Understand the relationship
The reasoning behind the result
The transfer ellipse touches each circular orbit
at=(r1+r2)/2; et=|r2−r1|/(r1+r2)
An ideal Hohmann transfer connects two circular, coplanar, concentric orbits with an ellipse tangent to both. Its near and far radii are the smaller and larger of the entered circle radii. The spacecraft follows half of that ellipse between the impulses.
The initial and final directions of tangential motion have the same sense. A change of orbital plane or reversal of orbital direction is a separate maneuver. The model ignores burn duration, drag, finite-body effects and all gravitational perturbations.
Each burn is a signed difference between two velocities
Δv1=√(μ/r1)[√(2r2/(r1+r2))−1]; Δv2=√(μ/r2)[1−√(2r1/(r1+r2))]
At departure, subtract the initial circular speed from the transfer speed at that radius. At arrival, subtract the transfer speed from the final circular speed. Outward transfers have positive prograde changes at both ends; inward transfers have negative retrograde changes.
The ideal delta-v budget is |Δv1|+|Δv2|, not their algebraic sum. The ledger keeps the signs and endpoint speeds so a small burn cannot be confused with the much larger speed already present in orbit. Delta-v is not propellant mass; converting it requires a separately defined propulsion and vehicle-mass model.
The coast takes half the transfer ellipse's period
ttransfer=π√(at³/μ)
The departure and arrival tangent points are opposite each other about the focus. Moving between them takes half a period in the ideal two-body model. Reversing the same pair of radii preserves the coast time and total burn budget while reversing the burn signs.
When the radii are identical, no orbit transfer is required: this tool reports zero burns and zero required transfer time. It separately retains the half-circle travel time. Reaching an object at another phase on the same circle is a phasing problem and is not solved by declaring a half orbit to be a transfer.
Target phase is a geometric condition, not a schedule
φdeparture=π−n2 ttransfer (mod 2π); n2=√(μ/r2³)
For an optional target travelling in the final circle in the same sense, the target advances by n2 times the coast duration. It must begin at the angle that makes its arrival direction coincide with the spacecraft's opposite tangent point. Positive displayed phase means ahead of the departure direction; negative means behind, using the equivalent angle within ±180°.
The repeat interval follows 2π/|n1−n2| for two distinct circular radii. No current positions, dates, launch site, sphere-of-influence transitions, capture or operational rendezvous are included. Equal radii have no relative circular phase drift, so no finite repeat interval is inferred.
Follow the numbers
A unit-gravity outward transfer
- Use an ideal mathematical reference with μ=1 m³/s², r1=1 m and r2=2 m. The transfer semimajor axis is 1.5 m and eccentricity is 1/3.
- Initial circular speed is 1 m/s; departure transfer speed is √(4/3)=1.154700538 m/s. Burn 1 is +0.154700538 m/s.
- Arrival transfer speed is √(1/3)=0.577350269 m/s; final circular speed is √(1/2)=0.707106781 m/s. Burn 2 is +0.129756512 m/s.
- Total ideal delta-v is 0.284457050 m/s. Coast time is π√(1.5³)=5.771474236 s.
The two positive burns raise the orbit. Reversing the radii gives retrograde burns of the same combined magnitude and the same coast time.
Quick guide
How to use this calculator
- Enter central gravity and choose radii or surface heights. The spacecraft and optional target are negligible test masses in the same central field.
- Add named initial/final circular-orbit pairs. Both circles must have the same plane and sense of motion.
- Read departure and arrival circular speeds, transfer endpoint speeds and the signed changes at each burn. Add the absolute burn magnitudes for the ideal budget.
- Optional phase describes where a same-sense circular target would need to be at departure in this ideal geometry. It is not a launch date or a rendezvous solution.
Calculation method
Calculation and interpretation
Turn entered departure and arrival radii into a complete two-impulse budget, with the intermediate velocities and timing needed to verify each burn.
at=(r1+r2)/2; vc=√(μ/r); vt=√[μ(2/r−1/at)]; Δv1=vt1−vc1; Δv2=vc2−vt2; transfer time=π√(at³/μ).
Worked example
A unit-gravity outward transfer
The two positive burns raise the orbit. Reversing the radii gives retrograde burns of the same combined magnitude and the same coast time.
at=(r1+r2)/2; vc=√(μ/r); vt=√[μ(2/r−1/at)]; Δv1=vt1−vc1; Δv2=vc2−vt2; transfer time=π√(at³/μ).
Supported inputs
Precision and limits
Ideal two impulses
This model assumes instantaneous tangential burns, circular endpoints, one plane and one sense of motion. It does not model thrust limits, finite burns, losses or spacecraft capabilities.
No universal optimum claim
A Hohmann comparison does not establish the best trajectory among all possible transfers. Bi-elliptic, low-thrust, noncoplanar and time-constrained alternatives are outside this workflow.
No current target or navigation data
Target phase is optional ideal geometry only. This is not a launch window, live ephemeris, collision assessment, encounter plan or mission design.
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