Pictured above is Robert Farquhar's route between EML2 and LEO. It's time reversible so it could be to or from EML2.
A .15 km/s burn at EML2 will drop a spacecraft to a perilune 111 km from the moon's surface. At this perilune the spacecraft is traveling nearly lunar escape velocity with regard to the moon and so enjoys an Oberth benefit. A .19 km/s perilune burn suffices to send the spacecraft earthward to a perigee deep in earth's gravity.
At perigee the spacecraft is traveling about 10.8 km/s, just a hair under earth escape. A burn at this perigee enjoys a huge Oberth benefit. A .6 km/s burn would suffice for Trans Mars Injection (TMI). So delta V from EML2 to TMI is (.15 + .19 + .6) km/s. I will round .94 km/s up to 1 km/s to give a little margin and also 1 is an easier number to type.
After TMI the EDS as well as it's payload is moving 11.5 km/s. To reuse the EDS we would need to return it to EML2.
Farquhar notes the trip from perilune to perigee takes about 140 hours. In that time the moon will advance 76º and the space craft 180º. So in my shotgun orbit simulator I set the perigee 104º ahead of the moon. My first try had pellets ranging from 10.7 to 10.9 km/s and then I'd narrow the blast to the pellets coming closest to the moon. After a few iterations I arrived at a perigee velocity of about 10.85 km/s. This gives an apogee of about 396,000 km and a period close to 2/5 that of the moon. After 50 days, the pellets return to a near moon fly by:
Thus braking about .6 km/s drop the EDS hyperbolic path to a trajectory where it will do a near moon fly-by after 50 days. At the near moon fly by it can do a .14 km/s burn for lunar capture. Then when it reaches an apolune near EML2, a .19 km/s burn to park at EML2.
Thus the EDS' delta V for returning to EML2 will be about 1 km/s.
This page still a work in progress, I'm getting good comments and info from a NASA spaceflight thread. Cryogenic boil off was an issue raised in that thread. A sixty day round trip goes well beyond what present hydrogen/oxygen upper stages do.
Revisiting Kirk Sorensen's EML2 thread I noticed he had posted a Farquhar illustration suggesting reusable booster stages could be returned to EML2!
The United Launch Alliance has done work on hydrogen/oxygen upper stages that could do longer missions. See Advanced Cryogenic Evolved Stage (ACES). Cyrogenic boil off might be mitigated by Multi Layer Insulation (MLI). Another cooling device is a Thermodynamic Vent System (TVS). Those who live in the southwest are familiar with "swamp coolers" where water soaked pads cool by evaporation. In a similar fashion hydrogen boil-off can be used to cool the cryogens. The hydrogen boil off can be vented in a specific directions and used for station keeping or attitude control.
Besides these passive thermal control systems the ACES might also utilize a two stage turboBrayton cryocooler.
"This design was based on the Creare NICMOS cooler that has been flying on the Hubble Space Telescope for the last ~4 years. The turboBrayton cycle uses GHe as the working fluid and this cooled gas can be easily distributed to the loads (i.e. the 22K and 95K shields). The ACES cryocooler configuration, shown in Figure 3-1, has 3 compressors in series and 2 expansion turbines in parallel, one for the 22K load, and one for the 95K load".
In this ULA pdf an ACES 41 propellant tanker has 5 tonnes dry mass and 41 tonnes propellent. I will be much more conservative in my hypothetical reusable EDS. A Centaur has 2.25 tonnes dry mass, 21 tonnes propellent and 99.2 kilo newtons. I will use the same but boost the dry mass to 5 tonnes for MLI, cryocooling, solar arrays, etc.
Specs for Hop's EDS
5 tonnes dry mass
21 tonnes hydrogen/oxygen
99.2 kilo newtons thrust
After the EDS sends the payload on its way, it will need 1 km/s of propellent of delta V to return to EML2. Exhaust velocity of hydrogen and oxygen is about 4.4 km/s. Exp(1/4.4) - 1 is about .255. To get back the EDS' 5 tonnes of dry mass we'd need 1.3 tonnes of propellent.
So for the first leg of the trip we have (21 - 1.3) tonnes of propellent or 19.7 tonnes. The first leg is also a 1 km/s delta V budget. With a 1 km/s delta V budget, 19.7 tonnes of propellent can do 19.7tonnes/.255. That's about 77 tonnes. But recall 6.3 tonnes is EDS dry mass plus propellent for the return trip. That's (77 - 6.3) tonnes of propellent available for payload. Let's call that 70 tonnes.
This little EDS could impart Trans Mars Insertion (TMI) to 70 tonne payload. Two of these EDS stages could send a 140 tonne payload on its way to Mars. Wilson and Clarke imagine a Mars Transfer Vehicle (MTV) of 130 tonnes.
Of the MTVs 130 tonnes, about 60 tonnes is propellent and consumables. If propellent, water and air are available from an asteroid or lunar volatiles, it would only be necessary to send the MTV's 70 tonne dry mass to EML2.
Wilson and Clarke also call for two EDS stages (they call them TMS -- Trans Mars Stages). Their stages are 110 tonnes and not reusable.
130 + 2*110 = 350. 350 tonnes to LEO for each (non reusable) conventional MTV. Vs 70 tonnes to LEO for an MTV that relies on extra terrestrial propellent and consumables. And an MTV departing from and returning to EML2 would have a much lower delta V budget. Making the MTV reusable would be much more doable.
The EDS would zoom through the perigee neighborhood very quickly. Would it have enough time to do the burns and enjoy an Oberth benefit?
The EDS and payload would spend about 54 minutes in the shaded region above.
A 70 tonne payload plus a 26 tonne EDS total 96 tonnes. The thrust of the engine is 99.2 kilonewtons. Acceleration is newtons/kilograms. 96/99.2 is ~.96. .96 meters/second^2 is about a tenth of a g.
Delta V imparted is acceleration * time of burn. Recall the perigee burn is about .6 km/s or 600 meters/second. We solve for t.
a * t = v
.96 m/s^2 * t = 600 m/s
t = 600/.96 seconds = ~620 seconds, a little over 10 minutes. The ten minute neighborhood just preceding perigee is all close to 10.8 km/s.
After separating from payload, the EDS and it's return propellent mass 6.3 tonnes. 99.2 kilonewtons divided by 6.3 tonnes is 15.75 meters/second^2 or nearly two g's. The deceleration burn to brake the hyperbolic orbit to an elliptical capture orbit would take about 40 seconds.
Near Earth Asteroid Retrieval
The Near Earth Asteroid retrieval described in the Keck Report uses xenon as a propellent. The exhaust velocity would be 30 km/s. What possible use could an EDS with a measly 4.4 km/s exhaust velocity be for such a vehicle?
Along with xenon's high exhaust velocity comes very low thrust. It would take nearly two years to spiral from Low Earth Orbit (LEO) to escape velocity. A good part of that long spiral would be spent in the Van Allen Belts. Low Earth Orbit also has a relatively high debris density.
Low thrust rockets don't enjoy any Oberth benefit. So the spiral from LEO to C3=0 would take about 7 km/s. Recall the exhaust velocity of the xenon rockets is around 30 km/s. Exp(7/30) - 1 is .26. Using an EDS would leave the asteroid fetcher with about 33% more xenon.
Many NEAs are much closer than Mars in terms of delta V. So perigee burn would be much less than .6 km/s for TMI, probably more often in the neighborhood of .2 or .3 km/s.
