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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 ...
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 ...
For this the gravitational force, i.e. the gradient of the potential, must be computed. Efficient recursive algorithms have been designed to compute the gravitational force for any N z {\displaystyle N_{z}} and N t {\displaystyle N_{t}} (the max degree of zonal and tesseral terms) and such algorithms are used in standard orbit propagation software.
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]
For example, a hollow sphere does not produce any net gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the resultant field is that of all masses not including the sphere, which can be inside and outside the sphere).
Assuming a constant density , the masses of a shell and the sphere inside it are: = and = The required energy for a shell is the negative of the gravitational potential energy: d U = − G m s h e l l m i n t e r i o r r {\displaystyle dU=-G{\frac {m_{\mathrm {shell} }m_{\mathrm {interior} }}{r}}}
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.
Geopotential is the potential of the Earth's gravity field.For convenience it is often defined as the negative of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation.