Editing Elliptic orbit (section)

==Energy==
Under standard assumptions, the [[specific orbital energy]] (<math>\epsilon</math>) of an elliptic orbit is negative and the orbital energy conservation equation (the [[Vis-viva equation]]) for this orbit can take the form:<ref>{{cite book |first1=Roger R. |last1=Bate |first2=Donald D. |last2=Mueller |first3=Jerry E. |last3=White |url=https://books.google.com/books?id=UtJK8cetqGkC&pg=PA27 |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |pages=27–28 |edition=First}}</ref> 
:<math>{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0</math>
where:
*<math>v\,</math> is the [[orbital speed]] of the orbiting body,
*<math>r\,</math> is the distance of the orbiting body from the [[central body]],
*<math>a\,</math> is the length of 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]] to find:
*the time-average of the specific potential energy is equal to −2ε
**the time-average of ''r''<sup>−1</sup> is ''a''<sup>−1</sup>
*the time-average of the specific kinetic energy is equal to ε

=== Energy in terms of semi major axis ===
It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The total energy of the orbit is given by
:<math>E = - G \frac{M m}{2a}</math>,

where a is the semi major axis.

==== Derivation ====
Since gravity is a central force, the angular momentum is constant:
:<math>\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0</math>

At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore:
:<math>L = r p = r m v</math>.

The total energy of the orbit is given by<ref>{{cite book |first1=Roger R. |last1=Bate |first2=Donald D. |last2=Mueller |first3=Jerry E. |last3=White |url=https://books.google.com/books?id=UtJK8cetqGkC&pg=PA15 |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |page=15 |edition=First}}</ref> 
:<math>E = \frac{1}{2}m v^2 - G \frac{Mm}{r}</math>.

Substituting for v, the equation becomes
:<math>E = \frac{1}{2}\frac{L^2}{mr^2} - G \frac{Mm}{r}</math>.

This is true for r being the closest / furthest distance so two simultaneous equations are made, which when solved for E:
:<math>E = - G \frac{Mm}{r_1 + r_2}</math>

Since <math display="inline">r_1 = a + a \epsilon</math> and <math>r_2 = a - a \epsilon</math>, where epsilon is the eccentricity of the orbit, the stated result is reached.