Gaussian gravitational constant

Carl Friedrich Gauss expressed the gravitational constant in units of the solar system rather than SI units. The benefit is that the motion of the planets can be accurately described, without exact knowledge of the scale of the solar system or the masses of the Sun and planets expressed in mundane units like those of the SI system.

Gauss used the following units:

• length A: astronomical unit (the mean axis of the orbit of the Earth around the Sun).
• time D: mean solar day (the mean rotation period of the Earth around its axis, with respect to the Sun).
• mass S: the mass of the Sun.

From Kepler's 3rd law applied to the motion of the Earth, he derived his gravitational constant:

k = 0.01720209895 A^3/2S^-1/2D^-1

In 1939 the International Astronomical Union adopted the above value as a defining constant in astronomy. The value of the astronomical unit is derived from it, and is no longer defined by the actual orbit of the Earth. In modern ephemerides, the mean orbital axis of the Earth is slightly longer than 1 A.U., and the sidereal year is slightly shorter than 1 Gaussian year.

Gauss was not fully aware of the secular increase in the length of the mean solar day and unaware of the relativistic differences in the rate of clocks. The day defined by this constant was later understood as the basis of the rate of Ephemeris Time, and in modern usage this day is understood to be measured in units of Barycentric Dynamical Time (TDB). 1/86400 of the day defined by this constant was known as the ephemeris second, and the length of the ephemeris second as measured by clocks on the surface of the earth was adopted as the SI second.