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| The '''surface gravity''', ''g'', of an [[astronomical object|astronomical]] or other object is the [[gravitational acceleration]] experienced at its surface. The surface gravity may be thought of as the acceleration 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.
| | Bryan is a superstar inside the generating as well as vocation expansion to start with second to his third resort album, & , is the proof. He burst on the scene in 2006 regarding his unique mix of downward-residence convenience, video star very good appearance and lines, is defined t within a major way. The new record Top around the nation graph and #2 on the put graphs, producing it the next top first appearance in those days of 1999 for any land musician. <br><br>The boy of the , is aware patience and determination are important [http://lukebryantickets.iczmpbangladesh.org mumford and sons tickets] elements when it comes to a prosperous job- . His first recording, Continue to be Me, produced the best hits “All My Friends Say” and “Country Gentleman,” whilst his effort, Doin’ Issue, discovered the singer-a few directly No. 2 men and women: More Calling Is often a Good Point.”<br><br>In the tumble of 2004, Tour: Bryan And that had an outstanding listing of , which include Downtown. “It’s much like you are acquiring a endorsement to look to a higher level, states these artists that had been a part of the Tourabove into a greater level of musicians.” It packaged as the most [http://lukebryantickets.pyhgy.com luke bryan cheap tickets] successful tours in their 10-calendar year historical past.<br><br>[http://www.banburycrossonline.com tim mcgraw concert] Here is my webpage :: [http://lukebryantickets.neodga.com luke brian tickets] |
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| Surface gravity is measured in units of [[acceleration]], which, in the [[SI units|SI]] system, are [[metre per second squared|meters per second squared]]. It may also be expressed as a multiple of the [[Earth]]'s [[standard gravity|standard surface gravity]], ''g'' = 9.80665 m/s<sup>2</sup>.<ref>p. 29, [http://physics.nist.gov/Document/sp330.pdf The International System of Units (SI)], ed. Barry N. Taylor, NIST Special Publication 330, 2001.</ref> In [[astrophysics]], the surface gravity may be expressed as log ''g'', which is obtained by first expressing the gravity in [[cgs unit]]s, where the unit of acceleration is [[centimeter]]s per second squared, and then taking the base 10 [[logarithm]].<ref>{{cite web
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| | last = Smalley | first = B. | date = 2006-07-13
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| | url =http://www.astro.keele.ac.uk/~bs/publs/review_text.html
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| | title =The Determination of T''eff'' and log ''g'' for B to G stars
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| | publisher = Keele University | accessdate = 2007-05-31 }}</ref> Therefore, as gravity affects all things equally, regardless of their mass in grams or kilograms, and because 1 m/s<sup>2</sup> = 100 cm/s<sup>2</sup>, the surface gravity of Earth could be expressed in cgs units as 980.665 cm/s<sup>2</sup> and at base 10 logarithm (log ''g'') as 2.992.
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| The surface gravity of a [[white dwarf]] is very high, and of a [[neutron star]] even more. The neutron star's compactness gives it a surface gravity of up to 7{{e|12}} m/s² with typical values of a few {{e|12}} m/s² (that is more than 10<sup>11</sup> times that of [[Earth]]). One measure of such immense gravity is the fact that neutron stars have an [[escape velocity]] of around [[Orders of magnitude (speed)#8|100,000 km/s]], about a third of the [[speed of light]].
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| ==Mass, radius and surface gravity==
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| {| class="wikitable sortable" border="1" align="right"
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| |+Surface gravity of{{Break}}some Solar System bodies{{Break}}compared with Earth<ref>{{Cite book|title=The Collapsing Universe|author=Isaac Asimov|publisher=Corgi|date=1978|ISBN=0-552-10884-7|page=44}}</ref>
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| ! scope="col" | Name
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| ! scope="col" | Surface gravity
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| | [[Mercury (planet)|Mercury]] || 0.38
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| | [[Venus]] || 0.90
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| | [[Earth]] || 1.00
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| | [[Moon]] || 0.165
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| | [[Mars]] || 0.38
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| | [[Jupiter]] || 2.53
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| | [[Saturn]] || 1.07
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| | [[Uranus]] || 0.89
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| | [[Neptune]] || 1.14
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| | [[Pluto]] || 0.067
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| |}
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| In the [[Newtonian gravity|Newtonian]] theory of [[gravity]], the [[gravitational 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 an inverse 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 of [[light]], which also follows an inverse square law: with relation to distance, light exponentially becomes less visible.
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| A large object, such as a [[planet]] or [[star]], will usually be approximately round, approaching [[hydrostatic equilibrium]] (where all points on the surface have the same amount of [[gravitational 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.<ref>[http://www.newton.dep.anl.gov/askasci/gen99/gen99251.htm Why is the Earth round?], at Ask A Scientist, accessed online May 27, 2007.</ref> For most celestial objects, the result is that the planet or star in question can be treated as a near-perfect [[sphere]] when the rotation rate is low. However, for young, massive stars, the equatorial [[azimuth]]al velocity can be quite high—up to 200 km/s or more—causing a significant amount of [[equatorial bulge]]. Examples of such [[stellar rotation|rapidly rotating stars]] include [[Achernar]], [[Altair]], [[Regulus|Regulus A]] and [[Vega]].
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| The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. The gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by [[Sir Isaac Newton]].<ref>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.</ref> Therefore, the surface gravity of a [[planet]] or [[star]] with a given mass will be approximately inversely proportional to the square of its [[radius]], 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,<ref>[http://www.eso.org/public/outreach/press-rel/pr-2007/pr-22-07.html Astronomers Find First Earth-like Planet in Habitable Zone], ESO 22/07, press release from the [[European Southern Observatory]], April 25, 2007</ref> and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth.<ref>[http://adsabs.harvard.edu/abs/2007arXiv0704.3841U The HARPS search for southern extra-solar planets XI. Super-Earths (5 & 8 M_Earth) in a 3-planet system], S. Udry, X. Bonfils), X. Delfosse, T. Forveille, M. Mayor, C. Perrier, F. Bouchy, C. Lovis, F. Pepe, D. Queloz, and J.-L. Bertaux. arXiv:astro-ph/0704.3841.</ref><ref name="model">[http://adsabs.harvard.edu/abs/2007arXiv0704.3454V Detailed Models of super-Earths: How well can we infer bulk properties?], Diana Valencia, Dimitar D. Sasselov, and Richard J. O'Connell, arXiv:astro-ph/0704.3454.</ref> 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.<ref name="model" />
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| These proportionalities may be expressed by the formula ''g'' = ''m''/''r''<sup>2</sup>, where ''g'' is the surface gravity of an object, expressed as a multiple of the [[Earth]]'s, ''m'' is its mass, expressed as a multiple of the [[Earth]]'s mass (5.976·10<sup>24</sup> [[kilogram|kg]]) and ''r'' its radius, expressed as a multiple of the Earth's (mean) radius (6,371 [[kilometer|km]]).<ref>[http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_4.html 2.7.4 Physical properties of the Earth], web page, accessed on line May 27, 2007.</ref> For instance, [[Mars]] has a mass of 6.4185·10<sup>23</sup> [[kilogram|kg]] = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii.<ref>[http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html Mars Fact Sheet], web page at NASA NSSDC, accessed May 27, 2007.</ref> The surface gravity of Mars is therefore approximately
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| :<math>\frac{0.107}{0.532^2} = 0.38</math>
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| times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from [[Newton's Law of Gravitation]], which gives the formula
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| :<math>g = {\frac{GM}{r^2}}</math>
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| where ''M'' is the mass of the object, ''r'' is its radius, and ''G'' is the [[gravitational constant]].
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| If we let ρ = ''m''/''V'' denote the mean [[density]] of the object, we can also write this as
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| :<math>g = {\frac{4\pi}{3} G \rho r}</math>
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| so that, for fixed mean density, the surface gravity ''g'' is proportional to the radius ''r''.
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| Since gravity is inversely proportional to the square of the distance, a space station 100 miles above the Earth feels almost the same gravitational force as we do on the Earth's surface. The reason a space station does not plummet to the ground is not that it is not subject to [[gravity]], but that it is in a [[free-fall]] [[orbit]].
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| ==Non-spherically symmetric objects==
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| Most real astronomical objects are not absolutely spherically symmetric. One reason for this is that they are often rotating, which means that they are affected by the combined effects of [[gravity|gravitational force]] and [[centrifugal force]]. This causes stars and planets to be [[oblateness|oblate]], which means that their surface gravity is smaller at the equator than at the poles. This effect was exploited by [[Hal Clement]] in his SF novel ''[[Mission of Gravity]]'', dealing with a massive, fast-spinning planet where gravity was much higher at the poles than at the equator.
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| To the extent that an object's internal distribution of mass differs from a symmetric model, we may use the measured surface gravity to deduce things about the object's internal structure. This fact has been put to practical use since 1915–1916, when [[Roland Eötvös]]'s [[torsion balance]] was used to prospect for [[oil]] near the city of [[Egbell]] (now [[Gbely]], [[Slovakia]].)<ref>[http://www.lct.com/technical-pages/pdf/Li_G_Tut.pdf Ellipsoid, geoid, gravity, geodesy, and geophysics], Xiong Li and Hans-Jürgen Götze, ''Geophysics'', '''66''', #6 (November–December 2001), pp. 1660–1668. DOI [http://dx.doi.org/10.1190/1.1487109 10.1190/1.1487109].</ref><sup>, p. 1663;</sup><ref name="hung">[http://www.pp.bme.hu/ci/2002_2/pdf/ci2002_2_09.pdf Prediction by Eötvös' torsion balance data in Hungary], Gyula Tóth, ''Periodica Polytechnica Ser. Civ. Eng.'' '''46''', #2 (2002), pp. 221–229.</ref><sup>, p. 223.</sup> In 1924, the torsion balance was used to locate the [[Nash Dome]] oil fields in [[Texas]].<ref name="hung" /><sup>, p. 223.</sup>
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| 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.
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| ==Surface gravity of a black hole==
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| 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. This is 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 the gravitational redshift factor (which goes to zero at the event horizon). For the Schwarzschild case, this value is mathematically well behaved for all non-zero values of r and M.
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| 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.
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| The surface gravity <math>\kappa</math> of a static [[Killing horizon]] is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if <math>k^a</math> is a suitably normalized [[Killing vector]], then the surface gravity is defined by
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| :<math>k^a \nabla_a k^b = \kappa k^b</math>,
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| where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that <math>k^a k_a \rightarrow -1</math> as <math>r\rightarrow\infty</math>, and so that <math>\kappa \geq 0</math>. For the Schwarzschild solution, we take <math>k^a</math> to be the time translation [[Killing vector]] <math>k^a\partial_a = \frac{\partial}{\partial t}</math>, and more generally for the Kerr-Newman solution we take <math>k^a\partial_a = \frac{\partial}{\partial t}+\Omega\frac{\partial}{\partial\phi}</math>, the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, where <math>\Omega</math> is the angular velocity.
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| ===The Schwarzschild solution===
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| Since <math>k^a</math> is a Killing vector <math>k^a \nabla_a k^b = \kappa k^b </math> implies <math> -k^a \nabla^b k_a = \kappa k^b</math>. In <math>(t,r,\theta,\phi)</math> coordinates <math>k^{a}=(1,0,0,0)</math>. Performing a coordinate change to the advanced Eddington-Finklestein coordinates <math>v = t+r+2M\ln |r-2M|</math> causes the metric to take the form <math>ds^{2} = -\left(1-\frac{2M}{r}\right)dv^{2}+2dvdr+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).</math>
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| Under a general change of coordinates the Killing vector transforms as <math>k^{v} = A_{t}^{v}k^{t}</math> giving the vectors <math>k^{a'}=(1,0,0,0)</math> and <math>k_{a'} = \left(-1+\frac{2M}{r},1,0,0\right).</math>
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| Considering the b=v entry for <math>k^a \nabla_a k^b = \kappa k^b</math> gives the differential equation <math>-\frac{1}{2}\frac{\partial}{\partial r}\left(-1+\frac{2M}{r}\right) = \kappa.</math>
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| Therefore the surface gravity for the [[Schwarzschild solution]] with mass <math>M</math> is <math>\kappa = \frac{1}{4M}.</math>
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| ===The Kerr-Newman solution===
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| The surface gravity for the [[Kerr-Newman solution]] is
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| :<math>\kappa = \frac{r_+-r_-}{2(r_+^2+a^2)} = \frac{\sqrt{M^2-Q^2-J^2/M^2}}{2M^2-Q^2+2M\sqrt{M^2-Q^2-J^2/M^2}}</math>, | |
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| where <math>Q</math> is the electric charge, <math>J</math> is the angular momentum, we define <math>r_\pm := M \pm \sqrt{M^2-Q^2-J^2/M^2}</math> to be the locations of the two horizons and <math>a := J/M</math>.
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| ===Dynamical black holes===
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| Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing.<ref>{{cite book|last=Wald|first=Robert|title=General Relativity|year=1984|publisher=University Of Chicago Press|isbn=978-0-226-87033-5}}</ref> Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a [[Killing vector field|Killing vector (field)]].<ref>{{cite journal|last=Nielsen|first=Alex|coauthors=Yoon|title=Dynamical Surface Gravity|journal=Classical Quantum Gravity|year=2008|volume=25}}</ref> Several definitions have been proposed over the years by various authors. As of current, there is no consensus or agreement of which definition, if any, is correct.<ref>{{cite journal|last=Pielahn|first=Mathias|coauthors=G. Kunstatter, A. B. Nielsen|title=Dynamical surface gravity in spherically symmetric black hole formation|journal=Physical Review D|date=November 2011|volume=84|issue=10|pages=104008(11)|doi=10.1103/PhysRevD.84.104008|bibcode = 2011PhRvD..84j4008P |arxiv = 1103.0750 }}</ref>
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://farside.ph.utexas.edu/teaching/301/lectures/node152.html Newtonian surface gravity]
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| [[Category:Gravitation]]
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| [[Category:Black holes]]
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| [[Category:General relativity]]
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