# Reasons for using a bi-elliptic transfer for Soyuz-ISS rendezvous

According to the ESA movie Soyuz rendezvous and docking explained, Soyuz uses a bi-elliptic transfer to move from the phasing orbit to the ISS orbit.

Why is this more complex transfer used instead of a Hohmann transfer?

• Have you really played the clip to the end? It's explained lucidly in the video: Hohmann transfer has the final burn $\Delta V$ too large and too risky in case of main propulsion failure. – Deer Hunter Dec 22 '14 at 21:30
• A related paper by Russkies' ballistics luminaries (Mourtazine, Petroff): dx.doi.org/10.1016/j.actaastro.2012.03.019 – Deer Hunter Dec 22 '14 at 21:35

First off, Hohmann transfers are a fiction. A very useful fiction, but a fiction nonetheless. Hohmann transfers are for transferring from one perfectly circular Keplerian orbit about a planet (or star) with a perfectly spherical mass distribution to another perfectly circular Keplerian orbit. There is no such thing as a perfectly circular Keplerian orbit. Orbits always have some eccentricity, and there are always perturbing gravitational bodies about. Moreover, the Earth does not have a spherical mass distribution; it most notably has an equatorial bulge. This alone pretty much ruins the idea of a Hohmann transfer in low Earth orbit.

Another problem with Hohmann transfers is that while they are great (in theory) for transferring to another orbit, rendezvous demands more than just reaching the ISS's orbit. The transfer has to reach an exact point on that orbit at an exact point in time. This is a rather demanding requirement. A Hohmann transfer (assuming they exist) would have to be performed at a very specific point in time, with very little margin for error.

A generalization of the concept of a Hohmann transfer is that of a two burn transfer. There are targeting algorithms (Lambert's problem) that solve for reaching an exact point in space at an exact time. Suppose the first burn is off by a little. A correction burn halfway through the transfer to ensure that the vehicle will still reach the target can be rather expensive. There are always correction burns because nothing in space ever goes exactly according to plan.

A three burn transfer is a tiny bit more expensive in terms of delta-V compared to the non-corrected two burn transfer. However, that midpoint burn can absorb a lot of slop without a huge delta-V penalty. This is one reason to prefer this slightly suboptimal solution (and it's only slightly suboptimal).

Another reason, as noted in the comments, is that the final burn of a two burn transfer can be rather large. Adding that intermediate point, coupled with making the transfer take a full orbit rather than half an orbit, can significantly reduce the final burn. The Space Station is a very, very expensive international asset. Expending a bit extra fuel to ensure the safety of the ISS is a worthwhile expense.

The ESA movie is wrong not only in the rendezvous part, but also in that the escape tower is separated AFTER booster separation. For details of Soyuz-ISS rendezvous see https://ac.els-cdn.com/S0094576512000914/1-s2.0-S0094576512000914-main.pdf?_tid=b6e06c3b-e4a3-45d0-b673-ae26d62e20bf&acdnat=1541585740_c7bc7567121073fd95235ad18c57d415