Space Mechanics: two-body orbits, manoeuvres and orbital elements
Space Mechanics covers the two-body problem, orbital elements, impulsive manoeuvres and elementary orbital perturbations - the standard content for orbit determination and mission delta-V budgeting.
Core relations
- Vis-viva: v² = μ(2/r − 1/a). Circular speed v_c = √(μ/r); escape speed = √(2μ/r) = √2 v_c; escape delta-V from a circular orbit is (√2 − 1)v_c ≈ 0.414 v_c.
- Orbital period T = 2π√(a³/μ). For Earth μ = 398 600 km³/s²; use radius from the centre, not altitude.
- Kepler's equation M = E − e sin E links mean and eccentric anomaly; time of flight on an ellipse follows from the eccentric anomaly.
- Hohmann transfer between circular radii r1 and r2: Δv1 = √(μ/r1)(√(2r2/(r1+r2)) − 1), and a symmetric expression at r2; transfer time is half the period of the ellipse with a = (r1+r2)/2.
- A pure plane change Δv = 2v sin(Δi/2) is cheapest at high altitude and fast speed is the enemy.
- J2 perturbation causes nodal regression and apsidal rotation; sun-synchronous orbits exploit nodal regression.
Worked example
Circular speed at r = 7000 km: v = √(398600/7000) = √56.94 = 7.546 km/s. The period is 2π√(7000³/398600) ≈ 5828 s ≈ 97 min, a typical low-Earth-orbit value.
Traps
- Inserting altitude instead of radius (add 6378 km for Earth).
- Units: μ in km³/s² pairs with km and km/s; mixing metres and kilometres gives answers off by 10³.
- Assuming a Hohmann transfer is always optimal - for large radius ratios a bi-elliptic transfer with a very high apogee can need less delta-V.
- Forgetting that combined plane change and raising is cheaper than doing them sequentially.