Orbital angular momentum calculator

This orbital angular momentum calculator returns the specific angular momentum of an orbiting body from three instantaneous quantities: its speed, its distance from the centre of the primary body, and its flight-path angle. The angle is measured from the local horizontal, making zero degrees a purely transverse direction. Enter radius in metres, speed in metres per second, and the signed path angle in degrees to receive specific angular momentum in square metres per second. The calculation is deterministic and does not require the orbiting body's mass.

What specific orbital angular momentum represents

Specific orbital angular momentum describes orbital angular momentum per unit mass. It is commonly written as h and has units of square metres per second. The quantity depends on the cross product of the position and velocity vectors, so only the component of velocity perpendicular to the radius contributes.

That geometric fact is why this calculator asks for a flight-path angle as well as speed and radius. A path angle of zero degrees means motion along the local horizontal, perpendicular to the radius, and gives the largest magnitude available for the supplied speed and radius. As the path becomes more radial, the perpendicular velocity component becomes smaller.

At plus or minus ninety degrees the motion is radial and the calculated specific angular momentum is ideally zero, apart from ordinary floating-point precision. Because mass has been divided out, the result can characterize the trajectory without knowing whether the orbiting object is a spacecraft, satellite, asteroid, or planet. This capability calculates the scalar signed by the cosine convention; it does not return the three-dimensional angular-momentum vector or its orientation in an inertial coordinate frame.

How to enter radius, speed, and path angle

Use the instantaneous distance from the central body's centre for radius, not altitude above its surface. For example, a spacecraft altitude must be added to the body's relevant reference radius before it is entered. Supply that radius in metres and speed in metres per second.

The flight-path angle is entered in degrees and is measured from the local horizontal: zero is transverse motion, a positive value conventionally indicates motion away from the central body, and a negative value indicates motion toward it. The calculation uses h = r times v times cosine of the path angle. Since cosine is even, equal positive and negative path angles produce the same scalar value here; the sign distinguishes ascent from descent but does not change the magnitude of the transverse component.

All three fields are required and must be finite numbers. Speed must not be negative, and radius may be zero but must never be negative. Keeping one unit system throughout is essential.

If kilometres and kilometres per second are used outside this interface, their product would be square kilometres per second, but this calculator's declared inputs are SI metres and metres per second so its output is consistently square metres per second.

Interpreting and checking the result

For motion governed by an ideal central force, specific angular momentum remains constant along the orbit. That makes the result useful for checking whether two state descriptions could belong to the same unperturbed trajectory, or for carrying an instantaneous state into later orbital calculations. A quick reasonableness check is to compare the answer with r multiplied by v.

The magnitude cannot exceed that product when the supplied speed and radius are non-negative, because the absolute value of cosine cannot exceed one. A zero result can mean zero radius, zero speed, or a purely radial path. It does not by itself mean that the body is stationary.

Real trajectories can experience thrust, atmospheric drag, nonspherical gravity, and third-body perturbations, so values derived from measurements at different times may not match exactly. Measurement uncertainty also matters: uncertainty in radius and speed scales the answer directly, while angle uncertainty has a stronger relative effect near radial flight. This calculator performs the geometric computation only.

It does not infer an orbit, calculate total angular momentum, apply a gravitational parameter, propagate a state, or decide whether a trajectory is bound. Multiply specific angular momentum by mass separately if total angular momentum is needed and the mass is known.