The perihelion and aphelion distances in an elliptical orbit are defined by the formulas a(1-e) and a(1+e), respectively, where 'a' represents the semi-major axis (SMA) and 'e' is the eccentricity. This relationship is derived from the definition of eccentricity, which states that the focal points of an ellipse are located at a distance of ±ea from the center. The sum of the distances from any point on the ellipse to the two foci equals 2a, confirming these distance formulas. Classical mechanics textbooks provide foundational insights into these derivations.
PREREQUISITESAstronomy students, physicists, and anyone interested in understanding the mechanics of planetary orbits and the mathematical foundations of elliptical shapes.
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 ##.bruhh said: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.