Kepler's Equation Solver
Solve M = E − e·sin(E) for E (the eccentric anomaly) given a mean anomaly M and an eccentricity e. The calculator runs Newton–Raphson client-side and shows the converged E, the corresponding true anomaly ν, and the iteration count for the inputs you pick.
- Eccentric anomaly E
- 72.29°
- True anomaly ν
- 103.35°
- Iterations to converge
- 5 (converged)
How it works
Kepler's equation is transcendental — there's no closed-form solution for E given M and e. Newton–Raphson on f(E) = E − e·sin(E) − M converges quadratically, so even at high eccentricities (Halley, e ≈ 0.967) you converge inside 6–8 iterations. Below e = 0.05 (most moons, Earth), it's a single step.
The visualization shows the elliptical orbit, the auxiliary eccentric circle (dashed), the focus where the central body sits, and the position of the orbiting body at the current (M, e). The slider for e tops out at 0.97 — the same horizon we use in production for Halley.