Gaussian gravitational constant

Piazzi's discovery of Ceres, described in his book Della scoperta del nuovo pianeta Cerere Ferdinandea, demonstrated the utility of the Gaussian gravitation constant in predicting the positions of objects within the Solar System.

The Gaussian gravitational constant (symbol k) is an astronomical constant first proposed by German polymath Carl Friedrich Gauss in his 1809 work Theoria motus corporum coelestium in sectionibus conicis solem ambientum ("Theory of Motion of the Celestial Bodies Moving in Conic Sections around the Sun"), although he had already used the concept to great success in predicting the orbit of Ceres in 1801. It is equal to the square root of GM where G is the Newtonian gravitational constant and M is the solar mass and roughly equal to the mean angular velocity of the Earth in orbit around the Sun. The Gaussian gravitational constant is related to an expression which is the same for all bodies orbiting the Sun. A different constant is needed for the objects in orbit about another body.

Its value was measured to great precision by Canadian-American astronomer Simon Newcomb in his Tables of the Sun (1895): his numerical value of 0.017 202 098 95 in the astronomical system of units is the value that was calculated by Gauss and still used today. It formed the basis of the definition of the international second from 1956 to 1967, and has been a defining constant in the astronomical system of units since 1952.

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,

$h=r^2\dot{\theta}=2\frac{\Delta A}{\Delta t}=\sqrt{pGM\left(1+m\right)}$

On dividing by the variable quantities on the right associated with the orbiting body we get,

$\frac{2}{\sqrt{p\left(1+m\right)}}\frac{\Delta A}{\Delta t}=\sqrt{GM}=k$

For a 1AU circular orbit p = 1AU, 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 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 simple function the semi-major axis, a, in AU.

$n=k \sqrt{ \frac{1+m}{a^3}}$

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.

Later definitions

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. His original constant was not empirically measured for a full year.

When Canadian-American astronomer Simon Newcomb was appointed director of the Naval Almanac Office of the United States Naval Observatory in 1877, he set about a program of redetermination of the astronomical constants with George William Hill. Their efforts led to the preparation of Newcomb's Tables of the Sun in 1895, which were based on a value of the Gaussian gravitation constant of 0.017 202 098 950 000 A^3/2 S^−1/2 D⁻¹, where A is the length of the semi-major axis of the Earth's orbit, S is the solar mass and D is the mean solar day at J1900.0. In 1938, the International Astronomical Union (IAU) adopted the above value for all future ephemerides.

When Ephemeris Time was adopted in 1952, the length of the ephemeris second was defined to be consistent with Newcomb's value of k, so the length of the ephemeris day was exactly 86400 ephemeris seconds. Although the definition of the ephemeris second (and later the international second in 1956 and the SI second in 1960) refers to a fixed fraction of the tropical year at J1900.0, the only measure of the tropical year at that epoch was Newcomb's Tables, based on his measured value of the Gaussian gravitational constant. In effect, the second was redefined to better agree with Newcomb's Tables and hence with his value for k.

Newcomb was aware of the secular variation in the length of the mean solar day caused by tidal acceleration, but he does not appear to have fully corrected for it. By extrapolating from modern measurements, the date on which the mean solar day would have been exactly 86400 seconds long was about 1820, neatly in the middle of the data (from 1750–1890) which Newcomb used to prepare his Tables.

The astronomical system of units was redefined in 1976 to fix the value of k at precisely 0.017 202 098 95 A^3/2 S^−1/2 D⁻¹. The value of the astronomical unit is no longer defined as the semi-major axis of the Earth's orbit, but instead is that length which give exactly Newcomb's 1895 value of the Gaussian gravitational constant. In modern ephemerides, the mean orbital axis of the Earth is slightly longer than 1 AU, and the sidereal year is slightly shorter than 1 Gaussian year. The day was also redefined to be exactly 86400 SI seconds when measured at mean sea level on the Earth: in practice, it is measured in Barycentric Dynamical Time (TDB).

External links

• Gaussian Gravitational Constant, Wolfram ScienceWorld