Calculate an orbit from energy
Use kilometer-based units. The calculator applies the two-body gravity model.
Example input data
These cases use Earth parameters and demonstrate common trajectory classes.
| Case | Radius | Speed | Flight angle | Expected path |
|---|---|---|---|---|
| Low circular orbit | 7000 km | 7.546 km/s | 0° | Near-circular ellipse |
| Escape boundary | 7000 km | 10.672 km/s | 0° | Parabolic |
| Escape departure | 7000 km | 11.200 km/s | 10° | Hyperbolic |
| Elliptical transfer | 7000 km | 9.000 km/s | 0° | Elliptic transfer |
Formula used
Specific orbital energy controls the broad shape of a two-body orbit.
Use the equation above when speed and radius are known.
Use this form when specific energy and radius are known.
Semimajor axis applies when the energy is not zero.
Angular momentum uses the tangential velocity component.
Eccentricity identifies the detailed conic shape.
Apoapsis is finite only for elliptic paths.
Period exists only for a closed elliptic orbit.
How to use this calculator
- Choose a central body preset, or enter custom gravitational values.
- Enter the current radius from the body's center.
- Select speed mode or direct specific-energy mode.
- Enter the flight path angle from local horizontal.
- Select Calculate orbit to view the classification and elements.
- Check periapsis altitude before treating the path as physically safe.
- Download CSV or PDF when you need a saved result.
Understanding orbital energy
Orbital energy describes how a body moves under gravity. It combines kinetic energy with gravitational potential energy. The calculator uses specific orbital energy. This means energy per unit mass. That approach removes spacecraft mass from the core equation.
A negative specific energy creates a bound ellipse. The object returns after one orbital period. More negative energy produces a smaller semimajor axis. A circular orbit is a special ellipse. Its energy equals minus mu divided by twice radius.
Zero specific energy marks the escape boundary. This is a parabolic path. The object has exactly escape speed at that radius. It continues outward forever, while its speed approaches zero far away. Parabolic cases are ideal mathematical limits.
Positive specific energy produces a hyperbolic trajectory. The object escapes with surplus speed. Engineers call that distant residual speed hyperbolic excess velocity. It is useful for interplanetary departure and planetary flyby design. A higher positive energy means a faster distant departure.
Energy alone identifies the broad orbit type. It does not fully define its orientation. The current radius and flight path angle provide angular momentum information. Angular momentum determines the eccentricity and turning points. With these values, the calculator estimates periapsis, apoapsis, semimajor axis, and speed limits.
The flight path angle is measured from local horizontal. A zero degree angle is fully tangential. Positive angles point outward. Negative angles point inward. A near radial path has very little angular momentum. Its periapsis can fall near the central body's center. Treat those cases carefully.
Use consistent units throughout the calculation. This page uses kilometers, seconds, and square kilometers per second squared. The gravitational parameter must match those units. Earth has a standard parameter near 398600.4418 km³/s². Its equatorial radius is near 6378.137 km.
Check the physical warning beneath the results. A calculated periapsis below the body's radius indicates impact. An elliptical orbit can still be mathematically valid while crossing terrain or atmosphere. Low periapsis also causes drag around planets with atmospheres. Mission planning needs additional environmental constraints.
The reported period applies only to elliptic orbits. Parabolic and hyperbolic paths do not repeat. Their apoapsis is unbounded. Numerical rounding can make a nearly parabolic result look slightly elliptic or hyperbolic. Use adequate input precision when working near escape speed.
This calculator supports direct energy input and speed-based input. Choose direct energy when a mission analysis provides specific orbital energy. Choose speed when you know local velocity. Enter a reasonable flight path angle for the most useful element estimates. Review the output before using it for navigation decisions.
Energy is conserved when gravity is the only important force. Thrust, drag, solar pressure, and third-body effects alter real paths. Use this tool for two-body estimates. Apply detailed propagation models for final mission design. Include station keeping, close approaches, safety margins, and operational performance checks when accuracy matters. Extra checks greatly improve mission planning confidence.
Frequently asked questions
1. What does negative orbital energy mean?
Negative specific orbital energy means the object is bound to the central body. In an ideal two-body model, it follows an elliptic path and can return after a finite orbital period.
2. What does zero orbital energy mean?
Zero specific energy represents a parabolic trajectory. It is the exact escape boundary. The object reaches zero speed only at an infinite distance from the central body.
3. What does positive orbital energy mean?
Positive specific energy indicates a hyperbolic escape trajectory. The object leaves the central body's gravity field with a remaining speed called hyperbolic excess velocity.
4. Why is spacecraft mass not required?
The calculator uses specific orbital energy, which is energy divided by mass. Mass cancels in the two-body equations, so the same path properties apply to any test mass.
5. Is radius the same as altitude?
No. Radius is measured from the central body's center. Altitude is measured above its surface. The calculator uses radius for the equations and reports altitude using the entered body radius.
6. Why do I need a flight path angle?
Energy gives the orbit family, but not angular momentum. The flight path angle separates speed into radial and tangential components. That lets the calculator estimate eccentricity, periapsis, and apoapsis.
7. Can this calculate a circular orbit?
Yes. Enter a tangential flight path angle of zero degrees. Use the local circular speed, which equals the square root of gravitational parameter divided by radius.
8. Why is the apoapsis unbounded?
Parabolic and hyperbolic paths do not turn back under ideal gravity. Their distance can increase without limit, so they have no finite apoapsis or repeating period.
9. Does the calculator include atmospheric drag?
No. It uses an ideal two-body gravity model. Atmospheric drag, thrust, oblateness, solar pressure, and third-body gravity require a more detailed numerical propagation model.
10. What happens when periapsis is below the body radius?
The trajectory intersects the central body in this simplified model. The mathematical conic may still exist, but it is not a collision-free physical orbit around a solid planet or moon.
11. Which units should I use?
Use kilometers for distance, seconds for time, kilometers per second for speed, and km³/s² for gravitational parameter. Specific orbital energy is then km²/s².