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General InfoEdit

Ellipse2

WhatEdit

An ellipse is mathematically defined as the set of points that satisfies the equation

$ r + r' = 2a $

Semi Major AxisEdit

The semi major axis $ a $ is half the length of the long, or major axis of the ellipse

Semi Minor AxisEdit

The semi minor axis $ b $ is half the length of the short, or minor axis of the ellipse

Focal PointEdit

The focal points are denoted by $ F_1 $ and $ F_2 $ and the distance to from the focal points to the point $ X $ on the ellipse is denoted as $ r $ and $ r^' $

EccentricityEdit

The eccentricity $ e $ is defined as the distance between the two foci divided by the major axis of the ellipse. The distance from the center to either foci can be expressed as $ ae $

For a circle, $ e=0 $

PerihelionEdit

The point on the ellipse that is closest to the principal focus is called perihelion

The distance from the principal focus at perihelion is:

$ d = a - ea = a(1 - e) $

AphelionEdit

The point on the ellipse that is farthest from the principal focus is called aphelion

The distance from the principal focus at aphelion is:

$ d = a + ea = a(1 + e) $

RadiusEdit

The radius can be defined as:

$ r = \frac {a(1-e^2)}{1 + e \textrm{cos}\theta} $

Relationship Between Major and Semi Major AxisEdit

$ b^2 = a^2(1-e^2) $

Total Energy of a Binary OrbitEdit

$ E = -G \frac {m_1 m_2}{2a} $