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The Hohmann Transfer is clear enough; Accelerate on your (circular) orbit at the right time, raising the apoapsis until it meets the target orbit. Once there, burn again to raise the periapsis - matching the target orbit, and as result speed of your target. If you picked the moment right, you're really close to the target, and ready for final approach and proximity operations. The only really tricky part is finding the right moment to start, so that the moment you're at your new periapsis the target is there too.

Now, how is the orbital transfer performed if raising the apoapsis can't be easily done in one short, neat burn? Say, you use a ion engine that will need two hours to create the delta-v necessary for the Hohmann transfer orbit, and your whole orbital period currently is forty minutes?

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You can change orbits by slowly spiraling out, but then it isn't a Hohmann transfer anymore. In the extreme case of very low thrust, this will mean your orbit is almost circular at all times. You'll also burn more delta-v than with the Hohmann transfer, because on average you add orbital energy while your speed is lower.

Wikipedia has this to say:

It can be shown that going from one circular orbit to another by gradually changing the radius costs a delta-v of simply the absolute value of the difference between the two speeds....

Such a low-thrust maneuver requires more delta-v than a 2-burn Hohmann transfer maneuver, requiring more fuel for a given engine design. However, if only low-thrust maneuvers are required on a mission, then continuously firing a low-thrust, but very high-efficiency (high effective exhaust velocity) engine might generate this higher delta-v using less propellant mass than a high-thrust engine using an otherwise more efficient Hohmann transfer maneuver.

Transfer orbit using Electrical Propulsion or Low Thrust enginees optimize the transfer time to reach the final orbit and not the delta-v as in the Hohmann transfer orbit. For geostationary orbit, the initial orbit is set to be supersynchronous and by thrusting continuously in the direction of the velocity at Apogee, the transfer orbit transforms to a circular geosynchronous one. This method however takes much longer to achieve due to the low thrust injected into the orbit.

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  • $\begingroup$ I was wondering about multiple accelerations when close to periapsis. It would take a long time, as the extending orbit period increases, and calculating the right time for rendezvous would be a nightmare, but it should be doable. $\endgroup$ – SF. Jun 21 '15 at 21:49
  • $\begingroup$ Accelerating only at periapsis should give a similar delta-v requirement to a Hohmann transfer, since energy is added at the same speeds, but as for time calculation you've reached the limit of my (mostly Kerbal Space Program-derived) knowledge. I suppose that for cargo for which delta-v is expensive even for a low-thrust engine but doesn't need to be delivered on a tight schedule (many tons of platinum ore mined from an asteroid?) periapsis kicks might be the optimal maneuver. $\endgroup$ – lirtosiast Jun 21 '15 at 22:00
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    $\begingroup$ This answer doesn't address the issue of rendezvous at all, which seems to be the main part of the question. $\endgroup$ – Chris Jun 22 '15 at 14:03
  • $\begingroup$ Chris, that's a good point. I assumed it was about orbital transfer based on "Now, how is the orbital transfer performed if raising the apoapsis can't be done in one short, neat burn?" but I will try to expand this answer to include differences in rendevous techniques. $\endgroup$ – lirtosiast Jun 22 '15 at 14:10

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