Orbital Energy Calculator
Enter the primary mass and orbit radius to compute specific orbital energy, velocity, period and escape velocity.
Input Data
Results
At a glance:For a test mass in a circular orbit of radius r about a primary of mass M, the gravitational parameter μ = GM governs the motion. The orbital (circular) speed is v = √(μ/r), the specific orbital energy (energy per unit mass) is ε = −μ/(2r) = v²/2 − μ/r (negative, bound orbit), the period is T = 2π√(r³/μ) (Kepler's third law), and the escape velocity from that radius is v_esc = √(2μ/r) = √2·v. The total energy E = ε·m; a bound orbit has ε < 0, a parabolic escape trajectory ε = 0, hyperbolic ε > 0. This tool returns v, T, ε and v_esc from M and r.
Formula
μ = GM
v = √(μ/r)
ε = −μ/(2r)
T = 2π√(r³/μ)
v_esc = √(2μ/r) = √2·v
$$v = \sqrt{\frac{GM}{r}}, \quad T = 2\pi\sqrt{\frac{r^3}{GM}}, \quad \varepsilon = -\frac{GM}{2r}, \quad v_{esc} = \sqrt{\frac{2GM}{r}} = \sqrt{2} v$$How to Use
- Enter the primary mass M (kg).
- Enter the orbit radius r (m).
- The calculator returns v, T, ε and v_esc.
Case Studies
Low Earth orbit
M = 5.97e24 kg, r = 6.77e6 m (400 km altitude).
v ≈ 7670 m/s, T ≈ 5550 s (92.5 min).
v_esc ≈ 10850 m/s (√2·v).
FAQ
Why is orbital energy negative?
The zero of energy is at infinite separation; a bound orbit has less energy than a free particle at rest at infinity, so ε < 0. More negative = more tightly bound.
How is period related to radius?
T = 2π√(r³/μ): period grows with r^(3/2) — farther orbits are slower and take longer (Kepler's third law).
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.