How to use it
- Choose the central body, or enter a custom mass.
- Enter the distance r from the centre of the body (not the altitude: add the body’s radius to an altitude).
- Enter the semi-major axis a. For a circular orbit a = r; for an ellipse a is half the sum of the nearest and farthest distances.
- Read the speed, then the circular and escape speeds at that distance, the period and the energy.
Formula
v = √(μ × (2/r − 1/a))
- circular speed at r: √(μ/r); escape speed: √(2μ/r);
- period T = 2π√(a³/μ) for a > 0;
- specific energy ε = −μ ÷ (2a).
μ values: Earth 3.986004418 × 10¹⁴ m³/s², the Sun 1.32712440018 × 10²⁰ m³/s²; a custom body uses G = 6.6743 × 10⁻¹¹ m³/(kg·s²).
Worked example
A circular orbit around Earth with r = a = 6778 km (about 407 km up):
- v = √(3.986004418 × 10¹⁴ ÷ 6,778,000) = 7.6686 km/s, which is 7,668.6 m/s or 27,607 km/h;
- escape speed there 10.8451 km/s;
- period 2π√(6,778,000³ ÷ μ) = 92.56 min;
- energy −μ ÷ (2 × 6,778,000) = −29.404 MJ/kg.
At the perigee of a transfer orbit, r = 6671 km and a = 24,414 km, the speed is √(μ × (2/6,671,000 − 1/24,414,000)) = 10.1575 km/s.
Tips
- Distances are from the centre, so for Earth add 6371 km to the altitude.
- a from the two ends of an ellipse: a = (r_near + r_far) ÷ 2, both measured from the centre.
- The equation assumes a small body around a much heavier one, with no drag or thrust.