Orbits and Kepler's laws simulator

What’s happening

Kepler’s three laws describe how planets move:

1. Orbits are ellipses with the star at one focus. The shape is set by the eccentricity $e$: the closest distance is $r_p = a(1-e)$ and the farthest is $r_a = a(1+e)$.
2. A planet sweeps out equal areas in equal times, so it moves fastest near the star.
3. The period depends only on the semi-major axis and the central mass. In units of years, AU and solar masses: $P = \sqrt{a^3/M}$.

To find where a planet is at a given time, solve Kepler’s equation for the eccentric anomaly $E$:

$$M = E - e\sin E,$$

where the mean anomaly $M = 2\pi t/P$ grows uniformly with time. There is no closed-form solution, but Newton’s method converges quickly from the starting guess $E_0 = M + e\sin M$. Then $r = a(1 - e\cos E)$, and the speed follows from the vis-viva equation:

$$v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right).$$

Assumptions and limits. Two bodies only, with the planet’s mass negligible compared with the star’s. Other planets cause small perturbations that this model ignores.

Try this

1. Mars. Mars orbits the Sun at $a = 1.524$ AU. What is its period?
   Answer

   $P = 1.524^{3/2} \approx 1.88$ years.
2. Earth’s speed. Earth has $a = 1$ AU and $e = 0.0167$. Find its speed at perihelion and aphelion.
   Answer

   $r_p = 0.9833$ AU gives $v_p \approx 30.3$ km/s; $r_a = 1.0167$ AU gives $v_a \approx 29.3$ km/s.
3. Solve Kepler’s equation. For $e = 0.5$ and $M = 1$ rad, find $E$ and $r/a$.
   Answer

   $E \approx 1.4987$ rad, so $r/a = 1 - 0.5\cos E \approx 0.964$. The true anomaly is about $116.4°$.
4. Heavier star. A planet orbits a $4\,M_\odot$ star at 1 AU. How long is its year?
   Answer

   $P = \sqrt{1/4} = 0.5$ years.