Derivation for Aphelion and Perihelion Distances

[bruhh]

I know the formulas for perihelion and aphelion distances in an orbit with SMA a and eccentricity e are a(1-e) and a(1+e), respectively. However, why is this? I can't seem to find any derivations for this anywhere and also cannot figure this out myself.

 

[Ibix]

Not sure what you mean by "why" here. The focus-directrix definition of an ellipse gives you this more or less instantly, but you may have to work harder with other definitions. Or are you trying to prove it from $F=GMm/r^2$?

Basically, you have told us where you want to go but not where you are. That makes it tricky to give directions.

 

[DrSteve]

See any classical mechanics textbook

 

[WernerQH]

I know the formulas for perihelion and aphelion distances in an orbit with SMA a and eccentricity e are a(1-e) and a(1+e), respectively. However, why is this? I can't seem to find any derivations for this anywhere and also cannot figure this out myself.

It follows from the definition of the eccentricity: the focal points of an ellipse are at a distance $\pm e a$ from the center. And for perihelion and aphelion, as for any points on the ellipse, the sum of the distances to the foci must be $2a$.