Surface gravity

thumb|Astronaut [[John Young (astronaut)|John Young jumping on the Moon, illustrating that the gravitational 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 and by the suit's pressurization resisting the bending of the suit, as needed for jumping.]]

The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. Surface gravity may be understood as the acceleration due to gravity experienced by a hypothetical test particle located very close to the object's surface, which has negligible mass so as not to disturb the system. For objects where the surface lies deep within an atmosphere and the radius is not well defined, 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 SI system, are meters per second squared. It may also be expressed as a multiple of the Earth's standard surface gravity, which is equal to

In astrophysics, the surface gravity may be expressed as <math>\log g</math>, which is obtained by first expressing the gravity in cgs units, where the unit of acceleration and surface gravity is centimeters per second squared (cm/s<sup>2</sup>), and then taking the base-10 logarithm of the cgs value of the surface gravity. Therefore, the surface gravity of Earth could be expressed in cgs units as , and then taking the base-10 logarithm ("log g") of 980.665, giving 2.992 as "log g".

The surface gravity of a white dwarf is very high, and that of a neutron star is even higher. A white dwarf's surface gravity is around 100,000 g (), while the compactness of a neutron star gives it a surface gravity of up to , with typical values on the order of . This is more than 10<sup>11</sup> times that of Earth. One consequence of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about one-third of the speed of light. Since black holes do not have a surface, their surface gravity is not defined.

Relationship of surface gravity to mass and radius

{| class="wikitable sortable" style="float:right; clear:right; margin-left:1em"

|+ Surface gravity of various<br />Solar System bodies<br/><div style="font-size:70%; line-height:110%">(1 g = 9.80665 m/s<sup>2</sup>, the average surface gravitational acceleration on Earth)</div>

! scope="col" | Name

! scope="col" data-sort-type=number | Surface gravity

|- style="background:#FF8B8B"

| Sun || 28.02 g

|- style="background:#EEFFFF"

| Mercury || 0.377 g

|- style="background:#FDFFFF"

| Venus || 0.905 g

|- style="background:#FFFFFF"

| Earth || 1 g (midlatitudes)

|- style="background:#E0FFFF"

| Moon || 0.165 7 g (average)

|- style="background:#EEFFFF"

| Mars || 0.379 g (midlatitudes)

|- style="background:#7EFFFF"

| Phobos || 0.000 581 g

|- style="background:#72FFFF"

| Deimos || 0.000 306 g

|- style="background:#BDFFFF"

| Pallas || 0.022 g (equator)

|- style="background:#BFFFFF"

| Vesta || 0.025 g (equator)

|- style="background:#C2FFFF"

| Ceres || 0.029 g

|- style="background:#FFDFDF"

| Jupiter || 2.528 g (midlatitudes)

|- style="background:#E2FFFF"

| Io || 0.183 g

|- style="background:#DCFFFF"

| Europa || 0.134 g

|- style="background:#DEFFFF"

| Ganymede || 0.146 g

|- style="background:#DBFFFF"

| Callisto || 0.126 g

|- style="background:#FDFFFF"

| Saturn || 1.065 g (midlatitudes)

|- style="background:#A7FFFF"

| Mimas || 0.006 48 g

|- style="background:#B1FFFF"

| Enceladus || 0.011 5 g

|- style="background:#B5FFFF"

| Tethys || 0.014 9 g

|- style="background:#BDFFFF"

| Dione || 0.023 7 g

|- style="background:#C0FFFF"

| Rhea || 0.026 9 g

|- style="background:#DDFFFF"

| Titan || 0.138 g

|- style="background:#BDFFFF"

| Iapetus || 0.022 8 g

|- style="background:#A1FFFF"

| Phoebe || 0.003 9–0.005 1 g

|- style="background:#FDFFFF"

| Uranus || 0.886 g (equator)

|- style="background:#A9FFFF"

| Miranda || 0.007 9 g

|- style="background:#BFFFFF"

| Ariel || 0.025 4 g

|- style="background:#BDFFFF"

| Umbriel || 0.023 g

|- style="background:#C6FFFF"

| Titania || 0.037 2 g

|- style="background:#C5FFFF"

| Oberon || 0.036 1 g

|- style="background:#FFFAFA"

| Neptune || 1.137 g (midlatitudes)

|- style="background:#A9FFFF"

| Proteus || 0.007 g

|- style="background:#D3FFFF"

| Triton || 0.079 4 g

|- style="background:#CFFFFF"

| Pluto || 0.063 g

|- style="background:#C2FFFF"

| Charon || 0.029 4 g

|- style="background:#D4FFFF"

| Eris || 0.084 g

|- style="background:#BFFFFF"

| Haumea || 0.0247 g (equator)

|- style="background:#40FFFF"

| 67P-CG || 0.000 017 g

In the 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 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 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. 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 azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega.

The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell 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 by Sir Isaac Newton. 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, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. 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. For instance, Mars has a mass of  = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. The surface gravity of Mars is therefore approximately

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 universal gravitation, which gives the formula

where <math>M</math> is the mass of the object, <math>r</math> is its radius, and <math>G</math> is the gravitational constant. If <math>\rho = M/V</math> denote the mean density of the object, this can also be written as

so that, for fixed mean density, the surface gravity <math>g</math> is proportional to the radius <math>r</math>. Solving for mass, this equation can be written as

<math display="block">g = G \left ( \frac{4\pi \rho}{3} \right ) ^{2/3} M^{1/3}</math>

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:

<math display="block">g\propto M^{1/2}</math>

here with <math>g</math> in times Earth's surface gravity and <math>M</math> in times Earth's mass. In fact, the exoplanets found fulfilling the former relationship have been found to be rocky planets. It has been found that for giant planets with masses up to 100 times that of Earth, surface gravity is nevertheless very similar and close to 1<math>g</math>, a region known as the gravity plateau. In 1924, the torsion balance was used to locate the Nash Dome oil fields in Texas.

Kerr solution

The surface gravity for the uncharged, rotating black hole is, simply

<math display="block">\kappa = g - k , </math>

where <math display="inline">g = \frac 1 {4M}</math> is the Schwarzschild surface gravity, and <math>k := M \Omega_+^2 </math> is the spring constant of the rotating black hole. <math>\Omega_+</math> is the angular velocity at the event horizon. This expression gives a simple Hawking temperature of <math> 2\pi T = g - k </math>.

Kerr–Newman solution

The surface gravity for the Kerr–Newman solution is

<math display="block">\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,</math>

where <math>Q</math> is the electric charge, <math>J</math> is the angular momentum, define <math display="inline">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>.

References

External links

Newtonian surface gravity
Exploratorium – Your Weight on Other Worlds