Bi-Elliptic Transfer Calculator
Compare a three-burn bi-elliptic transfer against a standard Hohmann transfer to find the most fuel-efficient maneuver.
Last reviewed: April 2026This bi-elliptic transfer calculator computes the total delta-v for a three-burn orbital maneuver between two circular orbits and compares it directly against the equivalent two-burn Hohmann transfer. Enter your starting orbit, target orbit, and the intermediate apoapsis to see Δv₁, Δv₂, and Δv₃, the total mission Δv, and whether bi-elliptic or Hohmann wins for your geometry. The sections below explain the vis-viva derivation, the 11.94 critical ratio that separates the two regimes, a fully worked LEO-to-lunar-distance example, and how this idealized model relates to real mission design.
What This Calculator Does
Three-Burn Bi-Elliptic Profile
A bi-elliptic transfer moves a spacecraft between two circular orbits using three discrete impulsive burns rather than the two burns of a Hohmann transfer. Burn 1 raises apoapsis from the starting circular orbit r₁ out to a distant intermediate point r₃, placing the spacecraft on a highly elliptical transfer ellipse. Burn 2 occurs at r₃ and raises the perihelion of that ellipse from r₁ up to the target orbit radius r₂, putting the spacecraft on a second descending ellipse. Burn 3, executed when the spacecraft re-enters the vicinity of r₂, circularizes the orbit at the target altitude. The calculator computes each burn magnitude using the vis-viva equation and sums them to deliver total mission Δv.
Direct Comparison to Hohmann
For every input geometry, the tool also computes the standard two-burn Hohmann Δv as a baseline and reports the percentage savings (or penalty). When the radius ratio r₂/r₁ exceeds roughly 11.94, bi-elliptic begins to outperform Hohmann; below that threshold, Hohmann is always more fuel-efficient. The comparison is visible at a glance in the result panel above, with green-bordered boxes highlighting fuel savings when bi-elliptic wins. This single side-by-side view is the calculator's central value: rather than guessing whether the extra mid-course burn is worth the propellant savings, you see the numerical verdict immediately.
Assumptions Built Into This Model
The calculator assumes the restricted two-body problem (only one gravitating central body), impulsive instantaneous burns, circular initial and final orbits, no atmospheric drag, no third-body perturbations from the Sun or Moon, and no oblateness effects from the central body's non-spherical mass distribution. These idealizations make the math tractable and the results physically meaningful for first-order trade studies, but operational mission design requires layering in finite-burn corrections, gravity-loss accounting, the J₂ perturbation, solar-radiation pressure, and patched-conic transitions when the trajectory crosses between sphere-of-influence boundaries.
How to Use This Calculator
Input Field Reference
Enter r₁ as the starting orbital radius measured from the central body's center (not surface altitude). For an Earth-centered example, a 400 km altitude LEO has r₁ = 6,371 + 400 = 6,771 km. Enter r₂ as the target circular orbit radius, again from the center. Geostationary orbit sits at r₂ = 42,164 km. Enter r₃ as the chosen intermediate apoapsis — this must exceed both r₁ and r₂, and the larger r₃ is, the more bi-elliptic deviates from the Hohmann profile and (above the critical ratio) the cheaper it becomes. Enter μ as the central body's gravitational parameter; Earth's value is μ = 398,600 km³/s², the Moon's is 4,902 km³/s², and the Sun's is 1.327 × 10¹¹ km³/s².
Interpreting the Three Δv Outputs
Δv₁ is the magnitude of the first prograde burn at r₁ that raises apoapsis from r₁ to r₃. Δv₂ is the second prograde burn at the distant intermediate point r₃ that raises perihelion of the second ellipse from r₁ to r₂. Δv₃ is typically a retrograde burn at r₂ that circularizes the orbit by decelerating the spacecraft to the local circular velocity. All three are reported in km/s. The total Δv shown in the headline box is simply Δv₁ + Δv₂ + Δv₃, and it represents the total propellant budget you must allocate from your spacecraft's mass through the Tsiolkovsky rocket equation.
Comparing to Hohmann
The Hohmann row shows what the same r₁ → r₂ transfer would cost with the simpler two-burn profile. If the fuel-savings box reports a positive percentage, bi-elliptic wins for your geometry; if it reads "Hohmann wins," the extra mid-course complication of bi-elliptic costs more propellant than it saves and you should use Hohmann instead. The orbit ratio output makes the threshold immediately visible: ratios below 11.94 always favor Hohmann; ratios above favor bi-elliptic when r₃ is far enough out.
Sharing and Resetting
The share URL field encodes your current input set, letting you bookmark or send a specific scenario to a colleague. The Reset button restores the LEO-to-lunar-distance default scenario; Print produces a clean output suitable for embedding in a mission-trade-study document.
Worked Example: LEO to GEO via 100,000 km, and LEO to Lunar Distance
Scenario 1 — LEO (400 km altitude) to GEO (35,786 km altitude)
Take r₁ = 6,771 km (400 km altitude + 6,371 km Earth radius) and r₂ = 42,164 km (GEO radius). Pick intermediate apoapsis r₃ = 100,000 km, and use μ_Earth = 398,600 km³/s². The radius ratio r₂/r₁ = 42,164 / 6,771 ≈ 6.23, which is well below the 11.94 critical ratio.
Running the calculator (or computing by hand using vis-viva):
- Semi-major axis a₁ = (r₁ + r₃) / 2
- (6,771 + 100,000) / 2 = 53,386 km
- Semi-major axis a₂ = (r₂ + r₃) / 2
- (42,164 + 100,000) / 2 = 71,082 km
- Δv₁ at r₁
- |√(2μ/r₁ − μ/a₁) − √(μ/r₁)| ≈ |10.50 − 7.67| ≈ 2.83 km/s
- Δv₂ at r₃
- |√(2μ/r₃ − μ/a₂) − √(2μ/r₃ − μ/a₁)| ≈ |1.54 − 0.71| ≈ 0.83 km/s
- Δv₃ at r₂
- |√(μ/r₂) − √(2μ/r₂ − μ/a₂)| ≈ |3.07 − 3.65| ≈ 0.57 km/s
- Bi-elliptic total Δv
- ≈ 4.23 km/s
- Hohmann total Δv (for same r₁ → r₂)
- ≈ 2.40 + 1.46 ≈ 3.86 km/s
Verdict: Hohmann wins by approximately 0.4 km/s (about 9% savings). This is exactly the expected outcome — at a radius ratio of 6.23, well below 11.94, the simpler two-burn profile dominates. Any operational LEO-to-GEO transfer should use Hohmann or a Hohmann-with-plane-change combined burn, never bi-elliptic.
Scenario 2 — LEO (400 km) to Lunar Distance (~384,000 km)
Same r₁ = 6,771 km, but now r₂ = 384,400 km (mean Earth-Moon distance). The radius ratio jumps to 384,400 / 6,771 ≈ 56.8 — far above the 11.94 threshold, deep in bi-elliptic-favored territory. Choose r₃ = 1,000,000 km as the intermediate apoapsis.
- a₁ = (6,771 + 1,000,000) / 2
- 503,386 km
- a₂ = (384,400 + 1,000,000) / 2
- 692,200 km
- Δv₁ at r₁
- ≈ 3.14 km/s
- Δv₂ at r₃
- ≈ 0.40 km/s
- Δv₃ at r₂ (retrograde, circularize)
- ≈ 0.21 km/s
- Bi-elliptic total Δv
- ≈ 3.74 km/s
- Hohmann total Δv
- ≈ 3.91 km/s
Verdict: Bi-elliptic beats Hohmann by roughly 0.17 km/s (about 4%), even though the chosen r₃ = 1,000,000 km is only 2.6× the target r₂. Pushing r₃ out further — say to 5,000,000 km — shrinks Δv₂ toward zero and widens bi-elliptic's advantage, but the transit time grows from weeks to months. For an uncrewed cargo flight where time is cheap, this trade can be worth it; for any crewed or time-sensitive payload, the Hohmann profile remains the operational choice.
Key Takeaway From Both Scenarios
Bi-elliptic's fuel advantage only materializes when the radius ratio is genuinely extreme — beyond about 11.94 — and the intermediate apoapsis r₃ is pushed far enough out to let the second burn approach zero magnitude. In practice, that combination is rare in operational missions. The most common use case is end-of-life disposal of geostationary satellites and large repositioning maneuvers within highly elliptical orbits.
When Bi-Elliptic Wins: The 11.94 Critical Ratio
Hoelker-Silber 1959 Analytical Result
The critical radius ratio above which bi-elliptic outperforms Hohmann (for very large r₃) was first derived analytically by Friedrich Hoelker and Roland Silber in their 1959 NASA technical report. By setting the total Δv of the bi-elliptic transfer equal to the total Δv of the Hohmann transfer and taking the limit as r₃ → ∞, the boundary collapses to the cubic root of a transcendental polynomial whose real solution is approximately R* = 11.9387. Below this ratio, no choice of r₃ can make bi-elliptic cheaper than Hohmann; above it, bi-elliptic begins to dominate for sufficiently large r₃.
The 15.58 Always-Wins Threshold
Between r₂/r₁ ≈ 11.94 and r₂/r₁ ≈ 15.58, bi-elliptic only outperforms Hohmann when r₃ is chosen larger than some specific minimum threshold that depends on the exact ratio. Above r₂/r₁ ≈ 15.58, bi-elliptic outperforms Hohmann for any r₃ greater than r₂. This second threshold is the cleaner operational dividing line: above it, the bi-elliptic profile is essentially always cheaper in propellant if you can accept the longer transit time.
The Time Trade-Off
The transfer time for a Hohmann transfer between r₁ and r₂ is half the period of the transfer ellipse: T_Hohmann = π × √(a³/μ), where a = (r₁ + r₂)/2. For a bi-elliptic with intermediate r₃, the transit time is the half-periods of both transfer ellipses summed: T_BE = π × [√(((r₁ + r₃)/2)³/μ) + √(((r₂ + r₃)/2)³/μ)]. Because both half-periods scale with the 3/2 power of semi-major axis, pushing r₃ outward to gain fuel efficiency lengthens transit nonlinearly. A bi-elliptic between LEO and lunar distance through 5,000,000 km can take 60–90 days versus roughly 5 days for the equivalent Hohmann — a 12–18× time penalty for a 2–4% propellant gain.
Practical Decision Framework
Use Hohmann when: r₂/r₁ < 11.94, or transit time is critical, or the spacecraft carries crew, or you need a simple flight profile. Use bi-elliptic when: r₂/r₁ > 15.58, you have ample time, the payload is uncrewed cargo or a propellant-constrained disposal stage, and you need every kilogram of propellant savings to fit within the launch vehicle's capacity.
Common Use Cases
End-of-Life Satellite Disposal
Geostationary communications satellites are required by ITU regulation to be moved to a "graveyard" orbit roughly 300 km above GEO at end of mission, freeing the operational slot for replacement spacecraft. The radius ratio is small (GEO at 42,164 km versus graveyard at ~42,500 km), so bi-elliptic does not pay off for this specific transfer. However, satellites being repositioned across vastly different regimes — for example, from a high-elliptical orbit to a circular disposal orbit — can benefit from a bi-elliptic-like profile to stretch limited remaining propellant. Mission planners model the trade explicitly using calculators like this one to decide.
Lunar Mission Staging
Crewed lunar architectures from Apollo through Artemis have generally chosen Hohmann-like trans-lunar injection because the 5-day transit fits within life-support consumables and the radius ratio (Earth-LEO to lunar orbit, ratio ~56) is in bi-elliptic-favored territory but with diminishing returns. For uncrewed cargo deliveries to lunar gateway stations or surface depots, a bi-elliptic profile through a high apoapsis can save 5–10% of total mission propellant, increasing payload-to-the-Moon by an equivalent percentage. NASA's commercial lunar payload services contracts have considered this profile for low-priority cargo where transit time is flexible.
Interplanetary Mission Design
For interplanetary missions, the relevant transfers are heliocentric (Sun-centered) rather than Earth-centered, but the same bi-elliptic mathematics applies with μ replaced by the Sun's gravitational parameter. The Galileo mission to Jupiter and the Cassini mission to Saturn both used gravity assists rather than pure bi-elliptic profiles to reach their destinations, but mission designers regularly use bi-elliptic as a baseline reference trajectory against which gravity-assist tours are evaluated for propellant savings.
Fuel-Efficient End-of-Life Repositioning
Aging satellites with low remaining propellant occasionally use bi-elliptic-like maneuvers to reach disposal orbits when a direct Hohmann would exceed remaining tank capacity. The mid-course burn at distant apoapsis is small in Δv terms, making total propellant mass smaller even though three burns are required. This use case is increasingly common as constellation operators retire first-generation hardware.
Gravity-Assist Planning Baseline
Modern interplanetary missions almost universally use planetary flybys to gain energy "for free" rather than spending propellant. But every gravity-assist tour is benchmarked against a reference no-assist trajectory — usually a Hohmann or a bi-elliptic — to quantify the savings. This calculator gives mission designers the bi-elliptic baseline number in seconds; the gravity-assist tour optimization happens in dedicated software like JPL's MIDAS or ESA's PaGMO.
Behind the Physics: Vis-Viva, Oberth, and Lagrange Optimization
Hermann Oberth's 1929 Formulation
Hermann Oberth first described the bi-elliptic transfer in his 1929 book Wege zur Raumschiffahrt ("Ways to Spaceflight"), expanding on Walter Hohmann's 1925 two-burn analysis from Die Erreichbarkeit der Himmelskörper. Oberth recognized that for very large orbit transitions, splitting the maneuver into three burns and routing through a distant intermediate point could reduce total propellant. The full quantitative analysis with the 11.94 critical ratio was completed by Hoelker and Silber thirty years later, using calculus-of-variations techniques to confirm and refine Oberth's qualitative insight.
The Vis-Viva Equation
Every burn in this calculator is sized using the vis-viva equation: v² = μ × (2/r − 1/a). This expresses the kinetic energy of a spacecraft on any conic orbit in terms of its current distance r, the semi-major axis a of the orbit, and the central body's gravitational parameter μ. The derivation falls out of the conservation of orbital energy: total energy ε = ½v² − μ/r = −μ/(2a) is constant on a Keplerian orbit, so v² = μ × (2/r − 1/a) directly. Solving for v gives the orbital speed at any point.
Δv Computation for Each Burn
For Burn 1 at r₁: the spacecraft is on a circular orbit with speed v_circ = √(μ/r₁), then transitions onto transfer ellipse 1 with semi-major axis a₁ = (r₁ + r₃)/2. Speed on the ellipse at perihelion (r₁) is v_p1 = √(2μ/r₁ − μ/a₁). Δv₁ = |v_p1 − v_circ|. For Burn 2 at r₃: spacecraft is at apoapsis of ellipse 1 with speed v_a1 = √(2μ/r₃ − μ/a₁), and must transition to ellipse 2 with a₂ = (r₂ + r₃)/2 at speed v_a2 = √(2μ/r₃ − μ/a₂). Δv₂ = |v_a2 − v_a1|. For Burn 3 at r₂: spacecraft arrives at perihelion of ellipse 2 with speed v_p2 = √(2μ/r₂ − μ/a₂) and must decelerate to local circular speed v_circ_target = √(μ/r₂). Δv₃ = |v_circ_target − v_p2|.
Lagrange Optimization and the 11.94 Derivation
The 11.94 critical ratio falls out of the calculus-of-variations problem: minimize total Δv_BE(r₃) over r₃ for fixed r₁ and r₂, then compare against Δv_Hohmann. Setting ∂Δv_BE/∂r₃ = 0 yields a transcendental equation whose limit as r₃ → ∞ defines the critical ratio. The exact value 11.9387 emerges from solving 3 − R − 4√R + R√R = 0 numerically.
Patched-Conic Approximation
For interplanetary or trans-lunar transfers, the simple two-body model breaks down when the spacecraft crosses between spheres of influence (Earth's into the Moon's, or Earth's into the Sun's). The patched-conic method handles this by treating the trajectory as a series of two-body arcs stitched together at sphere-of-influence boundaries. This calculator does not implement patched-conic stitching — its outputs are first-order trade-study numbers, not operational trajectories. Real mission design begins with patched-conic and refines with full numerical integration of the perturbed equations of motion.
Limitations of This Idealized Model
Impulsive vs. Finite Burns
The calculator assumes instantaneous changes in velocity — a mathematical abstraction. Real chemical engines deliver finite thrust over burn durations of seconds to minutes. During a finite burn at low altitude, the spacecraft moves along its orbit while the engine fires, so part of the velocity change is wasted against gravity (gravity loss) and part is misdirected because the orbit's tangent vector has rotated during the burn. Operational mission design adds 1–5% to impulsive Δv estimates to cover these losses, with the exact penalty depending on thrust-to-weight ratio and burn altitude.
No Atmosphere, No Drag
This model ignores atmospheric drag. Real LEO transfers below roughly 800 km altitude must account for drag, which removes energy from the orbit continuously and increases propellant requirements for station-keeping. For altitudes above 1,000 km, drag is essentially negligible on mission-design timescales and the impulsive-burn calculator output is adequate.
No Third-Body Perturbations
The two-body assumption ignores gravitational pulls from the Sun, Moon, and other planets. For most LEO and GEO transfers, these are minor over the timescale of a single transfer maneuver. For lunar transfers and any trajectory that ventures beyond Earth's sphere of influence, third-body effects dominate and the simple two-body Δv is only a rough order-of-magnitude estimate.
No Oblateness Effects
Earth is not a sphere — its equatorial bulge (J₂ perturbation) causes orbital precession and small but measurable changes in orbital elements over time. For multi-orbit phasing maneuvers, J₂ effects can shift the perigee location by degrees per day, affecting burn timing and direction. This calculator's outputs assume a spherically symmetric central body.
No Navigation or Execution Errors
Real spacecraft have navigation uncertainties (position errors of meters to kilometers depending on tracking quality) and burn execution errors (typically 1–3% magnitude error, 0.1–1° pointing error). Mission designers reserve 5–15% additional Δv for trajectory correction maneuvers to clean up execution errors. The calculator's nominal Δv numbers should be inflated by this margin for any real budgeting exercise.
Comparison: Hohmann vs. Bi-Elliptic vs. Gravity Assist vs. Low-Thrust
Hohmann Transfer (Two-Burn)
The baseline for nearly every mission analysis. Two impulsive burns, half an elliptical period of transit time. Optimal in Δv for radius ratios below 11.94. Operationally simple and well-understood. The default choice for crewed missions, LEO-to-GEO commercial deployments, and any time-constrained transfer.
Bi-Elliptic Transfer (Three-Burn)
The subject of this calculator. Three impulsive burns through a distant intermediate apoapsis. Cheaper than Hohmann only above radius ratio 11.94 (and consistently above 15.58). Transit times 3–20× longer than Hohmann depending on chosen r₃. Used selectively for high-radius-ratio cargo deliveries and end-of-life disposal.
Gravity Assist (Flyby)
A trajectory that uses a planetary flyby to gain (or lose) energy without expending propellant. Voyager 1 and 2 are the canonical examples — their tours of the outer solar system would have been impossible without Jupiter, Saturn, Uranus, and Neptune flybys. Modern missions like JUICE and Europa Clipper rely heavily on Venus-Earth-Earth gravity assists. Gravity assists trade calendar time and trajectory complexity for massive Δv savings. Used wherever the geometry allows.
Low-Thrust Spirals (Electric Propulsion)
Ion engines (xenon, krypton, or argon Hall thrusters) deliver continuous low thrust at very high specific impulse (3,000–5,000 s versus ~450 s for chemical). The spacecraft spirals outward gradually over weeks to months rather than executing discrete burns. NASA's Dawn, Deep Space 1, and ESA's BepiColombo all used electric propulsion for primary cruise propulsion. Total Δv is higher than chemical impulsive, but propellant mass is dramatically lower, allowing missions to destinations that would be unreachable chemically. The 11.94 ratio doesn't apply — Edelbaum's continuous-thrust transfer theory replaces it.
Choosing the Right Maneuver
Time-critical or crewed: Hohmann. High radius ratio and time-flexible cargo: bi-elliptic. Outer planet missions: gravity assist. Mass-constrained, propellant-limited, low-thrust capable: ion. Real missions often layer these — a launch into LEO, a Hohmann-like burn to leave Earth, a series of planetary gravity assists, and final orbit insertion at the target — with bi-elliptic appearing only in specific phases where the geometry favors it.
Frequently Asked Questions
Quick reference
| Aspect | Detail |
|---|---|
| Purpose | Bi-Elliptic Transfer Calculator is a free browser calculator for fast, repeatable estimates. |
| Runs in | Any modern browser — no install, no signup. |
| Privacy | Inputs are computed locally; nothing is uploaded. |
| Best for | Quick estimates, daily reference, study, planning. |