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Elliptic orbit
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====Using vectors==== The general equation of an ellipse under these assumptions using vectors is: :<math> |\mathbf{F2} - \mathbf{p}| + |\mathbf{p}| = 2a \qquad\mid z=0</math> where: *<math>a\,\!</math> is the length of the [[semi-major axis]]. *<math>\mathbf{F2} = \left(f_x,f_y\right)</math> is the second ("empty") focus. *<math>\mathbf{p} = \left(x,y\right)</math> is any (x,y) value satisfying the equation. The semi-major axis length (a) can be calculated as: :<math>a = \frac{\mu |\mathbf{r}|}{2\mu - |\mathbf{r}| \mathbf{v}^2}</math> where <math>\mu\ = Gm_1</math> is the [[standard gravitational parameter]]. The empty focus (<math>\mathbf{F2} = \left(f_x,f_y\right)</math>) can be found by first determining the [[Eccentricity vector]]: :<math>\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu}</math> Where <math>\mathbf{h}</math> is the specific angular momentum of the orbiting body:<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=PA17 |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |page=17 |edition=First}}</ref> :<math>\mathbf{h} = \mathbf{r} \times \mathbf{v}</math> Then :<math>\mathbf{F2} = -2a\mathbf{e}</math>
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