constant

The Gaussian gravitational constant (symbol k) is a parameter used in the orbital mechanics of the Solar System. It relates the orbital period to the orbit's semi-major axis and the mass of the orbiting body in Solar masses.

The value of k historically expresses the mean angular velocity of the system of Earth+Moon and the Sun considered as a two body problem, with a value of about 0.986 degrees per day, or about 0.0172 radians per day. As a consequence of the law of gravitation and Kepler's third law, k is directly proportional to the square root of the standard gravitational parameter of the Sun, and its value in radians per day follows by setting Earth's semi-major axis (the astronomical unit, au) to unity, k:(rad/d) = (GM_☉)^0.5·au^−1.5.

A value of k = 0.01720209895 rad/day was determined by Carl Friedrich Gauss in his 1809 work Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum ("Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections").^[1] Gauss' value was introduced as a fixed, defined value by the IAU (adopted in 1938, formally defined in 1964), which detached it from its immediate representation of the (observable) mean angular velocity of the Sun–Earth system. Instead, the astronomical unit now became a measurable quantity slightly different from unity. This was useful in 20th-century celestial mechanics to prevent the constant adaptation of orbital parameters to updated measured values, but it came at the expense of intuitiveness, as the astronomical unit, ostensibly a unit of length, was now dependent on the measurement of the strength of the gravitational force.

The IAU abandoned the defined value of k in 2012 in favour of a defined value of the astronomical unit of 1.49597870700×10¹¹ m exactly, while the strength of the gravitational force is now to be expressed in the separate standard gravitational parameter GM_☉, measured in SI units of m³⋅s⁻².^[2]