Specific Orbital Angular Momentum Calculator
Solve the specific orbital angular momentum relation h = r · v · cos(γ) for any of its four unknowns, with step-by-step math, a live orbit schematic, and companion orbital metrics including semi-latus rectum, eccentricity, semi-major axis, and Δv for plane changes.
A constant of motion
Like energy, h does not change along an unperturbed two-body orbit. You can compute it at periapsis, apoapsis, or anywhere in between and get the same value. This makes it one of the most useful invariants in orbital mechanics, alongside specific energy.
Determines the semi-latus rectum
h² = μ · p, where p is the semi-latus rectum — the distance from the focus to the orbit at the true anomaly of 90°. p defines the "width" of the orbit at right angles to the apse line and is the simplest way to connect h to the geometric shape of the orbit.
γ is the flight path angle
The flight path angle measures how much of the velocity is "horizontal" (transverse) versus "vertical" (radial). At γ = 0°, the velocity is fully tangential and h is maximum for a given r·v. At γ = ±90°, all velocity is radial and h = 0 — a purely radial trajectory passes through the central body.
Drives plane-change Δv
The Δv required to change the orbital plane by an angle Δi is Δv = 2·v·sin(Δi/2). Because h is what defines the plane, mission designers use h to compute plane-change costs. This is why plane changes are cheapest at high altitude (low v) and expensive at low altitude.
How to Use This Calculator
The specific angular momentum relation links four quantities: the momentum itself, the radius, the velocity, and the flight path angle. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick h, r, v, or γ — the tool rearranges the relation automatically for whichever you choose. Solving for γ uses the inverse cosine with a domain check on h/(r·v).
2. Use realistic presets
h presets span 10¹⁰ to 5.6×10¹¹ m²/s. Radius presets cover LEO/ISS/GEO/lunar. Velocity presets span 1,000 to 11,000 m/s. γ presets cover −30° to +60°, including the tangential case γ = 0°.
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the arithmetic plus the transverse velocity component v·cos(γ) and the classification of the orbit geometry.
4. Check the orbit schematic
The diagram shows the central body, the orbit path, the radius vector, the velocity vector rotated by γ from the local horizontal, and the out-of-plane h vector — all updating live.
5. Switch to Advanced for depth
Advanced mode adds a chart of h vs. radius, a 1% sensitivity check, and companion metrics including semi-latus rectum, eccentricity, semi-major axis, periapsis and apoapsis radii, plane-change Δv, and escape velocity.
6. Cross-check with the reference table
The reference section lists h for typical Earth orbits, planetary escape trajectories, and heliocentric transfers, plus a lookup grid across the r·v envelope.
What This Tool Offers
Built around the fundamental angular momentum invariant of two-body orbital motion, used throughout astrodynamics and mission design.
Four-way solving
Solve for angular momentum, radius, velocity, or flight path angle — whichever the problem gives you.
Live orbit schematic
A central body with the orbit path, radius vector, velocity vector rotated by γ, and the out-of-plane h vector, all updating live.
Dual comparison charts
h vs. flight path angle at three r·v values, plus h vs. radius at three velocity values.
Companion orbital metrics
Advanced mode reports transverse velocity, radial velocity, semi-latus rectum, eccentricity, semi-major axis, periapsis/apoapsis radii, orbital period, and plane-change Δv.
Built-in sensitivity check
See how much h moves for a 1% change in r, v, or γ. The cos(γ) dependence means h is least sensitive to γ at γ = 0°.
Reference tables
h values for LEO, ISS, sun-synchronous, GPS, GEO, and lunar orbits, plus escape conditions and a lookup grid across the r·v envelope.
Reference Tables
Specific angular momentum across typical missions, plus a lookup grid across the r·v envelope.
Specific angular momentum quick lookup (10⁹ m²/s, γ = 0°)
h = r · v · cos(γ) for various orbital radii and velocities (tangential case, γ = 0°).
| v (m/s) | r = 6,600 km | r = 7,000 km | r = 12,000 km | r = 26,560 km | r = 42,164 km |
|---|
Typical Values by Mission Class
Earth-centred orbits at γ = 0° (circular or tangential). Values are per unit mass.
| Mission | Radius r (km) | Velocity v (m/s) | h (10⁹ m²/s) | v_transverse |
|---|---|---|---|---|
| LEO 200 km circular | 6,571 | 7,788 | 51.2 | 7,788 m/s |
| ISS 400 km circular | 6,778 | 7,672 | 52.0 | 7,672 m/s |
| Sun-synchronous 700 km | 7,071 | 7,505 | 53.1 | 7,505 m/s |
| GPS constellation | 26,560 | 3,874 | 102.9 | 3,874 m/s |
| GEO | 42,164 | 3,074 | 129.6 | 3,074 m/s |
| Moon's orbit | 384,400 | 1,022 | 392.8 | 1,022 m/s |
Gravitational Parameters & Related Quantities
| Body | μ (m³/s²) | hLEO (10⁹ m²/s) | vesc at rLEO | Plane-change Δv at 45° |
|---|---|---|---|---|
| Moon | 4.905×10¹² | — | — | — |
| Mars | 4.283×10¹³ | — | — | — |
| Earth | 3.986×10¹⁴ | 52.0 | 10,850 m/s | 5,869 m/s |
| Jupiter | 1.267×10¹⁷ | — | — | — |
| Sun | 1.327×10²⁰ | — | — | — |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| h | Specific orbital angular momentum | 10⁹ – 10¹² | m²/s |
| r | Radius from central body | 6.4×10⁶ – 4×10⁸ | m |
| v | Orbital velocity (magnitude) | 1,000 – 15,000 | m/s |
| γ | Flight path angle from local horizontal | −90° – +90° | degrees |
| p | Semi-latus rectum = h²/μ | 6.4×10⁶ – 1×10⁹ | m |
What Does the Specific Orbital Angular Momentum Measure?
Specific orbital angular momentum is the angular momentum of an orbiting body divided by its mass. It is a vector quantity, perpendicular to the orbital plane, and its magnitude is what determines the "shape" and orientation of the orbit. For any two-body motion it is a constant — it does not change from point to point along the trajectory — which makes it one of the most powerful invariants in orbital mechanics. Combined with specific orbital energy, h fully determines the size and shape of the orbit (a and e), and the direction of h determines the plane.
Specific Orbital Angular Momentum Formula
The defining relationship is:
In vector form this is h = r × v, and its magnitude is r·v·cos(γ) where γ is the flight path angle. Equivalently, if θ is the angle between the position and velocity vectors, h = r·v·sin(θ) — the two forms are related by γ + θ = 90°. Two very useful consequences:
Rearranged to solve for r, v, or γ:
Angular Momentum Variables
| Symbol | Meaning | Notes |
|---|---|---|
| h | Specific orbital angular momentum | Constant along the orbit; magnitude determines p |
| r | Radius from central body | Distance from focus to current position |
| v | Orbital speed (magnitude of velocity) | Largest at periapsis, smallest at apoapsis |
| γ | Flight path angle | 0° for circular/tangential, ±90° for radial |
| p | Semi-latus rectum | p = h²/μ; equals r at true anomaly = 90° |
How Each Parameter Moves the Angular Momentum
The relation is linear in r and v and depends on γ through cos(γ). A 1% increase in radius or velocity raises h by 1%. The dependence on γ is strongest near γ = ±90°, where a small change in γ produces a large fractional change in h — the sensitivity dh/dγ = −r·v·sin(γ) peaks at ±90°. At γ = 0°, h is at its maximum for a given r·v, and the derivative dh/dγ = 0, so h is insensitive to small γ errors. This is why near-circular orbits (γ ≈ 0) have well-defined h even when the flight path angle fluctuates slightly, while radial trajectories are extremely sensitive to any transverse component. Mission designers exploit this by performing orbit insertion burns tangentially — it is the most "efficient" way to build up h.
Specific Orbital Angular Momentum Calculation Examples
cos(0°) = 1
h = 6,778,000 × 7,672 × 1 = 5.200 × 10¹⁰ m²/s
Semi-latus rectum: p = h²/μ = (5.200×10¹⁰)² / 3.986×10¹⁴
= 2.704×10²¹ / 3.986×10¹⁴ = 6,778,000 m (equal to r, as expected for a circle)
v = h / (r · cos(γ)) = 5.2×10¹⁰ / (6,778,000 × 1)
= 7,672 m/s
r · v = 6,778,000 × 8,200 = 5.558×10¹⁰ m²/s
cos(γ) = h / (r · v) = 5.2×10¹⁰ / 5.558×10¹⁰ = 0.9356
γ = arccos(0.9356) = ±20.66°
Sign depends on whether the spacecraft is ascending (γ > 0) or descending (γ < 0).
Frequently Asked Questions
Because gravity is a central force — it always points from the spacecraft directly toward the central body, along the radius vector. A force that is always parallel to the position vector produces no torque about the central body, and with zero torque, angular momentum is conserved. This is the same reason a spinning ice skater speeds up when they pull their arms in: with no external torque, h = r·v_transverse is invariant, so as r decreases, v_transverse must increase.
For a satellite, h represents the "swirl" of its motion around the central body, per unit mass. Its direction is perpendicular to the orbital plane (this is what defines the plane), and its magnitude is a measure of how fast the spacecraft sweeps out area. Kepler's second law — equal areas in equal times — is a direct consequence: dA/dt = h/2. So h is literally twice the rate at which the radius vector sweeps out area. This geometric interpretation makes h one of the most intuitive of all orbital parameters.
The flight path angle γ is the angle between the velocity vector and the local horizontal, measured at the spacecraft's current position. The true anomaly ν is the angle between the radius vector and the periapsis direction, measured from the central body. They are related by tan(γ) = e·sin(ν) / (1 + e·cos(ν)), where e is the eccentricity. For a circular orbit (e = 0), γ is always zero. For an eccentric orbit, γ is zero at periapsis and apoapsis (the only points where v is perpendicular to r) and reaches maxima at intermediate points.
Three big uses: (1) Orbit determination — given two position and velocity vectors, h = r × v immediately gives you the orbital plane, and its magnitude gives you the semi-latus rectum. (2) Plane changes — the Δv to change inclination by Δi is 2·v·sin(Δi/2), which is derived directly from the change in the direction of h. Because plane changes are cheap only when v is small, mission designers schedule them at high altitude. (3) Lambert's problem — solving for the orbit that connects two position vectors in a given time uses h as an intermediate quantity. Nearly every orbit-determination and targeting algorithm uses h as a primary variable.
Because a purely radial trajectory has no transverse component of velocity, and the radius vector and velocity vector are parallel (or anti-parallel). The cross product of two parallel vectors is zero, so h = |r × v| = 0. Physically, a spacecraft on a radial trajectory is moving directly toward or away from the central body, sweeping out no area, so dA/dt = 0. Such trajectories are unstable in practice — any small tangential perturbation causes h to become nonzero, and the orbit bends into a conic section. Radial trajectories are also physically meaningful only for a limited time, because they terminate at the central body.
The angular momentum determines the semi-latus rectum p = h²/μ, which is the distance from the focus to the orbit at true anomaly 90°. The eccentricity then follows from the specific orbital energy ε and h: e = √(1 + 2ε·h²/μ²). Together, ε and h completely determine the size and shape of the orbit. The direction of h (a unit vector) determines the orbital plane. Together with the argument of periapsis and the true anomaly, these form the complete set of six orbital elements — but ε and h are two of the most fundamental because they come directly from the conservation laws.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| h | Specific angular momentum | m2/s |
| r | Radial distance | m |
| v | Orbital speed | m/s |
| γ | Flight path angle | deg |
What it means
Specific angular momentum is a conserved quantity in central force motion, equal to the cross product of position and velocity. It defines the orbital plane and the shape of the orbit (semi‑latus rectum p = h²/μ). It is used in orbit determination and in the calculation of orbital elements. Understanding h is essential for orbital mechanics and for analysing perturbed orbits.
Worked example
Specific Orbital Angular Momentum – Two Examples
Real‑World| Parameter | Value |
|---|---|
| r | 6.678×10⁶ |
| v | 7726 |
| γ | 0° |
| Parameter | Value |
|---|---|
| r | 4.216×10⁷ |
| v | 3075 |
Common mistakes
- Specific orbital angular momentum: h = r·v·cos(γ).
- γ: Flight path angle (angle between velocity and local horizontal).
- For circular orbit, γ=0, so h = r·v.
- Units: m²/s.
- Constant for a given orbit.
Applications
Specific orbital angular momentum, h = r·v·cosγ, is the angular momentum per unit mass, a conserved quantity in central force motion. It determines the shape and orientation of the orbit. Engineers use h to compute the semi‑latus rectum and eccentricity, and to design orbital manoeuvres. It is essential for determining the orbit's geometry and for targeting. By understanding angular momentum, aerospace engineers can predict the path of spacecraft and plan orbit changes with precision.
- Orbit determination and parameter estimation
- Trajectory design and targeting
- Computation of orbital elements from position and velocity
- Orbital rendezvous and interplanetary navigation
- Analysis of perturbations and orbital decay
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