Editing Orbit equation (section)

== Central, inverse-square law force ==

Consider a [[two-body system]] consisting of a central body of mass ''M'' and a much smaller, orbiting body of mass <math>m</math>, and suppose the two bodies interact via a [[central force|central]], [[inverse-square law]] force (such as [[gravitation]]). In [[polar coordinates]], the orbit equation can be written as<ref name="fetter13">{{cite book| last1=Fetter|first1=Alexander|last2=Walecka|first2=John|title=Theoretical Mechanics of Particles and Continua| date=2003|publisher=[[Dover Publications]]|pages=13&ndash;22}}</ref>
<math display="block">r = \frac{\ell^2}{m^2\mu}\frac{1}{1+e\cos\theta}</math>
where 
* <math>r</math> is the separation distance between the two bodies and
* <math>\theta</math> is the angle that <math>\mathbf{r}</math> makes with the axis of [[periapsis]] (also called the ''[[true anomaly]]'').
* The parameter <math>\ell</math> is the [[angular momentum]] of the orbiting body about the central body, and is equal to <math>mr^2\dot{\theta}</math>, or the mass multiplied by the magnitude of the cross product of the relative position and velocity vectors of the two bodies.<ref group="note">There is a related parameter, known as the [[specific relative angular momentum]], <math>h</math>. It is related to <math>\ell</math> by <math>h = \ell/m</math>.</ref>
* The parameter <math>\mu</math> is the constant for which <math>\mu/r^2</math> equals the acceleration of the smaller body (for gravitation, <math>\mu</math> is the [[standard gravitational parameter]], <math>-GM</math>). For a given orbit, the larger <math>\mu</math>, the faster the orbiting body moves in it: twice as fast if the attraction is four times as strong.
* The parameter <math>e</math> is the [[eccentricity (mathematics)|eccentricity]] of the orbit, and is given by<ref name="fetter13" />
*:<math>e = \sqrt{1+\frac{2E\ell^2}{m^3\mu^2}}</math>
*:where <math>E</math> is the energy of the orbit.

The above relation between <math>r</math> and <math>\theta</math> describes a [[conic section]].<ref name="fetter13" /> The value of <math>e</math> controls what kind of conic section the orbit is:
* when <math>e<1</math>, the orbit is [[elliptic orbit|elliptic]] (circles are ellipses with <math>e=0</math>);
* when <math>e=1</math>, the orbit is [[parabolic orbit|parabolic]];
* when <math>e>1</math>, the orbit is [[hyperbolic orbit|hyperbolic]].

The minimum value of <math>r</math> in the equation is:
<math display="block">r={{\ell^2}\over{m^2\mu}}{{1}\over{1+e}}</math>
while, if <math>e<1</math>, the maximum value is:
<math display="block">r={{\ell^2}\over{m^2\mu}}{{1}\over{1-e}}</math>

If the maximum is less than the radius of the central body, then the conic section is an ellipse which is fully inside the central body and no part of it is a possible trajectory. If the maximum is more, but the minimum is less than the radius, part of the trajectory is possible:
*if the energy is non-negative (parabolic or hyperbolic orbit): the motion is either away from the central body, or towards it.
*if the energy is negative: the motion can be first away from the central body, up to <math display="block">r={{\ell^2}\over{m^2\mu}}{{1}\over{1-e}}</math> after which the object falls back.

If <math>r</math> becomes such that the orbiting body enters an atmosphere, then the standard assumptions no longer apply, as in [[atmospheric reentry]].