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Astronomy

Kepler Orbit Solver

· Last updated：2026-06-18

Numerically solve Kepler’s equation E − e·sin(E) = M for eccentric anomaly E, true anomaly ν, orbital radius and relative speed, with Newton/bisection/fixed-point methods, convergence history and an SVG orbit diagram

Details

What this tool helps you do

Given mean anomaly M (degrees), eccentricity e, and optional semi-major axis a (AU), this tool solves Kepler’s transcendental equation M = E − e·sin(E) for the eccentric anomaly E, then derives the true anomaly ν, orbital radius r, and relative speed. Three solver methods (Newton-Raphson, bisection, fixed-point) are available with full iteration history. The report renders an SVG orbit diagram and classifies the orbit (circular / elliptical / parabolic / hyperbolic).

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Written by the Elysia Tools team
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Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Astronomy
Input types
number, select
Output type
html
Sample coverage
4
API ready
Yes

Overview

The Kepler Orbit Solver is a numerical calculator designed to solve Kepler's transcendental equation E − e·sin(E) = M for the eccentric anomaly. By inputting the mean anomaly, eccentricity, and semi-major axis, you can instantly compute the true anomaly, orbital radius, and relative speed, complete with an interactive SVG orbit diagram and detailed solver convergence history.

When to use

  • When calculating the precise position and velocity of a celestial body or satellite in an elliptical orbit from its mean anomaly.
  • When comparing the convergence speed and numerical stability of Newton-Raphson, bisection, and fixed-point iteration methods.
  • When generating visual SVG orbit diagrams showing the focus, periapsis, apoapsis, and current orbital radius vector.

How it works

  1. 1Input the mean anomaly (M) in degrees, the orbital eccentricity (e), and optionally the semi-major axis (a) in astronomical units.
  2. 2Select your preferred numerical solver method (Newton-Raphson, bisection, or fixed-point iteration) along with the maximum iteration limit and convergence tolerance.
  3. 3The tool numerically solves for the eccentric anomaly (E), computes the true anomaly (ν), orbital radius (r), and relative speed, and renders the SVG orbit visualization.

Use cases

  • Orbital mechanics homework verification for astronomy students studying the two-body problem.
  • Satellite mission planning to determine instantaneous orbital positions and relative velocities.
  • Numerical analysis studies comparing the convergence rates of different root-finding algorithms on transcendental equations.

Examples

1. Analyzing a Moderate Eccentricity Orbit

Aerospace Engineering Student

Background

A student needs to find the exact position of a satellite in an elliptical orbit with an eccentricity of 0.3 at a mean anomaly of 45 degrees.

Problem

Manually solving the transcendental Kepler equation is highly tedious and requires multiple manual iterations.

How to use

Enter 45 for Mean Anomaly, 0.3 for Eccentricity, and 1.0 for Semi-major Axis. Select the Newton solver method with a tolerance of 1e-12.

M = 45, e = 0.3, a = 1.0, method = newton, tol = 1e-12

Outcome

The tool calculates the eccentric anomaly as approximately 59.68 degrees, the true anomaly as 75.54 degrees, and displays the convergence path in 4 iterations alongside an SVG orbit diagram.

2. Comparing Solver Convergence for High Eccentricity Orbits

Celestial Mechanics Researcher

Background

A researcher is studying comet-like orbits with high eccentricity (e = 0.8) and wants to compare how different numerical methods converge.

Problem

Fixed-point iteration often fails or converges extremely slowly for high eccentricities near periapsis.

FAQ

What is Kepler's equation?

Kepler's equation, E − e·sin(E) = M, relates the geometric position of a body in an elliptical orbit (eccentric anomaly E) to the time elapsed since periapsis (mean anomaly M).

Which solver method should I choose?

The Newton-Raphson method is recommended for most orbits due to its quadratic convergence, though bisection is preferred for extreme eccentricities where fixed-point iteration may diverge.

How is the relative speed calculated?

It is derived using the vis-viva equation, expressing the orbital speed relative to the circular orbit speed at that distance.

Can this tool solve parabolic or hyperbolic orbits?

The tool classifies the orbit based on eccentricity, but Kepler's standard equation E − e·sin(E) = M specifically solves bound elliptical (0 ≤ e < 1) and circular (e = 0) orbits.

What happens if I leave the semi-major axis (a) as 0?

If the semi-major axis is set to 0, the tool calculates the orbital radius (r) in relative units of a rather than absolute Astronomical Units (AU).

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How to use

Set Mean Anomaly to 90, Eccentricity to 0.8, and run the solver first with Newton-Raphson, then with Bisection.

M = 90, e = 0.8, method = bisection, maxIter = 100

Outcome

The tool demonstrates that while Newton-Raphson converges rapidly in a few steps, the Bisection method takes significantly more iterations but guarantees convergence within the specified tolerance.