# Kepler Orbit Solver 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 > Canonical page: https://elysiatools.com/en/tools/kepler-orbit-solver - **Category:** Astronomy - **Keywords:** kepler, kepler equation, orbit, eccentric anomaly, true anomaly, mean anomaly, celestial mechanics, newton raphson, two-body problem, astronomy ## Overview 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). ## Inputs - **Mean Anomaly (M, degrees)** (number): e.g. 45 - **Eccentricity (e)** (number): 0.0 (circle) to 1.0 (parabola); e.g. 0.3 - **Semi-major Axis (a, AU)** (number): e.g. 1.0 (Earth); leave 0 to express r in units of a - **Solver Method** (select) - **Max iterations** (number) - **Tolerance** (select) ## 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 - Input the mean anomaly (M) in degrees, the orbital eccentricity (e), and optionally the semi-major axis (a) in astronomical units. - Select your preferred numerical solver method (Newton-Raphson, bisection, or fixed-point iteration) along with the maximum iteration limit and convergence tolerance. - The 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. ## Frequently asked questions ### 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). ## Related tools - [Normal Distribution Plotter](https://elysiatools.com/en/tools/normal-distribution-plotter): Plot a normal-distribution bell curve as SVG with z-score markers, interval or tail-probability shading, and 68/95/99.7 standard-deviation bands. - [Jovian Galilean Moons, Europa Flyby, Cassini State and Resonance Solver](https://elysiatools.com/en/tools/jovian-galilean-moons-europa-flyby-cassini-state-and-resonance-solver): Solve Galilean satellite positions with Meeus/Lieske E5 series, chart the Laplace-resonance libration, scan mutual events and eclipses, model a Europa Clipper-style flyby with patched-conic Δv, and solve Jupiter Cassini states from J2 oblateness precession. - [Biomass Pyramid & Energy Flow (10% Law)](https://elysiatools.com/en/tools/biomass-pyramid): Trophic pyramid from producer energy and transfer efficiency: per-level energy, % of base, losses, and why energy pyramids stay upright. - [Chinese Stroke Order Decomposer](https://elysiatools.com/en/tools/chinese-stroke-order-decomposition): Decompose a Chinese character into its stroke sequence and radical structure, and render an SVG stroke-order demonstration with component coloring. - [CSS Gradient Mesh Generator (Conic + Radial Blend)](https://elysiatools.com/en/tools/css-gradient-mesh-generator): Layered radial-gradient "mesh" backgrounds with conic underlays, blend modes, grain and seeded layouts — live preview, CSS and Tailwind export. - [CSV Histogram, Scatter and Box Plot Chart Studio](https://elysiatools.com/en/tools/csv-histogram-scatter-box-plot-chart-studio): Upload or paste a CSV and chart it instantly: histograms with Sturges / Freedman-Diaconis / Scott binning, scatter plots with Pearson r and a least-squares fit, and Tukey box plots — every chart downloadable as PNG and SVG. - [Glassmorphism CSS Snippet Builder](https://elysiatools.com/en/tools/glassmorphism-css-snippet-builder): End-to-end frosted-glass / neumorphism CSS snippet generator: blur+saturate backdrop, tint, border, layered shadows, grain overlay, @supports fallback, -webkit- prefix, Tailwind mapping and live preview. - [Kakuro / Fillomino / Masyu / Yajilin / Hashiwokakero / Nurikabe / Slitherlink / Star Battle Puzzle Generator with Uniqueness Prover](https://elysiatools.com/en/tools/kakuro-fillomino-masyu-yajilin-hashiwokakero-nurikabe-slitherlink-and-star-battle-puzzle-generator-with-uniqueness-prover): Deterministically seeded generator for eight Nikoli logic-puzzle families — each board is grown solution-first, rendered as puzzle + solution SVGs, and verified by an exhaustive backtracking solver that counts solutions to two, so “unique” is a proof (and an exhausted budget is reported as unproven, never guessed). ## Samples - [SVG Samples](https://elysiatools.com/en/samples/svg-samples): Scalable Vector Graphics (SVG) samples demonstrating various SVG features and techniques - [Web Image Processing Python Samples](https://elysiatools.com/en/samples/web-image-processing-python): Web Python image processing examples using PIL/Pillow including reading, saving, resizing, and format conversion - [D3.js Data Visualization Samples](https://elysiatools.com/en/samples/d3js-data-visualization): Comprehensive D3.js data visualization examples including charts, maps, animations, and interactive visualizations - [SVG Viewer Samples](https://elysiatools.com/en/samples/svg-viewer-samples): Sample SVG vector graphics (scene, icon set, diagram) for the in-browser SVG Viewer ## Related content - [Astronomical Coordinate, Time, and Orbit Workflows](https://elysiatools.com/en/hubs/astronomy-coordinate-position-workflows): Convert celestial coordinates, lock the observation time scale, and solve orbital or sky-position questions with explicit precision limits.