Gaussian gravitational constant: Information from Answers.com

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.^[1] It is equal to the square root of the Newtonian gravitational constant G when this is expressed in a certain system of units: it is also roughly equal to the mean angular velocity of the Earth in orbit around the Sun.

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 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.

1 Derivation
2 Later definitions
3 References
4 External links

Derivation

Kepler's 3rd law states that:

"The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

Symbolically:

$P^2 \propto r^3\,$

The centripetal acceleration of a planet in its orbit is rω² (ω is the planet's angular velocity): according to Newton's law of gravitation, the gravitational acceleration on a planet due to the Sun is GS/r², where S is the mass of the Sun and G is the Newtonian gravitational constant. Hence:

$r\omega^2 = \frac{GS}{r^2}$

Neither G nor S can be measured accurately, but Gauss realised that only their product was needed to define planetary motion and, if this was measured for the Earth, the same constant could be used for the other planets. The orbital period is related to the angular velocity by

$P = \frac{2\pi}{\omega}$

Hence:

$\frac{4\pi^2}{P^2} = \frac{GS}{r^3} \Rightarrow P^2 = \frac{4\pi^2}{GS}r^3$

as Kepler's Third Law requires.

It is useful to transform back from orbital periods to angular velocity, remembering that the Earth's angular velocity is 2π radians per sidereal year (the sidereal year is the Earth's orbital period). The length of the semi-major axis of the Earth's orbit can be denoted by A (it later came to be known as the astronomical unit). For the Earth:

$\omega^2 = \frac{GS}{A^3}$

The angular velocity of the Earth can be measured by plotting the apparent position of the Sun: a convenient unit is radians per day (rad/d) which gives for (365.2425 days per Gregorian year)

$\omega = \frac{2\pi}{365.2425} \approx 0.017\,{\rm rad/d}$

The Earth's angular velocity is not constant – the Earth moves slightly faster when it it is closest to the Sun (perihelion) and slightly slower when it is furthest away (apihelion) – but it is an observable quantity whose mean value can be calculated. It is this mean value in radians per day which was the original Gaussian gravitational constant, k.

The term "gravitational constant" comes from the fact that k² is the Newtonian gravitational constant expressed in in a system of measurement where masses are measured in solar masses, time is measured in days and length is measured in half-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.^[2]

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.^[3]

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.^[4] 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:^[4] in practice, it is measured in Barycentric Dynamical Time (TDB).

References

^ Forbes, Eric G. (1971), "Gauss and the Discovery of Ceres", J. Hist. Astron. 2: 195–99, http://adsabs.harvard.edu/abs/1971JHA.....2..195F
^ Resolution of the VIth General Assembly of the International Astronomical Union, Stockholm, 1983.
^ Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, 1961, p. 70 .
^ ^a ^b Resolution No. 10 of the XVIth General Assembly of the International Astronomical Union, Grenoble, 1976.

External links

Gaussian Gravitational Constant, Wolfram ScienceWorld

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Gaussian gravitational constant

Contents

Derivation

Later definitions

References

External links