Astronomers
 
have found it convenient to use data connected with the Earth’s orbit and the Sun as their units of time, distance and mass. Newton’s precise statement of Kepler’s third law for a planet of mass m₂ revolving about the Sun of mass m₁ may be written in the form 
k²(m₁ + m₂)T ² = 4π²a³.
Taking the solar mass, the mean solar day and the Earth’s mean distance from the Sun as the units of mass, time and distance respectively, the equation becomes
k²(1 + m₂)T ² = 4π²a³
where k₂ is written for G, the gravitational constant, and m₂, T and a are all in the units defined earlier. The quantity k is called the Gaussian constant of gravitation.
If, as was done by Gauss, the planet is taken to be the Earth and T given the value of 365·256 383 5 mean solar days (the length of the sidereal day adopted by Gauss), while m2 is taken to be 1/354 710 solar masses, k is found to have the value 0·017 202 098 95, the value of a being, of course, unity. This distance, the semi-major axis of the Earth’s orbit, was called the astronomical unit (AU).
The concept has since been refined. From time to time, various quantities have been determined more accurately but to avoid having to re-compute k and other related quantities every time, astronomers retain the original value of k as absolutely correct. This means that the Earth is treated like any other planet. The unit of time is now the ephemeris day. The Earth’s mean distance from the Sun is now taken as 1·000 000 03 astronomical units while the Earth–Moon system’s mass is 1/328 912 solar masses.
We may note that the precise definition of the astronomical unit is given by Kepler’s third law,
k²(1 + m)T ² = 4π²a³
with the Sun’s mass taken to be unity, k = 0·017 202 098 95, and the unit of time taken to be one ephemeris day. It is the radius of a circular orbit in which a body of negligible mass, undisturbed by the gravitational attractions of all other bodies, will revolve about the Sun in one Gaussian year of 2π/k ephemeris days. It has a value of 1·496 00 × 10¹¹ m.