| 360° | ||||||||||||
| [ | -e sin Θ | + | 2 | tan⁻¹ | ( | √ | 1-e | · tan | Θ | ) | ] | |
| 1+e cos Θ | (1-e²)½ | 1+e | 2 | |||||||||
| 0 | ||||||||||||
which is twice the area of the ellipse.
The energy in the ellipse during an interval dt is
| 1 | mv²dt = | 1 | mµ | ( | 2 | - | 1 | ) | dt, |
| 2 | 2 | r | a |
from the well-known equation for the velocity in a focal conic. The integral of this for the whole ellipse is
| ⌠T | 1 | mv²dt = | ⌠360° | 1 | mµ | ( | 2r - | r² | ) | dΘ | |
| ⌡0 | 2 | ⌡0 | 2 | h | a |
= mµ½πa½.
Since
| ⌠ | r dΘ = | ⌠ | a · 1 - e² | dΘ = | 2a · 1 - e² | tan⁻¹ | ( | √ | 1 - e | tan | Θ | ) |
| ⌡ | ⌡ | 1 + e cos Θ | (1 - e²)½ | 1 + e | 2 |