Specific Orbital Energy Calculator
Solve the specific orbital energy relation ε = v²/2 − μ/r = −μ/(2a) for any of its five unknowns, with step-by-step math, a live orbit schematic, and companion orbital metrics including semi-major axis, eccentricity, velocity at periapsis and apoapsis, orbital period, and escape velocity.
ε is a constant of motion
For an unperturbed two-body orbit, ε does not change along the trajectory — it is the same at periapsis, apoapsis, and everywhere in between. This is why a single number completely determines the size (semi-major axis) of the orbit, regardless of where the spacecraft happens to be on that orbit.
Sign says everything
ε < 0 → elliptical (bound). ε = 0 → parabolic (marginal escape). ε > 0 → hyperbolic (unbound). The escape condition is exactly v = √(2μ/r), which makes the kinetic term cancel the potential term. All interplanetary trajectories have ε > 0.
ε scales with v², not v
Doubling the velocity at a fixed radius quadruples the kinetic term, but the potential term stays the same. A small velocity error at periapsis translates into a huge change in energy — which is why small launch errors can cause big orbit insertion errors. This also means raising the orbit costs more energy at low altitude (where v is high) than at high altitude.
Energy is what you pay for
Every orbital manoeuvre is fundamentally an energy change. A Hohmann transfer from LEO to GEO requires about 24 MJ/kg of energy change. Escape from Earth orbit requires about 30 MJ/kg. Interplanetary transfers cost tens to hundreds of MJ/kg. When mission designers talk about Δv budgets, they are really talking about ways to change ε efficiently.
How to Use This Calculator
The specific orbital energy relation links five quantities: the energy itself, the orbital velocity, the gravitational parameter, the radius, and the semi-major axis. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick ε, v, μ, r, or a — the tool rearranges the relation automatically for whichever you choose. Solving for a uses the inverse form a = −μ/(2ε).
2. Use realistic presets
Energy presets span −60 MJ/kg to +20 MJ/kg (all orbit classes). Velocity presets span 1,000 to 11,000 m/s. Gravitational parameter presets include Moon, Mars, Earth, Jupiter, and Sun.
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 orbit type classification (elliptical, parabolic, hyperbolic) and the implied semi-major axis.
4. Check the orbit schematic
The diagram shows the central body, the orbit ellipse scaled to the semi-major axis, the spacecraft position, and the velocity vector — all updating live with the current values.
5. Switch to Advanced for depth
Advanced mode adds a chart of ε vs. radius, a 1% sensitivity check, and companion metrics including escape velocity, circular velocity, periapsis and apoapsis velocities, orbital period, eccentricity, and specific angular momentum.
6. Cross-check with the reference table
The reference section lists ε for typical Earth orbits, planetary escape energies, and heliocentric transfer orbits, plus a lookup grid across the velocity–radius envelope.
What This Tool Offers
Built around the fundamental energy invariant of two-body orbital motion used throughout astrodynamics and mission design.
Five-way solving
Solve for specific energy, velocity, gravitational parameter, radius, or semi-major axis — whichever the problem gives you.
Live orbit schematic
A central body with a scaled orbit ellipse, the spacecraft position on it, and a velocity vector at the spacecraft, all updating live.
Dual comparison charts
ε vs. velocity at three radii, plus ε vs. radius at three velocities.
Companion orbital metrics
Advanced mode reports escape velocity, circular velocity, periapsis and apoapsis radii and velocities, orbital period, eccentricity, and specific angular momentum.
Built-in sensitivity check
See how much ε moves for a 1% change in velocity, radius, or μ. The v² and 1/r dependencies are shown explicitly.
Reference tables
ε values for typical Earth orbits (LEO, ISS, GPS, GEO, lunar), escape conditions for each planet, and a lookup grid across the velocity–radius envelope.
Reference Tables
Specific orbital energy across typical missions, plus a lookup grid across the velocity–radius envelope.
Specific energy quick lookup (MJ/kg, Earth μ = 3.986×10¹⁴ m³/s²)
ε = v²/2 − μ/r for various velocities and orbital radii around Earth.
| 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 unless noted. ε is per unit mass and does not depend on spacecraft mass.
| Mission | Radius r (km) | Velocity v (m/s) | ε (MJ/kg) | Orbit type |
|---|---|---|---|---|
| LEO 200 km circular | 6,571 | 7,788 | −30.32 | Circular |
| ISS 400 km circular | 6,778 | 7,672 | −29.41 | Circular |
| Sun-synchronous 700 km | 7,071 | 7,505 | −28.18 | Quasi-circular |
| GPS constellation | 26,560 | 3,874 | −7.50 | Quasi-circular |
| GEO | 42,164 | 3,074 | −4.73 | Circular |
| Earth escape (from LEO) | 6,778 | 11,180 | 0 | Parabolic |
Gravitational Parameters & Escape Velocities
| Body | μ (m³/s²) | Radius (km) | vesc at surface (m/s) | εesc |
|---|---|---|---|---|
| Moon | 4.905×10¹² | 1,737 | 2,376 | 0 |
| Mars | 4.283×10¹³ | 3,390 | 5,027 | 0 |
| Earth | 3.986×10¹⁴ | 6,371 | 11,186 | 0 |
| Jupiter | 1.267×10¹⁷ | 71,492 | 59,540 | 0 |
| Sun | 1.327×10²⁰ | 695,700 | 617,600 | 0 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| ε | Specific orbital energy | −60 to +100 | MJ/kg |
| v | Orbital velocity | 1,000 – 15,000 | m/s |
| μ | Gravitational parameter | 4.9×10¹² – 1.3×10²⁰ | m³/s² |
| r | Radius from centre | 6.4×10⁶ – 3.8×10⁸ | m |
| a | Semi-major axis | 6.4×10⁶ – 1×10⁹ | m |
What Does the Specific Orbital Energy Measure?
Specific orbital energy is the total mechanical energy (kinetic plus gravitational potential) of an orbiting body, divided by its mass. It is a scalar invariant: for any two-body orbit it has the same value at every point along the trajectory, and it depends only on the size of the orbit, not on the shape. The relation ε = v²/2 − μ/r = −μ/(2a) is the fundamental link between the instantaneous state (v, r) and the geometric orbit (a). When engineers talk about "how much energy it takes to get to orbit" or "the energy cost of a manoeuvre", they are really talking about changes in this quantity.
Specific Orbital Energy Formula
The defining relationship is:
Rearranged to solve for velocity, radius, or semi-major axis:
The escape condition is ε = 0, giving:
Specific Orbital Energy Variables
| Symbol | Meaning | Notes |
|---|---|---|
| ε | Specific orbital energy | Constant along the orbit; sign determines orbit class |
| v | Orbital velocity at the current point | Highest at periapsis, lowest at apoapsis |
| μ | Gravitational parameter of the central body | μ = G·M; Earth 3.986×10¹⁴ m³/s² |
| r | Current distance from the centre | Half the distance between periapsis and apoapsis at the orbit's focus |
| a | Semi-major axis | Positive for ellipse, infinite for parabola, negative for hyperbola |
How Each Parameter Moves the Specific Energy
The kinetic term v²/2 dominates in the low-orbit regime where v is largest; the potential term μ/r dominates at large r. Because v² grows quadratically with velocity, a 1% increase in v changes the kinetic term by 2%, whereas a 1% increase in r changes the potential term by only 1%. Over the range of typical orbital speeds (v = 1,000 to 15,000 m/s at Earth), the specific energy spans from −30 MJ/kg (LEO) to +80 MJ/kg (hyperbolic escape). Interplanetary transfers require specific energies of +100 to +1,000 MJ/kg relative to the Sun, which is why planetary missions need such large velocity increments.
Specific Orbital Energy Calculation Examples
v²/2 = 7,726² / 2 = 29,845,076 / 2 = 14,922,538 J/kg
μ/r = 3.986×10¹⁴ / 6,778,000 = 58,809,089 J/kg
ε = 14,922,538 − 58,809,089 = −43,886,551 J/kg ≈ −43.9 MJ/kg
This corresponds to a = −μ/(2ε) = −3.986×10¹⁴ / (−8.777×10⁷) = 4,541,000 m
Wait — let's check: for a circular orbit at r = 6,778 km, vcirc = √(μ/r) = 7,672 m/s. Since v = 7,726 m/s > vcirc, this is actually an elliptical orbit (periapsis at this point). Let's use vcirc to check ε = −μ/(2a): a = 6,778,000 m, ε = −3.986×10¹⁴ / (2 × 6,778,000) = −29,394,000 J/kg = −29.4 MJ/kg.
a = −μ / (2ε) = −3.986×10¹⁴ / (2 × −30,000,000)
= −3.986×10¹⁴ / (−6×10⁷)
= 6,643,333 m = 6,643 km
That is roughly a 270 km LEO orbit.
Frequently Asked Questions
Because the gravitational potential energy is defined to be zero at infinity and negative everywhere closer to the central body. A spacecraft bound in an orbit has less total energy than it would need to escape to infinity, so its total energy is negative. A positive energy would mean the spacecraft has more than enough energy to escape; a zero energy exactly matches the escape condition. The zero of potential energy is arbitrary, but choosing it at infinity makes the sign of ε physically meaningful: negative = bound, zero = marginally unbound, positive = unbound.
Because gravity is a conservative force. The work done by gravity around any closed path is zero, so the total mechanical energy is conserved. This is why you can compute ε at periapsis, at apoapsis, or at any intermediate point, and always get the same value. It is one of the most powerful simplifications in orbital mechanics: rather than tracking the velocity and position separately, you can track the single scalar ε, and everything else follows from the geometry of the orbit.
Specific energy is total energy divided by the mass of the spacecraft. Because orbital dynamics are mass-independent (in the two-body problem), it is more convenient to work with the specific quantity. Multiplying ε by the spacecraft mass gives the total orbital energy: E = m·ε. Two spacecraft with different masses on the same orbit have the same specific energy but different total energies. Any manoeuvre that changes the orbit changes ε regardless of mass, so ε is the natural "cost" variable in mission design.
Every rocket burn imparts a Δv at some point on the orbit, changing the velocity and therefore the specific energy. If the burn is prograde (in the direction of motion), the new velocity is higher and ε increases. If it is retrograde, ε decreases. The Δε = v·Δv + Δv²/2 for the burn, evaluated at the burn point. This is why the same Δv produces a larger change in ε when applied at periapsis (where v is high) than at apoapsis (where v is low) — the well-known Oberth effect.
For hyperbolic orbits, the semi-major axis is defined to be negative by convention, so the formula still gives a positive specific energy. Geometrically, a hyperbola is like an ellipse with a very large eccentricity — its semi-major axis is still the distance from the centre to the vertex (in the hyperbolic branch), but the focus lies on the other side of the vertex and the sign convention makes a negative. The relationship ε = −μ/(2a) therefore holds for all conic sections with the appropriate sign convention. For parabolic orbits, a is infinite and ε = 0 exactly.
The Oberth effect is the observation that a rocket burn produces a larger change in specific energy when the rocket is moving fast (near periapsis) than when it is moving slowly (near apoapsis). It follows directly from the relation Δε = v·Δv + Δv²/2: for the same Δv, the term v·Δv is larger when v is larger. This is why interplanetary missions always perform their escape burns as close to the Earth as possible, and why deep-space manoeuvres are less efficient than they look on paper.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| ε | Specific orbital energy | J/kg |
| v | Orbital speed | m/s |
| μ | Gravitational parameter | m3/s2 |
| r | Radial distance | m |
| a | Semi-major axis | m |
What it means
Specific orbital energy is the sum of kinetic and potential energy per unit mass. For a bound orbit, it is negative. It is a conserved quantity in the two‑body problem. The vis‑viva equation is derived from this. This energy determines the orbital parameters and is used in trajectory design and in calculating Δv requirements. Understanding specific energy is essential for orbital mechanics and for mission analysis.
Worked example
Specific Orbital Energy – Two Examples
Real‑World| Parameter | Value |
|---|---|
| μ | 3.986×10¹⁴ |
| a | 6.678×10⁶ |
| Parameter | Value |
|---|---|
| a | 4.216×10⁷ |
Common mistakes
- Specific orbital energy: ε = v²/2 − μ/r = −μ/(2a).
- v: Speed (m/s).
- μ: Gravitational parameter.
- r: Radial distance (m).
- a: Semi‑major axis (m).
- Units: J/kg (m²/s²).
- Negative for bound orbits, zero for parabolic, positive for hyperbolic.
Applications
Specific orbital energy, ε = v²/2 − μ/r = −μ/(2a), is the total energy per unit mass of an orbiting body. It determines the size of the orbit (semi‑major axis). Engineers use this to calculate the energy required for orbital transfers and to assess the orbit type (elliptical, parabolic, hyperbolic). By understanding specific energy, aerospace engineers can plan interplanetary missions, compute launch energy, and design orbit insertion manoeuvres. It is a fundamental parameter in astrodynamics.
- Orbit determination and tracking
- Orbital transfer design (Hohmann, bi‑elliptic)
- Launch vehicle energy requirements
- Interplanetary trajectory design (energy balance)
- Computation of orbital elements from state vectors
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