Formula & Calculator
Specific Orbital Energy
Total mechanical energy per unit mass of an orbiting body, constant along a given orbit.
Interpretation
Specific orbital energy: ε = v²/2 − μ/r = −μ/(2a), where μ is gravitational parameter, r radial distance, a semi‑major axis. It is the total energy per unit mass. Example: For a circular orbit at r=7000 km, v≈7.55 km/s, ε = (7.55e3)²/2 − 3.986e14/7e6 ≈ 2.85e7 − 5.69e7 = −2.84e7 J/kg.
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
Frequently Asked Questions
It is the total mechanical energy per unit mass of an orbiting body, constant along a given orbit. It is used to classify orbits and compute orbital parameters.
ε = specific orbital energy (J/kg)
v = orbital speed (m/s)
μ = gravitational parameter (m³/s²)
r = orbital radius (m)
a = semi‑major axis (m)
It determines the type of orbit: ε < 0 for bound orbits (elliptical), ε = 0 for parabolic, ε > 0 for hyperbolic.
- Forgetting that specific orbital energy is negative for bound (elliptical/circular) orbits and only zero or positive for parabolic/hyperbolic trajectories.
- Using the wrong sign for μ.
- Confusing specific energy with total energy (which includes mass).
For a circular orbit at r = 7,000 km, v = 7.5 km/s, μ = 3.986e14. ε = 7500²/2 − 3.986e14/7e6 = 28,125,000 − 56,942,857 = −28,817,857 J/kg. Also, −μ/(2a) = −3.986e14/(2×7e6) = −28,471,429 J/kg (small difference due to rounding).
The vis‑viva equation is derived from the specific energy: v²/2 − μ/r = ε.
It corresponds to the escape trajectory (parabolic), where the object has just enough kinetic energy to escape the gravitational field.
For a circular orbit, ε = −μ/(2r); as r increases, ε becomes less negative (approaches zero).
At r = 42,164 km, ε = −3.986e14/(2×4.2164e7) ≈ −4.727e6 J/kg.
a = −μ/(2ε). For bound orbits, ε is negative, so a is positive.