Engineering
Hohmann Transfer Orbit formulas: Perigee Burn, Apogee Burn & Coast Time
Calculate signed tangential velocity changes and coast time for an ideal two-impulse transfer between coplanar circular orbits.
Ideal two-body transfer between circular orbits
Assume circular coplanar initial and final orbits around one spherical gravitating body, instantaneous tangential burns, and two-body dynamics. The transfer ellipse has semi-major axis (r1+r2)/2. Radii are measured from the central body, not from its surface; use km with mu in km³/s².
The example raises a 6678 km radius orbit to 42164 km. It omits launch, atmospheric drag, plane changes, finite burn losses and perturbations. A circular orbit at GEO radius is not automatically geostationary: inclination and direction must also be appropriate.
Both formulas below give signed tangential changes. For an inward transfer they are negative (retrograde); total maneuver delta-v is the sum of their absolute magnitudes. These formulas do not optimize over all possible transfer strategies or compute propellant mass. For equal radii no transfer is required; the worksheet reports zero burns and zero required transfer time.
| Orbital Maneuver Quantity | Formula | LEO (6,678 km) -> GEO (42,164 km) | Result |
|---|---|---|---|
| Transfer Semi-Major Axis a_t | (B3 + B4) / 2 | (6,678 + 42,164) / 2 | 24,421.0 km |
| Burn 1: GTO Perigee Kick | SQRT(mu/r1) * (SQRT(2*r2/(r1+r2)) - 1) | Enters elliptical GTO | 2.4258 km/s |
| Burn 2: GEO Apogee Kick | SQRT(mu/r2) * (1 - SQRT(2*r1/(r1+r2))) | Circularizes at 35,786 km alt | 1.4668 km/s (Total 3.8926) |
| Half-Ellipse Coast Duration | PI() * SQRT(a_t^3 / mu) / 3600 | Perigee to Apogee coast | 5.28 hours |
Departure burn: signed tangential delta-v
Positive for outward transfer and negative for inward transfer, in km/s.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0),"Invalid inputs",ROUND(SQRT(B2 / B3) * (SQRT((2 * B4) / (B3 + B4)) - 1), 4)))Half-ellipse coast time in hours
Use half the transfer ellipse period for distinct radii. Equal radii return zero because no orbital transfer is requested; half a circular orbital period would describe a different phasing task.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0),"Invalid inputs",IF(B3=B4,0,ROUND((PI()*SQRT((((B3+B4)/2)^3)/B2))/3600,2))))Arrival burn: signed tangential delta-v
This circularizes the transfer ellipse at the destination radius. For an inward transfer arrival is at periapsis rather than apoapsis.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0),"Invalid inputs",ROUND(SQRT(B2 / B4) * (1 - SQRT((2 * B3) / (B3 + B4))), 4)))Try the example
Use the original values or change them to test your understanding.