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| [[File:Mass potential well increasing mass.gif|thumb|The gravitational potential well of an increasing mass, ''M'', given by <math>\Phi(\mathbf{x}) = -G\frac{M}{|\mathbf{x}|}</math>.<br> | |||
| The ] of the continuum's matter increases as the matter spirals into its own gravitational well.<ref>Nemiroff, R. J. ♦ ''American Journal of Physics'', 61, 619 (1993) ♦ ''"Photons climbing out of a gravitating object become less energetic. This loss of energy is known as a 'redshifting', as photons in the visible spectrum would appear more red. Similarly, photons falling into a gravitational field become more energetic and exhibit a blueshifting."''</ref> As a result, the gravitational well deepens, undergoing an exponentially accelerating ].]] | |||
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| A '''gravity well''' or '''gravitational well''' is a conceptual model of the gravitational field surrounding a body in space. The more massive the body the deeper and more extensive the gravity well associated with it. The Sun has a far-reaching and deep gravity well. Asteroids and small moons have much shallower gravity wells. Anything on the surface of a planet or moon is considered to be at the bottom of the gravity well. Entering space from the surface of a planet or moon means climbing out of the gravity well. The deeper a planet or moon's gravity well is, the more energy it takes to achieve ]. | |||
| In ], a '''gravity well''' is specifically the ] field around a massive body. Other types of ]s include electrical and magnetic potential wells. Physical models of gravity wells are sometimes used to illustrate ]. Gravity wells are frequently confused with ] ]s, but the two concepts are unrelated. | |||
| ==Details== | |||
| ] | |||
| The external gravitational potential of a spherically symmetric body of mass ''M'' is given by the formula: | |||
| :<math>\Phi(\mathbf{x}) = -\frac{GM}{|\mathbf{x}|}</math> | |||
| A plot of this function in two dimensions is shown in the figure. This plot has been completed with an interior potential proportional to |x|<sup>2</sup>, corresponding to an object of uniform density, but this interior potential is generally irrelevant since the orbit of a test particle cannot intersect the body. | |||
| The potential function has a ] cross section; the sudden dip in the center is the origin of the name "gravity well." A black hole would not have this "closing" dip. | |||
| ==Physical gravity wells== | |||
| In a uniform gravitational field, the gravitational potential at a point is proportional to the height. Thus if the ] of a gravitational potential Φ(x,y) is constructed as a physical surface and placed in a uniform gravitational field so that the actual field points in the −Φ direction, then each point on the surface will have an actual gravitational potential proportional to the value of Φ at that point. As a result, an object constrained to move on the surface will have roughly the same equation of motion as an object moving in the potential field Φ itself. Gravity wells constructed on this principle can be found in many science museums. | |||
| There are several sources of inaccuracy in this model: | |||
| *The friction between the object and the surface has no analogue in vacuum. This effect can be reduced by using a rolling ball instead of a sliding block. | |||
| *The object's vertical motion contributes to its kinetic energy, and has no analogue in vacuum. This effect can be reduced by making the gravity well shallower (i.e. by choosing a smaller scaling factor for the Φ axis). | |||
| *A rolling ball's ] has no analogue in vacuum. This effect can be reduced by concentrating the ball's mass near its center so that the moment of inertia is small compared to ''mr''². | |||
| *A ball's center of mass is not located on the surface but at a fixed distance ''r'', which changes its potential energy by an amount depending on the slope of the surface at that point. For balls of a fixed size, this effect can be eliminated by constructing the surface so that the center of the ball, rather than the surface itself, lies on the graph of Φ. | |||
| ==The rubber-sheet model== | |||
| Consider an idealized rubber sheet suspended in a uniform gravitational field normal to the sheet. In equilibrium, the elastic ] in each part of the sheet must be equal and opposite to the gravitational pull on that part of the sheet; that is, | |||
| :<math>k \nabla^2 h = -g \rho</math> | |||
| where ''k'' is the ] of the rubber, h(x) is the upward displacement of the sheet (assumed to be small), ''g'' is the strength of the gravitational field, and ρ(x) is the mass density of the sheet. The mass density may be viewed as intrinsic to the sheet or as belonging to objects resting on top of the sheet. | |||
| This equilibrium condition is identical in form to the gravitational ] | |||
| :<math>\nabla^2 \Phi = - 4 \pi G \rho</math> | |||
| where Φ(x) is the gravitational potential and ρ(x) is the mass density. Thus, to a first approximation, a massive object placed on a rubber sheet will deform the sheet into a correctly shaped gravity well, and (as in the rigid case) a second test object placed near the first will gravitate toward it in an approximation of the correct force law. | |||
| ==Gravity wells and general relativity== | |||
| ] | |||
| Both the rigid gravity well and the rubber-sheet model are frequently misidentified as models of ], due to an accidental resemblance to general relativistic ]s,{{Citation needed|date=January 2011}} and perhaps Einstein's employment of gravitational "curvature" bending the path of light, which he described as a prediction of general relativity. In particular, the embedding diagram most commonly{{Citation needed|date=January 2011}} found in textbooks (an ] of a constant-time equatorial slice of the ] in Euclidean 3-dimensional space) superficially resembles a gravity well. | |||
| Embedding diagrams are, however, fundamentally different from gravity wells in a number of ways. Most importantly, an embedding is merely a shape, while a potential plot has a distinguished "downward" direction; thus turning a gravity well "upside down" (by negating the potential) turns the attractive force into a repulsive force, while turning a Schwarzschild embedding upside down (by rotating it) has no effect, since it leaves its intrinsic geometry unchanged. Geodesics on the Schwarzschild surface do bend toward the central mass like a ball rolling in a gravity well, but for entirely different reasons. There is no analogue of the Schwarzschild embedding for a repulsive field: while such a field can be modeled in general relativity, the spatial geometry cannot be embedded in three dimensions.{{Citation needed|date=January 2011}} | |||
| The Schwarzschild embedding is commonly drawn with a hyperbolic cross section like the potential well, but in fact it has a ] cross section which, unlike the gravity well, does not approach a planar ]. See ].{{Citation needed|date=January 2011}} | |||
| ==See also== | |||
| *] | |||
| ==References== | |||
| <references/> | |||
| ==External links== | |||
| * | |||
| {{DEFAULTSORT:Gravity well}} | |||
| ] | |||
| ] | |||
| ] | |||
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