What makes an orbit circular?

For a given radius, a purely tangential speed v_circ = √(GM/r) makes centripetal requirement match gravity (eccentricity ≈ 0). Too slow or a radial velocity component opens an ellipse that may hit the central body; at or above v_esc = √2 v_circ the specific energy is non-negative and the path is unbound.

Is this a full two-body problem?

No. The central mass is fixed in place — a common teaching approximation when one body is much heavier. Mutual two-body motion about the barycenter is not simulated.

What do E and e mean here?

Specific mechanical energy E = ½v² − GM/r is negative for bound ellipses and non-negative for escape. Eccentricity e comes from the Laplace–Runge–Lenz vector: e ≈ 0 is a circle, 0 < e < 1 an ellipse, e ≥ 1 an unbound conic.

Orbit Simulator

Fixed central mass: tune speed and launch angle for circular, elliptical, or escape trajectories.

About this model

Restricted Kepler problem: a massless test particle moves in the fixed inverse-square field of a stationary central mass (the primary does not move). Launch from radius r₀ with speed k·√(GM/r₀) and a velocity angle; 90° is tangential. Circular orbits need k = 1 and pure tangential launch; bound ellipses have specific energy E < 0; escape occurs when E ≥ 0 (roughly |v| ≥ √2 v_circ). Readouts show E, eccentricity e, and orbit type. Trajectories stop on the central body and the view auto-zooms.

Who it's for: Mechanics and astronomy; orbital energy, eccentricity, and escape speed.

Key terms

orbit
Kepler
eccentricity
escape velocity
specific energy
fixed central mass

How it works

A massless test particle moves in the fixed central field of a point mass (restricted two-body / Kepler problem — the primary does not move). Acceleration is a = −GM/r² toward the center. Launch from (r₀, 0) with speed |v| = k √(GM/r₀) and a chosen velocity angle: 90° is tangential, so k = 1 gives a circular orbit; smaller k yields ellipses that may hit the central body; k ≥ √2 (with enough tangential component) unbound escape. Specific energy E = ½v² − GM/r and eccentricity e classify the conic. Integration uses velocity Verlet; trajectories stop on the central body radius, and the view auto-zooms to keep the trail in frame.

Key equations

a = −GM/r² · r̂ (fixed central mass)

v_circ = √(GM/r), v_esc = √2 v_circ

E = ½v² − GM/r · e from Laplace–Runge–Lenz

Frequently asked questions

What makes an orbit circular?: For a given radius, a purely tangential speed v_circ = √(GM/r) makes centripetal requirement match gravity (eccentricity ≈ 0). Too slow or a radial velocity component opens an ellipse that may hit the central body; at or above v_esc = √2 v_circ the specific energy is non-negative and the path is unbound.
Is this a full two-body problem?: No. The central mass is fixed in place — a common teaching approximation when one body is much heavier. Mutual two-body motion about the barycenter is not simulated.
What do E and e mean here?: Specific mechanical energy E = ½v² − GM/r is negative for bound ellipses and non-negative for escape. Eccentricity e comes from the Laplace–Runge–Lenz vector: e ≈ 0 is a circle, 0 < e < 1 an ellipse, e ≥ 1 an unbound conic.

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