Gaussian Gravitational Constant - Derivation

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, ΔA, swept out by the radius divided by the time, Δt, and also related to the parameter, p = h2/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 ΔA = π AU2 and Gauss sets Δt = 365.2563835, the sidereal period, and the mass of the Earth, m equal to 1/354710 solar masses 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 ω2a3 = 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 = h2/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 k2 is related to the standard gravitational parameter expressed in a system of measurement where masses are measured in solar masses, 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.