Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Surface gravity

From Wikipedia, the free encyclopedia
Standard surface gravity
Part of a series on
Astrodynamics
Efficiency measures
AstronautJohn Young jumping on the Moon, illustrating that thegravitational pull of the Moon is approximately 1/6 of Earth's. The jumping height is limited by the EVA space suit's weight on the Moon of about 13.6 kg (30 lb) and by the suit's pressurization resisting the bending of the suit, as needed for jumping.[1][2]

Thesurface gravity,g, of anastronomical object is thegravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as theacceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere.

Surface gravity is measured in units of acceleration, which, in theSI system, aremeters per second squared. It may also be expressed as a multiple of theEarth'sstandard surface gravity, which is equal to[3]

g =9.80665 m/s2

Inastrophysics, the surface gravity may be expressed aslog⁡g{\displaystyle \log g}, which is obtained by first expressing the gravity incgs units, where the unit of acceleration and surface gravity iscentimeters per second squared (cm/s2), and then taking the base-10logarithm of the cgs value of the surface gravity.[4] Therefore, the surface gravity of Earth could be expressed in cgs units as980.665 cm/s2, and then taking the base-10logarithm ("log g") of 980.665, giving 2.992 as "log g".

The surface gravity of awhite dwarf is very high, and of aneutron star even higher. A white dwarf's surface gravity is around 100,000g (106 m/s2) whilst the neutron star's compactness gives it a surface gravity of up to7×1012 m/s2 with typical values of order1012 m/s2 (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have anescape velocity of around100,000 km/s, about a third of thespeed of light. Since black holes do not have a surface, the surface gravity is not defined.

Relationship of surface gravity to mass and radius

[edit]
Surface gravity of various
Solar System bodies[5]
(1 g = 9.80665 m/s2, the average surface gravitational acceleration on Earth)
NameSurface gravity
Sun28.02g
Mercury0.377g
Venus0.905g
Earth1g (midlatitudes)
Moon0.165 7g (average)
Mars0.379g (midlatitudes)
Phobos0.000 581g
Deimos0.000 306g
Pallas0.022g (equator)
Vesta0.025g (equator)
Ceres0.029g
Jupiter2.528g (midlatitudes)
Io0.183g
Europa0.134g
Ganymede0.146g
Callisto0.126g
Saturn1.065g (midlatitudes)
Mimas0.006 48g
Enceladus0.011 5g
Tethys0.014 9g
Dione0.023 7g
Rhea0.026 9g
Titan0.138g
Iapetus0.022 8g
Phoebe0.003 9–0.005 1g
Uranus0.886g (equator)
Miranda0.007 9g
Ariel0.025 4g
Umbriel0.023g
Titania0.037 2g
Oberon0.036 1g
Neptune1.137g (midlatitudes)
Proteus0.007g
Triton0.079 4g
Pluto0.063g
Charon0.029 4g
Eris0.084g
Haumea0.0247g (equator)
67P-CG0.000 017g

In theNewtonian theory ofgravity, thegravitational force exerted by an object is proportional to its mass: an object with twice the mass-produces twice as much force. Newtonian gravity also follows aninverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity oflight, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.

A large object, such as aplanet orstar, will usually be approximately round, approachinghydrostatic equilibrium (where all points on the surface have the same amount ofgravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached.[6] For most celestial objects, the result is that the planet or star in question can be treated as a near-perfectsphere when the rotation rate is low. However, for young, massive stars, the equatorialazimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount ofequatorial bulge. Examples of suchrapidly rotating stars includeAchernar,Altair,Regulus A andVega.

The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to theshell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established bySir Isaac Newton.[7] Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of itsradius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet,Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected,[8] and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth.[9][10] Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's.[10]

These proportionalities may be expressed by the formula:g∝mr2{\displaystyle g\propto {\frac {m}{r^{2}}}}whereg{\displaystyle g} is the surface gravity of an object, expressed as a multiple of the Earth's,m{\displaystyle m} is its mass, expressed as a multiple of theEarth's mass (5.976×1024 kg) andr{\displaystyle r} its radius, expressed as a multiple of the Earth's (mean) radius (6,371 km).[11] For instance,Mars has a mass of6.4185×1023 kg = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii.[12] The surfacegravity of Mars is therefore approximately0.1070.5322=0.38{\displaystyle {\frac {0.107}{0.532^{2}}}=0.38}times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly fromNewton's law of universal gravitation, which gives the formulag=GMr2{\displaystyle g={\frac {GM}{r^{2}}}}whereM{\displaystyle M} is the mass of the object,r{\displaystyle r} is its radius, andG{\displaystyle G} is thegravitational constant. Ifρ=M/V{\displaystyle \rho =M/V} denote the meandensity of the object, this can also be written asg=4π3Gρr{\displaystyle g={\frac {4\pi }{3}}G\rho r}so that, for fixed mean density, the surface gravityg{\displaystyle g} is proportional to the radius r{\displaystyle r}. Solving for mass, this equation can be written asg=G(4πρ3)2/3M1/3{\displaystyle g=G\left({\frac {4\pi \rho }{3}}\right)^{2/3}M^{1/3}}But density is not constant, but increases as the planet grows in size, as they are not incompressible bodies. That is why the experimental relationship between surface gravity and mass does not grow as 1/3 but as 1/2:[13]g∝M1/2{\displaystyle g\propto M^{1/2}}here withg{\displaystyle g} in times Earth's surface gravity andM{\displaystyle M} in times Earth's mass. In fact, the exoplanets found fulfilling the former relationship have been found to be rocky planets.[13] Thus, for rocky planets, density grows with mass asρ∝M1/4{\displaystyle \rho \propto M^{1/4}}.

Gas giants

[edit]

For gas giant planets such as Jupiter, Saturn, Uranus, and Neptune, the surface gravity is given at the 1 bar pressure level in the atmosphere.[14] It has been found that for giant planets with masses in the range up to 100 times Earth's mass, their surface gravity is nevertheless very similar and close to 1g{\displaystyle g}, a region named thegravity plateau.[13]

Non-spherically-symmetric objects

[edit]

Most real astronomical objects are not perfectly spherically symmetric. One reason for this is that they are often rotating, which means that they are affected by the combined effects ofgravitational force andcentrifugal force. This causes stars and planets to beoblate, which means that their surface gravity is smaller at the equator than at the poles. This effect was exploited byHal Clement in his SF novelMission of Gravity, dealing with a massive, fast-spinning planet where gravity was much higher at the poles than at the equator.

To the extent that an object's internal distribution of mass differs from a symmetric model, the measured surface gravity may be used to deduce things about the object's internal structure. This fact has been put to practical use since 1915–1916, whenRoland Eötvös'storsion balance was used to prospect foroil near the city ofEgbell (nowGbely,Slovakia.)[15]: 1663 [16]: 223  In 1924, the torsion balance was used to locate theNash Dome oil fields inTexas.[16]: 223 

It is sometimes useful to calculate the surface gravity of simple hypothetical objects which are not found in nature. The surface gravity of infinite planes, tubes, lines, hollow shells, cones, and even more unrealistic structures may be used to provide insights into the behavior of real structures.

Black holes

[edit]

In relativity, the Newtonian concept of acceleration turns out not to be clear cut. For a black hole, which must be treated relativistically, one cannot define a surface gravity as the acceleration experienced by a test body at the object's surface because there is no surface, although the event horizon is a natural alternative candidate, but this still presents a problem because the acceleration of a test body at the event horizon of a black hole turns out to be infinite in relativity. Because of this, a renormalized value is used that corresponds to the Newtonian value in the non-relativistic limit. The value used is generally the local proper acceleration (which diverges at the event horizon) multiplied by thegravitational time dilation factor (which goes to zero at the event horizon). For the Schwarzschild case, this value is mathematically well behaved for all non-zero values ofr{\displaystyle r} and M{\displaystyle M}.

When one talks about the surface gravity of a black hole, one is defining a notion that behaves analogously to the Newtonian surface gravity, but is not the same thing. In fact, the surface gravity of a general black hole is not well defined. However, one can define the surface gravity for a black hole whose event horizon is a Killing horizon.

The surface gravityκ{\displaystyle \kappa } of a staticKilling horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, ifka{\displaystyle k^{a}} is a suitably normalizedKilling vector, then the surface gravity is defined byka∇akb=κkb,{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},}where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so thatkaka→−1{\displaystyle k^{a}k_{a}\to -1} asr→∞{\displaystyle r\to \infty }, and so thatκ≥0{\displaystyle \kappa \geq 0}. For the Schwarzschild solution, takeka{\displaystyle k^{a}} to be thetime translationKilling vectorka∂a=∂∂t{\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}}, and more generally for theKerr–Newman solution takeka∂a=∂∂t+Ω∂∂φ{\textstyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}}, the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, whereΩ{\displaystyle \Omega } is the angular velocity.

Schwarzschild solution

[edit]

Sinceka{\displaystyle k^{a}} is a Killing vectorka∇akb=κkb{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}} implies−ka∇bka=κkb{\displaystyle -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}}. In(t,r,θ,φ){\displaystyle (t,r,\theta ,\varphi )} coordinateska=(1,0,0,0){\displaystyle k^{a}=(1,0,0,0)}. Performing a coordinate change to the advanced Eddington–Finklestein coordinatesv=t+r+2Mln⁡|r−2M|{\textstyle v=t+r+2M\ln |r-2M|} causes the metric to take the formds2=−(1−2Mr)dv2+(dvdr+drdv)+r2(dθ2+sin2⁡θdφ2).{\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+\left(dv\,dr+\,dr\,dv\right)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}

Under a general change of coordinates the Killing vector transforms askv=Atvkt{\displaystyle k^{v}=A_{t}^{v}k^{t}} giving the vectorska′=δva′=(1,0,0,0){\displaystyle k^{a'}=\delta _{v}^{a'}=(1,0,0,0)} andka′=ga′v=(−1+2Mr,1,0,0).{\textstyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).}

Considering theb=v{\displaystyle b=v} entry forka∇akb=κkb{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}} gives the differential equation−12∂∂r(−1+2Mr)=κ.{\textstyle -{\frac {1}{2}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .}

Therefore, the surface gravity for theSchwarzschild solution with massM{\displaystyle M} isκ=14M{\displaystyle \kappa ={\frac {1}{4M}}} (κ=c4/4GM{\displaystyle \kappa ={c^{4}}/{4GM}} in SI units).[17]

Kerr solution

[edit]

The surface gravity for the uncharged, rotating black hole is, simplyκ=g−k,{\displaystyle \kappa =g-k,}whereg=14M{\textstyle g={\frac {1}{4M}}} is the Schwarzschild surface gravity, andk:=MΩ+2{\displaystyle k:=M\Omega _{+}^{2}} is the spring constant of the rotating black hole.Ω+{\displaystyle \Omega _{+}} is the angular velocity at the event horizon. This expression gives a simple Hawking temperature of2πT=g−k{\displaystyle 2\pi T=g-k}.[18]

Kerr–Newman solution

[edit]

The surface gravity for theKerr–Newman solution isκ=r+−r−2(r+2+a2)=M2−Q2−J2/M22M2−Q2+2MM2−Q2−J2/M2,{\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2\left(r_{+}^{2}+a^{2}\right)}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},}whereQ{\displaystyle Q} is the electric charge,J{\displaystyle J} is the angular momentum, definer±:=M±M2−Q2−J2/M2{\textstyle r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}} to be the locations of the two horizons anda:=J/M{\displaystyle a:=J/M}.[19]

Dynamical black holes

[edit]

Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing.[20] Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a timelikeKilling vector (field).[21] Several definitions have been proposed over the years by various authors, such as peeling surface gravity and Kodama surface gravity.[22] As of current, there is no consensus or agreement on which definition, if any, is correct.[23]Semiclassical results indicate that the peeling surface gravity is ill-defined for transient objects formed in finite time of a distant observer.[24]

References

[edit]
  1. ^Kluger, Jeffrey (October 12, 2018)."How Neil Armstrong's Moon Spacesuit Was Preserved for Centuries to Come".Time.Archived from the original on December 3, 2023. RetrievedNovember 29, 2023.
  2. ^"How Do You Pick Up Something on the Moon?".WIRED. December 9, 2013.Archived from the original on December 3, 2023. RetrievedNovember 29, 2023.
  3. ^Taylor, Barry N., ed. (2001). "The International System of Units (SI)".NIST Special Publication 330(PDF). United States Department of Commerce: National Institute of Standards and Technology. p. 29. Retrieved2012-03-08.{{cite book}}: CS1 maint: publisher location (link)
  4. ^Smalley, B. (13 July 2006)."The Determination of Teff and log g for B to G stars".Keele University. Retrieved31 May 2007.
  5. ^Isaac Asimov (1978).The Collapsing Universe. Corgi. p. 44.ISBN 978-0-552-10884-3.
  6. ^"Why is the Earth round?".Ask A Scientist. Argonne National Laboratory, Division of Educational Programs. Archived fromthe original on 21 September 2008.
  7. ^Book I, §XII, pp. 218–226,Newton's Principia: The Mathematical Principles of Natural Philosophy, Sir Isaac Newton, tr. Andrew Motte, ed. N. W. Chittenden. New York: Daniel Adee, 1848. First American edition.
  8. ^Astronomers Find First Earth-like Planet in Habitable ZoneArchived 2009-06-17 at theWayback Machine, ESO 22/07, press release from theEuropean Southern Observatory, April 25, 2007
  9. ^Udry, Stéphane; Bonfils, Xavier; Delfosse, Xavier; Forveille, Thierry; Mayor, Michel; Perrier, Christian; Bouchy, François; Lovis, Christophe; Pepe, Francesco; Queloz, Didier; Bertaux, Jean-Loup (2007)."The HARPS search for southern extra-solar planets XI. Super-Earths (5 and 8 M🜨) in a 3-planet system"(PDF).Astronomy & Astrophysics.469 (3):L43–L47.arXiv:0704.3841.Bibcode:2007A&A...469L..43U.doi:10.1051/0004-6361:20077612.S2CID 119144195. Archived fromthe original(PDF) on October 8, 2010.
  10. ^abValencia, Diana; Sasselov, Dimitar D; O'Connell, Richard J (2007). "Detailed Models of super-Earths: How well can we infer bulk properties?".The Astrophysical Journal.665 (2):1413–1420.arXiv:0704.3454.Bibcode:2007ApJ...665.1413V.doi:10.1086/519554.S2CID 15605519.
  11. ^2.7.4 Physical properties of the Earth, web page, accessed on line May 27, 2007.
  12. ^Mars Fact Sheet, web page at NASA NSSDC, accessed May 27, 2007.
  13. ^abcBallesteros, Fernando; Luque, Bartolo (2016). "Walking on exoplanets: Is Star Wars right?".Astrobiology.16 (5):1–3.arXiv:1604.07725.Bibcode:2016AsBio..16..325B.doi:10.1089/ast.2016.1475.PMID 27104945.
  14. ^"Planetary Fact Sheet Notes".
  15. ^Li, Xiong; Götze, Hans-Jürgen (2001). "Ellipsoid, geoid, gravity, geodesy, and geophysics".Geophysics.66 (6):1660–1668.Bibcode:2001Geop...66.1660L.doi:10.1190/1.1487109.
  16. ^abPrediction by Eötvös' torsion balance data in HungaryArchived 2007-11-28 at theWayback Machine, Gyula Tóth,Periodica Polytechnica Ser. Civ. Eng.46, #2 (2002), pp. 221–229.
  17. ^Raine, Derek J.; Thomas, Edwin George (2010).Black Holes: An Introduction (illustrated ed.).Imperial College Press. p. 44.ISBN 978-1-84816-382-9.Extract of page 44
  18. ^Good, Michael; Yen Chin Ong (February 2015). "Are Black Holes Springlike?".Physical Review D.91 (4) 044031.arXiv:1412.5432.Bibcode:2015PhRvD..91d4031G.doi:10.1103/PhysRevD.91.044031.S2CID 117749566.
  19. ^Ruiz, O.; Molina, U.; Viloria, P. (2019). "Thermodynamic analysis of Kerr-Newman black holes".Journal of Physics: Conference Series.1219 (1) 012016.Bibcode:2019JPhCS1219a2016R.doi:10.1088/1742-6596/1219/1/012016.hdl:11323/6160.
  20. ^Wald, Robert (1984).General Relativity. University Of Chicago Press.ISBN 978-0-226-87033-5.
  21. ^A. B. Nielsen; J. H. Yoon (2008). "Dynamical Surface Gravity".Classical and Quantum Gravity.25 (8) 085010.arXiv:0711.1445.Bibcode:2008CQGra..25h5010N.doi:10.1088/0264-9381/25/8/085010.S2CID 15438397.
  22. ^H. Kodama (1980)."Conserved Energy Flux for the Spherically Symmetric System and the Backreaction Problem in the Black Hole Evaporation".Progress of Theoretical Physics.63 (4): 1217.Bibcode:1980PThPh..63.1217K.doi:10.1143/PTP.63.1217.S2CID 122827579.
  23. ^Pielahn, Mathias; G. Kunstatter; A. B. Nielsen (November 2011). "Dynamical surface gravity in spherically symmetric black hole formation".Physical Review D.84 (10): 104008(11).arXiv:1103.0750.Bibcode:2011PhRvD..84j4008P.doi:10.1103/PhysRevD.84.104008.S2CID 119015033.
  24. ^R. B. Mann; S. Murk; D. R. Terno (2022). "Surface gravity and the information loss problem".Physical Review D.105 (12) 124032.arXiv:2109.13939.Bibcode:2022PhRvD.105l4032M.doi:10.1103/PhysRevD.105.124032.S2CID 249799593.

External links

[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Surface_gravity&oldid=1330643943"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2026 Movatter.jp