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Aerospace Engineering · Orbital Mechanics · ⏱ ~13 min read

Hohmann Transfer Orbit — The Fuel-Optimal Two-Impulse Manoeuvre

Walter Hohmann proved in 1925 that the most fuel-efficient way to travel between two circular coplanar orbits is a single elliptical transfer arc, requiring just two brief engine firings. Despite its age — and the rise of electric propulsion and gravity assists — the Hohmann transfer remains the reference for comparing orbital manoeuvres and appears directly in satellite station-keeping, Geosynchronous Transfer Orbits (GTO), and lunar transit design.

1. Orbital Mechanics Foundations

Newton's gravity + inertia → conic section orbits (Kepler) Gravitational parameter: μ = GM (μ_Earth = 3.986×10¹⁴ m³/s²) Circular orbit velocity at radius r: v_c = √(μ/r) LEO (r=6571 km): v_c = 7.78 km/s GEO (r=42164 km): v_c = 3.07 km/s Moon orbit (r=384400 km): v_c = 1.02 km/s Orbital energy (specific): ε = −μ/(2a) e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola) Semi-major axis a: average of periapsis and apoapsis a = (r_p + r_a) / 2

2. Vis-Viva Equation

Conservation of energy: KE + PE = constant = ε v² / 2 − μ/r = −μ/(2a) → Vis-viva: v² = μ · (2/r − 1/a) Special cases: Circular orbit (r=a): v = √(μ/r) ✓ Apoapsis (r=r_a, v=v_a): v_a² = μ(2/r_a − 1/a) (slower) Periapsis (r=r_p, v=v_p): v_p² = μ(2/r_p − 1/a) (faster) Escape velocity from circular orbit: Adding Δv = v_e − v_c where v_e = √(2μ/r) = v_c·√2 Δv_esc = v_c(√2 − 1) ≈ 0.414 · v_c From LEO: Δv_esc ≈ 3.22 km/s on top of v_c = 7.78 km/s

3. Hohmann Transfer Derivation

Setup: transfer from circular orbit r₁ to circular orbit r₂ (r₂ > r₁) Transfer ellipse semi-major axis: a_t = (r₁ + r₂) / 2 Step 1 — Δv₁ at periapsis (departure from r₁): v_t,peri = √(μ(2/r₁ − 1/a_t)) (transfer orbit speed at r₁) v_c1 = √(μ/r₁) (circular speed at r₁) Δv₁ = v_t,peri − v_c1 (prograde: + adds speed) Step 2 — Δv₂ at apoapsis (arrival at r₂): v_t,apo = √(μ(2/r₂ − 1/a_t)) (transfer orbit speed at r₂) v_c2 = √(μ/r₂) (circular speed at r₂) Δv₂ = v_c2 − v_t,apo (prograde: circularise) Total mission Δv: Δv_total = |Δv₁| + |Δv₂| Example Earth LEO (400 km) → GEO: r₁ = 6778 km, r₂ = 42164 km Δv₁ = 2.43 km/s, Δv₂ = 1.47 km/s Δv_total ≈ 3.90 km/s (actual GTO missions also add ~1.5 km/s plane change to 0°i)

4. Transfer Time and Phase Angle

Kepler's third law: T² ∝ a³ → T = 2π√(a³/μ) Transfer time = half the period of the transfer ellipse: t_H = π√(a_t³/μ) = π√((r₁+r₂)³/8μ) LEO → GEO transfer time: a_t = (6778 + 42164)/2 = 24471 km t_H = π√((24.471×10⁶)³ / 3.986×10¹⁴) ≈ 19070 s ≈ 5.3 hours Phase angle θ required at departure: Target must be ahead by ωt_H − π radians: θ = π(1 − √((r₁+r₂)³/(8r₂³))) [radians] LEO → GEO: θ ≈ 105° before target satellite in GEO Launch windows recur with period T_syn = (1/T₁ − 1/T₂)⁻¹

5. Bi-Elliptic Transfer Comparison

Bi-elliptic: three burns — fly out to r_b >> r₂, then come back Burn 1: r₁ → transfer₁ perigee (Δv₁) Burn 2: at r_b → enter transfer₂ (Δv₂) Burn 3: circularise at r₂ (Δv₃) When is bi-elliptic better? Savings occur when r₂/r₁ > ~11.94 (Hohmann is optimal below this) For r₂/r₁ ≈ 15.58+: bi-elliptic always wins regardless of r_b For 11.94 < ratio < 15.58: depends on r_b Example r₁=LEO, r₂=r₁×20 (≫ GEO): Hohmann Δv ≈ 6.20 km/s Bi-elliptic (r_b=∞): Δv ≈ 5.77 km/s — saves 0.43 km/s Trade-off: flight time can be years if r_b is very large Practical use: bi-elliptic saves propellant for deep-space insertion at the cost of significantly longer transfer time.

6. Plane Changes and Combined Manoeuvres

Pure plane change (inclination change Δi at speed v): Δv_plane = 2v · sin(Δi/2) At LEO (v=7.78 km/s), Δi=28.5° (equatorial to polar-ish): Δv_plane = 2 × 7.78 × sin(14.25°) ≈ 3.83 km/s — very expensive! Combined manoeuvre (plane change + velocity change at apoapsis): Δv = √(v₁² + v₂² − 2v₁v₂cos(Δi)) Optimal: perform plane change at apoapsis where v is minimum That's why GTO launches use r_a near GEO — Δv_plane there costs ~50% less Interplanetary C3: C3 = v_∞² (hyperbolic excess energy, km²/s²) Δv from LEO needed = √(v_e² + C3) − v_LEO Earth→Mars (Hohmann): C3 ≈ 8.7 km²/s² → Δv≈3.6 km/s from LEO Earth→Jupiter: C3 ≈ 80 km²/s² → Δv≈6.3 km/s from LEO

7. JavaScript Hohmann Calculator

// Hohmann transfer orbit calculator
const MU_EARTH = 3.986004418e14; // m³/s²
const R_EARTH  = 6371000;         // m

function hohmann(r1, r2, mu = MU_EARTH) {
  const at = (r1 + r2) / 2;
  const vc1 = Math.sqrt(mu / r1);
  const vc2 = Math.sqrt(mu / r2);
  const vt_peri = Math.sqrt(mu * (2/r1 - 1/at));
  const vt_apo  = Math.sqrt(mu * (2/r2 - 1/at));
  const dv1 = vt_peri - vc1;
  const dv2 = vc2 - vt_apo;
  const tof = Math.PI * Math.sqrt(at**3 / mu); // seconds
  return { dv1, dv2, dvTotal: Math.abs(dv1) + Math.abs(dv2), tof, at };
}

function biElliptic(r1, r2, rb, mu = MU_EARTH) {
  const a1 = (r1 + rb) / 2;
  const a2 = (rb + r2) / 2;
  const vc1 = Math.sqrt(mu/r1);
  const vc2 = Math.sqrt(mu/r2);
  const v1p  = Math.sqrt(mu*(2/r1 - 1/a1));
  const v1a  = Math.sqrt(mu*(2/rb - 1/a1));
  const v2p  = Math.sqrt(mu*(2/rb - 1/a2));
  const v2a  = Math.sqrt(mu*(2/r2 - 1/a2));
  const dv1 = Math.abs(v1p - vc1);
  const dv2 = Math.abs(v2p - v1a);
  const dv3 = Math.abs(vc2 - v2a);
  return { dv1, dv2, dv3, dvTotal: dv1+dv2+dv3 };
}

// LEO (400 km alt) → GEO (35786 km alt)
const r1 = R_EARTH + 400e3;
const r2 = R_EARTH + 35786e3;
const h = hohmann(r1, r2);
console.log(`Δv₁: ${(h.dv1/1000).toFixed(3)} km/s`);  // ~2.43
console.log(`Δv₂: ${(h.dv2/1000).toFixed(3)} km/s`);  // ~1.47
console.log(`ToF: ${(h.tof/3600).toFixed(2)} hours`);  // ~5.26

8. Real Missions — GTO, Lunar, and Interplanetary

GTO → GEO

Commercial satellites launch to GTO (400×35786 km), then onboard apogee kick motor provides ~1.5 km/s to circularise at GEO. Takes ~3 manoeuvres over days.

Trans-Lunar Injection

Apollo TLI: ~3.2 km/s from LEO. Hohmann to Moon's orbit would take 5 days — actual crews took ~3 days using a slightly non-Hohmann fast trajectory.

Mars Mission

Hohmann Earth–Mars: Δv ≈ 5.6 km/s, 8.5 month flight. Launch windows occur every 26 months (Earth-Mars synodic period). Curiosity and Perseverance used this arc.

Gravity Assists

Voyager, Cassini, New Horizons — not Hohmann orbits. Gravity assists from Jupiter, Saturn provide essentially free Δv by stealing from planet orbital energy.

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