standard gravitational parameter

{{solar_system_orbital_period_vs_semimajor_axis.svg}}

The central body in an orbital system can be defined as the one whose mass (M) is much larger than the mass of the orbiting body (m), or {{nowrap|M ≫ m}}. This approximation is standard for planets orbiting the Sun or most moons and greatly simplifies equations. Under Newton's law of universal gravitation, if the distance between the bodies is r, the force exerted on the smaller body is:

$$F\; =\; \backslash frac\{G\; M\; m\}\{r^2\}\; =\; \backslash frac\{\backslash mu\; m\}\{r^2\}$$

Thus only the product of G and M is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product, μ, not G and M separately. The gravitational constant, G, is difficult to measure with high accuracy,{{Citation |author=George T. Gillies |title=The Newtonian gravitational constant: recent measurements and related studies |journal=Reports on Progress in Physics |date=1997 |volume=60 |issue=2 |pages= 151–225 |url=https://iopscience.iop.org/article/10.1088/0034-4885/60/2/001/pdf |doi=10.1088/0034-4885/60/2/001 | bibcode = 1997RPPh...60..151G |s2cid=250810284 |url-access=subscription }}. A lengthy, detailed review. while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision.

For a circular orbit around a central body, where the centripetal force provided by gravity is {{nowrap|1=F = mv{{sup|2}}r{{sup|−1}}}}:

$$\backslash mu\; =\; rv^2\; =\; r^3\backslash omega^2\; =\; \backslash frac\{4\backslash pi^2r^3\}\{T^2\}\; ,$$

where r is the orbit radius, v is the orbital speed, ω is the angular speed, and T is the orbital period.

This can be generalized for elliptic orbits:

$$\backslash mu\; =\; \backslash frac\{4\backslash pi^2a^3\}\{T^2\}\; ,$$

where a is the semi-major axis, which is Kepler's third law.

For parabolic trajectories rv² is constant and equal to 2μ. For elliptic and hyperbolic orbits magnitude of μ = 2 times the magnitude of a times the magnitude of ε, where a is the semi-major axis and ε is the specific orbital energy.

In the more general case where the bodies need not be a large one and a small one, e.g. a binary star system, we define:

• the vector r is the position of one body relative to the other
• r, v, and in the case of an elliptic orbit, the semi-major axis a, are defined accordingly (hence r is the distance)
• μ = Gm₁ + Gm₂ = μ₁ + μ₂, where m₁ and m₂ are the masses of the two bodies.

Then:

• for circular orbits, rv² = r³ω² = 4π²r³/T² = μ
• for elliptic orbits, {{nowrap|1=4π²a³/T² = μ}} (with a expressed in AU; T in years and M the total mass relative to that of the Sun, we get {{nowrap|1=a³/T² = M}})
• for parabolic trajectories, rv² is constant and equal to 2μ
• for elliptic and hyperbolic orbits, μ is twice the semi-major axis times the negative of the specific orbital energy, where the latter is defined as the total energy of the system divided by the reduced mass.

The standard gravitational parameter can be determined using a pendulum oscillating above the surface of a body as:

{{citation

| last1 = Lewalle

| first1 = Philippe

| last2 = Dimino

| first2 = Tony

| year = 2014

| title = Measuring Earth's Gravitational Constant with a Pendulum

| page = 1

| url = http://teacher.pas.rochester.edu/phy141/Laboratory/SampleReports/Sample141lab_pendulumg.pdf

}}

$$\backslash mu\; \backslash approx\; \backslash frac\{4\; \backslash pi^2\; r^2\; L\}\{T^2\}$$

where r is the radius of the gravitating body, L is the length of the pendulum, and T is the period of the pendulum (for the reason of the approximation see Pendulum in mechanics).

{{Further|Gaussian gravitational constant}}

{{Further|Earth mass}}

G{{Earth mass}}, the gravitational parameter for the Earth as the central body, is called the geocentric gravitational constant. It equals {{val|3.986004418|0.000000008|e=14|u=m³⋅s⁻²}}.{{cite web|title=IAU Astronomical Constants: Current Best Estimates|url=https://iau-a3.gitlab.io/NSFA/NSFA_cbe.html#GME2009|website=iau-a2.gitlab.io|publisher=IAU Division I Working Group on Numerical Standards for Fundamental Astronomy|access-date=25 June 2021}}, citing Ries, J. C., Eanes, R. J., Shum, C. K., and Watkins, M. M., 1992, "Progress in the Determination of the Gravitational Coefficient of the Earth," Geophys. Res. Lett., 19(6), pp. 529-531.

The value of this constant became important with the beginning of spaceflight in the 1950s, and great effort was expended to determine it as accurately as possible during the 1960s. Sagitov (1969) cites a range of values reported from 1960s high-precision measurements, with a relative uncertainty of the order of 10⁻⁶.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.

During the 1970s to 1980s, the increasing number of artificial satellites in Earth orbit further facilitated high-precision measurements,

and the relative uncertainty was decreased by another three orders of magnitude, to about {{val|2|e=-9}} (1 in 500 million) as of 1992.

Measurement involves observations of the distances from the satellite to Earth stations at different times, which can be obtained to high accuracy using radar or laser ranging.{{cite journal|last1=Lerch|first1=Francis J.|last2=Laubscher|first2=Roy E.|last3=Klosko|first3=Steven M.|last4=Smith|first4=David E.|last5=Kolenkiewicz|first5=Ronald|last6=Putney|first6=Barbara H.|last7=Marsh|first7=James G.|last8=Brownd|first8=Joseph E.|title=Determination of the geocentric gravitational constant from laser ranging on near-Earth satellites|journal=Geophysical Research Letters|date=December 1978|volume=5|issue=12|pages=1031–1034|doi=10.1029/GL005i012p01031|bibcode=1978GeoRL...5.1031L}}

{{Further|Solar mass}}

G{{Solar mass}}, the gravitational parameter for the Sun as the central body,

is called the heliocentric gravitational constant or geopotential of the Sun and equals {{nowrap|{{val|1.32712440042|0.0000000001|e=20|u=m³⋅s⁻²}}.{{cite journal|last1=Pitjeva|first1=E. V.|title=Determination of the Value of the Heliocentric Gravitational Constant from Modern Observations of Planets and Spacecraft|journal=Journal of Physical and Chemical Reference Data|date=September 2015|volume=44|issue=3|pages=031210|doi=10.1063/1.4921980|bibcode=2015JPCRD..44c1210P}}}}

The relative uncertainty in G{{Solar mass}}, cited at below 10⁻¹⁰ as of 2015, is smaller than the uncertainty in G{{Earth mass}}

because G{{Solar mass}} is derived from the ranging of interplanetary probes, and the absolute error of the distance measures to them is about the same as the earth satellite ranging measures, while the absolute distances involved are much bigger.{{citation needed|date=August 2016}}

• Astronomical system of units
• Planetary mass

{{reflist|30em

| refs =

{{cite journal

|last1=Anderson |first1=John D.

|last2=Colombo |first2=Giuseppe

|last3=Esposito |first3=Pasquale B.

|last4=Lau |first4=Eunice L.

|last5=Trager |first5=Gayle B.

|title=The mass, gravity field, and ephemeris of Mercury

|journal=Icarus

|date=September 1987

|volume=71

|issue=3

|pages=337–349

|doi=10.1016/0019-1035(87)90033-9

|bibcode=1987Icar...71..337A

}}

{{cite journal

|last1=Konopliv |first1=Alex S.

|last2=Banerdt |first2=W. Bruce

|last3=Sjogren |first3=William L.

|title=Venus Gravity: 180th degree and order model

|journal=Icarus

|date=May 1999

|volume=139

|issue=1

|pages=3-18

|doi=10.1006/icar.1999.6086

|bibcode=1999Icar..139....3K

}}

{{cite web

| title = Astrodynamic Constants

| date = 27 February 2009

| publisher = NASA/JPL

| url = http://ssd.jpl.nasa.gov/?constants

| access-date = 27 July 2009

}}

{{cite journal

| doi = 10.1007/s11208-005-0033-2

| author = E.V. Pitjeva

| date = 2005

| title = High-Precision Ephemerides of Planets — EPM and Determination of Some Astronomical Constants

| url = http://iau-comm4.jpl.nasa.gov/EPM2004.pdf

| archive-url = https://web.archive.org/web/20060822150010/http://iau-comm4.jpl.nasa.gov/EPM2004.pdf

| url-status = dead

| archive-date = 2006-08-22

| journal = Solar System Research

| volume = 39

| issue=3

| pages=176–186

| bibcode = 2005SoSyR..39..176P

| s2cid = 120467483

}}

{{cite book

| author = D. T. Britt| author2 = D. Yeomans| author3 = K. Housen| author4 = G. Consolmagno

| date = 2002

| chapter = Asteroid density, porosity, and structure

| chapter-url = http://www.lpi.usra.edu/books/AsteroidsIII/pdf/3022.pdf

| title = Asteroids III

| editor = W. Bottke

| editor2 = A. Cellino

| editor3 = P. Paolicchi

| editor4 = R.P. Binzel

| page = 488

| publisher = University of Arizona Press

| url = http://www.lpi.usra.edu/books/AsteroidsIII/download.html

}}

{{cite report

|version=Version 0.5

|date=October 16, 2015

|first1=Carol|last1=Raymond|first2=Boris|last2=Semenov

|title=Asteroid Ceres P_constants (PcK) SPICE kernel file

|url=http://naif.jpl.nasa.gov/pub/naif/DAWN/kernels/pck/dawn_ceres_v05.tpc

}}

{{cite journal

| doi = 10.1086/116211

| author = R.A. Jacobson

| author2 = J.K. Campbell

| author3 = A.H. Taylor

| author4 = S.P. Synnott

| date = 1992

| title = The masses of Uranus and its major satellites from Voyager tracking data and Earth-based Uranian satellite data

| journal = Astronomical Journal

| volume = 103

| issue = 6

| pages = 2068–2078

| bibcode = 1992AJ....103.2068J

}}

{{cite journal

| doi = 10.1086/504422

| display-authors = 4

| author = M.W. Buie

| author2 = W.M. Grundy

| author3 = E.F. Young

| author4 = L.A. Young

| author5 = S.A. Stern

| date = 2006

| title = Orbits and photometry of Pluto's satellites: Charon, S/2005 P1, and S/2005 P2

| journal = Astronomical Journal

| volume = 132

| issue = 1

| pages = 290–298

| bibcode = 2006AJ....132..290B

| arxiv = astro-ph/0512491

| s2cid = 119386667

}}

{{cite journal

| doi = 10.1126/science.1139415

| author = M.E. Brown

| author2 = E.L. Schaller

| date = 2007

| title = The Mass of Dwarf Planet Eris

| journal = Science

| volume = 316

| issue = 5831

| pages = 1586

| bibcode = 2007Sci...316.1585B

| pmid=17569855

| s2cid = 21468196

| url = https://resolver.caltech.edu/CaltechAUTHORS:20121001-135149660

}}

}}

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