Kepler Sector Calculator

Name: Kepler Sector - Geometry Calculator
Author: Jürgen Kummer

Calculations at a Kepler sector. A Kepler sector is an elliptical sector that originates not from the center of the ellipse, but from one of its focal points. At the beginning of the 17th century, Johannes Kepler discovered the laws of celestial mechanics named after him. Kepler's first law states that the planets move in elliptical orbits around the Sun, which is located at one focal point of this ellipse. Kepler's second law states that a planet sweeps out equal areas in equal time along this orbit. These areas are called Kepler sectors.
Enter both semi-axes and both angles, round if necessary, and click Calculate. Please enter angles in degrees, here you can convert angle units. The angles in the following sketch have values of approximately α=45° and β=-68°.

A Kepler sector in an ellipse, the two focal points are marked on the semi-major axis.

Formulas:

ε = \sqrt{1 - \frac{b^{2}}{a^{2}}}

A = π a b

A_{1} = \frac{a b}{2} \cdot | {2 \cdot \arctan [\sqrt{\frac{(1 - ε)}{(1 + ε)}} \cdot \tan (\frac{β}{2})] - ε \cdot \sqrt{1 - ε^{2}} \cdot \frac{\sin (β)}{[1 + ε \cdot \cos (β)]}} - {2 \cdot \arctan [\sqrt{\frac{(1 - ε)}{(1 + ε)}} \cdot \tan (\frac{α}{2})] - ε \cdot \sqrt{1 - ε^{2}} \cdot \frac{\sin (α)}{[1 + ε \cdot \cos (α)]}} |

A_{2} = A - A_{1}

The semi axes have a one-dimensional unit (e.g. meter, or in this case rather astronomical unit), the areas have this unit squared (e.g. square meter or AU²). Numerical eccentricity is dimensionless. |...| is the absolute value.

The first sector area is the gray shaded area in the sketch. The second sector area is the remainder of the ellipse's surface without the first sector area. Adding both together gives the total area of the ellipse. Which of the two sector areas is the one that is needed depends, of course, on the context.

According to Kepler's second law, a planet must orbit faster the closer it is to the Sun (or its star). This is because as the distance decreases, a greater way must be covered in the same amount of time to sweep out the same area. By the way, the planets in our solar system have orbits around the Sun that are nearly circular and not as highly eccentric as in this sketch. However, planets with such eccentric orbits are certainly possible.

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