Editing Gaussian gravitational constant (section)

{{Short description|Constant used in orbital mechanics}}
[[File:GAUSS JPG.jpg|thumb|[[Carl Friedrich Gauss]] introduced his constant to the world in his 1809 ''Theoria Motus''.]]
[[File:Cerere Ferdinandea.gif|thumb|[[Giuseppe Piazzi|Piazzi's]] discovery of [[Ceres (dwarf planet)|Ceres]], described in his book '' the discovery a new planet Ceres Ferdinandea'', demonstrated the utility of the Gaussian gravitation constant in predicting the positions of objects within the Solar System.]]

The '''Gaussian gravitational constant''' (symbol {{mvar|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 mass]]es.

The value of {{mvar|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 [[degree (angle)|degrees]] per [[day]], or about 0.0172 [[radian]]s per day. As a consequence of the [[law of gravitation]] and [[Kepler's third law]], 
{{mvar|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, {{mvar|k}}:(rad/d) {{=}} ({{mvar|G}}{{solar mass}})<sup>0.5</sup>·au<sup>−1.5</sup>.

A value of {{mvar|k}} {{=}} {{val|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").<ref name="Gauss">{{cite book | last1 = Gauss | first1= Carl Friedrich | first2 = Charles Henry | last2 = Davis | title = Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections | publisher = Little, Brown and Company | location = Boston | year = 1857 | url = https://archive.org/details/bub_gb_1TIAAAAAQAAJ| page = [https://archive.org/details/bub_gb_1TIAAAAAQAAJ/page/n22 2] }}</ref>
Gauss's 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 {{mvar|k}} in 2012 in favour of a defined value of the astronomical unit of {{val|1.49597870700|e=11|u=m}}  exactly, while the strength of the gravitational force  is now to be expressed in the separate [[standard gravitational parameter]] {{mvar|G}}{{solar mass}},  measured in [[SI units]] of m<sup>3</sup>⋅s<sup>−2</sup>.<ref name="Smart53">{{cite book | last = Smart | first = W. M. | title = Celestial Mechanics | publisher = Longmans, Green and Co. | location = London | year = 1953 | page = 4}}</ref>