Keplerian Orbit Simulator

Keplerian Orbit Simulator models a planet’s elliptical orbit around a star, showing period and motion per Kepler’s laws.

Central Body Mass \\(M\\):

Orbiting Body:

Semi-major Axis \\(a\\) (AU):

Eccentricity \\(e\\):

Formulas Used in Keplerian Orbit Simulator

The simulator models an elliptical orbit and computes the orbital period:

Orbital Period (Kepler’s Third Law):

\\[ T = \sqrt{\frac{4\pi^2 a^3}{G M}} \\]

Elliptical Orbit:

\\[ x = a \cos(\theta) – a e, \quad y = a \sqrt{1 – e^2} \sin(\theta) \\]

Where:

\\(T\\): Orbital period (years)
\\(a\\): Semi-major axis (m)
\\(e\\): Eccentricity (0 ≤ e < 1)
\\(G\\): Gravitational constant (\\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\\))
\\(M\\): Central body mass (kg)
\\(x, y\\): Coordinates of the planet relative to the focus
\\(\theta\\): Angle parameter (radians)

Example: Earth orbiting the Sun (\\(M = 1.989 \times 10^{30} \, \text{kg}, a = 1 \, \text{AU}, e = 0.0167\\))

\\[ T = \sqrt{\frac{4\pi^2 (1.496 \times 10^{11})^3}{6.67430 \times 10^{-11} \times 1.989 \times 10^{30}}} \approx 1 \, \text{year} \\]