Orbital Mechanics · Space Physics
📅 Березень 2026⏱ 20 хв читання🚀 Aerospace

Orbital Transfers — Hohmann, Bi-Elliptic & Gravity Assist

How does a spacecraft go from a low Earth orbit to Mars, Jupiter, or interstellar space while burning as little fuel as possible? The answer lies in Kepler's laws and the vis-viva equation — a single formula that determines the velocity needed at every point in any orbit.

1. The Vis-Viva Equation

Every orbital mechanics calculation reduces to one equation — the conservation of energy in a Keplerian orbit. Given a spacecraft at distance r from the central body in an ellipse with semi-major axis a:

v² = GM · (2/r − 1/a) v = orbital speed at distance r (m/s) GM = gravitational parameter (3.986 × 10¹⁴ m³/s² for Earth) r = current distance from centre (m) a = semi-major axis of the orbit (m) Special cases: Circular orbit (r = a): v_c = √(GM/r) Escape velocity: v_e = √(2GM/r) = v_c · √2

The delta-V (Δv) for any manoeuvre is simply the difference in orbital velocity at the burn point. A rocket equation then converts Δv to propellant mass.

2. Hohmann Transfer

The Hohmann transfer is the minimum-energy two-burn transfer between two circular, coplanar orbits. It uses a single elliptical transfer orbit connecting the two circular orbits at its periapsis and apoapsis.

Transfer orbit semi-major axis: a_t = (r₁ + r₂) / 2 Burn 1 (periapsis, raise apoapsis): v_t1 = √(GM · (2/r₁ − 1/a_t)) v_c1 = √(GM/r₁) Δv₁ = v_t1 − v_c1 Burn 2 (apoapsis, circularise): v_t2 = √(GM · (2/r₂ − 1/a_t)) v_c2 = √(GM/r₂) Δv₂ = v_c2 − v_t2 Total: Δv_total = |Δv₁| + |Δv₂| Transfer time (half-period of ellipse): t = π · √(a_t³ / GM)

Earth to Mars (approximate): r₁ = 1.0 AU, r₂ = 1.524 AU. Total Δv ≈ 5.6 km/s (from Earth surface to trans-Mars injection, including escaping Earth gravity). Transfer time ≈ 259 days (8.5 months) — exactly how long Mars missions typically take.

Why is the Hohmann minimum-energy? Adding velocity at periapsis raises apoapsis most efficiently; removing at apoapsis circularises most efficiently. Any other 2-burn strategy does more work because it applies thrust at less optimal points.

3. Bi-Elliptic Transfer

Surprisingly, for very large orbit ratio changes (r₂/r₁ > 11.94), a three-burn bi-elliptic transfer uses less total Δv than Hohmann, despite covering a longer path. The route goes: r₁ → very high apoapsis r_b → r₂.

Burns: Δv₁ = v at r₁ in transfer orbit 1 − v_c1 (at r₁, raise to r_b) Δv₂ = v at r_b in transfer orbit 2 − v at r_b in orbit 1 (at r_b, target r₂) Δv₃ = v_c2 − v at r₂ in transfer orbit 2 (circularise) Transfer time: t = (T₁ + T₂) / 2 T₁ = π√((r₁ + r_b)³ / (8GM)), T₂ = π√((r₂ + r_b)³ / (8GM))

The catch: travel time is much longer (years for large r_b). The bi-elliptic is therefore only used when fuel is the primary constraint and time is not — for example, large satellite constellation slot changes or interplanetary probes given enough time.

Orbit ratio r₂/r₁ Winner Savings
< 11.94 Hohmann Hohmann is cheaper by a few %
= 11.94 Tie Identical with r_b → ∞
> 15.58 Bi-elliptic always wins Up to ~8% savings for r₂/r₁ = 1000

4. Gravity Assist (Swing-By)

A gravity assist uses a planet's gravity well and orbital velocity to deflect and accelerate a spacecraft — with zero propellant. In the planet's reference frame, the spacecraft enters and exits at the same speed (elastic scattering). In the Sun's reference frame, the spacecraft gains (or loses) kinetic energy equal to the work done by the planet's gravity across the flyby arc.

Maximum speed gain (direct flyby, aligned with planet velocity): Δv_max ≈ 2 · v_planet · sin(δ/2) v_planet = planet's orbital velocity around Sun δ = deflection angle = 2·arcsin(r_periapsis / (r_periapsis + b)) b = hyperbolic excess velocity parameter r_soi × (1 + 2GM_planet / (r_soi · v_inf²)) Simplified: for a spacecraft entering at v_inf (hyperbolic excess speed): v_exit = v_inf + 2·v_planet for a perfectly aligned prograde flyby (theoretical maximum)

Voyager 1 used a Jupiter gravity assist in 1979 to reach Saturn, then another at Saturn for escape. Voyager 2 used Jupiter, Saturn, Uranus, and Neptune (the Grand Tour). The energy came entirely from the planets, which were slowed by an imperceptibly tiny amount.

JUICE (ESA, 2023) uses a complex sequence: Earth–Moon flyby → Venus → Earth → Earth flyby again, accelerating to reach Jupiter in 2031. The Venus flyby is counter-intuitive (decelerating) but allows a more efficient trajectory geometry.

5. The Oberth Effect

A rocket burn at high velocity produces more useful kinetic energy than the same burn at low velocity. The spacecraft's kinetic energy is proportional to v² — so the same Δv added at high speed gains more absolute energy.

ΔKE = ½m(v + Δv)² − ½mv² = m·v·Δv + ½m·(Δv)² For Δv ≪ v: ΔKE ≈ m·v·Δv (proportional to current velocity!) Oberth benefit of periapsis burn vs apoapsis burn for same Δv: Ratio ≈ v_periapsis / v_apoapsis = a_result(1+e) / a_result(1-e)

This is why trans-lunar injection burns happen close to Earth (fastest point of parking orbit), and why a rocket heading to Jupiter fires its engine at closest approach to the Sun. Every km/s of Δv applied at periapsis does more orbit-raising work than the same Δv applied farther away.

Oberth + gravity assist combined: An Oberth manoeuvre inside a planet's gravity well (burning at closest approach during a gravity assist) combines both effects for maximum efficiency. Used in lunar flyby missions and proposed for Pluto orbiters.

6. Delta-V Budgets

Manoeuvre Δv (km/s) Notes
Surface to LEO (200 km) 9.4 Includes gravity drag and air drag (~1.5 km/s overhead)
LEO → GEO 4.2 Two Hohmann burns; 3× Falcon 9 flight time
LEO → Trans-Mars Injection 3.6 Departs at ~11.2 km/s total from Earth
Mars orbit insertion 0.9 Aerobraking saves most; rockets use ~1.5
LEO → Trans-Jupiter Injection 6.3 Usually requires gravity assists in practice
Earth escape (C3 = 0) 3.22 from LEO Total geocentric: 11.2 km/s from r=6578 km
Lunar orbit insertion 0.8–1.0 Depends on lunar altitude and approach angle

7. Real Mission Examples