- Bi-elliptic transfer
In
astronautics andaerospace engineering , the bi-elliptic transfer is anorbital maneuver that moves aspacecraft from oneorbit to another and may, in certain situations, require lessdelta-v than a Hohmann transfer.The bi-elliptic transfer consists of two half
elliptic orbit s. From the initial orbit, a delta-v is applied boosting the spacecraft into the first transfer orbit with anapoapsis at some point r_b away from thecentral body . At this point, a second delta-v is applied sending the spacecraft into the second elliptical orbit withperiapsis at the radius of the final desired orbit where a third delta-v is performed injecting the spacecraft into the desired orbit.While it requires one more burn than a Hohmann transfer and generally requires a greater period of time, the bi-elliptic transfer may require a lower amount of total delta-v than a Hohmann transfer in situations where the ratio of final to the initial
semi-major axis is greater than 11.94 [Cite book | last = Vallado | first = David Anthony | title = Fundamentals of Astrodynamics and Applications | page = 317 | publisher = Springer | date = 2001 | isbn = 0792369033 | url = http://books.google.com/books?id=PJLlWzMBKjkC&printsec] .Calculation
Delta-v
Utilizing the "vis viva" equation where,:v^2 = mu left( frac{2}{r} - frac{1}{a} ight) where:
* v ,! is the speed of an orbiting body
*mu = GM,! is thestandard gravitational parameter of the primary body
* r ,! is the distance of the orbiting body from the primary
* a ,! is thesemi-major axis of the body's orbitThe magnitude of the first delta-v at the initial
circular orbit with radius r_0 is::Delta v_1 = sqrt{ frac{2 mu}{r_0} - frac{mu}{a_1 - sqrt{frac{mu}{r_0At r_b the delta-v is::Delta v_2 = sqrt{ frac{2 mu}{r_b} - frac{mu}{a_2 - sqrt{ frac{2 mu}{r_b} - frac{mu}{a_1
The final delta-v at the final circular orbit with radius r_f::Delta v_3 = sqrt{frac{mu}{r_f - sqrt{ frac{2 mu}{r_f} - frac{mu}{a_2
Where a_1 and a_2 are the semimajor axes of the two elliptical transfer orbits and are given by::a_1 = frac{r_0+r_b}{2}:a_2 = frac{r_f+r_b}{2}
Transfer time
Like the Hohmann transfer, both transfer orbits used in the bi-elliptic transfer constitute exactly one half of an elliptic orbit. This means that the time required to execute each phase of the transfer is simply half the orbital period of each transfer ellipse.
Using the equation for the
orbital period and the notation from above, we have::T = 2 pi sqrt{frac{a^3}{mu
The total transfer time t is simply the sum of the time required for each half orbit Therefore we have:
:t_1 = pi sqrt{frac{a_1^3}{mu quad and quad t_2 = pi sqrt{frac{a_2^3}{mu
and finally:
:t = t_1 + t_2 ;
Example
For example, to transfer from circular low earth orbit with r_0=6700 km to a new circular orbit with r_1=14 r_0=93800 km using
Hohmann transfer orbit requires delta-v of 2824.34+1308.38=4132.72 m/s. However if spaceship first accelerates 3060.31 m/s, thus getting in elliptic orbit with apogee at r_2=40 r_0=268000 km, then in apogee accelerates another 608.679 m/s, which places it in new orbit with perigee at r_1=14 r_0=93800, and, finally, in perigee slows down by 447.554 m/s, placing itself in final circular orbit, then total delta-v will be only 4116.54, which is 16.18 m/s less.*"green|Forward applied ΔV"
*"red|Reverse applied ΔV"Evidently, the bi-elliptic orbit spends more of its delta-V early on (in the first burn). This yields a higher contribution to the
specific orbital energy and, due to theOberth effect , is responsible for the net reduction in required delta-V.ee also
*
Delta-v budget
*The Oberth effect References
External links
Wikimedia Foundation. 2010.