Gaussian Constant

one of the fundamental astronomical constants (designated k). It was first defined by K. Gauss as the approximate value of the square root of the gravitational constant k², which is found in the formula of the two-body problem (in celestial mechanics):

This formula relates the sun’s mass m_s, the earth’s mass m_T, and the moon’s mass m_L to the period of revolution P of the earth-moon system along an elliptical orbit around the sun and to the semimajor axis a of this orbit. Gauss took the mass of the sun and the indicated semimajor axis a as the units of mass and length and the mean solar day as the unit of time. Using the values of P and the ratios m_T/m_s and m_L/m_T current in his day, Gauss found

k = 0.01720209895

This value of k (which is considered accurate) is part of the modern system of fundamental astronomical constants and is called the Gaussian constant. Provided that the ephemeris day is used as the unit of time, the unit of distance corresponding to this value of k and to formula (1) is called the astronomical unit (a.u.). This differs somewhat from the semimajor axis a of the orbit of the earth-moon system, which according to formula (1) and the contemporary values of P, m_T/m_S, and m_L/m_T is 1.000000032 a.u.