Ellipse

General Info

What

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

$r + r' = 2a$

Semi Major Axis

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

Semi Minor Axis

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

Focal Point

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 $r^{'}$

Eccentricity

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, $e = 0$

Perihelion

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

The distance from the principal focus at perihelion is:

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

Aphelion

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

The distance from the principal focus at aphelion is:

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

Radius

The radius can be defined as:

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

Relationship Between Major and Semi Major Axis

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

Total Energy of a Binary Orbit

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