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I have an ellipse of a known eccentricity $e$. I also know the length of the perihelion, $d_p$ (the closest distance of the orbiting body to the focus $F_1$). I need to find the length of the aphelion $d_a$ (the farthest distance of the orbiting body to the same focus) given only $e$ and $d_p$.

I understand that $e = \frac{c}{a}$, where $c$ is the distance from the center of the ellipse to a focus and $a$ is the distance from that focus to a vertex on the minor axis. But as this is a ratio, I obviously can't take $c$ and $a$ as lengths as-is. I know that if I could get the distance from the focus to the center of the ellipse, I could then determine the length of the major axis and from there subtract $d_p$ to obtain $d_a$.

I am unsure of how to proceed. Any guidance would be very much appreciated.

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Yes, you can take $a$ and $c$ as actual distances. The perihelion is $d_p=a-c$ and the aphelion is $d_a=a+c$, so $$d_a=2c+d_p=2ea+d_p\\d_a=2a-d_p\\ed_a=2ae-ed_p\\(1-e)d_a=(1+e)d_p\\d_a=\frac {1+e}{1-e}d_p$$

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