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The principal reasoning is that Newton's law of gravitation yields an acceleration ¨ = /; if the product of gravitational constant and attractive mass at the center of the orbit are known, position and velocity are the initial values for that second order differential equation for () which has a unique solution.
To properly use this formula, the units must be consistent; for example, must be in kilograms, and must be in meters. The answer will be in meters per second. The answer will be in meters per second. The quantity G M {\displaystyle GM} is often termed the standard gravitational parameter , which has a different value for every planet or moon in ...
In astrodynamics, an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time.Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e. circular ...
The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity.
As is the case with spherical coordinates and spherical harmonics, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate spheroidal coordinate.
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation.
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Series-expanded equation of the center as a function of mean anomaly for various eccentricities, with the equation of the center truncated at e 7 for all curves. Note that the truncated equation fails at high eccentricity and produces an oscillating curve. But this is because the coefficients of the Fourier series are inaccurate due to ...