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Concept
Orbit equation
Summary
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 orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory, or radial trajectory) with the central body located at one of the two foci, or the focus (Kepler's first law).
If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness).
Consider a two-body system consisting of a central body of mass M and a much smaller, orbiting body of mass , and suppose the two bodies interact via a central, inverse-square law force (such as gravitation). In polar coordinates, the orbit equation can be written as
where
is the separation distance between the two bodies and
is the angle that makes with the axis of periapsis (also called the true anomaly).
The parameter is the angular momentum of the orbiting body about the central body, and is equal to , or the mass multiplied by the magnitude of the cross product of the relative position and velocity vectors of the two bodies.
The parameter is the constant for which equals the acceleration of the smaller body (for gravitation, is the standard gravitational parameter, ). For a given orbit, the larger , the faster the orbiting body moves in it: twice as fast if the attraction is four times as strong.
The parameter is the eccentricity of the orbit, and is given by
where is the energy of the orbit.
The above relation between and describes a conic section. The value of controls what kind of conic section the orbit is:
when , the orbit is elliptic (circles are ellipses with );
when , the orbit is parabolic;
when , the orbit is hyperbolic.
Official source
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