View source for Gaussian gravitational constant

{{Short description|Constant used in orbital mechanics}}
[[File:GAUSS JPG.jpg|thumb|[[Carl Friedrich Gauss]] introduced his constant to the world in his 1809 ''Theoria Motus''.]]
[[File:Cerere Ferdinandea.gif|thumb|[[Giuseppe Piazzi|Piazzi's]] discovery of [[Ceres (dwarf planet)|Ceres]], described in his book '' the discovery a new planet Ceres Ferdinandea'', demonstrated the utility of the Gaussian gravitation constant in predicting the positions of objects within the Solar System.]]

The '''Gaussian gravitational constant''' (symbol {{mvar|k}}) is a parameter used in the [[orbital mechanics]] of the [[Solar System]].
It relates the orbital period to the orbit's [[semi-major axis]] and the [[mass]] of the orbiting body in [[Solar mass]]es.

The value of {{mvar|k}} historically expresses the mean [[angular velocity]] of the system of Earth+Moon and the Sun considered as a [[two body problem]],
with a value of about 0.986 [[degree (angle)|degrees]] per [[day]], or about 0.0172 [[radian]]s per day. As a consequence of the [[law of gravitation]] and [[Kepler's third law]], 
{{mvar|k}} is directly proportional to the square root of the [[standard gravitational parameter]] of the [[Sun]], and its value in radians per day follows by setting Earth's semi-major axis (the [[astronomical unit]], au) to unity, {{mvar|k}}:(rad/d) {{=}} ({{mvar|G}}{{solar mass}})<sup>0.5</sup>·au<sup>−1.5</sup>.

A value of {{mvar|k}} {{=}} {{val|0.01720209895}} rad/day was determined 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").<ref name="Gauss">{{cite book | last1 = Gauss | first1= Carl Friedrich | first2 = Charles Henry | last2 = Davis | title = Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections | publisher = Little, Brown and Company | location = Boston | year = 1857 | url = https://archive.org/details/bub_gb_1TIAAAAAQAAJ| page = [https://archive.org/details/bub_gb_1TIAAAAAQAAJ/page/n22 2] }}</ref>
Gauss's value was introduced as a fixed, defined value by the [[IAU]] (adopted in 1938, formally defined in 1964), which detached it from its immediate representation of the (observable) mean angular velocity of the Sun–Earth system. Instead, the [[astronomical unit]] now became a measurable quantity slightly different from unity. This was useful in 20th-century celestial mechanics to prevent the constant adaptation of orbital parameters to updated measured values, but it came at the expense of intuitiveness, as the astronomical unit, ostensibly a unit of length, was now dependent on the measurement of the strength of the [[gravitational force]].

The IAU abandoned the defined value of {{mvar|k}} in 2012 in favour of a defined value of the astronomical unit of {{val|1.49597870700|e=11|u=m}}  exactly, while the strength of the gravitational force  is now to be expressed in the separate [[standard gravitational parameter]] {{mvar|G}}{{solar mass}},  measured in [[SI units]] of m<sup>3</sup>⋅s<sup>−2</sup>.<ref name="Smart53">{{cite book | last = Smart | first = W. M. | title = Celestial Mechanics | publisher = Longmans, Green and Co. | location = London | year = 1953 | page = 4}}</ref>

==Discussion==
<!-- here is an 1803 edition of Principia in English, 
{{cite book | last1 = Newton | first1 = Isaac | editor1-last = Davis | editor1-first = William | title = The Mathematical Principles of Natural Philosophy | publisher = H. D. Symonds | location = London | translator-first = Andrew | translator-last = Motte | date = 1803 | url = https://archive.org/details/bub_gb_exwAAAAAQAAJ}}
it isn't clear what is being referenced by it.-->

Gauss's constant is derived from the application of [[Kepler's laws of planetary motion|Kepler's third law]] to the 
system of Earth+Moon and the Sun considered as a [[two-body problem]],
relating the period of revolution ({{mvar|P}}) to the major semi-axis of the orbit ({{mvar|a}}) and the total mass of the orbiting bodies ({{mvar|M}}).
Its numerical value was obtained by setting the major semi-axis and the mass of the Sun to unity and measuring the period in mean solar days:
:{{mvar|k}} {{=}} 2{{pi}} {{sqrt|{{mvar|a}}<sup>3</sup> }}  / ({{mvar|P}} {{sqrt|{{mvar|M}}}} ) ≈ 0.0172021 [rad], where:
: {{mvar|P}} ≈ 365.256  [days], {{mvar|M}} = ({{solar mass}}+{{earth mass}}+{{Lunar mass|sym=yes}}) ≈ 1.00000304 [{{solar mass}}], and {{mvar|a}} = 1 by definition.

The value represents the  [[mean motion|mean angular motion]] of the Earth-Sun system, in [[radian]]s per [[day]], equivalent to a value just below [[Degree (angle)|one degree]] (the division of the circle into 360 degrees in [[Babylonian astronomy]] was likely intended as approximating the number of days in a solar year<ref>David H. Kelley, Eugene F. Milone, ''Exploring Ancient Skies: A Survey of Ancient and Cultural Astronomy'' (2011), [https://books.google.com/books?id=ILBuYcGASxcC&pg=PA219 p. 219]</ref>). 
The correction due to the division by the square root of {{mvar|M}} reflects the fact that the Earth–Moon system is not orbiting the Sun itself, but the [[center of mass]] of the system.

[[Isaac Newton]] himself determined a value of this constant which agreed with Gauss's value to six significant digits.<ref>"The numerical value of the Gaussian constant was determined by Newton himself 120 years prior to Gauss. It agrees with the modern value to six significant figures. Hence the name 'Gaussian constant' should be regarded as a tribute to Gauss' services to celestial mechanics as a whole, instead of indicating priority in determining the numerical value of the gravitational constant used in celestial mechanics, as is sometimes considered in referring to his work." Sagitov (1970:713). [This claim is questionable since Sagitov does not give a citation to where Newton computed this value.]</ref>
Gauss (1809) gave the value with nine significant digits, as 3548.18761 [[arc second]]s.

Since all involved parameters, the [[year|orbital period]], the Earth-to-Sun [[Earth mass|mass ratio]], the [[astronomical unit|semi-major axis]] and the length of the [[day|mean solar day]], are subject to increasingly refined measurement, the precise value of the constant would have to be revised over time. 
But since the constant is involved in determining the orbital parameters of all other bodies in the Solar System, it was found to be more convenient to set it to a fixed value, by definition, implying that the value of {{mvar|a}} would deviate from unity.
The fixed value of {{mvar|k}} {{=}} 0.01720209895 [rad] was taken to be the one set by Gauss (converted from degrees to [[radian]]), so that 
{{mvar|a}} {{=}} 4{{pi}}<sup>2</sup>:({{mvar|k}}<sup>2</sup> {{mvar|P}}<sup>2</sup> {{mvar|M}}) ≈ 1.<ref name=Sagitov>Sagitov, M. U., "Current Status of Determinations of the Gravitational Constant and the Mass of the Earth", Soviet Astronomy, Vol. 13 (1970), 712–718, translated from ''Astronomicheskii Zhurnal'' Vol. 46, No. 4 (July–August 1969), 907–915.</ref>

Gauss's 1809 value of the constant was thus used as an authoritative reference value for the [[orbital mechanics]] of the [[Solar System]] for two centuries.
From its introduction until 1938 it was considered a measured quantity, and from 1938 until 2012 it was used as a defined quantity, with measurement uncertainty delegated to the value of the [[astronomical unit]]. 
The defined value of {{mvar|k}} was abandoned by the [[IAU]] in 2012, and the use of {{mvar|k}} was deprecated, to be replaced by a fixed value of the astronomical unit, and the (measured) quantity of the [[standard gravitational parameter]] {{mvar|G}}{{solar mass}}.

===Role as a defining constant of Solar System dynamics===

Gauss himself stated the constant in [[arc second]]s, with nine significant digits, as {{mvar|k}}  {{=}}  {{gaps|3548″.187|61}}.
In the late 19th century, this value was adopted, and converted to [[radian]], by [[Simon Newcomb]], as {{mvar|k}}  {{=}} {{gaps|0.017|202|098|95}}.<ref name="Clemence65">
{{cite journal| last1 = Clemence | first1 = G. M. | title = The System of Astronomical Constants | journal = Annual Review of Astronomy and Astrophysics | year = 1965| volume = 3| page = 93|bibcode=1965ARA&A...3...93C|doi = 10.1146/annurev.aa.03.090165.000521 }}</ref> 
and the constant appears in this form in  his  ''[[Newcomb's Tables of the Sun|Tables of the Sun]]'', published in 1898.<ref>
"The adopted value of the Gaussian constant is that of Gauss himself, namely: {{mvar|k}} {{=}} {{gaps|3548″.187|61}} {{=}} {{gaps|0.017|202|098|95}}".
{{cite book | last = Newcomb | first = Simon | title = Astronomical Papers Prepared for the use of the American Ephemeris and Nautical Almanac | publisher = Bureau of Equipment, Navy Department | year = 1898 | volume = VI|chapter=I, Tables of the Motion of the Earth on Its Axis and Around the Sun|page=10 |url=https://books.google.com/books?id=bEw0AQAAIAAJ}}</ref>

Newcomb's work was widely accepted as the best then available<ref>{{cite journal| last1 = de Sitter | first1 = W.| last2 = Brouwer | first2 = D. | title = On the system of astronomical constants | journal = Bulletin of the Astronomical Institutes of the Netherlands | year = 1938| volume = 8| page = 213|bibcode=1938BAN.....8..213D}}</ref> 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 [[International Astronomical Union|IAU]] declared,

{{quote|We adopt for the constant of Gauss, the value 
<blockquote>{{mvar|k}} {{=}} {{gaps|0.01720|20989|50000}}</blockquote>
the unit of time is the mean solar day of 1900.0<ref>{{cite web|url=http://www.iau.org/static/resolutions/IAU1938_French.pdf |title=Resolutions of the VIth General Assembly of the International Astronomical Union, Stockholm, 1938}}.
Before the 1940s, the [[second]] itself was defined as a fraction of the mean solar day, so that the mean solar day was 86,400&nbsp;s by definition (since the re-definition of the second, the mean solar day has been a measured quantity, fluctuating between 86,400.000 and 86,400.003&nbsp;s), see [[Day]].</ref>}}

However, no further effort toward establishing a set of constants was forthcoming until 1950.<ref>{{cite journal| last1 = Wilkins | first1 = G. A. | title = The System of Astronomical Constants. Part I | journal = Quarterly Journal of the Royal Astronomical Society | year = 1964| volume = 5| page = 23|bibcode=1964QJRAS...5...23W}}</ref> An IAU symposium on the system of constants was held in Paris in 1963, partially in response to recent developments in space exploration.<ref name=" Clemence65"/> The attendees finally decided at that time to establish a consistent set of constants. Resolution 1 stated that

{{quote|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

{{quote|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

{{quote|The gaussian constant of gravitation, as defined by the VIth General Assembly of the I.A.U. in 1938, having the value 0.017202098950000.<ref name=" Clemence65"/>}}

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

{{quote|Gaussian gravitational constant, defining the au &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {{mvar|k}} {{=}} 0.01720209895<ref name="Clemence65"/>}}

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.<ref>{{cite web|url=http://www.iau.org/static/resolutions/IAU1964_French.pdf|title=Resolutions of the XIIth General Assembly of the International Astronomical Union, Hamburg, Germany, 1964}}</ref>

====Definition of the astronomical unit====
Gauss intended his constant to be defined using a mean distance<ref group=note>Historically,{{citation needed|date=June 2018}} the term ''mean distance'' was used interchangeably with the elliptical parameter the ''[[semi-major axis]]''. It does not refer to an actual average distance.</ref> of Earth from the Sun of 1 [[astronomical unit]] precisely.<ref name=" Clemence65"/> 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 {{val|86400}} seconds). This transferred the uncertainty from the gravitational constant to uncertainty in the semi-major axis of the Earth-Sun system, which was no longer exactly one au (the au being defined as depending on the value of the gravitational constant).
The astronomical unit thus became a measured quantity rather than a defined, fixed one.<ref name="Herrick65">{{cite journal| last1 = Herrick | first1 = Samuel | title = The fixing of the gaussian gravitational constant and the corresponding geocentric gravitational constant | journal = Proceedings of the IAU Symposium No. 21 | year = 1965| volume = 21 | page = 95|bibcode=1965IAUS...21...95H}}</ref>

In 1976, the IAU reconfirmed the Gaussian constant's status at the XVIth General Assembly in Grenoble,<ref>{{cite web|url=http://www.iau.org/static/resolutions/IAU1976_French.pdf|title=Resolutions of the XVIth General Assembly of the International Astronomical Union, Grenoble, France, 1976}}</ref> declaring it to be a defining constant, and that

{{quote|The astronomical unit of length is that length ({{mvar|A}}) for which the Gaussian gravitational constant ({{mvar|k}}) takes the value {{val|0.01720209895}} when the units of measurement are the astronomical units of length, mass and time. The dimensions of {{math|''k''<sup>2</sup>}} are those of the constant of gravitation ({{mvar|G}}), i.e., {{dimanalysis|length=3|mass=−1|time=−2}}. The term "unit distance" is also used for the length ({{mvar|A}}).}}

From this definition, the mean distance of Earth from the Sun works out to 1.000 000 03 au, but with perturbations by the other planets, which do not average to zero over time, the average distance is 1.000 000 20 au.<ref name=" Clemence65"/>

====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<ref>{{cite web|url=http://www.iau.org/static/resolutions/IAU2012_English.pdf|title=Resolutions of the XXVIIIth General Assembly of the International Astronomical Union, 2012}}</ref> 
{{quote|a conventional unit of length equal to {{val|149597870700|u=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

{{quote|that the Gaussian gravitational constant {{mvar|k}} be deleted from the system of astronomical constants.}}

The value of ''k'' based on the defined value for the astronomical unit would now be subject to the measurement uncertainty of the [[standard gravitational parameter]],
<math>k =  \sqrt{G M_\odot }  \cdot \text{au}^{-1.5} \cdot \text{d}  =  {1.32712440018(9)}^{0.5} \cdot 1.495978707^{-1.5} \cdot 8.64 \cdot 10^{-2.5} = 0.0172020989484(6).</math>

===Units and dimensions===
{{mvar|k}} is given as a unit-less fraction of the order of 1.7%, but it can be considered equivalent to the square root of the [[gravitational constant]],<ref>{{cite book  | last1 = U.S. Naval Observatory | first1=Nautical Almanac Office | last2 = H.M. Nautical Almanac Office | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac | publisher = H.M. Stationery Office |location=London | year = 1961 | page = 493}}</ref> in which case it has the [[Units of measurement|units]] of au<sup>{{frac|3|2}}</sup>⋅d<sup>−1</sup>⋅{{solar mass}}<sup>−{{frac|1|2}}</sup>,<ref name="Clemence65"/>  where
:au is the [[distance]] for which {{mvar|k}} takes its value as defined by Gauss—the distance of the [[Perturbation (astronomy)|unperturbed]] [[circular orbit]] of a hypothetical, massless body whose [[orbital period]] is {{math|{{sfrac|2π|''k''}}}} days,<ref name="Herrick65"/>
:d is the [[mean solar day]] (86,400 seconds),
:{{solar mass}} is the [[mass]] of the [[solar mass|Sun]].
Therefore, the [[Dimensional analysis|dimensions]] of {{mvar|k}} are<ref>{{cite book|last1 = Brouwer|first1 = Dirk|last2 = Clemence|first2 = Gerald M.| title = Methods of Celestial Mechanics|url = https://archive.org/details/methodsofcelesti00brou|url-access = registration|publisher = Academic Press |location=New York and London|date=1961|page=[https://archive.org/details/methodsofcelesti00brou/page/58 58]}}</ref>
:length<sup>{{frac|3|2}}</sup> time<sup>−1</sup> mass<sup>−{{frac|1|2}}</sup> or {{math|L<sup>{{frac|3|2}}</sup> T<sup>−1</sup> M<sup>−{{frac|1|2}}</sup>}}.

In spite of this {{mvar|k}} is known to much greater accuracy than {{mvar|G}} (or the square root of {{mvar|G}}).
The absolute value of {{mvar|G}} is known to an accuracy of about 10<sup>−4</sup>, but the product {{math|''G''{{solar mass}}}} (the gravitational parameter of the Sun) is known to an accuracy better than 10<sup>−10</sup>.

== Derivation ==

===Gauss's original===
Gauss begins his ''Theoria Motus'' by presenting without proof several laws concerning the motion of bodies about the Sun.<ref name="Gauss"/> Later in the text, he mentions that [[Pierre-Simon Laplace]] treats these in detail in his ''Mécanique Céleste''.<ref>{{cite book | last1 = Laplace | first1 = Pierre Simon|last2 = Bowditch|first2 = Nathaniel | title = Mécanique Céleste | publisher = Hilliard, Gray, Little and Wilkins|location= Boston | year = 1829|url=https://archive.org/details/mcaniquecles01laplrich}}</ref> Gauss's 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 [[Johannes Kepler|Kepler]]'s [[Kepler's laws of planetary motion#Second law|second law of planetary motion]].
* The [[Square (algebra)|square]] of this quotient is proportional to the parameter (that is, the [[Ellipse#Latus rectum|latus rectum]]) of the [[orbit]] and the [[addition|sum]] of the [[mass]] of the Sun and the body. This is a modified form of [[Kepler's laws of planetary motion#Third law|Kepler's third law]].

He next defines:

*{{math|2''p''}} as the parameter (i.e., the [[latus rectum]]) of a body's orbit,
*{{mvar|μ}} as the mass of the body, where the mass of the Sun = 1,
*{{math|{{sfrac|1|2}}''g''}} as the area swept out by a line joining the Sun and the body,
*{{mvar|t}} as the time in which this area is swept,

and declares that 
:<math>\frac{g}{t\sqrt{p}\sqrt{1+\mu}}</math>
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" {{math|π{{sqrt|''p''}}}}, and uses this to simplify his constant to
:<math>\frac{2\pi}{t\sqrt{1+\mu}}.</math>
Here, he names the constant {{mvar|k}} and plugging in some measured values, {{mvar|t}} = {{val|365.2563835}} days, {{mvar|μ}} = {{sfrac|1|{{val|354710}}}} solar masses, achieves the result {{math|''k''}} = {{val|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

:<math>h=2\frac{dA}{dt},</math>

where<ref name="Smart77-100">{{cite book | last = Smart | first = W. M. | title = Textbook on Spherical Astronomy | url = https://archive.org/details/textbookonspheri00smar | url-access = limited | publisher = Cambridge University Press |location=Cambridge | year = 1977|edition=6th |isbn=0-521-29180-1|page=[https://archive.org/details/textbookonspheri00smar/page/n99 100]}}</ref>

*{{math|{{sfrac|''dA''|''dt''}}}} is the time rate of sweep of [[area]] by a body in its [[orbit]], a constant according to [[Johannes Kepler|Kepler]]'s [[Kepler's laws of planetary motion#Second law|second law]], and
*{{mvar|h}} is the [[Specific relative angular momentum|specific angular momentum]], one of the constants of [[Two-body problem|two-body motion]].

Next define

:<math>h^2=\mu p,</math>

where<ref name="Smart77-101">Smart, W. M. (1977). p. 101.</ref>
*{{math|''μ'' {{=}} ''G''(''M'' + ''m'')}}, a [[Standard gravitational parameter|gravitational parameter]],<ref group=note>Do not confuse {{mvar|μ}} the gravitational parameter with Gauss's notation for the mass of the body.</ref> where
**{{mvar|G}} is [[Isaac Newton|Newton's]] [[gravitational constant]],
**{{mvar|M}} is the [[mass]] of the primary body (i.e., the [[Sun]]),
**{{mvar|m}} is the mass of the secondary body (i.e., a [[planet]]), and
*{{mvar|p}} is the semi-parameter (the [[Ellipse#Latus rectum|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,

:<math>\left(2\frac{dA}{dt}\right)^2=G(M+m)p,</math>

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

:<math>2\frac{dA}{dt}=\sqrt{G}\sqrt{M+m}\sqrt{p},</math>

and solving for {{math|{{sqrt|''G''}}}},

:<math>\sqrt{G}=\frac{2dA}{dt\sqrt{M+m}\sqrt{p}}.</math>

At this point, define {{math|''k'' ≡ {{sqrt|''G''}}}}.<ref name="Smart53"/> Let {{mvar|dA}} be the entire area swept out by the body as it orbits, hence {{math|''dA'' {{=}} π''ab''}}, the area of an [[ellipse]], where {{mvar|a}} is the [[semi-major axis]] and {{mvar|b}} is the [[semi-minor axis]]. Let {{math|''dt'' {{=}} ''P''}}, the time for the body to complete one orbit. Thus,

:<math>k=\frac{2\pi ab}{P\sqrt{M+m}\sqrt{p}}.</math>

Here, Gauss decides to use Earth to solve for {{mvar|k}}. From the geometry of an [[ellipse]], {{math|''p'' {{=}} {{sfrac|''b''<sup>2</sup>|''a''}}}}.<ref name="Smart77-99">Smart, W. M. (1977). p. 99.</ref> &nbsp; By setting Earth's semi-major axis, {{math|''a'' {{=}} 1}}, {{mvar|p}} reduces to {{math|''b''<sup>2</sup>}} and {{math|{{sqrt|''p''}} {{=}} ''b''}}. Substituting, the area of the ellipse "is evidently" {{math|π{{sqrt|''p''}}}}, rather than {{math|π''ab''}}. Putting this into the [[numerator]] of the equation for {{mvar|k}} and reducing,

:<math>k=\frac{2\pi}{P\sqrt{M+m}}.</math>

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,

:<math>k=\frac{2\pi}{P\sqrt{1+m}},</math>

where now {{mvar|m}} is in [[solar mass]]es. What is left are two quantities: {{mvar|P}}, the [[orbital period|period]] of Earth's orbit or the [[sidereal year]], a quantity known precisely by measurement over centuries, and {{mvar|m}}, the mass of the Earth–Moon system. 
Again plugging in the measured values as they were known in Gauss's time, {{mvar|P}} = {{val|365.2563835}} days, {{mvar|m}} = {{sfrac|{{val|354710}}}} solar masses,{{clarify|date=June 2018}}<!--surprisingly far from the modern value of 1:328900, needs an explanation or at least a citation, how can k still be accurate if P was correct but M was off by >7%?--> yielding the result {{mvar|k}} = {{val|0.01720209895}}.

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

:<math>k=\frac{2\pi ab}{P\sqrt{M+m}\sqrt{p}},</math>

where
*{{mvar|a}} is the [[semi-major axis]] of the [[elliptical orbit]],
*{{mvar|b}} is the [[semi-minor axis]] of the elliptical orbit,
*{{mvar|P}} is the [[orbital period]],
*{{mvar|M}} is the [[mass]] of the primary body,
*{{mvar|m}} is the mass of the secondary body, and
*{{mvar|p}} is the [[Ellipse#Latus rectum|semi-latus rectum]] of the elliptical orbit.

From the geometry of an [[ellipse]], the semi-latus rectum, {{mvar|p}} can be expressed in terms of {{mvar|a}} and {{mvar|b}} thus: {{mvar|''p'' {{=}} {{sfrac|''b''<sup>2</sup>|''a''}}}}.<ref name="Smart77-99"/> &nbsp; Therefore,

:<math>\sqrt{p}=\frac{b}{\sqrt{a}}.</math>

Substituting and reducing, Gauss's constant becomes

:<math>k=\frac{2\pi}{P}\sqrt{\frac{a^3}{M+m}}.</math>

From [[orbital mechanics]], {{math|{{sfrac|2π|''P''}}}} is just {{mvar|n}}, the [[mean motion]] of the body in its orbit.<ref name="Smart77-100"/> Hence,

:<math>\begin{align}
k&=n\sqrt{\frac{a^3}{M+m}},\\[8pt]
k^2&=\frac{n^2a^3}{M+m},\\[8pt]
k^2(M+m)&=n^2a^3,
\end{align}</math>

which is the definition of Kepler's third law.<ref name="Smart77-101"/><ref>{{cite book | last = Vallado | first = David A. | title = Fundamentals of Astrodynamics and Applications | publisher = Microcosm Press |location = El Segundo, CA | year = 2001 | isbn = 1-881883-12-4 | edition=2nd|page=31}}
</ref> In this form, it is often seen with {{mvar|G}}, the [[gravitational constant|Newtonian gravitational constant]] in place of {{math|''k''<sup>2</sup>}}.

Setting {{math|''a'' {{=}} 1}}, {{math|''M'' {{=}} 1}}, {{math|''m'' ≪ ''M''}}, and {{mvar|n}} in [[radians]] per [[day]] results in {{math|''k'' ≈ ''n''}}, also in units of radians per day, about which see the relevant section of the [[Mean motion#Mean motion and the gravitational constants|mean motion]] article.

==Other definitions==
{{see|Gravitational constant}}
The value of Gauss's constant, exactly as he derived it, had been used since Gauss's time because it was held to be a fundamental constant, as described above. The [[solar mass]], [[Solar time#Mean solar time|mean solar day]] and [[sidereal year]] with which Gauss defined his constant are all slowly changing in value. If modern{{clarify|date=June 2018}} values were inserted into the defining equation, a value of {{val|0.01720209789}} would result.{{dubious|date=June 2018}}<ref name="Danby88">{{cite book  | last = Danby | first = J. M. A. | title = Fundamentals of Celestial Mechanics | publisher = Willmann-Bell | location = Richmond, VA | year = 1988 | isbn=0-943396-20-4|page=146}}</ref> <!--this is at best outdated (1988), and obsolete at least since the 2012 redefinition of the AU-->

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 {{math|2π}}. These are often called ''canonical units''.<ref name="Danby88"/>

==See also==
* [[Gravitational constant]]
* [[Standard gravitational parameter]]
* [[Kepler's laws of planetary motion]]
* [[Mean motion]]

==Notes==
{{reflist|group=note}}

==References==
{{reflist|30em}}

==Further reading==
*{{cite journal|last=Brumfiel|first=Geoff|title=The astronomical unit gets fixed: Earth–Sun distance changes from slippery equation to single number.|journal=Nature|date=14 September 2012|doi=10.1038/nature.2012.11416|s2cid=123424704 |url=http://www.nature.com/news/the-astronomical-unit-gets-fixed-1.11416|access-date=14 September 2012}}
*{{cite journal|last=Seares|first=Frederick H.|title=The Constant of Attraction|journal=Publications of the Astronomical Society of the Pacific|volume=11|number=66|date=February 1899|page=22 |bibcode=1899PASP...11...22S|doi = 10.1086/121298 |doi-access=free}}

==External links==
{{Commonscat}}
*[http://asa.usno.navy.mil/SecM/Glossary.html#_G Glossary entry ''Gaussian gravitational constant''] {{Webarchive|url=https://web.archive.org/web/20170819204825/http://asa.usno.navy.mil/SecM/Glossary.html#_G |date=2017-08-19 }} at the [[US Naval Observatory]]'s [http://asa.usno.navy.mil/index.html ''Astronomical Almanac Online''] {{Webarchive|url=https://web.archive.org/web/20150420225915/http://asa.usno.navy.mil/index.html |date=2015-04-20 }}
*[http://scienceworld.wolfram.com/physics/GaussianGravitationalConstant.html Gaussian Gravitational Constant, Wolfram ScienceWorld]

[[Category:Physical constants]]