In celestial mechanics, the standard gravitational parameterμ of a celestial body is the product of the gravitational constantG and the mass M of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when one body is much larger than the other.
For several objects in the Solar System, the value of μ is known to greater accuracy than either G or M.[11] The SI units of the standard gravitational parameter are m3s−2. However, units of km3s−2 are frequently used in the scientific literature and in spacecraft navigation.
Log-log plot of period T vs semi-major axis a (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that a³/T² is constant (green line)
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 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:
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,[12] while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision.
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.
In a pendulum
The standard gravitational parameter can be determined using a pendulum oscillating above the surface of a body as:[13]
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).
GMEarth, the gravitational parameter for the Earth as the central body, is called the geocentric gravitational constant. It equals (3.986004418±0.000000008)×1014 m3 s−2.[3]
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.[14]
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 2×10−9 (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.[15]
GM☉, the gravitational parameter for the Sun as the central body, is called the heliocentric gravitational constant or geopotential of the Sun and equals (1.32712440042±0.0000000001)×1020 m3 s−2.[16]
The relative uncertainty in GM☉, cited at below 10−10 as of 2015, is smaller than the uncertainty in GMEarth because GM☉ 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]
^ "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.
^ Anderson, John D.; Colombo, Giuseppe; Esposito, Pasquale B.; Lau, Eunice L.; Trager, Gayle B. (September 1987). "The mass, gravity field, and ephemeris of Mercury". Icarus. 71 (3): 337–349. Bibcode:1987Icar...71..337A. doi:10.1016/0019-1035(87)90033-9.
^ ab"IAU Astronomical Constants: Current Best Estimates". iau-a2.gitlab.io. IAU Division I Working Group on Numerical Standards for Fundamental Astronomy. Retrieved 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.
^. Western Australian Geodesy Group. Archived from the original on 2013-04-10.
^ E.V. Pitjeva (2005). (PDF). Solar System Research. 39 (3): 176–186. Bibcode:2005SoSyR..39..176P. doi:10.1007/s11208-005-0033-2. S2CID 120467483. Archived from the original (PDF) on 2006-08-22.
^ D. T. Britt; D. Yeomans; K. Housen; G. Consolmagno (2002). "Asteroid density, porosity, and structure" (PDF). In W. Bottke; A. Cellino; P. Paolicchi; R.P. Binzel (eds.). Asteroids III. University of Arizona Press. p. 488.
^ R.A. Jacobson; J.K. Campbell; A.H. Taylor; S.P. Synnott (1992). "The masses of Uranus and its major satellites from Voyager tracking data and Earth-based Uranian satellite data". Astronomical Journal. 103 (6): 2068–2078. Bibcode:1992AJ....103.2068J. doi:10.1086/116211.
^ M.W. Buie; W.M. Grundy; E.F. Young; L.A. Young; et al. (2006). "Orbits and photometry of Pluto's satellites: Charon, S/2005 P1, and S/2005 P2". Astronomical Journal. 132 (1): 290–298. arXiv:astro-ph/0512491. Bibcode:2006AJ....132..290B. doi:10.1086/504422. S2CID 119386667.
^ M.E. Brown; E.L. Schaller (2007). "The Mass of Dwarf Planet Eris". Science. 316 (5831): 1586. Bibcode:2007Sci...316.1585B. doi:10.1126/science.1139415. PMID 17569855. S2CID 21468196.
^This is mostly because μ can be measured by observational astronomy alone, as it has been for centuries. Decoupling it into G and M must be done by measuring the force of gravity in sensitive laboratory conditions, as first done in the Cavendish experiment.
^George T. Gillies (1997), "The Newtonian gravitational constant: recent measurements and related studies", Reports on Progress in Physics, 60 (2): 151–225, Bibcode:1997RPPh...60..151G, doi:10.1088/0034-4885/60/2/001, S2CID 250810284. A lengthy, detailed review.
^Lewalle, Philippe; Dimino, Tony (2014), Measuring Earth's Gravitational Constant with a Pendulum(PDF), p. 1
^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.
^Lerch, Francis J.; Laubscher, Roy E.; Klosko, Steven M.; Smith, David E.; Kolenkiewicz, Ronald; Putney, Barbara H.; Marsh, James G.; Brownd, Joseph E. (December 1978). "Determination of the geocentric gravitational constant from laser ranging on near-Earth satellites". Geophysical Research Letters. 5 (12): 1031–1034. Bibcode:1978GeoRL...5.1031L. doi:10.1029/GL005i012p01031.
^Pitjeva, E. V. (September 2015). "Determination of the Value of the Heliocentric Gravitational Constant from Modern Observations of Planets and Spacecraft". Journal of Physical and Chemical Reference Data. 44 (3): 031210. Bibcode:2015JPCRD..44c1210P. doi:10.1063/1.4921980.
standard, gravitational, parameter, body, 327124, 1020, mercury, 2032, 1013, venus, 24859, 1014earth, 986004, 1014, moon, 9048695, 1012mars, 282837, 1013, ceres, 26325, 1010, jupiter, 266865, 1017saturn, 7931187, 1016uranus, 793939, 1015, neptune, 836529, 1015. Body m m3 s 2 Sun 1 327124 400 18 9 1020 1 Mercury 2 2032 9 1013 2 Venus 3 24859 9 1014Earth 3 986004 418 8 1014 3 Moon 4 9048695 9 1012Mars 4 282837 2 1013 4 Ceres 6 26325 1010 5 6 7 Jupiter 1 266865 34 9 1017Saturn 3 7931187 9 1016Uranus 5 793939 9 1015 8 Neptune 6 836529 9 1015Pluto 8 71 9 1011 9 Eris 1 108 9 1012 10 In celestial mechanics the standard gravitational parameter m of a celestial body is the product of the gravitational constant G and the mass M of the bodies For two bodies the parameter may be expressed as G m1 m2 or as GM when one body is much larger than the other m G M displaystyle mu GM For several objects in the Solar System the value of m is known to greater accuracy than either G or M 11 The SI units of the standard gravitational parameter are m3 s 2 However units of km3 s 2 are frequently used in the scientific literature and in spacecraft navigation Contents 1 Definition 1 1 Small body orbiting a central body 1 2 General case 1 3 In a pendulum 2 Solar system 2 1 Geocentric gravitational constant 2 2 Heliocentric gravitational constant 3 See also 4 ReferencesDefinition EditSmall body orbiting a central body Edit Log log plot of period T vs semi major axis a average of aphelion and perihelion of some Solar System orbits crosses denoting Kepler s values showing that a T is constant green line 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 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 G M m r 2 m m r 2 displaystyle F frac GMm r 2 frac 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 m not G and M separately The gravitational constant G is difficult to measure with high accuracy 12 while orbits at least in the solar system can be measured with great precision and used to determine m with similar precision For a circular orbit around a central body m r v 2 r 3 w 2 4 p 2 r 3 T 2 displaystyle mu rv 2 r 3 omega 2 frac 4 pi 2 r 3 T 2 where r is the orbit radius v is the orbital speed w is the angular speed and T is the orbital period This can be generalized for elliptic orbits m 4 p 2 a 3 T 2 displaystyle mu frac 4 pi 2 a 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 2m For elliptic and hyperbolic orbits m 2a e where e is the specific orbital energy General case Edit 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 m Gm1 Gm2 m1 m2 where m1 and m2 are the masses of the two bodies Then for circular orbits rv2 r3w2 4p2r3 T2 m for elliptic orbits 4p2a3 T2 m with a expressed in AU T in years and M the total mass relative to that of the Sun we get a3 T2 M for parabolic trajectories rv2 is constant and equal to 2m for elliptic and hyperbolic orbits m 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 In a pendulum Edit The standard gravitational parameter can be determined using a pendulum oscillating above the surface of a body as 13 m 4 p 2 r 2 L T 2 displaystyle mu approx frac 4 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 Solar system EditFurther information Gaussian gravitational constant Geocentric gravitational constant Edit Further information Earth mass G MEarth the gravitational parameter for the Earth as the central body is called the geocentric gravitational constant It equals 3 986004 418 0 000000 008 1014 m3 s 2 3 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 14 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 2 10 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 15 Heliocentric gravitational constant Edit Further information Solar mass G M the gravitational parameter for the Sun as the central body is called the heliocentric gravitational constant or geopotential of the Sun and equals 1 327124 400 42 0 000000 0001 1020 m3 s 2 16 The relative uncertainty in G M cited at below 10 10 as of 2015 is smaller than the uncertainty in G MEarth because G M 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 See also EditAstronomical system of units Planetary massReferences Edit Astrodynamic Constants NASA JPL 27 February 2009 Retrieved 27 July 2009 Anderson John D Colombo Giuseppe Esposito Pasquale B Lau Eunice L Trager Gayle B September 1987 The mass gravity field and ephemeris of Mercury Icarus 71 3 337 349 Bibcode 1987Icar 71 337A doi 10 1016 0019 1035 87 90033 9 a b IAU Astronomical Constants Current Best Estimates iau a2 gitlab io IAU Division I Working Group on Numerical Standards for Fundamental Astronomy Retrieved 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 Mars Gravity Model 2011 MGM2011 Western Australian Geodesy Group Archived from the original on 2013 04 10 Asteroid Ceres P constants PcK SPICE kernel file Retrieved 5 November 2015 E V Pitjeva 2005 High Precision Ephemerides of Planets EPM and Determination of Some Astronomical Constants PDF Solar System Research 39 3 176 186 Bibcode 2005SoSyR 39 176P doi 10 1007 s11208 005 0033 2 S2CID 120467483 Archived from the original PDF on 2006 08 22 D T Britt D Yeomans K Housen G Consolmagno 2002 Asteroid density porosity and structure PDF In W Bottke A Cellino P Paolicchi R P Binzel eds Asteroids III University of Arizona Press p 488 R A Jacobson J K Campbell A H Taylor S P Synnott 1992 The masses of Uranus and its major satellites from Voyager tracking data and Earth based Uranian satellite data Astronomical Journal 103 6 2068 2078 Bibcode 1992AJ 103 2068J doi 10 1086 116211 M W Buie W M Grundy E F Young L A Young et al 2006 Orbits and photometry of Pluto s satellites Charon S 2005 P1 and S 2005 P2 Astronomical Journal 132 1 290 298 arXiv astro ph 0512491 Bibcode 2006AJ 132 290B doi 10 1086 504422 S2CID 119386667 M E Brown E L Schaller 2007 The Mass of Dwarf Planet Eris Science 316 5831 1586 Bibcode 2007Sci 316 1585B doi 10 1126 science 1139415 PMID 17569855 S2CID 21468196 This is mostly because m can be measured by observational astronomy alone as it has been for centuries Decoupling it into G and M must be done by measuring the force of gravity in sensitive laboratory conditions as first done in the Cavendish experiment George T Gillies 1997 The Newtonian gravitational constant recent measurements and related studies Reports on Progress in Physics 60 2 151 225 Bibcode 1997RPPh 60 151G doi 10 1088 0034 4885 60 2 001 S2CID 250810284 A lengthy detailed review Lewalle Philippe Dimino Tony 2014 Measuring Earth s Gravitational Constant with a Pendulum PDF p 1 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 Lerch Francis J Laubscher Roy E Klosko Steven M Smith David E Kolenkiewicz Ronald Putney Barbara H Marsh James G Brownd Joseph E December 1978 Determination of the geocentric gravitational constant from laser ranging on near Earth satellites Geophysical Research Letters 5 12 1031 1034 Bibcode 1978GeoRL 5 1031L doi 10 1029 GL005i012p01031 Pitjeva E V September 2015 Determination of the Value of the Heliocentric Gravitational Constant from Modern Observations of Planets and Spacecraft Journal of Physical and Chemical Reference Data 44 3 031210 Bibcode 2015JPCRD 44c1210P doi 10 1063 1 4921980 Portals Physics Astronomy Stars Spaceflight Outer space Solar System Retrieved from https en wikipedia org w index php title Standard gravitational parameter amp oldid 1131371515, wikipedia, wiki, book, books, library,
