How to use it
- Choose the body the orbits go around, or enter a custom mass (distances then count from its centre).
- Choose whether you enter altitudes above the surface (mean radius) or distances from the centre.
- Enter the starting and the target orbit in km or astronomical units.
- Read the total delta-v, each burn, the transfer time and the phase angle.
Formulas
With μ = GM of the central body and orbit radii r₁, r₂:
- circular speeds v₁ = √(μ/r₁), v₂ = √(μ/r₂);
- transfer ellipse a = (r₁ + r₂) ÷ 2;
- Δv₁ = |√(μ(2/r₁ − 1/a)) − v₁|, Δv₂ = |v₂ − √(μ(2/r₂ − 1/a))|;
- transfer time t = π√(a³/μ), half the period of the ellipse;
- phase angle φ = 180° × (1 − (a/r₂)^1.5).
Earth uses μ = 3.986004418 × 10¹⁴ m³/s² and a mean radius of 6371 km; the Sun 1.32712440018 × 10²⁰ m³/s².
Worked example
From a 300 km orbit to geostationary altitude, 35,786 km above Earth (r₁ = 6671 km, r₂ = 42,157 km):
- v₁ = 7.73 km/s, speed at the low end of the ellipse 10.158 km/s, so Δv₁ = 2.428 km/s;
- v₂ = 3.075 km/s, speed at the high end 1.607 km/s, so Δv₂ = 1.468 km/s;
- total 3.895 km/s; the coast takes π√(24,414³ km³ ÷ μ) = 5.27 h;
- the phase angle is 100.67°.
Around the Sun from 1 AU to 1.524 AU the transfer needs 2.946 + 2.65 = 5.596 km/s and 258.92 days.
Tips
- Going down costs the same as going up between the same two orbits; the burns are just swapped and point backwards.
- Very distant targets can be cheaper with three burns (a bi-elliptic transfer) when the radius ratio is above about 11.9.
- The figures are ideal: instant burns, circular coplanar orbits, no atmosphere and no third body.