Surface gravity
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The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. 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. 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 the
In
The surface gravity of a white dwarf is very high, and of a neutron star even higher. A white dwarf's surface gravity is around 100,000 g (106 m/s2) whilst the neutron star's compactness gives it a surface gravity of up to 7×1012 m/s2 with typical values of order 1012 m/s2 (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.
Relationship of surface gravity to mass and radius
Name | Surface gravity |
---|---|
Sun | 28.02 g |
Mercury | 0.377 g |
Venus | 0.905 g |
Earth | 1 g (midlatitudes) |
Moon | 0.165 7 g (average) |
Mars | 0.379 g (midlatitudes) |
Phobos | 0.000 581 g |
Deimos | 0.000 306 g |
Pallas | 0.022 g (equator) |
Vesta | 0.025 g (equator) |
Ceres | 0.029 g |
Jupiter | 2.528 g (midlatitudes) |
Io | 0.183 g |
Europa | 0.134 g |
Ganymede | 0.146 g |
Callisto | 0.126 g |
Saturn | 1.065 g (midlatitudes) |
Mimas |
0.006 48 g |
Enceladus | 0.011 5 g |
Tethys | 0.014 9 g |
Dione | 0.023 7 g |
Rhea | 0.026 9 g |
Titan | 0.138 g |
Iapetus | 0.022 8 g |
Phoebe | 0.003 9–0.005 1 g |
Uranus | 0.886 g (equator) |
Miranda | 0.007 9 g |
Ariel | 0.025 4 g |
Umbriel | 0.023 g |
Titania | 0.037 2 g |
Oberon | 0.036 1 g |
Neptune | 1.137 g (midlatitudes) |
Proteus | 0.007 g |
Triton | 0.079 4 g |
Pluto | 0.063 g |
Charon | 0.029 4 g |
Eris | 0.084 g |
Haumea | 0.0247 g (equator) |
67P-CG | 0.000 017 g |
In the
A large object, such as a
The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the
These proportionalities may be expressed by the formula:
Since gravity is inversely proportional to the square of the distance, a space station 400 km above the Earth feels almost the same gravitational force as we do on the Earth's surface. A space station does not plummet to the ground because it is in an orbit around the Earth.
Gas giants
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.[11]
Non-spherically symmetric objects
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 of
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
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
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. 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 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 of r and 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 gravity of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if is a suitably normalized
Schwarzschild solution
Since is a Killing vector implies . In coordinates . Performing a coordinate change to the advanced Eddington–Finklestein coordinates causes the metric to take the form
Under a general change of coordinates the Killing vector transforms as giving the vectors and
Considering the b = entry for gives the differential equation
Therefore, the surface gravity for the
Kerr solution
The surface gravity for the uncharged, rotating black hole is, simply
Kerr–Newman solution
The surface gravity for the
Dynamical black holes
Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing.[16] Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a timelike Killing vector (field).[17] Several definitions have been proposed over the years by various authors, such as peeling surface gravity and Kodama surface gravity.[18] As of current, there is no consensus or agreement on which definition, if any, is correct.[19] Semiclassical results indicate that the peeling surface gravity is ill-defined for transient objects formed in finite time of a distant observer.[20]
References
- ^ Taylor, Barry N., ed. (2001). The International System of Units (SI) (PDF). United States Department of Commerce: National Institute of Standards and Technology. p. 29. Retrieved 2012-03-08.
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ignored (help) - ^ Smalley, B. (13 July 2006). "The Determination of Teff and log g for B to G stars". Keele University. Retrieved 31 May 2007.
- ISBN 978-0-552-10884-3.
- ^ "Why is the Earth round?". Ask A Scientist. Argonne National Laboratory, Division of Educational Programs. Archived from the original on 21 September 2008.
- ^ 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.
- ^ Astronomers Find First Earth-like Planet in Habitable Zone Archived 2009-06-17 at the Wayback Machine, ESO 22/07, press release from the European Southern Observatory, April 25, 2007
- S2CID 119144195. Archived from the original(PDF) on October 8, 2010.
- ^ S2CID 15605519.
- ^ 2.7.4 Physical properties of the Earth, web page, accessed on line May 27, 2007.
- ^ Mars Fact Sheet, web page at NASA NSSDC, accessed May 27, 2007.
- ^ "Planetary Fact Sheet Notes".
- .
- ^ a b Prediction by Eötvös' torsion balance data in Hungary Archived 2007-11-28 at the Wayback Machine, Gyula Tóth, Periodica Polytechnica Ser. Civ. Eng. 46, #2 (2002), pp. 221–229.
- S2CID 117749566.
- ISBN 978-0-226-87033-5.
- S2CID 15438397.
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