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Within a uniform spherical body of radius R, density ρ, and mass m, the gravitational force g inside the sphere varies linearly with distance r from the center, giving the gravitational potential inside the sphere, which is [7] [8] = [] = [],, which differentiably connects to the potential function for the outside of the sphere (see the figure ...
It has precisely balanced positive kinetic energy and negative gravitational potential energy; [a] it will always be slowing down, asymptotically approaching zero speed, but never quite stop. [1] Escape velocity calculations are typically used to determine whether an object will remain in the gravitational sphere of influence of a given
A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of mass. This can be seen as follows: take a point within such a sphere, at a distance from the center of the sphere. Then you can ignore all of the ...
The Hill sphere is a common model for the calculation of a gravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of an astronomical body (m) in which it dominates over the gravitational influence of other bodies, particularly a primary (M). [1]
The amount of energy released when the sphere contracts from radius to radius is obtained by differentiation this expression to , so =. The critical mass is attained as soon as the released gravitational energy is equal to the work done on the gas: M 2 R 2 = n T R 2 . {\displaystyle {\frac {M^{2}}{R^{2}}}=nTR^{2}.}
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study.The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
A gravitationally bound system has a lower (i.e., more negative) gravitational potential energy than the sum of the energies of its parts when these are completely separated—this is what keeps the system aggregated in accordance with the minimum total potential energy principle.
The Plummer model or Plummer sphere is a ... The corresponding potential ... where G is Newton's gravitational constant. The velocity dispersion is = ...