Surface gravity

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The surface gravity of an astronomical object (planet, star, etc.) is the gravitational acceleration experienced at its surface. The surface gravity depends on the mass of the object and its radius. It is often expressed as a ratio to the value for Earth.

Other, more technical, uses of the term (such as the definition given for black holes given below) are natural extensions of this concept.

1 Physical basis
2 Measurement of surface gravity
3 Calculation of surface gravity
4 Surface gravity of a black hole
- 4.1 The Schwarzschild solution
- 4.2 The Kerr-Newman solution
5 External links

[edit] Physical basis

All objects attract one another due to gravity.

The acceleration of each object produced by this force is proportional to the mass of the other, so that a large body may significantly affect a smaller body, while being relatively unaffected itself. For this reason, it is often convenient to speak of the forces on a hypothetic "test particle" of negligible mass, as a way of describing the gravitational field of a single object.

The closer the test particle is to the mass in question, the greater the forces on it, typically falling off with the inverse square of the distance between their centers of mass. Since the force of gravity also tends to impose a roughly spherical shape on gravitationally significant bodies (moons, planets, etc.) it is frequently useful to consider the largest force such a body could exert, which would be on a test particle situated on the body's surface. This force, typically measured as a ratio to the value for Earth, is called the surface gravity of the body.

By honing this basic definition, a variety of more technical definitions may be obtained.

[edit] Measurement of surface gravity

[edit] Pendulum equipment

[edit] Torsion balances

Developed cccvby Hungarian physicist Vásárosnaményi Báró Eötvös Loránd (aka Roland Eötvös).

[edit] Gravimeter

[edit] Remote determination (via satellites)

[edit] Calculation of surface gravity

[edit] For spherically symmetric bodies

The surface gravity of a spherically symmetric body $B$ may be calculated from Newton's Law of Gravitation, by taking the ratio of the force on a test particle near the surface of $B$ to the force on an identical test particle near the surface of the Earth. The properties of the test particle and the gravitational constant cancel out, leaving a simple ratio of the relative mass to the square of the relative radius:

$G_{surface} = \frac{G \cdot \frac{m_B \cdot m_{test} }{r_B^2}}{G \cdot \frac{m_{Earth} \cdot m_{test} }{r_{Earth}^2}} = \frac{m_B/m_{Earth}}{(r_B/r_{Earth})^2}$

where

m B

is the mass of

B

r B

is the radius of

B

m E a r t h

and

r E a r t h

are the mass and radius of the Earth

[edit] For abstract bodies

It is sometimes useful to calculate the surface gravity of hypothetical objects which are not found in nature. For example, the surface gravity of a uniform infinite plane is zero, regardless of the mass of the plane, because (at the surface) all the components of the gravitational forces are horizontal and cancel out. This observation is useful in building an understanding of real structures that approximate the properties of a uniform infinite plane, such as the disk of a galaxy; it allows us to see immediately that stars near the galactic equator should, on average, experience no net gravitational forces normal to the plane of the galaxy.

Likewise, the surface gravity of infinite tubes, lines, hollow shells, cones, and even more unrealistic structures may be used to provide insights into the behavior of real structures.

[edit] For irregular objects

Most real astronomical bodies are not, in fact, spherically symmetric. To the extent that their internal distribution of mass differs from the spherically symmetric approximation, 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, when Roland Eötvös's torsion balance was used to prospect for oil in Czechoslovakia. In 1924 it was used to locate the Nash dome reserves in Texas.

[edit] Surface gravity of a black hole

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The surface gravity $κ$ of a Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if $k a$ is a suitably normalized Killing vector, then the surface gravity is defined by

$k^a \nabla_a k^b = \kappa k^b$ ,

where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that $k^a k_a \rightarrow -1$ as $r\rightarrow\infty$ , and so that $\kappa \geq 0$ . For the Schwarzschild solution, we take $k a$ to be the time translation Killing vector $k^a\partial_a = \frac{\partial}{\partial t}$ , and more generally for the Kerr-Newman solution we take $k^a\partial_a = \frac{\partial}{\partial t}+\Omega\frac{\partial}{\partial\phi}$ , the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, where $Ω$ is the angular velocity.

[edit] The Schwarzschild solution

Since $k a$ is a Killing vector $k^a \nabla_a k^b = \kappa k^b$ implies $-k^a \nabla^b k_a = \kappa k^b$ . In $(t, r,θ,φ)$ coordinates $k a = (1,0,0,0)$ . Performing a coordinate change to the advanced Eddington-Finklestein coordinates $v = t + r + 2 M ln | r - 2 M |$ causes the metric to take the form $ds^{2} = -\left(1-\frac{2M}{r}\right)dv^{2}+2dvdr+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$ .

Under a general change of coordinates the Killing vector transforms as $k^{v} = A_{t}^{v}k^{t}$ giving the vectors $k a' = (1,0,0,0)$ and $k_{a'} = \left(-1+\frac{2M}{r},1,0,0\right)$

Considering the b=r entry for $-k^a \nabla^b k_a = \kappa k^b$ gives the differential equation $-\frac{1}{2}\frac{\partial}{\partial r}\left(-1+\frac{2M}{r}\right) = \kappa$

Therefore the surface gravity for the Schwarzschild solution with mass $M$ is $\kappa = \frac{1}{4M}$ .

[edit] The Kerr-Newman solution

The surface gravity for the Kerr-Newman solution is

$\kappa = \frac{r_+-r_-}{2(r_+^2+a^2)} = \frac{\sqrt{M^2-Q^2-J^2/M^2}}{2M^2-Q^2+2M\sqrt{M^2-Q^2-J^2/M^2}}$ ,

where $Q$ is the electric charge, $J$ is the angular velocity, we define $r_\pm := M \pm \sqrt{M^2-Q^2-J^2/M^2}$ to be the locations of the two horizons and $a : = J / M$ .