Editing Orbital mechanics (section)

===Elliptical orbits===
If <math>0 < e < 1</math>, then the denominator of the equation of free orbits varies with the true anomaly <math>\theta</math>, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude at periapsis <math>r_p</math>, which is given by:
:<math>r_p=\frac{p}{1+e}</math>

The maximum value <math>r</math> is reached when <math>\theta = 180^\circ</math>. This point is called the apoapsis, and its radial coordinate, denoted <math>r_a</math>, is
:<math>r_a=\frac{p}{1-e}</math>

Let <math>2a</math> be the distance measured along the apse line from periapsis <math>P</math> to apoapsis <math>A</math>, as illustrated in the equation below:
:<math>2a=r_p+r_a</math>

Substituting the equations above, we get:
:<math>a=\frac{p}{1-e^2}</math>

a is the semimajor axis of the ellipse. Solving for <math>p</math>, and substituting the result in the conic section curve formula above, we get:
:<math>r=\frac{a(1-e^2)}{1+e\cos\theta}</math>

====Orbital period====
Under standard assumptions the [[orbital period]] (<math>T\,\!</math>) of a body traveling along an elliptic orbit can be computed as:
:<math>T=2\pi\sqrt{a^3\over{\mu}}</math>
where:
*<math>\mu\,</math> is the [[standard gravitational parameter]],
*<math>a\,\!</math> is the length of the [[semi-major axis]].
Conclusions:
*The orbital period is equal to that for a [[circular orbit]] with the orbit radius equal to the [[semi-major axis]] (<math>a\,\!</math>),
*For a given semi-major axis the orbital period does not depend on the eccentricity (See also: [[Kepler's laws of planetary motion#Third law|Kepler's third law]]).

====Velocity====
Under standard assumptions the [[orbital speed]] (<math>v\,</math>) of a body traveling along an '''elliptic orbit''' can be computed from the [[Vis-viva equation]] as:
:<math>v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}</math>
where:
*<math>\mu\,</math> is the [[standard gravitational parameter]],
*<math>r\,</math> is the distance between the orbiting bodies.
*<math>a\,\!</math> is the length of the [[semi-major axis]].

The velocity equation for a [[hyperbolic trajectory]] is <math>v=\sqrt{\mu\left({2\over{r}}+\left\vert {1\over{a}} \right\vert\right)}</math>.

====Energy====
Under standard assumptions, [[specific orbital energy]] (<math>\epsilon\,</math>) of elliptic orbit is negative and the orbital energy conservation equation (the [[Vis-viva equation]]) for this orbit can take the form:
:<math>{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0</math>
where:
*<math>v\,</math> is the speed of the orbiting body,
*<math>r\,</math> is the distance of the orbiting body from the center of mass of the [[central body]],
*<math>a\,</math> is the [[semi-major axis]],
*<math>\mu\,</math> is the [[standard gravitational parameter]].
Conclusions:
*For a given semi-major axis the specific orbital energy is independent of the eccentricity.

Using the [[virial theorem]] we find:
*the time-average of the specific potential energy is equal to <math>2\epsilon</math>
*the time-average of <math>r^{-1}</math> is <math>a^{-1}</math>
*the time-average of the specific kinetic energy is equal to <math>-\epsilon</math>