Apsis speed ratio calculator

Enter the eccentricity of an elliptical orbit and it evaluates (1 + e) / (1 - e), a dimensionless result that needs no mass, distance, or unit conversion. A circular orbit returns 1 because its speed is constant at every point. As eccentricity approaches 1, the contrast grows sharply. Values at or above 1 are rejected because those trajectories do not have both a periapsis and an apoapsis.

What the apsis speed ratio means

Periapsis is the point where an orbiting body is closest to the focus, while apoapsis is the point where it is farthest away. In a bound Keplerian orbit, the body travels fastest at periapsis and slowest at apoapsis. This calculator reports periapsis speed divided by apoapsis speed, so the result is always at least 1 for a valid elliptical orbit. A result of 3, for example, means the periapsis speed is three times the apoapsis speed; it does not mean either speed is three units per second. The ratio is dimensionless and describes contrast rather than an absolute velocity. Only eccentricity is needed because conservation of angular momentum makes the absolute scale and central mass cancel when the two apsis speeds are divided. That makes the result useful for comparing orbit shapes even when their sizes, periods, or central bodies differ. For a circular orbit, eccentricity is 0 and the two apsides coincide conceptually, producing a ratio of 1. Increasing eccentricity stretches the ellipse and increases the difference between the nearest-point and farthest-point speeds.

How the formula is obtained and evaluated

At either apsis, velocity is perpendicular to the radius vector. Specific angular momentum can therefore be written as periapsis radius times periapsis speed and also as apoapsis radius times apoapsis speed. Equating those expressions gives periapsis speed divided by apoapsis speed as apoapsis radius divided by periapsis radius. For an ellipse with semi-major axis a and eccentricity e, the apsis radii are a(1 + e) and a(1 - e). Dividing them cancels a and leaves the compact expression (1 + e) / (1 - e). The calculator validates the supplied eccentricity, performs that expression directly, and returns both the normalized input and the ratio. Numeric strings are accepted by the calculation layer for API convenience, although the published input contract uses a number. Results are rounded to fifteen significant digits so ordinary floating-point artifacts do not create unstable output while useful precision is retained. No ephemeris, gravitational parameter, or orbital distance is consulted. The computation is deterministic, uses no network request, and produces the same output for the same input in the browser and through the API.

Domain limits and interpreting extreme values

The accepted domain is 0 less than or equal to e and e less than 1. Negative eccentricity is not part of the standard geometric convention for Keplerian ellipses, so it is rejected rather than silently converted. An eccentricity of exactly 1 describes a parabolic trajectory, and a value greater than 1 describes a hyperbolic trajectory. Neither has a finite apoapsis, so a periapsis-to-apoapsis speed ratio is not defined for those cases and the calculator returns an input error. Near 1, the denominator 1 - e becomes very small and the ratio grows rapidly. This is mathematically expected, not an indication that an absolute speed has become infinite for a real engineered orbit. It means that the ideal two-body ellipse has a very large relative contrast between its apsis speeds. When using a value extremely close to 1, retain enough digits in the eccentricity because small rounding changes can materially affect the ratio. The formula also assumes an ideal two-body Keplerian ellipse. Perturbations, atmospheric drag, thrust, non-spherical gravity, and relativistic effects can change actual velocities, so mission-grade state prediction should use an appropriate orbit propagator rather than this shape-only ratio.