How to compute Hohmann transfer orbit: optimal orbital maneuver (minimum delta-v) to move from one circular orbit of radius r1 to another of radius r2 around the same gravitational body, via tangent transfer ellipse. First impulse formula: delta-v1 = sqrt(mu/r1) * (sqrt(2*r2/(r1+r2)) - 1). Second impulse at apogee: delta-v2 = sqrt(mu/r2) * (1 - sqrt(2*r1/(r1+r2))). Transfer time: T = pi * sqrt(a³/mu), where a=(r1+r2)/2 ellipse semi-major axis. Essential tool for astronautical engineering (NASA JPL Pasadena, SpaceX Hawthorne, Boeing Seattle), satellite missions (LEO->GEO transfer), interplanetary (Earth->Mars 8.5 months, Voyager 1-2 NASA), navigation (GPS USA 32 sats, Galileo EU 30 sats). Walter Hohmann (Germany 1880-1945) civil engineer Aachen published 1925 "Die Erreichbarkeit der Himmelskoerper". Bi-elliptic transfer more efficient for r2/r1 > 11.94 (Sternfeld 1934).

Hohmann transfer formulas

Initial and final circular velocities: v1 = sqrt(mu/r1), v2 = sqrt(mu/r2). mu = gravitational constant * mass (mu_Earth = 3.986e14 m³/s², mu_Sun = 1.327e20). Example LEO 400 km altitude (r1=6778 km): v1=7.67 km/s. GEO 35786 km altitude (r2=42164 km): v2=3.07 km/s.

Transfer ellipse: semi-major axis a = (r1+r2)/2. Velocity at perigee (near r1): v_peri = sqrt(mu*(2/r1 - 1/a)). Velocity at apogee (near r2): v_apo = sqrt(mu*(2/r2 - 1/a)). Example LEO->GEO: a=24471 km, v_peri=10.15 km/s (faster than v1), v_apo=1.61 km/s (slower than v2).

Delta-v first impulse: delta_v1 = v_peri - v1 (boost at perigee to widen orbit). Example LEO->GEO: 10.15 - 7.67 = 2.48 km/s. Delta-v second impulse: delta_v2 = v2 - v_apo (boost at apogee to circularize). 3.07 - 1.61 = 1.46 km/s. Delta_v total: 3.94 km/s (typical GTO->GEO from Falcon 9 SpaceX or Ariane 5).

Transfer time: T_Hohmann = pi * sqrt(a³/mu) = (1/2) * T_full_ellipse (half orbit). Example LEO->GEO: T = pi*sqrt(24471³/3.986e14) = 5.27 hours. Example Earth->Mars heliocentric: a=188.77 million km, T=8.52 months.

Ellipse eccentricity: e = (r2-r1)/(r1+r2). LEO->GEO: e=(42164-6778)/(48942)=0.723 highly elongated ellipse. Earth->Mars: e=(228-150)/(378)=0.206 moderate ellipse. e=0 circular (no transfer), e=1 parabolic (escape velocity).

Bi-elliptic transfer (Sternfeld 1934): for ratios r2/r1 > 11.94 double elliptical transfer (with very distant apogee r3 >> r2) is more efficient in delta-v than single Hohmann. Long time (T=2*T_ellipse), but lower delta-v. Used in outer Solar System missions (Voyager 1-2 1977 Saturn-Jupiter swing-by gravitational assist).

History and context

Walter Hohmann (Germany 1880-1945) civil engineer from Hardheim, obtained degree at TU Berlin 1904. Worked as construction supervisor in Essen Germany during World War I, after hours studied math and celestial mechanics independently. Published 1925 "Die Erreichbarkeit der Himmelskoerper" (Reachability of Celestial Bodies) Oldenbourg Munich Publisher, fundamental work of modern astronautics describing optimal orbital transfers. Inspired Wernher von Braun (Germany 1912-1977, USA 1955) Saturn V Apollo program 1969 Moon landing. Konstantin Tsiolkovsky (Russia 1857-1935) "Investigation of Outer Space by Means of Reaction Devices" 1903 mathematical founder of astronautics, rocket equation. Robert Goddard (USA 1882-1945) first liquid-fuel rocket 1926 Auburn MA. Hermann Oberth (Romania 1894-1989) "Die Rakete zu den Planetenraeumen" 1923. Ari Sternfeld (Poland 1905-1980) bi-elliptic transfer 1934 more efficient alternative for large ratios. USA: Robert Goddard Princeton NJ rocket pioneer, Wernher von Braun NASA Marshall Space Flight Center Huntsville AL Saturn V design 1962, JPL Jet Propulsion Laboratory Caltech Pasadena CA founded 1936 NASA HQ Greenbelt MD. Industry global aerospace 400+ billion USD/year: SpaceX Falcon 9 Hawthorne CA Elon Musk 2002 founded, Boeing Seattle WA, Lockheed Martin Bethesda MD, ULA United Launch Alliance Centennial CO Atlas V Delta IV, Northrop Grumman Falls Church VA, Aerojet Rocketdyne Sacramento CA, Blue Origin Kent WA Bezos.

NASA JPL Pasadena Mars rovers Spirit Opportunity Curiosity Perseverance, JSC Houston ISS operations, KSC Kennedy Space Center Cape Canaveral FL launches, MSFC Huntsville Saturn V design. SpaceX achievements: first reusable Falcon 9 booster landing 2015, Crew Dragon ISS 2020 first commercial astronaut transport, Starship Boca Chica TX Mars colonization vision. Constellation GPS USA 32 satellites MEO 20200 km navigation worldwide, Cape Canaveral launches. James Webb Space Telescope JWST 2021 NASA-ESA-CSA $10 billion L2 halo orbit infrared astronomy.

Typical Hohmann transfer missions table

• LEO 400 km altitude: r=6778 km, v=7.67 km/s, T=92 minutes
• ISS International Space Station (408 km): r=6786 km, v=7.66 km/s, T=92.8 min
• Hubble HST 547 km: r=6925 km, v=7.59 km/s, T=95 min
• SSO Sun-Synchronous 700 km: r=7078 km, v=7.51 km/s, T=99 min
• MEO Galileo EU constellation 23222 km: r=29622 km, v=3.67 km/s, T=14 hours
• MEO GPS USA 20200 km: r=26600 km, v=3.87 km/s, T=12 hours
• GTO Geostationary Transfer Orbit: ellipse 200x35786 km, perigee 7.7 km/s, apogee 1.6 km/s
• GEO Geostationary 35786 km altitude: r=42164 km, v=3.07 km/s, T=24 hours (synchronous)
• LEO->GEO Hohmann transfer: delta-v 3.94 km/s, T=5.27 hours
• Earth->Mars heliocentric transfer: delta-v 5.6 km/s, T=8.5 months
• Earth->Venus transfer: delta-v 5.2 km/s, T=4.9 months
• Earth->Moon transfer: delta-v 3.2 km/s, T=4.5 days
• Voyager 1 outer planets gravitational assist: 1977 launch, Saturn 1980, Jupiter 1979
• Bi-elliptic threshold r2/r1: 11.94 (Sternfeld 1934)

Worked numerical examples

Example 1 - LEO->GEO TLC satellite transfer. Initial orbit LEO 400 km (r1=6778 km). Final orbit GEO 35786 km (r2=42164 km). mu_Earth=3.986e14 m³/s². v1=sqrt(3.986e14/6.778e6)=7670 m/s=7.67 km/s. v2=sqrt(3.986e14/4.2164e7)=3070 m/s. Ellipse: a=(6.778+42.164)/2=24.471 million m. v_peri=sqrt(3.986e14*(2/6.778e6-1/2.447e7))=10150 m/s. v_apo=sqrt(3.986e14*(2/4.2164e7-1/2.447e7))=1610 m/s. delta_v1=10150-7670=2480 m/s. delta_v2=3070-1610=1460 m/s. Total=3940 m/s=3.94 km/s. Time: T=pi*sqrt(2.447e7³/3.986e14)=18960 s=5.27 hours. Typical Falcon 9 SpaceX or Ariane 5 GTO->GEO operational.

Example 2 - American ISS launch. From Cape Canaveral Florida USA or Baikonur Kazakhstan. Initial parking orbit 200 km (r1=6571 km). ISS orbit 400 km (r2=6771 km, ISS real 408 km). v1=7.79 km/s, v2=7.67 km/s. a=(6571+6771)/2=6671 km. v_peri=7.856 km/s, v_apo=7.621 km/s. delta_v1=7.856-7.79=66 m/s. delta_v2=7.67-7.621=49 m/s. Total=115 m/s. Transfer 45 minutes. NASA Crew Dragon SpaceX 2020 first commercial astronaut transport, Soyuz Roscosmos backup. American astronauts Chris Hadfield, Scott Kelly 340 days mission 2015-2016 longest single ISS stay.

Example 3 - American interplanetary mission Earth->Mars. Heliocentric (orbiting Sun). r1=Earth=149.6 million km, r2=Mars=227.94 million km. mu_Sun=1.327e20 m³/s². v1=29.78 km/s (Earth orbital), v2=24.13 km/s (Mars orbital). a=(149.6+227.94)/2=188.77 million km. v_peri=32.73 km/s (perihelion), v_apo=21.48 km/s (aphelion). delta_v1=32.73-29.78=2.95 km/s (Earth escape Lagrange + injection). delta_v2=24.13-21.48=2.65 km/s (Mars insertion orbit). Total=5.60 km/s. Time: T=pi*sqrt(188.77e9³/1.327e20)=2.235e7 s=259 days=8.5 months. NASA missions: Mars Pathfinder 1996-1997 Sojourner rover, Curiosity 2011-2012 MSL JPL, Perseverance 2020-2021 Mars 2020. Launch window every 26 months (Mars-Earth synodic period).

Example 4 - American Earth->Moon Apollo program. Earth-centric. r1=LEO 200 km (r1=6571 km). r2=Moon distance 384400 km. mu_Earth=3.986e14. v1=7.79 km/s, v2=1.02 km/s (Moon orbital). Ellipse: a=(6571+384400)/2=195485 km. v_peri=10.93 km/s (TLI Trans-Lunar Injection), v_apo=0.19 km/s. delta_v1=10.93-7.79=3.14 km/s (TLI Saturn V S-IVB stage). delta_v2=1.02-0.19=0.83 km/s (LOI Lunar Orbit Insertion). Total=3.97 km/s. Time: T=pi*sqrt(195.5e6³/3.986e14)=4.32 days. Apollo 11 1969 launch July 16 Cape Canaveral, Moon arrival July 19, landing July 20 Eagle module Sea of Tranquility 4 days journey. Neil Armstrong-Buzz Aldrin first humans Moon.

Example 5 - American GPS constellation MEO 20200 km. Transfer from LEO 400 km to MEO 20200 km altitude (r2=26580 km). v1=7.67 km/s, v2=3.87 km/s. a=(6778+26580)/2=16679 km. v_peri=8.65 km/s, v_apo=2.20 km/s. delta_v1=8.65-7.67=0.98 km/s. delta_v2=3.87-2.20=1.67 km/s. Total=2.65 km/s. Time: T=pi*sqrt(16679³/3.986e14)=3.67 hours. GPS constellation 32 satellites operational US Department of Defense, civilian use free worldwide. Boeing Lockheed Martin GPS III satellites Cape Canaveral. Delta IV ULA, Falcon 9 SpaceX launches GPS III SV01 2018 onwards.

Limits and considerations

1. Hohmann valid for coplanar circular orbits (same plane). Inclination change requires huge additional delta-v (combined inclination plane change).
2. Bi-elliptic transfer more efficient for r2/r1 > 11.94 (Sternfeld 1934), long time (>>T_Hohmann).
3. Initial elliptical orbits: use generalized vis-viva v²=mu*(2/r-1/a) formulas.
4. Patched conic approximation: in interplanetary missions ignores influence zones detailed simulation.
5. Gravitational assist (Voyager, Cassini, Solar Orbiter): use planets for free acceleration not Hohmann but swing-by.
6. Continuous low-thrust (electric propulsion, ion engines NEXT NASA, ESA SMART-1 lunar): no impulsive Hohmann, continuous spirals.
7. Maneuver errors: pointing precision (0.1° = 1% delta-v lost), ignition time, variable thrust combustion.
8. Atmospheric drag: low orbits <500 km require continuous station-keeping (ISS reboost every 6 months).
9. Solar radiation pressure: significant for large GEO satellites (>10 m² solar arrays), correction delta-v 100-200 m/s/year.
10. Three-body problem: Lagrange L1/L2 transfers (Solar Orbiter, JWST) require Lambert solver not classical Hohmann.

Common errors

1. Confusing altitude (h) with orbit radius (r=R_Earth+h): R_Earth=6378 km always added.
2. Neglecting inclination change: 28.5° (Cape Canaveral) -> 0° (equatorial) costs ~3.9 km/s extra.
3. Calculating delta-v in m/s vs km/s: 1000 factor error.
4. Confusing mu (gravitational parameter) with G (gravitational constant): mu = G*M.
5. Using single Hohmann for r2/r1 > 11.94: bi-elliptic more efficient in delta-v but slower.
6. Neglecting Oberth effect: burns delta-v more efficiently at high velocity (perigee).
7. Demanding classical Earth-Moon transfer: Moon not stationary, requires patched conic.
8. Calculating T_transfer in seconds vs minutes vs hours: 60 and 3600 factor errors.
9. Ignoring delta-v residual for propulsion (Tsiolkovsky equation): delta-v=Isp*g0*ln(m0/mf).
10. Confusing mu_Earth (3.986e14) with mu_Sun (1.327e20): 10⁵ factor error.

Numbered applications

1. TLC satellite launches: LEO->GEO transfer via GTO, Falcon 9 SpaceX, Ariane 6 ESA, Atlas V ULA.
2. GPS USA constellation: 32 satellites MEO 20200 km, US Department of Defense, civilian use worldwide.
3. Galileo EU constellation: 30 satellites MEO 23222 km, EU navigation alternative GPS.
4. Starlink SpaceX constellation: 5000+ satellites LEO 550 km, Falcon 9 multiple launches.
5. Interplanetary missions: NASA Mars Pathfinder Curiosity Perseverance, ESA Mars Express Trace Gas.
6. Lunar missions: Apollo 11 1969, Artemis 2025+ NASA-ESA, Chandrayaan-3 ISRO 2023.
7. Solar System exploration: Voyager 1-2 1977, Cassini-Huygens 1997, Juno 2011, Solar Orbiter 2020.
8. James Webb Space Telescope JWST: L2 halo orbit 1.5 mln km Earth-Sun Lagrange, NASA-ESA-CSA 2021.
9. International Space Station ISS: cargo Dragon SpaceX, Cygnus Northrop Grumman, Soyuz Russia.
10. Educational: astronautics courses MIT Cambridge, Caltech Pasadena, UT Austin, Princeton NJ.

For support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary orbital mechanics. For professional missions consult Hohmann W 1925 'Die Erreichbarkeit der Himmelskoerper' Oldenbourg Munich, Curtis 'Orbital Mechanics for Engineering Students' 2020 Elsevier, Vallado 'Fundamentals of Astrodynamics' 2013 Microcosm Press, Tsiolkovsky 1903 'Investigation Outer Space by Means Reaction Devices'. NASA JPL Pasadena CA mission planning. NASA Goddard Space Flight Center Greenbelt MD operations.

Updated July 2026