Kepler's Laws Simulator

The Law of Ellipses: The orbit of a planet is an ellipse with the Sun at one of the two foci.

r = a(1 - e^2) / (1 + e cos θ)

Orbit Parameters

Eccentricity (e) 0.50

0 = Circle, near 1 = Flat Ellipse

Visualization

Sun (Focus 1)

Empty Focus 2

Planet

Show Axes & Foci

The Law of Equal Areas: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

dA / dt = const

Visualization

The orbit is divided into 12 equal time intervals. Notice how the sectors near the Sun are short and wide, while those far away are long and narrow. Their areas are identical.

Eccentricity (e) 0.60 Simulation Speed 0.5x

Sector A (Time T/12)

Sector B (Time T/12)

The Law of Harmonies: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

T^2 ∝ a^3

Compare Planets

Planet 1 (Ref, a=10)

Planet 2 (Variable)

Axis Planet 2 (a) 15.0

P1 Laps: 0
P2 Laps: 0

                T1 = 1.00 yr

                T2 = 1.84 yr

                Ratio T2/T1 = 1.84

Speed 1.0x