Gaussian gravitational constant

Carl Friedrich Gauss introduced his constant to the world in his 1809 Theoria Motus.

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 a fixed, empirical number which relates the period of an orbit to the total mass of the orbiting bodies and their distance apart, first proposed by Carl Friedrich Gauss in his 1809 work Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum ("Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections").^[1]

Gauss' constant was formerly used in celestial mechanics theory in preference to $G$ , Newton's gravitational constant.^[2] It was officially a defining constant of Solar system mechanics from 1964 until 2012.

Discussion

Isaac Newton was primarily concerned with the laws of force, that is, what the forces between matter were and how they would cause the planets to orbit as they do. His 1687 formulation of the law of universal gravitation^[3] thus included a constant $G$ which was convenient for that investigation—one which quantified the relatively weak gravitational force between masses. Such weak forces were difficult to measure with any accuracy; indeed Henry Cavendish's (indirect) first measurement was published only in 1798, just a few years before Gauss' work.

Gauss developed a method of calculating the orbit of a body from observations and hence needed an accurate and easily derived gravitational constant suitable for Solar System celestial mechanics on an astronomical scale. In formulating it, he specified distance in astronomical units (approximately the radius of Earth's orbit). Orbital period was specified in days, and mass in Solar masses. The result was roughly equal to the mean angular motion of Earth in its orbit around the Sun. In early celestial mechanics, comparative measurements, such as the size of an orbit versus Earth's orbit, or the mass of a body versus the Sun's mass, were easier to make and known more accurately than the same values in standard units—for instance, kilometers or kilograms. For these reasons, $k$ was more appropriate for theoretical work.

Role as a defining constant of Solar System dynamics

In the late nineteenth century, Simon Newcomb undertook the enormous task of calculating the fundamental constants of astronomy from the accurate observations of numerous observatories over many decades, a work which occupied him and his associates for more than 20 years.^[4] He used Gauss' gravitational constant throughout his work as one of these constants, and stated so in his Tables of the Sun in 1898,

The adopted value of the Gaussian constant is that of Gauss himself, namely:
$k$ = 3548″.18761 = 0.01720209895^[5]

Newcomb's work was widely accepted as the best then available^[6] and his values of the constants were incorporated into a great quantity of astronomical research. Because of this, it became difficult to separate the constants from the research; new values of the constants would, at least partially, invalidate a large body of work. Hence, after the formation of the International Astronomical Union in 1919 certain constants came to be gradually accepted as "fundamental": defining constants from which all others were derived. In 1938, the VIth General Assembly of the IAU declared,

We adopt for the constant of Gauss, the value
$k$ = 0.017202098950000

the unit of time is the mean solar day of 1900.0^[7]

However, no further effort toward establishing a set of constants was forthcoming until 1950.^[8] An IAU symposium on the system of constants was held in Paris in 1963, partially in response to recent developments in space exploration.^[4] The attendees finally decided at that time to establish a consistent set of constants. Resolution 1 stated that

The new system shall be defined by a non-redundant set of fundamental constants, and by explicit relations between these and the constants derived from them.

Resolution 4 recommended

that the working group shall treat the following quantities as fundamental constants (in the sense of Resolution No. 1).

Included in the list of fundamental constants was

The gaussian constant of gravitation, as defined by the VIth General Assembly of the I.A.U. in 1938, having the value 0.017202098950000.^[4]

These resolutions were taken up by a working group of the IAU, who in their report recommended two defining constants, one of which was

Gaussian gravitational constant, defining the a.u. $k$ = 0.01720209895^[4]

For the first time, the Gaussian constant's role in the scale of the Solar System was officially recognized. The working group's recommendations were accepted at the XIIth General Assembly of the IAU at Hamburg, Germany in 1964.^[9]

Definition of the astronomical unit

Gauss intended his constant to be defined using a mean distance^{[note 1]} of Earth from the Sun of 1 astronomical unit precisely.^[4] With the acceptance of the 1964 resolutions, the IAU, in effect, did the opposite: defined the constant as fundamental, and the astronomical unit as derived, the other variables in the definition being already fixed: mass (of the Sun), and time (the day of 7004864000000000000♠86400 seconds). This transferred the uncertainty from the gravitational constant to an uncertainty in the mean distance of Earth from the Sun, which was no longer exactly one a.u. The Earth's mean distance became an observed quantity rather than a defined, fixed one.^[10]

In 1976, the IAU reconfirmed the Gaussian constant's status at the XVIth General Assembly in Grenoble,^[11] declaring it to be a defining constant, and that

The astronomical unit of length is that length ( $A$ ) for which the Gaussian gravitational constant ( $k$ ) takes the value 6998172020989500000♠0.01720209895 when the units of measurement are the astronomical units of length, mass and time. The dimensions of $k 2$ are those of the constant of gravitation ( $G$ ), i.e., L³M⁻¹T⁻². The term "unit distance" is also used for the length ( $A$ ).

From this definition, the mean distance of Earth from the Sun works out to 7000100000003000000♠1.00000003 a.u., but with perturbations by the other planets, which do not average to zero over time, the average distance is 7000100000019999999♠1.0000002 a.u.^[4]

Abandonment

In 2012, the IAU, as part of a new, self-consistent set of units and numerical standards for use in modern dynamical astronomy, redefined the astronomical unit as^[12]

a conventional unit of length equal to 7011149597870700000♠149597870700 m exactly, …
… considering that the accuracy of modern range measurements makes the use of distance ratios unnecessary

and hence abandoned the Gaussian constant as an indirect definition of scale in the Solar System, recommending

that the Gaussian gravitational constant $k$ be deleted from the system of astronomical constants.

Units and dimensions

The units of $k$ are^[4] (a.u.)^³⁄₂ (day)⁻¹ (solar mass)^{− ¹⁄₂}, where

(a.u.) is the distance for which

k

takes its value as defined by Gauss—the distance of the unperturbed circular orbit of a hypothetical, massless body whose orbital period is

2π / k

days,^[10]

(day) is the mean solar day,

(solar mass) is the mass of the Sun.

Therefore, the dimensions of $k$ are^[13]

length^³⁄₂ time⁻¹ mass^{− ¹⁄₂} or

L 3 ⁄ 2 T -1 M - 1 ⁄ 2

Because $k 2 = G$ ,^[14] the Newtonian gravitational constant, the dimensions of $k 2$ are the same as $G$ : $L 3 T -2 M -1$ .

Derivation

Gauss' original

Gauss begins his Theoria Motus by presenting without proof several laws concerning the motion of bodies about the Sun.^[1] Later in the text, he mentions that Pierre-Simon Laplace treats these in detail in his Mécanique Céleste.^[15] Gauss' final two laws are as follows:

The area swept by a line joining a body and the Sun divided by the time in which it is swept gives a constant quotient. This is Kepler's second law of planetary motion.
The square of this quotient is proportional to the parameter (that is, the latus rectum) of the orbit and the sum of the mass of the Sun and the body. This is a modified form of Kepler's third law.

He next defines

$2 p$ as the parameter (i.e., the latus rectum) of a body's orbit,
$μ$ as the mass of the body, where the mass of the Sun = 1,
$1 / 2 g$ as the area swept out by a line joining the Sun and the body,
$t$ as the time in which this area is swept,

and declares that

{\frac {g}{t{\sqrt {p}}{\sqrt {1+\mu }}}}

is "constant for all heavenly bodies". He continues, "it is of no importance which body we use for determining this number," and hence uses Earth, defining

unit distance = Earth's mean distance (that is, its semi-major axis) from the Sun,
unit time = one solar day.

He states that the area swept out by Earth in its orbit "will evidently be" $π \sqrt p$ , and uses this to simplify his constant to

{\frac {2\pi }{t{\sqrt {1+\mu }}}}.

Here, he names the constant $k$ and plugging in some measured values, $t$ = 7002365256383500000♠365.2563835 days, $μ$ = 1/7005354710000000000♠354710 solar masses, achieves the result $k$ = 6998172020989500000♠0.01720209895.

In modern terms

Gauss is notorious for leaving out details, and this derivation is no exception. It is here repeated in modern terms, filling out some of the details.

Define without proof

h=2{\frac {dA}{dt}},

where^[16]

$dA / dt$ is the time rate of sweep of area by a body in its orbit, a constant according to Kepler's second law, and
$h$ is the specific angular momentum, one of the constants of two-body motion.

Next define

h^{2}=\mu p,

where^[17]

μ = G(M + m), a gravitational parameter,^{[note 2]} where
- $G$ is Newton's gravitational constant,
- $M$ is the mass of the primary body (i.e., the Sun),
- $m$ is the mass of the secondary body (i.e., a planet), and
$p$ is the semi-parameter (the semi-latus rectum) of the body's orbit.

Note that every variable in the above equations is a constant for two-body motion. Combining these two definitions,

\left(2{\frac {dA}{dt}}\right)^{2}=G(M+m)p,

which is what Gauss was describing with the last of his laws. Taking the square root,

2{\frac {dA}{dt}}={\sqrt {G}}{\sqrt {M+m}}{\sqrt {p}},

and solving for $\sqrt G$ ,

{\sqrt {G}}={\frac {2dA}{dt{\sqrt {M+m}}{\sqrt {p}}}}.

At this point, define $k \equiv \sqrt G$ .^[2] Let $dA$ be the entire area swept out by the body as it orbits, hence $dA = π ab$ , the area of an ellipse, where $a$ is the semi-major axis and $b$ is the semi-minor axis. Let $dt = P$ , the time for the body to complete one orbit. Thus,

k={\frac {2\pi ab}{P{\sqrt {M+m}}{\sqrt {p}}}}.

Here, Gauss decides to use Earth to solve for $k$ . From the geometry of an ellipse, $p = b 2 / a$ .^[18] By setting Earth's semi-major axis, $a = 1$ , $p$ reduces to $b 2$ and $\sqrt p = b$ . Substituting, the area of the ellipse "is evidently" $π \sqrt p$ , rather than $π ab$ . Putting this into the numerator of the equation for $k$ and reducing,

k={\frac {2\pi }{P{\sqrt {M+m}}}}.

Note that Gauss, by normalizing the size of the orbit, has eliminated it completely from the equation. Normalizing further, set the mass of the Sun to 1,

k={\frac {2\pi }{P{\sqrt {1+m}}}},

where now $m$ is in solar masses. What is left are two quantities: $P$ , the period of Earth's orbit or the sidereal year, a quantity known precisely by measurement over centuries, and $m$ , the mass of the Earth–Moon system. Again plugging in the measured values as they were known in Gauss's time, $P$ = 7002365256383500000♠365.2563835 days, $m$ = 1/7005354710000000000♠354710 solar masses, yielding the result $k$ = 6998172020989500000♠0.01720209895.

Gauss' constant and Kepler's third law

The Gaussian constant is closely related to Kepler's third law of planetary motion, and one is easily derived from the other. Beginning with the full definition of Gauss' constant,

k={\frac {2\pi ab}{P{\sqrt {M+m}}{\sqrt {p}}}},

where

$a$ is the semi-major axis of the elliptical orbit,
$b$ is the semi-minor axis of the elliptical orbit,
$P$ is the orbital period,
$M$ is the mass of the primary body,
$m$ is the mass of the secondary body, and
$p$ is the semi-latus rectum of the elliptical orbit.

From the geometry of an ellipse, the semi-latus rectum, $p$ can be expressed in terms of $a$ and $b$ thus: $p = b 2 / a$ .^[18] Therefore,

{\sqrt {p}}={\frac {b}{\sqrt {a}}}.

Substituting and reducing, Gauss' constant becomes

k={\frac {2\pi }{P}}{\sqrt {\frac {a^{3}}{M+m}}}.

From orbital mechanics, $2π / P$ is just $n$ , the mean motion of the body in its orbit.^[16] Hence,

{\begin{aligned}k&=n{\sqrt {\frac {a^{3}}{M+m}}},\\[8pt]k^{2}&={\frac {n^{2}a^{3}}{M+m}},\\[8pt]k^{2}(M+m)&=n^{2}a^{3},\end{aligned}}

which is the definition of Kepler's third law.^[17]^[19] In this form, it is often seen with $G$ , the Newtonian gravitational constant in place of $k 2$ .

Setting $a = 1$ , $M = 1$ , $m ≪ M$ , and $n$ in radians per day results in $k \approx n$ , also in units of radians per day, about which see the relevant section of the mean motion article.

Other definitions

The value of Gauss' constant, exactly as he derived it, had been used since Gauss' time because it was held to be a fundamental constant, as described above. The solar mass, mean solar day and sidereal year with which Gauss defined his constant are all slowly changing in value. If modern values were inserted into the defining equation, a value of 6998172020978900000♠0.01720209789 would result.^[20] Such would be of little use unless the entire system of constants was changed to reflect the new gravitational constant.

It is also possible to set the gravitational constant, the mass of the Sun, and the astronomical unit to 1. This defines a unit of time with which the period of the resulting orbit is equal to $2π$ . These are often called canonical units.^[20] More on such conversions can be found at the Gravitational constant article.

Notes

↑ Historically, the term mean distance was used interchangeably with the elliptical parameter the semi-major axis. Neither refers to an actual average distance.
↑ Do not confuse $μ$ the gravitational parameter with Gauss' notation for the mass of the body.

References

1 2 Gauss, Carl Friedrich; Davis, Charles Henry (1857). Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. Boston: Little, Brown and Company.
1 2 Smart, W. M. (1953). Celestial Mechanics. London: Longmans, Green and Co. p. 4.
↑ Newton, Isaac (1803). Davis, William, ed. The Mathematical Principles of Natural Philosophy. Translated by Motte, Andrew. London: H. D. Symonds.
1 2 3 4 5 6 7 Clemence, G. M. (1965). "The System of Astronomical Constants". Annual Review of Astronomy and Astrophysics. 3: 93. Bibcode:1965ARA&A...3...93C. doi:10.1146/annurev.aa.03.090165.000521.
↑ Newcomb, Simon (1898). "I, Tables of the Motion of the Earth on Its Axis and Around the Sun". Astronomical Papers Prepared for the use of the American Ephemeris and Nautical Almanac. VI. Bureau of Equipment, Navy Department. p. 10.
↑ de Sitter, W.; Brouwer, D. (1938). "On the system of astronomical constants". Bulletin of the Astronomical Institutes of the Netherlands. 8: 213. Bibcode:1938BAN.....8..213D.
↑ "Resolutions of the VIth General Assembly of the International Astronomical Union, Stockholm, 1938" (PDF).
↑ Wilkins, G. A. (1964). "The System of Astronomical Constants. Part I". Quarterly Journal of the Royal Astronomical Society. 5: 23. Bibcode:1964QJRAS...5...23W.
↑ "Resolutions of the XIIth General Assembly of the International Astronomical Union, Hamburg, Germany, 1964" (PDF).
1 2 Herrick, Samuel (1965). "The fixing of the gaussian gravitational constant and the corresponding geocentric gravitational constant". Proceedings of the IAU Symposium no. 21: 95. Bibcode:1965IAUS...21...95H.
↑ "Resolutions of the XVIth General Assembly of the International Astronomical Union, Grenoble, France, 1976" (PDF).
↑ "Resolutions of the XXVIIIth General Assembly of the International Astronomical Union, 2012" (PDF).
↑ Brouwer, Dirk; Clemence, Gerald M. (1961). Methods of Celestial Mechanics. New York and London: Academic Press. p. 58.
↑ U.S. Naval Observatory, Nautical Almanac Office; H.M. Nautical Almanac Office (1961). Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. London: H.M. Stationery Office. p. 493.
↑ Laplace, Pierre Simon; Bowditch, Nathaniel (1829). Mécanique Céleste. Boston: Hilliard, Gray, Little and Wilkins.
1 2 Smart, W. M. (1977). Textbook on Spherical Astronomy (6th ed.). Cambridge: Cambridge University Press. p. 100. ISBN 0-521-29180-1.
1 2 Smart, W. M. (1977). p. 101.
1 2 Smart, W. M. (1977). p. 99.
↑ Vallado, David A. (2001). Fundamentals of Astrodynamics and Applications (2nd ed.). El Segundo, CA: Microcosm Press. p. 31. ISBN 1-881883-12-4.
1 2 Danby, J. M. A. (1988). Fundamentals of Celestial Mechanics. Richmond, VA: Willmann-Bell. p. 146. ISBN 0-943396-20-4.

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