Bi-elliptic transfer
The bi-elliptic transfer is an orbital maneuver that enables a spacecraft to move from one coplanar circular orbit to another, typically larger, circular orbit using three impulsive velocity changes (Δv) and two intermediate elliptical transfer orbits, and it can require less total Δv than the Hohmann transfer in specific scenarios involving large orbit size differences.[1][2] This technique exploits the Oberth effect, particularly through the initial burn at high velocity near the initial orbit's periapsis, allowing for efficient energy adjustment.[3] The maneuver begins with the first burn at the periapsis of the initial circular orbit, which raises the apoapsis to a highly eccentric elliptical orbit extending far beyond the target orbit's radius (denoted as r^*).[2][3] At this apoapsis, the second burn adjusts the trajectory to a second elliptical orbit sharing the same apoapsis but with its periapsis at the desired final orbit radius (r_f).[1] Finally, a third burn at the new periapsis circularizes the orbit to match r_f.[3] The total Δv is calculated using the vis-viva equation, v = \sqrt{\mu \left( \frac{2}{r} - \frac{1}{a} \right)}, applied at each burn point, where \mu is the gravitational parameter, r is the radial distance, and a is the semi-major axis.[2] Compared to the Hohmann transfer, which uses two burns and a single elliptical orbit tangent to both initial and final orbits, the bi-elliptic approach is less efficient for small orbit radius ratios (r_f / r_i < 11.94), roughly equivalent between 11.94 and 15.58 depending on the apoapsis height, and more efficient above 15.58 regardless of the intermediate apoapsis choice.[4][3] For example, transferring from an initial orbit at Earth's radius plus 191 km to a final orbit at 60 Earth radii requires approximately 3.9 km/s total Δv via bi-elliptic (with r^* = 80 Earth radii), versus 4.0 km/s for Hohmann.[3] However, the bi-elliptic transfer demands significantly longer flight times—often weeks or months—due to the extended path, making it suitable when fuel savings outweigh time constraints.[2][1]Description
Maneuver Sequence
The bi-elliptic transfer is a three-impulse maneuver designed to move a spacecraft between two circular, coplanar orbits, from an initial orbit of radius r_1 to a target orbit of radius r_2 where r_2 > r_1. This method assumes familiarity with basic orbital elements, such as circular orbits where the radius remains constant and elliptical orbits where the spacecraft alternates between periapsis (closest point to the central body) and apapsis (farthest point). Unlike the simpler Hohmann transfer, which relies on a single elliptical path and requires only two impulses, the bi-elliptic approach uses two successive elliptical transfer orbits to achieve the change, necessitating three burns for insertion, adjustment, and circularization.[5] The sequence commences in the initial circular orbit at radius r_1. The first burn, performed prograde (tangential and in the direction of motion) at periapsis—effectively any point on the circular orbit—raises the apapsis to a distant radius r_a, typically much larger than r_2, placing the spacecraft into the first highly eccentric elliptical transfer orbit. The spacecraft then coasts along this ellipse to its apapsis at r_a.[5][6] At apapsis, the second prograde burn adjusts the orbit by raising the periapsis from r_1 to r_2, transitioning to the second elliptical transfer orbit, which now has its periapsis at r_2 and retains the apapsis at r_a. The spacecraft coasts from apapsis through to the new periapsis at r_2.[5][6] The third and final burn occurs at this periapsis of the second ellipse. A retrograde burn (tangential and opposite to the direction of motion) reduces the velocity to lower the apapsis to r_2, circularizing the orbit at the target radius. All three burns are impulsive and tangential, exploiting the spacecraft's velocity vector to efficiently modify the orbital energy and shape. The additional burn compared to the Hohmann transfer accommodates the dual-ellipse path, enabling potential optimizations for large radius ratios.[5][6]Orbital Geometry
The bi-elliptic transfer maneuver employs a geometric configuration consisting of an initial circular orbit with radius r_1, followed by two successive elliptical transfer orbits that connect to a final circular orbit with radius r_2. The first elliptical orbit is tangent to the initial circle at its perigee, located at radius r_1, and extends to an apogee at radius r_a. The second elliptical orbit shares the same apogee at r_a and is tangent to the final circle at its perigee, located at radius r_2. This setup assumes all orbits are coplanar and share a common focus at the central body's center of mass.[3][1] The key parameters defining this geometry are r_1, the radius of the initial orbit; r_2, the radius of the target orbit (typically with r_2 > r_1); and r_a, the apogee radius, which serves as a design variable selected to optimize the overall transfer efficiency. The semi-major axis of the first elliptical orbit is given by a_1 = \frac{r_1 + r_a}{2}, while that of the second elliptical orbit is a_2 = \frac{r_2 + r_a}{2}. These relations highlight the interdependence of the orbits, with r_a influencing the eccentricity and extent of both ellipses.[3][1] For transfers involving large separations between r_1 and r_2, r_a is typically chosen to be much larger than r_2, often approaching infinity in limiting cases, to leverage the reduced velocity perturbations at greater distances from the central body. This configuration allows the transfer orbits to become highly eccentric, minimizing the energy required for the intermediate phases.[3][7] Textually, the orbital diagram depicts concentric initial and final circular orbits, with the first ellipse departing tangentially from a point on the initial orbit and reaching apogee, where the second ellipse begins, curving inward to tangentially intersect the final orbit at perigee; the burns occur at these tangent points, ensuring smooth transitions between orbits.[1][8]Analytical Model
Delta-v Requirements
The delta-v requirements for a bi-elliptic transfer are determined using the vis-viva equation, which gives the speed of a spacecraft in an elliptical orbit as v = \sqrt{\mu \left( \frac{2}{r} - \frac{1}{a} \right)}, where \mu is the standard gravitational parameter, r is the radial distance from the central body, and a is the semi-major axis of the orbit.[1] This equation is applied at the points of each impulsive burn, assuming a two-body problem with circular, coplanar initial and target orbits of radii r_1 and r_2 (where r_1 < r_2), and an intermediate apapsis radius r_a > r_2. The transfer involves three tangential burns: the first at r_1 to enter the initial elliptical transfer orbit with semi-major axis a_1 = (r_1 + r_a)/2, the second at r_a to switch to the second elliptical transfer orbit with semi-major axis a_2 = (r_2 + r_a)/2, and the third at r_2 to circularize the final orbit. All burns are assumed impulsive, with no atmospheric drag, gravitational perturbations, or non-tangential components. The delta-v for the first burn, \Delta v_1, is the difference between the initial circular orbital speed v_1 = \sqrt{\mu / r_1} and the perigee speed in the first transfer orbit: \Delta v_1 = \sqrt{\mu \left( \frac{2}{r_1} - \frac{1}{a_1} \right)} - \sqrt{\frac{\mu}{r_1}}. Substituting a_1 = (r_1 + r_a)/2 yields the closed-form expression \Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_a}{r_1 + r_a}} - 1 \right). This burn increases the spacecraft's velocity to raise the apapsis to r_a.[1][9] At the apapsis r_a, the second burn \Delta v_2 adjusts the perigee from r_1 to r_2 by changing the orbital energy. The speed in the first transfer orbit at apapsis is v_{a1} = \sqrt{\mu \left( 2/r_a - 1/a_1 \right)}, and in the second transfer orbit it is v_{a2} = \sqrt{\mu \left( 2/r_a - 1/a_2 \right)}. Since a_2 > a_1, v_{a2} > v_{a1}, so \Delta v_2 = \sqrt{\mu \left( \frac{2}{r_a} - \frac{1}{a_2} \right)} - \sqrt{\mu \left( \frac{2}{r_a} - \frac{1}{a_1} \right)}. The closed form is \Delta v_2 = \sqrt{\frac{\mu}{r_a}} \left[ \sqrt{\frac{2 r_2}{r_2 + r_a}} - \sqrt{\frac{2 r_1}{r_1 + r_a}} \right]. This positive delta-v accelerates the spacecraft tangentially at apapsis.[1][2] The third burn occurs at the perigee of the second transfer orbit, r_2, to circularize. The perigee speed is v_{p2} = \sqrt{\mu \left( 2/r_2 - 1/a_2 \right)}, which exceeds the target circular speed v_2 = \sqrt{\mu / r_2}, requiring a deceleration: \Delta v_3 = \sqrt{\frac{\mu}{r_2}} - \sqrt{\mu \left( \frac{2}{r_2} - \frac{1}{a_2} \right)}. The magnitude is |\Delta v_3| = \sqrt{\frac{\mu}{r_2}} \left( \sqrt{\frac{2 r_a}{r_2 + r_a}} - 1 \right). The total delta-v is \Delta v = \Delta v_1 + \Delta v_2 + |\Delta v_3|.[1][9] To minimize \Delta v, the parameter r_a is optimized, typically by numerical methods solving d(\Delta v)/dr_a = 0. For large orbit radius ratios r_2 / r_1 \gtrsim 12, the minimum occurs at very large r_a, approaching the bi-parabolic limit where \Delta v \approx (\sqrt{2} - 1) \left( \sqrt{\mu / r_1} + \sqrt{\mu / r_2} \right) as r_a \to \infty, with \Delta v_2 \to 0 and the first and third burns each approaching (\sqrt{2} - 1) times the local circular speed. Units for delta-v are typically in m/s, with \mu in m³/s² and radii in m.[1][2]Transfer Time
The transfer time for a bi-elliptic maneuver is determined using Kepler's third law, which relates the orbital period T of an ellipse to its semi-major axis a and the central body's gravitational parameter \mu via the formulaT = 2\pi \sqrt{\frac{a^3}{\mu}}. [3]
This law applies to each of the two elliptical transfer orbits in the bi-elliptic sequence.[10] The first phase of the transfer begins with a burn at the periapsis of the initial orbit (radius r_1) to enter the first elliptical orbit, which has an apoapsis at radius r_a. The semi-major axis of this orbit is a_1 = (r_1 + r_a)/2. The spacecraft coasts from periapsis to apoapsis, corresponding to a true anomaly change of 180 degrees. Due to the symmetry of elliptical orbits, this half-orbit traversal takes exactly half the full orbital period, yielding the time
t_1 = \pi \sqrt{\frac{ \left( \frac{r_1 + r_a}{2} \right)^3 }{\mu} }. [3]
Half-orbits are used because the impulsive burns occur precisely at periapsis and apoapsis locations, aligning the transfer phases with these symmetric segments of the ellipse.[10] The second phase follows a burn at the apoapsis (r_a) to enter the second elliptical orbit, with periapsis at the final orbit radius r_2 and semi-major axis a_2 = (r_2 + r_a)/2. The coast from apoapsis to periapsis again spans 180 degrees in true anomaly, taking half the period of this orbit:
t_2 = \pi \sqrt{\frac{ \left( \frac{r_2 + r_a}{2} \right)^3 }{\mu} }. [3]
The total transfer time is the sum of these phases,
t = t_1 + t_2,
which increases as the intermediate apoapsis radius r_a grows larger, since both semi-major axes scale with r_a.[10]
For instance, in a lunar transfer scenario from low Earth orbit (r_1 \approx 6678 km) to a lunar relay orbit (r_2 \approx 384{,}400 km), increasing r_a from $4 \times 10^5 km to $10^6 km raises the total time from approximately 15 days to 45 days.[10] These calculations assume instantaneous impulsive burns at the specified points, coplanar circular initial and final orbits, and no additional coasting phases or plane-change maneuvers, focusing solely on the radial transfer dynamics.[3]
The semi-major axes derive from the orbital geometry, while the choice of r_a indirectly influences time through its role in overall maneuver optimization.[10]
Comparison with Hohmann Transfer
Delta-v Efficiency
The delta-v efficiency of the bi-elliptic transfer relative to the Hohmann transfer is determined by comparing the total velocity change required to transition from an initial circular orbit of radius r_1 to a final circular orbit of radius r_2 > r_1, assuming coplanar orbits and impulsive burns. The Hohmann transfer, being a two-burn maneuver, provides a baseline for minimum energy in many cases, with its total delta-v given by \Delta v_H = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1 + r_2}} - 1 \right) + \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1 + r_2}} \right), where \mu is the standard gravitational parameter.[11] The bi-elliptic transfer, involving three burns, can achieve lower total delta-v than this Hohmann value under specific conditions, primarily when the orbital radius ratio \rho = r_2 / r_1 exceeds approximately 11.94. This break-even ratio is derived by setting the limiting-case bi-elliptic delta-v (as the intermediate apoapsis radius r_a \to \infty) equal to \Delta v_H and solving for \rho, marking the threshold beyond which even the infinite-r_a bi-elliptic configuration saves propellant.[12] For $11.94 < \rho < 15.58, savings occur only if r_a is chosen sufficiently large (approaching infinity at the lower end of this range); below 11.94, the Hohmann transfer is always more efficient.[2] In the asymptotic regime of very large \rho, the optimized bi-elliptic transfer's total delta-v approaches \Delta v_{BE} \approx (\sqrt{2} - 1) \left( \sqrt{\frac{\mu}{r_1}} + \sqrt{\frac{\mu}{r_2}} \right), derived from the limiting burns where the first burn nears escape from r_1, the intermediate burn vanishes, and the final circularization leverages a high entry speed at r_2. This is lower than the corresponding Hohmann limit \Delta v_H \approx (\sqrt{2} - 1) \sqrt{\frac{\mu}{r_1}} + \sqrt{\frac{\mu}{r_2}}, yielding asymptotic savings of approximately (2 - \sqrt{2}) \sqrt{\frac{\mu}{r_2}}. Graphical analyses of \Delta v versus \rho illustrate this crossover, with the bi-elliptic curve dipping below the Hohmann curve for \rho > 11.94 and remaining lower thereafter, though relative savings diminish as \rho grows large.[12][13] The choice of r_a critically affects efficiency, as the total bi-elliptic delta-v is minimized at an optimal finite value depending on r_1 and r_2; suboptimal r_a (e.g., too small or excessively large) can increase \Delta v_{BE} beyond \Delta v_H, negating any advantage. This sensitivity underscores the need for numerical optimization in mission design, often balancing delta-v savings against longer transfer times.[12]Duration Comparison
The duration of a Hohmann transfer between two coplanar circular orbits with radii r_1 and r_2 (where r_2 > r_1) is given by half the orbital period of the transfer ellipse, whose semi-major axis is a_H = (r_1 + r_2)/2. This yields the transfer time t_H = \pi \sqrt{ \frac{a_H^3}{\mu} }, where \mu is the standard gravitational parameter of the central body.[1] In contrast, the bi-elliptic transfer duration is the sum of the half-periods of its two elliptical arcs: the first from the initial orbit to the intermediate apogee at radius r_a, with semi-major axis a_1 = (r_1 + r_a)/2, and the second from r_a to the final orbit, with semi-major axis a_2 = (r_a + r_2)/2. Thus, t_{BE} = \pi \sqrt{ \frac{a_1^3}{\mu} } + \pi \sqrt{ \frac{a_2^3}{\mu} }. This results in t_{BE} > t_H for all practical cases, as the highly eccentric intermediate orbit extends the path length despite the two-stage structure.[1] The time penalty arises primarily from the elevated apogee radius r_a, which lengthens both transfer arcs compared to the more compact Hohmann ellipse; as r_a increases, t_{BE} grows significantly, often exceeding t_H by factors of 2–3 or more when r_2 / r_1 > 11.94, the threshold beyond which bi-elliptic transfers offer delta-v advantages over Hohmann.[1][5] This extended duration represents a key trade-off: while the bi-elliptic approach enables propellant savings by leveraging lower-velocity burns at apogee, mission planners must weigh the added time against operational constraints, such as communication windows or radiation exposure. Optimizing r_a minimizes total delta-v for a given r_2 / r_1 but at the cost of progressively longer transfer times, particularly for distant orbits where r_a must be substantially larger than r_2 to realize efficiency gains.[1][5]Break-even Conditions
The break-even condition for preferring a bi-elliptic transfer over a Hohmann transfer in terms of total \Delta v occurs when the ratio of the final orbit radius to the initial orbit radius, r_2 / r_1, exceeds approximately 11.937. This threshold is derived by equating the \Delta v requirement for a bi-elliptic transfer with the intermediate apogee radius r_a approaching infinity (effectively a bi-parabolic transfer) to that of the Hohmann transfer and solving for the radius ratio.[12] Above this value, the bi-elliptic transfer can achieve lower total \Delta v, with efficiency improving as r_2 / r_1 increases further.[12] However, significant limitations apply. For r_2 / r_1 < 11.94, the bi-elliptic transfer requires more [\Delta v](/page/Delta-v) than the Hohmann transfer for any choice of r_a, including the optimal.[12] These analyses assume coplanar circular initial and final orbits in a two-body central force field with impulsive maneuvers; real-world inefficiencies arise from non-optimal r_a selection, orbital perturbations, or non-impulsive burns.[3] A key advantage of the bi-elliptic transfer is its versatility for incorporating plane-change maneuvers at the intermediate apogee, where orbital velocity is minimized, thereby reducing the additional \Delta v needed for inclination adjustments compared to performing the change in a Hohmann transfer.[3] To sketch the derivation of the break-even ratio, one sets the total \Delta v for the bi-elliptic transfer in the limit r_a \to \infty equal to the Hohmann \Delta v and solves the resulting equation for r_2 / r_1, yielding \approx 11.937. This limit simplifies the middle maneuver to zero while leveraging the Oberth effect at perigee for the initial burn.[12]Practical Applications
Mission Examples
One illustrative hypothetical mission involves transferring a satellite from a low Earth orbit (LEO) at an altitude of 629 km (orbital radius r_1 = 7000 km) to a geostationary orbit (GEO) at an altitude of 35,786 km (orbital radius r_2 = 42,000 km), yielding a radius ratio of approximately 6, which is below the break-even threshold of 11.94 where bi-elliptic transfers become advantageous. Using Earth's gravitational parameter \mu = 3.986 \times 10^{14} m³/s², the Hohmann transfer requires a total \Delta v_H = 3.767 km/s, comprising \Delta v_{H1} = 2.332 km/s at injection and \Delta v_{H2} = 1.435 km/s at GEO insertion, with a transfer time of about 5.3 hours. In contrast, a bi-elliptic transfer with an optimized intermediate apogee radius r_a = 100,000 km demands a total \Delta v_{BE} = 4.156 km/s (\Delta v_1 = 2.769 km/s, \Delta v_2 = 0.813 km/s, \Delta v_3 = 0.574 km/s), exceeding the Hohmann by roughly 10%, while extending the transfer time to approximately 43 hours due to the longer intermediate legs.[3] For scenarios with larger orbital separations where the radius ratio exceeds 12, bi-elliptic transfers offer propellant savings. Consider a hypothetical deep-space probe departing from LEO (r_1 = 7000 km) to a circular orbit at approximately the Moon's distance (r_2 = 384,000 km, ratio ≈ 55). The Hohmann transfer requires \Delta v_H = 3.854 km/s (\Delta v_{H1} = 3.032 km/s, \Delta v_{H2} = 0.822 km/s), with a transfer time of roughly 4.8 days. An optimized bi-elliptic transfer, with intermediate apogee r_a \approx 1,000,000 km approaching the bi-parabolic limit, achieves a total \Delta v_{BE} \approx 3.548 km/s—a savings of about 8% over Hohmann—though the transfer time increases to over 10 days. Further optimization for such ratios can yield savings of 10-15% by leveraging the Oberth effect more effectively at the distant apogee.[3]| Parameter | Hohmann (LEO to GEO) | Bi-elliptic (LEO to GEO) | Hohmann (LEO to Moon Distance) | Bi-elliptic (LEO to Moon Distance) |
|---|---|---|---|---|
| Total \Delta v (km/s) | 3.767 | 4.156 | 3.854 | 3.548 |
| Transfer Time | 5.3 hours | 43 hours | 4.8 days | ~10 days |
| Radius Ratio | 6 | 6 | 55 | 55 |