Orbital Manoeuvres, Transfers & Δv Budgets

Moving between orbits efficiently. Learn impulsive burns, the Hohmann transfer, bi-elliptic transfers, plane changes and why they are expensive, combined manoeuvres, rendezvous and phasing, interplanetary transfers and gravity assists, low-thrust spirals, and building a mission Δv budget — with Python for a LEO-to-GEO transfer.

In space, 'distance' is measured in Δv — the velocity changes a spacecraft must make. Every burn costs propellant through the rocket equation, so mission designers look for the cheapest sequence of burns.

The Hohmann transfer (1925) is the classic two-burn transfer between circular orbits; plane changes are notoriously costly, which is why launch sites near the equator are valuable for geostationary missions.

An analogy

Changing orbit is like changing lanes on a circular racetrack by speeding up: speed up and you swing wide onto an outer ellipse, then speed up again at the far side to settle into the outer lane.

Burns change orbits

A prograde burn (along velocity) raises the opposite side of the orbit; a retrograde burn lowers it. Radial and normal burns rotate the orbit's shape or plane. Because burns are short compared with the orbit period, they are modelled as instantaneous (impulsive) velocity changes.

Hohmann and bi-elliptic transfers

The Hohmann transfer uses a tangential burn at the inner orbit to enter an ellipse touching the outer orbit, and a second tangential burn to circularise. It is the most efficient two-burn transfer for radius ratios below ~11.9; beyond ~15.6 a three-burn bi-elliptic transfer (going beyond the target, then coming back) needs less Δv, at the cost of time.

Rendezvous and phasing

To catch a target in the same orbit you do not simply speed up (that raises your orbit and slows you down). Instead you drop into a lower, faster phasing orbit, drift ahead, then raise back. Final approach uses Clohessy–Wiltshire relative-motion equations, as crew and cargo vehicles do at the ISS.

Interplanetary and low-thrust

Interplanetary missions use patched conics: escape Earth on a hyperbola with excess velocity v∞, coast on a heliocentric transfer (Hohmann-like, with launch windows e.g. every 26 months for Mars), and capture at the target. Gravity assists steal momentum from planets to change direction and speed for free (Voyager, Cassini). Electric propulsion spirals slowly, using more total Δv but far less propellant thanks to high I_sp.

Formulas

Worked example: Hohmann to GEO

Given: r₁ = 6678 km, r₂ = 42 164 km

Find: Total Δv.

Answer: 3.89 km/s, 5.28 h

Worked example: Plane change in LEO

Given: v = 7.73 km/s, Δi = 10°

Find: Δv.

Answer: 1.35 km/s — very expensive

Worked example: Propellant for Δv

Given: Satellite 3000 kg, Δv 1.83 km/s, I_sp 320 s

Find: Propellant.

Answer: ≈ 1326 kg (44% of mass)

LEO to GEO: Hohmann transfer and plane change

import math
mu, Re = 3.986004418e14, 6378.137e3
r1, r2 = Re + 300e3, 42164e3
a = (r1 + r2) / 2
v1 = math.sqrt(mu / r1); v2 = math.sqrt(mu / r2)
vp = math.sqrt(mu * (2 / r1 - 1 / a)); va = math.sqrt(mu * (2 / r2 - 1 / a))
dv1, dv2 = vp - v1, v2 - va
tof = math.pi * math.sqrt(a**3 / mu)
print(f"LEO->GEO Hohmann: dv1={dv1:.0f} m/s, dv2={dv2:.0f} m/s, total={dv1+dv2:.0f} m/s, time {tof/3600:.2f} h")
# Combined apogee burn with 28.5 deg plane change
di = math.radians(28.5)
dv2c = math.sqrt(va**2 + v2**2 - 2 * va * v2 * math.cos(di))
print(f"apogee burn incl. 28.5 deg plane change: {dv2c:.0f} m/s (total {dv1+dv2c:.0f} m/s)")
# Pure plane change in LEO
print(f"28.5 deg plane change in LEO alone: {2*v1*math.sin(di/2):.0f} m/s")
# LEO->GEO Hohmann: dv1=2426 m/s, dv2=1467 m/s, total=3893 m/s, time 5.28 h
# apogee burn incl. 28.5 deg plane change: 1830 m/s (total 4256 m/s)
# 28.5 deg plane change in LEO alone: 3803 m/s

Removing Cape Canaveral's 28.5° latitude inclination costs only 363 m/s extra when combined with the apogee burn (where the spacecraft is slow), versus 3.8 km/s if done in low orbit. Always change planes where velocity is lowest.

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