Orbital Velocity Calculator

How to use this orbital velocity calculator

What this calculator does

This calculator estimates the geometry, period, orbital energy, and speeds of an ideal Newtonian two-body elliptical orbit. It is designed for planets, moons, artificial satellites, and similar systems where one can approximate the motion as two masses interacting only through gravity. It does not model atmospheric drag, thrust, oblateness, third-body perturbations, relativity, or other mission-design effects. The core speed calculation uses the vis-viva relation, while the period follows Kepler's third-law form for two finite masses. For a concise derivation and physical explanation of orbital motion, see NASA's Gravity & Mechanics chapter.

When to use it

Use it to compare a planet's speed near periapsis and apoapsis, estimate the speed of a satellite at a known orbital radius, check how eccentricity changes the radial range of an orbit, or verify the period and total mechanical energy implied by a chosen semi-major axis and masses. It is also useful for classroom problems where the ideal two-body assumptions are explicitly stated.

How to calculate

1. The calculator opens with an Earth-around-Sun demonstration: a semi-major axis of 1 au, eccentricity 0.0167, one solar mass, one Earth mass, and a selected distance of 1 au. The first result and a validated example XLSX are ready immediately.
2. Replace the demonstration values with your orbit. You can edit either Eccentricity or Semi-minor axis; the other is recomputed so the ellipse remains internally consistent. Choose the Distance unit before entering a different scale if that is more convenient.
3. Enter Star mass and Satellite mass. Their unit selectors convert existing values. The linked Standard gravitational parameter updates from the masses; if you edit μ instead, the calculator solves the star mass while keeping the satellite mass fixed.
4. Enter the Distance at which you want the vis-viva speed. It must be between the computed periapsis and apoapsis distances. Choose the Speed unit, then read the live outputs and checkpoint table.
5. Select Download Excel to export the current canonical model as a real .xlsx workbook. Reset clears the demonstration and all semantic data. After reset, export is disabled until a complete valid state has been entered again.

Input guide

Semi-major axis is required and must be a positive decimal or scientific-notation value in the active distance unit; 1 au is the demonstration value. Increasing it generally lengthens the period and lowers characteristic orbital speeds for fixed masses. Do not enter altitude above a body's surface when the model calls for center-to-center orbital size. Semi-minor axis is also positive and cannot exceed the semi-major axis; the sample is 0.9998605453 au. It is linked to eccentricity, so editing it changes the implied orbital shape. Eccentricity is unitless, required through the linked shape definition, and must satisfy 0 ≤ e < 1 for a bound ellipse; the sample 0.0167 is nearly circular. A common mistake is entering 1.67 as if eccentricity were a percent.

Distance unit accepts au or km and converts the Semi-major axis, Semi-minor axis, and Distance fields in place. The exact astronomical unit is 149,597,870,700 m; NASA/JPL provides the formal astronomical-unit definition. Star mass is required and positive; enter a decimal or scientific-notation value in Solar masses or kg. The demonstration uses 1 Solar mass. Star mass unit converts the existing star-mass value between those units.

Satellite mass accepts zero or a positive value in Earth masses or kg; the demonstration uses 1 Earth mass. Setting it to zero creates the test-particle limit and makes the reported total energy zero because that output is the total mechanical energy of the orbiting mass. Satellite mass unit converts the existing value. Standard gravitational parameter is μ = G(M + m), entered here in km³/s²; it may be supplied directly as a positive decimal or scientific-notation value. If you edit μ, ensure it exceeds the contribution of the fixed satellite mass so a positive star mass can be recovered.

Distance is the required center-to-center radial distance at which to evaluate vis-viva speed. The sample is 1 au. For an ellipse it must fall between periapsis and apoapsis; entering a value outside that physical interval is rejected. Speed unit switches the displayed velocity values between km/s and mi/s without changing the underlying SI model.

Output guide

Velocity at selected distance is the vis-viva speed relative to the central body at the entered radial distance. It is an estimate from the ideal two-body model and is driven by μ, semi-major axis, and Distance. Distance at apoapsis and Distance at periapsis are the farthest and closest orbital radii, computed as a(1 + e) and a(1 – e). For a circular orbit they are identical. Velocity at apoapsis and Velocity at periapsis use those radii in the same vis-viva equation; in an eccentric bound orbit, periapsis speed is higher.

Period is the ideal orbital period derived from 2π√(a³/μ) and is shown in years or days depending on scale. Larger semi-major axis increases it strongly, while larger μ decreases it. Total energy is – GMm/(2a), the total Newtonian mechanical energy of the orbiting mass; a negative value indicates a bound elliptic orbit. The Orbital checkpoints table repeats three current-state rows – Periapsis, Selected distance, and Apoapsis – with Distance, Speed, and a short Interpretation. The table is a comparison, not a time history or trajectory plot.

Worked example

With a = 1 au and e = 0.0167, the calculator obtains a semi-minor axis of about 0.9998605453 au, periapsis 0.983300 au, and apoapsis 1.016700 au. Using one solar mass plus one Earth mass gives μ ≈ 1.327168518 × 10¹¹ km³/s². At r = 1 au, vis-viva gives 29.7852 km/s. The endpoint speeds are about 30.2868 km/s at periapsis and 29.2919 km/s at apoapsis, while the period is about 1.0000 year. These first-open values are the same values exported to the startup workbook. OpenStax's Satellite Orbits and Energy section provides additional context for circular/elliptical orbits and orbital energy.