The Bi-elliptic Transfer is an orbital maneuver used to transition a spacecraft from one orbit to another. It involves transitioning through two elliptical orbits, making it a three-step process. Although it requires more time than a Hohmann transfer, it can sometimes save fuel for transfers between orbits with a large difference in radii.
v: Speed of an orbiting body,
μ: Standard gravitational parameter of the primary body, here Earth is the primary object, so the value is 3.986004418*105 Km3/s2
r1: Radius of the initial circular orbit,
r2: Radius of the final circular orbit,
rb: Common apoapsis radius of the two transfer ellipses and is a free parameter of the maneuver,
a1&a2: Semimajor axes of the two elliptical transfer orbits, which are given by
a1=r1+rb/2
a2=r2+rb/2
Bi-elliptic Transfer involves two half-elliptical orbits. Starting from the initial orbit, the first burn increases the spacecraft's delta-v, propelling it into a transfer orbit with an apoapsis far from the central body. At this apoapsis, a second burn places the spacecraft into a second elliptical orbit with a periapsis matching the desired orbit's radius. A final burn at this point then positions the spacecraft into the target orbit.
Although the bi-elliptic transfer involves one additional engine burn and generally takes more time than a Hohmann transfer, it can be more fuel-efficient when the ratio of the final to initial semi-major axis exceeds 11.94, depending on the chosen intermediate semi-major axis.
ΔV1 =√(((2μ / r1) - (μ/a1)) )- √((μ / r1))
ΔV2=√(((2μ / rb) - (μ/a2)) )- √(((2μ / rb) - (μ/a1)) )
ΔV3 =√(((2μ / r2) - (μ/a2)) )- √((μ / r2))
ΔVBi-elliptic=ΔV1+ΔV2+ΔV3
For Example : If r1 be 100 Km, r2 be 200 Km, rb be 300 Km, a1 be 400 Km, and a2 be 500 Km, and μ is 3.986004418*105 Km3/s2 then bi-elliptic transfer , ΔVBi eliptical is
ΔV1 =√(((2μ / r1) - (μ/a1)) )- √((μ / r1))
ΔV1 =√(((2*3.986004418*105/ 100) - (3.986004418*105/400)) )- √ ((3.986004418*105 / 100)))
ΔV1 =20.38469364
ΔV2=√(((2μ / rb) - (μ/a2)) )- √(((2μ / rb) - (μ/a1)) )
ΔV2=√(((2*3.986004418*105/300) - (3.986004418*105/500)) )- √ (((2*3.986004418*105 / 300) - (3.986004418*105/400)) )
ΔV2 =2.37594142
ΔV3 =√(((2μ / r2) - (μ/a2)) )- √((μ / r2))
ΔV3 =√(((2*3.986004418*105/ 200) - (3.986004418*105/500) )- Sqrt((3.986004418*105/ 200))
ΔV3 =11.82643876
ΔVBi-elliptic=ΔV1+ΔV2+ΔV3
ΔVBi-elliptic=20.38469364+2.37594142+11.82643876
ΔVBi-elliptic= 34.58707381 Km/s
