Gaussian gravitational constant

The Gaussian gravitational constant (symbol k) is a parameter used in the orbital mechanics of the Solar System.

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").

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.

Its numerical value was obtained by setting the major semi-axis and the mass of the Sun to unity and measuring the period in mean solar days: The value represents the mean angular motion of the Earth-Sun system, in radians per day, equivalent to a value just below one degree (the division of the circle into 360 degrees in Babylonian astronomy was likely intended as approximating the number of days in a solar year[3]).

Since all involved parameters, the orbital period, the Earth-to-Sun mass ratio, the semi-major axis and the length of the mean solar day, are subject to increasingly refined measurement, the precise value of the constant would have to be revised over time.

[7] Newcomb's work was widely accepted as the best then available[8] and his values of the constants were incorporated into a great quantity of astronomical research.

[10] An IAU symposium on the system of constants was held in Paris in 1963, partially in response to recent developments in space exploration.

The working group's recommendations were accepted at the XIIth General Assembly of the IAU at Hamburg, Germany in 1964.

[11] Gauss intended his constant to be defined using a mean distance[note 1] of Earth from the Sun of 1 astronomical unit precisely.

[6] With the acceptance of the 1964 resolutions, the IAU, in effect, did the opposite: defined the constant as fundamental, and the astronomical unit as derived, the other variables in the definition being already fixed: mass (of the Sun), and time (the day of 86400 seconds).

[6] In 2012, the IAU, as part of a new, self-consistent set of units and numerical standards for use in modern dynamical astronomy, redefined the astronomical unit as[14] a conventional unit of length equal to 149597870700 m exactly, ... ... considering that the accuracy of modern range measurements makes the use of distance ratios unnecessaryand hence abandoned the Gaussian constant as an indirect definition of scale in the Solar System, recommending that the Gaussian gravitational constant k be deleted from the system of astronomical constants.The value of k based on the defined value for the astronomical unit would now be subject to the measurement uncertainty of the standard gravitational parameter,

Gauss begins his Theoria Motus by presenting without proof several laws concerning the motion of bodies about the Sun.

He continues, "it is of no importance which body we use for determining this number," and hence uses Earth, defining He states that the area swept out by Earth in its orbit "will evidently be" π√p, and uses this to simplify his constant to Here, he names the constant k and plugging in some measured values, t = 365.2563835 days, μ = ⁠1/354710⁠ solar masses, achieves the result k = 0.01720209895.

Again plugging in the measured values as they were known in Gauss's time, P = 365.2563835 days, m = ⁠1/354710⁠ solar masses,[clarification needed] yielding the result k = 0.01720209895.

The Gaussian constant is closely related to Kepler's third law of planetary motion, and one is easily derived from the other.

Beginning with the full definition of Gauss's constant, where From the geometry of an ellipse, the semi-latus rectum, p can be expressed in terms of a and b thus: p = ⁠b2/a⁠.

[dubious – discuss][22] It is also possible to set the gravitational constant, the mass of the Sun, and the astronomical unit to 1.