Editing Gaussian gravitational constant (section)

==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>.