### Author Topic: EML2  (Read 19373 times)

#### Hop_David

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##### EML2
« on: 05/08/2015 11:31 PM »
Due to EML2's proximity to the moon, I'm putting this in the moon section. I hope to expand on this as I have time.

An orbit's characteristic energy is

.5v2 - GM/r

Or sometimes C3 is used. C3 is twice the characteristic energy.

For this post, v is velocity wrt earth, GM is earth's gravitational parameter and r is distance from earth's center.

For earth surface I'm assuming from the equator which is moving about .5 km/s.

If nudged from the moon's Hill Sphere, an object from EML1 will fall into an ~100,000 km x 300,000 km ellipse. An EML2 object nudged from the moon's Hill Sphere will raise to an 440,000 x 1,800,000 km ellipse:

Characteristic energy is inversely proportional to semi-major axis. It can be seen from above pic that the ellipse from EML2 has a semi major axis about five times that EML1. So EML2 has about 1/5 the orbital energy of EML1. Both these have C3 much closer to zero than LEO or GEO. (LEO in this pic would be the size of a fly speck).

Being a Lagrange region, something parked at EML2 can stay there with very little station keeping.

Using a perilune burn, EML2 is about 9 days and 3.5 km/s from LEO:

EML2 is about 2.5 km/s from the lunar surface and about .4 km/s from an object in lunar DRO.

An object nudged from EML2 has an apogee of ~1.8 million km which is above earth's Hill Sphere (about 1.5 million km radius). Depending on timing of EML2 release and the nudge, the object might sail clear out of earth's influence into a heliocentric orbit. Or the the sun can bend the ellipse to a perigee deep in earth's gravity well:

Pellets in pic above are all nudged from EML2 with tiny burns varying by meters/sec. The blue pellets sail into heliocentric orbit.

Check out the #3 orange pellet -- it has a perigee deep in earth's gravity well. Using the sun to lower perigee, we can go from EML2 to a deep earth perigee with less than a .1 km/s nudge. An 6678 x 180000 ellipse has perigee velocity of 10.9 km/s. At this speed a .4 km/s suffices to send the pellet into an 11.3 km/s hyperbolic orbit for Trans Mars Insertion.

« Last Edit: 05/08/2015 11:42 PM by Hop_David »

#### gbaikie

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##### Re: EML2
« Reply #1 on: 05/08/2015 11:49 PM »
This seems to be about sending cargo [non crew] to Mars. Which is important- or most of the cost of manned
mars program.
But I like EML 1 or 2 in regards to getting crew fast to Mars. Which involves using oberth effect of swinging  near earth.
Can we have pretty pictures of that, too?

Edit: Oh, my mistake, I missed it, it seems this would take a long time for the crew- though I suppose one can dock with it on way in. Edit, edit, maybe crew could only go once through Van Allen belt on way to Mars- it's complicated though, not sure if possible or worth the risk. But with three passes thru belt could be launch crew to a highly elliptical and dock before passing thru Van Allen for second time, and third pass thru the belt on the outbound to leg to Mars.
In past I usually I think crew going to L-1 first. But thinking instead that crew launches from Earth and goes to say +120,000 km apogee and docks with in coming spacecraft before it reaches 30,000 km from Earth.
« Last Edit: 05/09/2015 12:29 AM by gbaikie »

#### sdsds

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##### Re: EML2
« Reply #2 on: 05/09/2015 04:22 AM »
Thanks very much for posting this! I have a question about the early part of your presentation, where you discuss the characteristic energy of various trajectories. It looks like your treatment includes only the gravitational potential due to the mass of the Earth. Is that realistic for Earth-Moon Lagrange points? What about the mutual potential energy of the object and the Moon?
-- sdsds --

#### Hop_David

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##### Re: EML2
« Reply #3 on: 05/09/2015 06:25 AM »
Thanks very much for posting this! I have a question about the early part of your presentation, where you discuss the characteristic energy of various trajectories. It looks like your treatment includes only the gravitational potential due to the mass of the Earth. Is that realistic for Earth-Moon Lagrange points? What about the mutual potential energy of the object and the Moon?

Once nudged loose, a payload fairly quickly gains distance from the moon and the moon's influence becomes negligible.

Nudged from EML1 payloads will speed ahead of the moon. So the moon will exert a backwards tug for a time. So the actual orbit has a slightly lower apogee but not a whole lot.

Nudged from EML2 payloads will lag behind the moon. So the moon will pull it forward for awhile resulting in a slightly higher orbit.

I've attached a screen shot from my orbit sim where pellets are nudge from from EML. The nudges range from 3 to 4 meters/sec.

Compare these orbits to what I pictured above and they're not too far off.

In my OP is a screen capture of payloads nudged from EML2. But these are hard to compare since the sun distorts the orbits a lot after they pass the Hill Sphere. Still, I think you can you can tell the apogees are about 1.8 million km.

#### Hop_David

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##### Re: EML2
« Reply #4 on: 05/09/2015 08:00 AM »
I like EML 1 or 2 in regards to getting crew fast to Mars.

Quick review of the Farquhar route:
.15 km/s drops payload at EML2 to an 111 km perilune.
.18 km/s perilune burn drops to a near earth perigee.
.5 km/s perigee burn does TMI. (11.3 hyperbola velocity - 10.8 perigee velocity)

EML2 to TMI is about 9 days and .9 km/s.

Let's imagine Impaler's MTV -- a 200 tonne Mars Transfer Vehicle (MTV) propelled by Hall Thrusters with acceleration of .000001 km/s^2. 11.6 days of acceleration gets 1 km/s delta V. It takes this ship about 3 months to spiral out of earth's gravity well. We don't want humans aboard, especially during the ~2 month spiral through the Van Allen Belts. It makes sense to send humans to dock with the MTV after it's climbed most the way out of earth's gravity well.

Radius of the MTV's spiral will eventually reach 327,000 km, the distance to the moon's Hill Sphere. So why not let the MTV spiral to EML1 and let the moon lend a hand getting the MTV to EML2? Park the MTV at EML2 where station keeping is inexpensive.

Send humans to EML2 to rendezvous with the MTV. From LEO to EML2 is 9 days and about 3.5 km/s.

As pictured in the OP, EML2 is pretty close to C3=0. But at a speed of 1.15 km/s, it doesn't enjoy much Oberth benefit. The MTV still needs to do another 3 km/s to reach a 1.52 aphelion (Mars' distance from the sun). At 1 km/s per 11.6 days, it will need about 34 days to achieve this 3 km/s.

Now let's imagine there's propellent and life support consumables at EML2. These might come from lunar cold traps or asteroids parked in a DROs. Stocking the MTV with EML2 water (for drinking, sanitation and radiation shielding) and EML2 oxygen to breathe would substantially reduce the gross lift off weight from earth's surface.

The MTV docks with a reusable Earth Departure Stage (EDS). This stage has a dry mass of 31 tonnes and carries 76 tonnes of lox/methane. At 107 tonnes the EDS is a little more than half the MTV's mass. Total mass at EML2 is now 307 tonnes.

EML2 to TMI is about .9 km/s and exhaust velocity of lox/methane is 3.6 km/s. e(.9/3.6) - 1 is .28. So getting this 307 tonne mass to TMI will take 68 tonnes of prop.

After TMI, the EDS separates from the MTV. It still carries about 9 tonnes of lox/methane. A .5 km/s braking burn slows it to an elliptical orbit with an apogee of about 1 lunar distance. After about 5 orbits the apogee nears the moon. At perilune .18 km/s injects it into a elliptical lunar orbit with apolune near EML2. At apolune .14 km/s parks it at EML2.

ELM2 to TMI is 9 days. TMI to the edge of the Hill Sphere is about 6 days. 34 - 15 = 19. 19 days isn't much of a savings but this is the outbound trip.

If I remember right, Impaler's MTV is reusable and returns to earth. Does it spiral back down to LEO? This would add another 7 km/s to it's delta V budget and 3 months to trip time. With the infrastructure I'm talking about it would only need to spiral to EML2 and then chemical rockets could take the astronauts from EML2 to LEO. To refuel the MTV, xenon would have to be delivered to EML2. But a xenon delivery tanker would be a small fraction of the mass of the MTV.

Impaler's MTV would be good for moving between heliocentric orbits but would suck at descending/ascending planetary gravity wells in a timely manner. It'd be well suited for traveling between EML2 and Deimos.

But I consider Mars colonization an unlikely fantasy if building space infra-structure gives no return on investment. Far more plausible in the near term is an asteroid retrieval vehicle as suggested in the Keck Report. At least Planetary Resources has a shot at enjoying ROI.

The Keck vehicles would take nearly two years to spiral from LEO to C3=0. For these more plausible vehicles the time savings could be huge.

#### Hop_David

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##### Re: EML2
« Reply #5 on: 05/09/2015 09:25 AM »
I've made a spreadsheet of Non-Hohmann transfers. Time can be shortened by shrinking the transfer ellipse's perihelion (which needs to be less than 1) or increasing transfer ellipse's aphelion (which needs to be more than 1.524).

Attached is a screen capture from a spread sheet where the Earth Departure Stage would do a 3.1 km/s perigee burn. Since falling from EML2 takes .4 km/s, the EDS would need a total delta V budget of about 3.5 km/s. This wouldn't be a reusable EDS.

Coming to Mars, a 4.67 km/s peri-aerion burn would capture the MTV into a 3697x23459 km ellipse. This is atmosphere grazing with apo-aerion at Deimos height. Orbital period is 13 hours, so about 2 periaerion drag passes per day. More drag passes per day as apo-aerion is lowered. I believe aerobraking could circularize the orbit within a week or so.

This transfer would take 3.4 months or about 103 days.

I've also attached the spreadsheet. A user can play with the transfer orbit by changing aphelion and perihelion.
« Last Edit: 05/09/2015 09:28 AM by Hop_David »

#### rklaehn

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##### Re: EML2
« Reply #6 on: 05/09/2015 10:06 AM »
Due to EML2's proximity to the moon, I'm putting this in the moon section. I hope to expand on this as I have time.

Pellets in pic above are all nudged from EML2 with tiny burns varying by meters/sec. The blue pellets sail into heliocentric orbit.

Check out the #3 orange pellet -- it has a perigee deep in earth's gravity well. Using the sun to lower perigee, we can go from EML2 to a deep earth perigee with less than a .1 km/s nudge. An 6678 x 180000 ellipse has perigee velocity of 10.9 km/s. At this speed a .4 km/s suffices to send the pellet into an 11.3 km/s hyperbolic orbit for Trans Mars Insertion.

Question: For a transfer from EML2 to TMI without chemical propulsion, would there still be some benefit from doing an earth flyby (orange trajectory), or would you just go straight to interplanetary space (blue trajectory)?

For the earth flyby, you would get at least some oberth effect for the part of the acceleration done in the earth gravity well. Also, you get to do all your acceleration at 1AU. But I am not sure if this is worth the additional complication and delay.

The orange trajectory is about 100 days, right?
Try the ISS 3D visualization at http://www.heavens-above.com/ISS_3D.aspx

#### Hop_David

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##### Re: EML2
« Reply #7 on: 05/09/2015 06:50 PM »
The orange trajectory is about 100 days, right?

I try to guesstimate these orbits using two body mechanics. First half of orange looks like a 450,000  x 1,800,000 ellipse which has a period of 140 days. Second half is a 6,678 x 1,800,000 ellipse which has a period of 100 days.

So I was going to tell you (100+140)/2 which is 120 days.

But then I thought why not focus the shot gun blast and do a screen capture as the pellets round perigee? The upper left of the square gives number of time increments. For this sim I had set the increments at 600 seconds.

The number in the upper left is 12111. 12111x600 seconds is 7266600 seconds or 84 days!

It looks like the sun has thrown these pellets back towards earth in a decidedly hyperbolic orbit.

To check I did another screen capture zoomed out. The way the pellets do a bee line out of earth's neighborhood suggest hyperbolas wrt earth.

Both screen captures attached.

#### sdsds

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##### Re: EML2
« Reply #8 on: 05/09/2015 07:41 PM »
That's a really neat class of trajectories! Does your pellet simulation tell you the end state of the pellets (i.e. position and velocity), so you could calculate their specific orbital energies to see if they are greater than zero?
-- sdsds --

#### Impaler

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##### Re: EML2
« Reply #9 on: 05/09/2015 09:00 PM »
Let's imagine Impaler's MTV -- a 200 tonne Mars Transfer Vehicle (MTV) propelled by Hall Thrusters with acceleration of .000001 km/s^2. 11.6 days of acceleration gets 1 km/s delta V. It takes this ship about 3 months to spiral out of earth's gravity well. We don't want humans aboard, especially during the ~2 month spiral through the Van Allen Belts. It makes sense to send humans to dock with the MTV after it's climbed most the way out of earth's gravity well.

Remember it was back and forth design, I provided the target acceleration rate, you specified ISP and targeted DeltaV to estimate 130 dry mass and 70 propellent, and a total thrust level of 200N, then I specified the X3 Hall thruster and the RAPDAR solar array which you found reasonable.  I believe the exact figures would work out too 100 tons payload, 22 tons for the dry propulsion systems, 70 ton of propellents, 8 tons for tank.  I do not recall if any propellent was allocated for returning from Mars.  Lastly I suspect a higher ISP then what was used may be desirable, though it will require more power to achieve the same acceleration the overall savings my be worth it.  Overall the vehicle concept is not unreasonable but could use more refinement and a more detailed analysis of it's propellent usage through the course of it's cycle between Earth and Mars.

Radius of the MTV's spiral will eventually reach 327,000 km, the distance to the moon's Hill Sphere. So why not let the MTV spiral to EML1 and let the moon lend a hand getting the MTV to EML2? Park the MTV at EML2 where station keeping is inexpensive.

Send humans to EML2 to rendezvous with the MTV. From LEO to EML2 is 9 days and about 3.5 km/s.

As pictured in the OP, EML2 is pretty close to C3=0. But at a speed of 1.15 km/s, it doesn't enjoy much Oberth benefit. The MTV still needs to do another 3 km/s to reach a 1.52 aphelion (Mars' distance from the sun). At 1 km/s per 11.6 days, it will need about 34 days to achieve this 3 km/s.

Now let's imagine there's propellent and life support consumables at EML2. These might come from lunar cold traps or asteroids parked in a DROs. Stocking the MTV with EML2 water (for drinking, sanitation and radiation shielding) and EML2 oxygen to breathe would substantially reduce the gross lift off weight from earth's surface.

The MTV docks with a reusable Earth Departure Stage (EDS). This stage has a dry mass of 31 tonnes and carries 76 tonnes of lox/methane. At 107 tonnes the EDS is a little more than half the MTV's mass. Total mass at EML2 is now 307 tonnes.

EML2 to TMI is about .9 km/s and exhaust velocity of lox/methane is 3.6 km/s. e(.9/3.6) - 1 is .28. So getting this 307 tonne mass to TMI will take 68 tonnes of prop.

After TMI, the EDS separates from the MTV. It still carries about 9 tonnes of lox/methane. A .5 km/s braking burn slows it to an elliptical orbit with an apogee of about 1 lunar distance. After about 5 orbits the apogee nears the moon. At perilune .18 km/s injects it into a elliptical lunar orbit with apolune near EML2. At apolune .14 km/s parks it at EML2.

ELM2 to TMI is 9 days. TMI to the edge of the Hill Sphere is about 6 days. 34 - 15 = 19. 19 days isn't much of a savings but this is the outbound trip.

I had assumed that hydro-lox would be the preferred propellent choice if some kind of lunar/asteroid ISPP is being used as water should be more abundant and easier to turn into propellents (simple electrolysis).  Carbon compounds while very likely to be present are likely to be in a whole slew of different forms requiring more complex synthesis.

Admittedly hydrocarbons are far easier to store so perhaps this is your motivation, but if the source is the lunar cold-traps then presumably hydrogen can be stored their passively with zero boil-off and the propellent delivery could be done via a 'launch on demand' directly to the waiting vehicle at EML2.

Finally the dry mass of the stage seems rather high, you said this stage is intended only for use in space so it should be free of most parasitic mass other then insulation to keep boil off to a reasonable level during the ~2 weeks in which it actively has propellent, I would think that 10% dry mass would be a perfectly reasonable figure, unless their are other demands being put on it that I don't know of.

I suspect that a simple increase in the propellent tank for the MTV will be able to fully replace the EDS.  At 3000s the MTV would need 10% propellent fraction to do 3 km/s.  Thus our 200 ton vehicle could just be 220 tons at EML2.  Now more propellent at LEO would be needed for it to push that larger mass up to EML2, as the DeltaV from LEO to EML2 is 7 km/s for low thrust vehicles that means a an additional 20% on top of the additional propellent to get the additional propellent to EML2, which is 5 tons.  So in totality the MTV needs 25 tons more IMLEO in propellent (Xenon) to replace the EDS.  And while I think the EDS presented above is too conservative and could be considerably lighter it will not be so light as to cost competitive with a simple enlargement of the MTV tanks, to speak nothing of the infrastructure costs of establishing in space propellent production.

If I remember right, Impaler's MTV is reusable and returns to earth. Does it spiral back down to LEO? This would add another 7 km/s to it's delta V budget and 3 months to trip time. With the infrastructure I'm talking about it would only need to spiral to EML2 and then chemical rockets could take the astronauts from EML2 to LEO. To refuel the MTV, xenon would have to be delivered to EML2. But a xenon delivery tanker would be a small fraction of the mass of the MTV.

Impaler's MTV would be good for moving between heliocentric orbits but would suck at descending/ascending planetary gravity wells in a timely manner. It'd be well suited for traveling between EML2 and Deimos.

But I consider Mars colonization an unlikely fantasy if building space infra-structure gives no return on investment. Far more plausible in the near term is an asteroid retrieval vehicle as suggested in the Keck Report. At least Planetary Resources has a shot at enjoying ROI.

The Keck vehicles would take nearly two years to spiral from LEO to C3=0. For these more plausible vehicles the time savings could be huge.

Upon return to Earth gravity well I would bring the vehicle back to EML2 and separate the 100 ton habitat leaving it at EML2, then take the drive-section (sooo Star Trek) and nearly empty tank down to LEO, re-propellent (yes I made that word up) it in LEO with all the propellent needed for the next journey and fly back to EML2 to rejoin the saucer section, I mean habitat.  This is functionally equivalent to the tanker you described but avoids the cost of another vehicle.  Crew would return to Earth via a fast taxi craft similar to what brought them to the MTV initially.

With regard to descending/ascending into planetary gravity wells, yes it is quite slow compared to chemical propulsion, but we would only carry the heavy habitat up the Earth gravity well once and unmanned as you described earlier.  Second the decent into Mars gravity well takes only ~1/3rd as long so at most a month is added to the journey each way for this maneuver.  Lastly we can choose to lower the ISP of the thrusters and increase thrust at the cost of propellent to get this maneuver done faster, it would be a trade off between extra propellent mass vs the same mass in shielding, which exposes the crew to less radiation.
« Last Edit: 05/09/2015 10:35 PM by Impaler »

#### gbaikie

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##### Re: EML2
« Reply #10 on: 05/09/2015 09:23 PM »
I've made a spreadsheet of Non-Hohmann transfers. Time can be shortened by shrinking the transfer ellipse's perihelion (which needs to be less than 1) or increasing transfer ellipse's aphelion (which needs to be more than 1.524).

Attached is a screen capture from a spread sheet where the Earth Departure Stage would do a 3.1 km/s perigee burn. Since falling from EML2 takes .4 km/s, the EDS would need a total delta V budget of about 3.5 km/s. This wouldn't be a reusable EDS.

Coming to Mars, a 4.67 km/s peri-aerion burn would capture the MTV into a 3697x23459 km ellipse. This is atmosphere grazing with apo-aerion at Deimos height. Orbital period is 13 hours, so about 2 periaerion drag passes per day. More drag passes per day as apo-aerion is lowered. I believe aerobraking could circularize the orbit within a week or so.

This transfer would take 3.4 months or about 103 days.

I've also attached the spreadsheet. A user can play with the transfer orbit by changing aphelion and perihelion.
Quote
Departure orbit radius   1   A.U.   Speed   6.283185307   A.U./yr
Destination orbit radius   1.524   A.U.   Speed   5.089643749   A.U./yr      4.740604351   AU/yr to km/s
Transfer orbit perihelion   0.7   A.U.
Transfer orbit aphelion   1.53   A.U.               2r1   2
Flight path angle at departure   0.36777381   Radians   21.07188715   degrees   Perigee burn (300 km altitude falling from high apogee)
Vinfinity at departure   2.375992013   A.U./yr   11.26363808   km/s   4.888082825   km/s

Flight path angle at destination   0.067838707   Radians   3.88687157   degrees   Periaerion burn to a 3697x23459 km ellipse
Vinfinity at destination   1.084314564   A.U./yr   5.140306341   km/s   2.568080059

Time of Flight   0.370249764   years   4.442997163   months   135.2337261   days

This appears to have flight time of 4.44 month, but I would guess that's it's at top of the perihelion orbit, and so one would do a patched conic, like you do with most Hohmann transfer to mars and thereby shorten the travel time, plus that would be adding some orbital speed so one is going closer to Mars velocity [mostly regarding it's orbital vector].
Anyways it's less delta-v than I expected.
And I think minimum travel time should be 3 months for crew.
Edit: I mean from the crew launching from Earth it should take 3 month or less to reach Mars [Though could  be less travel time but would take considerable more rocket fuel and it seem with enough shielding 3 months should fast enough so reduce microgravity effect and crew radiation exposure].

An alternative just do 1.524 [or less] and would be using more delta-v at Mars distance. So use a lot less at Earth and keep it to used later [using cryogenic rocket fuel a bit of problem] Or you look out right side window and mars is say 5 million km to your right and without rocket power Mars would stay ahead of you, so you using using rocket power to catch up. So being in the inside track, the rocket power not go ahead, but outwards to towards Mars [you will get closer and Mars gravity is helping a little]. Or this is simply a bigger patched conic adjustment. And carrying more fuel should reduce radiation effect more.
And as long as one does lose the rocket fuel somehow, it tends to give more abort options.
So one ends up using about same total delta-v, and one is just using it later. And should be about same time involved [about 3 months].
[[Edit perhaps one could use ion propulsion instead carrying a lot chemical fuel for last part of leg- one more than say a week of time to do the burn or don't need high thrust. That gives bonus of giving crew access to a lot of electrical energy. Though there seems like there is something wrong, or because now I am beginning to think maybe one could do whole thing with Ion engines. This is too close to a hohmann transfer delta-v and it seems it should cost more. There is cost to changing vector. And/or one has a "loss" or exchange of potential energy in regards to Sun's gravity- and no doubt in my mind the Hohmann is the most efficient transfer [in terms of delta-v, obviously, not referring to the amount of time it takes to get somewhere].
Or the only "answer" would seem to be one taking a lot orbital energy from Earth's orbit.
Always thought is was sort of like gravity assist, but more of a powered gravity assist.

Question re: the 2.321668688 km/sec  how much Oberth effect is added to the trajectory?

Quote
Transfer orbit perihelion   0.87   A.U.
Transfer orbit aphelion   1.524   A.U.               2r1   2
Flight path angle at departure   0.222899017   Radians   12.77117293   degrees   Perigee burn (300 km altitude falling from high apogee)
Vinfinity at departure   1.534703827   A.U./yr   7.275423642   km/s   2.321668688   km/s

Flight path angle at destination   0   Radians   0   degrees   Periaerion burn to a 3697x23459 km ellipse
Vinfinity at destination   0.750540983   A.U./yr   3.55801785   km/s   1.511760053
« Last Edit: 05/10/2015 01:10 AM by gbaikie »

#### Hop_David

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##### Re: EML2
« Reply #11 on: 05/09/2015 09:32 PM »
For the earth flyby, you would get at least some oberth effect for the part of the acceleration done in the earth gravity well.

Tried to do screen captures as red pellet #1 crosses lunar orbit on the way in and again as #1 crosses lunar orbit on the way out. My reflexes are slow so I reset time increment to 300 seconds. Capture attached.

4.86 days in cislunar space (the space between LEO and Lunar orbit). Rats, if crossing points are ~384,400 km, this looks like an ellipse with a mere 3 Lunar Distance (L.D.) apogee, still within the Hill Sphere. That doesn't make sense, maybe there's a flaw in my spreadsheets.

If crossing in the vicinity of lunar apogee looks like an ellipse with a 8 L.D. apogee or nearly double Hill Sphere radius. Hope that's the case.

So going with an ellipse a = 4 L.D., e=.9916. This would spend about 40 days within earth's Hill Sphere. Average velocity would be about .9 km/s.

Going with Impaler's 1 mm/s^2 acceleration, 20 days from edge of Hill Sphere to perigee would give about 1.7 km/s.

So let's look at the kinetic energy for 1/2 mv2… Screen capture attached.

Looks like the Oberth benefit gave us 1.53 mega-joules per kilogram.

#### Hop_David

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##### Re: EML2
« Reply #12 on: 05/09/2015 10:25 PM »
I've made a spreadsheet of Non-Hohmann transfers. Time can be shortened by shrinking the transfer ellipse's perihelion (which needs to be less than 1) or increasing transfer ellipse's aphelion (which needs to be more than 1.524).

Attached is a screen capture from a spread sheet where the Earth Departure Stage would do a 3.1 km/s perigee burn. Since falling from EML2 takes .4 km/s, the EDS would need a total delta V budget of about 3.5 km/s. This wouldn't be a reusable EDS.

Coming to Mars, a 4.67 km/s peri-aerion burn would capture the MTV into a 3697x23459 km ellipse. This is atmosphere grazing with apo-aerion at Deimos height. Orbital period is 13 hours, so about 2 periaerion drag passes per day. More drag passes per day as apo-aerion is lowered. I believe aerobraking could circularize the orbit within a week or so.

This transfer would take 3.4 months or about 103 days.

I've also attached the spreadsheet. A user can play with the transfer orbit by changing aphelion and perihelion.
Quote
Departure orbit radius   1   A.U.   Speed   6.283185307   A.U./yr
Destination orbit radius   1.524   A.U.   Speed   5.089643749   A.U./yr      4.740604351   AU/yr to km/s
Transfer orbit perihelion   0.7   A.U.
Transfer orbit aphelion   1.53   A.U.               2r1   2
Flight path angle at departure   0.36777381   Radians   21.07188715   degrees   Perigee burn (300 km altitude falling from high apogee)
Vinfinity at departure   2.375992013   A.U./yr   11.26363808   km/s   4.888082825   km/s

Flight path angle at destination   0.067838707   Radians   3.88687157   degrees   Periaerion burn to a 3697x23459 km ellipse
Vinfinity at destination   1.084314564   A.U./yr   5.140306341   km/s   2.568080059

Time of Flight   0.370249764   years   4.442997163   months   135.2337261   days

This appears to have flight time of 4.44 month, but I would guess that's it's at top of the perihelion orbit, and so one would do a patched conic, like you do with most Hohmann transfer to mars and thereby shorten the travel time, plus that would be adding some orbital speed so one is going closer to Mars velocity [mostly regarding it's orbital vector].
Anyways it's less delta-v than I expected.
And I think minimum travel time should be 3 months for crew.
Edit: I mean from the crew launching from Earth it should take 3 month or less to reach Mars [Though could  be less travel time but would take considerable more rocket fuel and it seem with enough shielding 3 months should fast enough so reduce microgravity effect and crew radiation exposure].

Indeed. I attached an illustration a transfer orbit with a .7 A.U. perihelion and a 1.53 aphelion. Vinfinities are indicated in red.

The aphelion is just a tad over Mars' radius so the flight path angle is only about 4 degrees.

The perihelion is way below earth's radius so the transfer orbit's flight path angle is 21º at intersection with earth orbit. The direction difference in these two vectors really jacks up Vinf. Ordinarily an 11.26 km/s vinf would be intolerable. But when dropping from a high apogee, perigee burn is only ~4.9 km/s. Kinda high but doable.

Turquoise is the area transfer orbit sweeps out from departure to destination, about 31.5% of the ellipse's area. The ellipse's semi-major axis is 1.115 A.U. It's period is 1.1153/2 years which is 1.177 years. 31.5% of 1.177 years is 4.44 months.

If you want something close to Hohmann, set perihelion at .999 and aphelion at 1.525.
« Last Edit: 05/09/2015 10:29 PM by Hop_David »

#### Hop_David

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##### Re: EML2
« Reply #13 on: 05/09/2015 11:16 PM »
Admittedly hydrocarbons are far easier to store so perhaps this is your motivation,

Yes. I looked for short routes back to EML2 that wouldn't take extra delta V. The shortest one I could find was 50 days. 8 or 9 tonnes of propellent would need to be stored until the final perilune and EML2 parking burns.

but if the source is the lunar cold-traps then presumably hydrogen can be stored their passively with zero boil-off and the propellent delivery could be done via a 'launch on demand' directly to the waiting vehicle at EML2.

If memory serves cold traps are 40 K and we need 20 K to stop hydrogen boil off. I do think the cold traps would drastically reduce the difficulty of refrigerating hydrogen though.

I like to imagine lox/hydrogen ships for delivering stuff from lunar surface to EML2 and back. But for longer trips I'm thinking methane. Unless the ULA propellent depot boys (Zegler, Kutter, & Barr) manage to reduce boil off. I give them better than even odds if they are funded. But perhaps I'm succumbing to the space cadet optimism I'm prone to.

Finally the dry mass of the stage seems rather high, you said this stage is intended only for use in space so it should be free of most parasitic mass other then insulation to keep boil off to a reasonable level during the ~2 weeks in which it actively has propellent, I would think that 10% dry mass would be a perfectly reasonable figure, unless their are other demands being put on it that I don't know of.

Thanks, you're gracious to say that. I was trying to err on the conservative side. Also high dry mass ratios make the vehicle sturdier and thus more amenable to re-use.

I suspect that a simple increase in the propellent tank for the MTV will be able to fully replace the EDS.  At 3000s the MTV would need 10% propellent fraction to do 3 km/s.  Thus our 200 ton vehicle could just be 220 tons at EML2.  Now more propellent at LEO would be needed for it to push that larger mass up to EML2, as the DeltaV from LEO to EML2 is 7 km/s for low thrust vehicles that means a an additional 20% on top of the additional propellent to get the additional propellent to EML2, which is 5 tons.  So in totality the MTV needs 25 tons more IMLEO in propellent (Xenon) to replace the EDS.

As mentioned, infra structure capable of making propellent could also provide water and oxygen. Drinking water, water for cleaning, radiation shielding, and oxygen to breathe could make up nearly half an MTV's mass.

So 125 tonnes more in IMLEO.

And while I think the EDS presented above is too conservative and could be considerably lighter it will not be so light as to cost competitive with a simple enlargement of the MTV tanks, to speak nothing of the infrastructure costs of establishing in space propellent production.

My EDS has ~30% mass fraction which makes it very durable. It never has to endure atmospheric re-entry. Round trip delta V budgets are .18 km/s.

A 30% dry mass fraction is more robust than the Falcon booster, lower delta V budget and it doesn't have to endure a 2 km/s re-entry. Rendezvous with prop depots and space craft could be slow and easy, no zooming towards a barge at terminal velocity and dealing with 9.8 meter/s^2 acceleration. This EDS is much more amenable for re-use than a booster.

An upper stage with payload will have about a 10% dry mass fraction, about as sturdy as a soda can. Now you're going to slam this foil cylinder into the atmosphere at 8 km/s. 16 times as much kinetic energy as a Falcon booster re-entry. Much, MUCH more difficult to re-use than a booster.

I don't expect reusable upper stages. I do believe Musk will be able to re-use boosters but there will be refurbishment and transportation costs.

If earth is the sole source of propellent, I expect Musk to cut launch costs by half, but not more than that.

#### Hop_David

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##### Re: EML2
« Reply #14 on: 05/09/2015 11:31 PM »
That's a really neat class of trajectories! Does your pellet simulation tell you the end state of the pellets (i.e. position and velocity), so you could calculate their specific orbital energies to see if they are greater than zero?

Sadly no. The two numbers in the upper right are number of time increments and orbital energy. Maybe I could do something with the second number but haven't figured out how.

My attempts at orbital simulations have been miserable failures. More knowledgeable folks have advised me I was using 1st order Runge Kutta methods when I should use 2nd or 3rd order.

The sims are JAVA that I stole from Bob Jenkins. Well, I didn't actually steal it as he gave me permission to use it. Jenkins' Java sims are one of my favorite toys.

Here is a page. My most recent operating system is paranoid of using JAVA from strangers. I had to jump through a few hoops setting system preferences so I could use my own pages, but it's doable.
« Last Edit: 05/09/2015 11:31 PM by Hop_David »

#### Impaler

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##### Re: EML2
« Reply #15 on: 05/10/2015 04:30 AM »
Unless the ULA propellent depot boys (Zegler, Kutter, & Barr) manage to reduce boil off. I give them better than even odds if they are funded. But perhaps I'm succumbing to the space cadet optimism I'm prone to.

I though the ULA integrated propellent management tech was a done deal, patented even.  It doesn't eliminate boil-off but it drops it a lot while also using the boil off as a power source and hugely reducing parasitic mass that is generally necessary to keep a stage 'alive'.  Add a bunch of sun-shades and your done, at least enough to do some kind of in-space tug which only holds propellent for a short time.  Now sending Hydrogen to Mars or creating a depot is another level of difficulty.

As mentioned, infra structure capable of making propellent could also provide water and oxygen. Drinking water, water for cleaning, radiation shielding, and oxygen to breathe could make up nearly half an MTV's mass.

So 125 tonnes more in IMLEO.

Life support closure is good and getting better all the time, see this paper on the current state of the art http://sites.nationalacademies.org/cs/groups/depssite/documents/webpage/deps_063596.pdf, it says the only 100 kg of make-up water is needed per person-year a very modest need (actually this may be out of date I recall that a very recent additional pyrolysis modules combines the CH4 and CO2 that are currently vented to make yet more water).  Conversely something like 700 kg of food, clothing and other manufactured goods are used so our consumables budget is already dominated by things that in-situ resource can not provide.  In fact it appears that learning to do laundry in space would reduce our consumable mass by more then getting water recycling to 100%, a very non-intuitive result.

Water is a good GRC shielding material per unit of mass, but because were effectively making it a structural part of the vehicle to remain there forever we only pay the cost to move this water from Earth once unlike propellents.  If the total habitat is going to mass 100 mT when fully loaded then I can't see more then a few tons of water for shielding because we should only try to shield from Solar Storm events, not GCR, even a full meter thickness of water will not do much to stop GCR's while having a tremendous mass.  Thus we would likely give our astronauts a lower total dose by just dropping the mass and going to the destination faster.  Overall the Radiation issue is one with a lot of zealotry around it driven by a combination of lack of data and Zubrin's constant attacks on NASA for not accepting higher risk.  I think the one thing people agree on is that were not going to use super massive thickness of shielding in space even if the mass was just sitting ready for us to use, we can't afford to push it around even with electric propulsion.

I don't expect reusable upper stages. I do believe Musk will be able to re-use boosters but there will be refurbishment and transportation costs.

If earth is the sole source of propellent, I expect Musk to cut launch costs by half, but not more than that.

I have the feeling your skepticism of upper stage reuse it driven by a need to justify the extraction of in-situ propellents.  In any case Musk already cut launch costs in half with F9, will cut them again with F9 Heavy, and will likely do it a third time with the 1st stage reuse that is almost assuredly going to happen.  Production of a super heavy class launcher even if it were to only have the 1st stage reused would likely drop costs again simply because so many launch costs do not scale with vehicle size making a larger launcher inherently more cost effective then a smaller one (so long as we compare commercial to commercial).
« Last Edit: 05/10/2015 07:18 AM by Impaler »

#### sdsds

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##### Re: EML2
« Reply #16 on: 05/10/2015 05:36 AM »
Let's imagine Impaler's MTV -- [...] with acceleration of .000001 km/s^2. 11.6 days of acceleration gets 1 km/s delta V. It takes this ship about 3 months to spiral out of earth's gravity well. [...]

Radius of the MTV's spiral will eventually reach 327,000 km, the distance to the moon's Hill Sphere.

I'd like to contribute to this thread a (still incomplete) analysis of the spiral out from LEO to the cislunar vicinity. (Long term I would like to be using something open source, but the free Mathematica for Raspberry Pi was sufficiently tempting.) Taking a hint from a discussion related to this on stackexchange I use a state vector of (r, v, gamma, phi} to make the acceleration in the direction of the velocity vector easy. The results below start from 400x400 km LEO and use the 0.001 m/s^2 constant acceleration mentioned above.

After 77 days the spacecraft has completed more than 345 revolutions around the Earth, reached an altitude of 323,657 km and a velocity of 1233.58 m/s. (The semi-major axis of the resulting orbit is 446,048 km, which certainly gets it within the Moon's Hill Sphere.)

[EDIT: 75 days is apparently enough. Added below is a revised trajectory showing 75 days of propulsion spiral and the resulting orbit, and also the (average) orbit of the Moon.]
« Last Edit: 05/10/2015 08:36 AM by sdsds »
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#### gbaikie

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##### Re: EML2
« Reply #17 on: 05/10/2015 08:58 AM »
Let's imagine Impaler's MTV -- [...] with acceleration of .000001 km/s^2. 11.6 days of acceleration gets 1 km/s delta V. It takes this ship about 3 months to spiral out of earth's gravity well. [...]

Radius of the MTV's spiral will eventually reach 327,000 km, the distance to the moon's Hill Sphere.

I'd like to contribute to this thread a (still incomplete) analysis of the spiral out from LEO to the cislunar vicinity. (Long term I would like to be using something open source, but the free Mathematica for Raspberry Pi was sufficiently tempting.) Taking a hint from a discussion related to this on stackexchange I use a state vector of (r, v, gamma, phi} to make the acceleration in the direction of the velocity vector easy. The results below start from 400x400 km LEO and use the 0.001 m/s^2 constant acceleration mentioned above.

After 77 days the spacecraft has completed more than 345 revolutions around the Earth, reached an altitude of 323,657 km and a velocity of 1233.58 m/s. (The semi-major axis of the resulting orbit is 446,048 km, which certainly gets it within the Moon's Hill Sphere.)

So it's delta-v of 6.6529 km/sec ?
77 * 24 * 3600 = 6.6529 million seconds.

Would that be a constant acceleration?
If so it requires storing electrical power when at night.
Unless one is in a Sun-synchronous orbit [polar orbit].
Or it's getting power beamed from Earth or something.

Without storing energy, getting beamed energy, having nuclear power, or Sun-synchronous,
Then when at lower orbit one only gets solar energy for about 60% of 24 hr day.
If thrust only during sunlight it might get some weird orbit. Instead you could burn for say
10 min of the 90 min orbit, which should roughly give an elliptical orbit- raising the apogee.
So .6 of 90 mins is 54 mins. And could charge batteries for say 44 mins and use solar power
and battery power for the 10 mins of thrust.
So in beginning it's one only burn for 1/9th the time, but it should raise the orbital height of
apogee in a more efficient manner. Or roughly take instead 9 times longer is would be about 5 times
longer [with same thrust level].
But after raise the apogee high enough, one will have more of the 24 hour day in sunlight. So
it could become more significant once got to say 1000 km height. So one could then switch back to
constant thrust at time during more daylight- say when apogee is say somewhere around 1000 to 2000 km.

If one could get constant thrust that would get you 300,000 km quicker. But if had only about 60% sunlight
at lower orbit then 10 min burn should rise almost as fast [or if had more thrust [the ion rocket could provide more thrust if provided with more electrical power solar + battery] then could be faster.  But even 10 min thrust period doesn't get one higher quicker orbit in time it would require less total delta-v [use less Xenon].
« Last Edit: 05/10/2015 09:05 AM by gbaikie »

#### Hop_David

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##### Re: EML2
« Reply #18 on: 05/10/2015 01:25 PM »
Let's imagine Impaler's MTV -- [...] with acceleration of .000001 km/s^2. 11.6 days of acceleration gets 1 km/s delta V. It takes this ship about 3 months to spiral out of earth's gravity well. [...]

Radius of the MTV's spiral will eventually reach 327,000 km, the distance to the moon's Hill Sphere.

I'd like to contribute to this thread a (still incomplete) analysis of the spiral out from LEO to the cislunar vicinity. (Long term I would like to be using something open source, but the free Mathematica for Raspberry Pi was sufficiently tempting.) Taking a hint from a discussion related to this on stackexchange I use a state vector of (r, v, gamma, phi} to make the acceleration in the direction of the velocity vector easy. The results below start from 400x400 km LEO and use the 0.001 m/s^2 constant acceleration mentioned above.

After 77 days the spacecraft has completed more than 345 revolutions around the Earth, reached an altitude of 323,657 km and a velocity of 1233.58 m/s. (The semi-major axis of the resulting orbit is 446,048 km, which certainly gets it within the Moon's Hill Sphere.)

[EDIT: 75 days is apparently enough. Added below is a revised trajectory showing 75 days of propulsion spiral and the resulting orbit, and also the (average) orbit of the Moon.]

Very interesting! By gamma you mean flight path angle? What's phi?

Could you explain how you did that? I would like to run similar analysis but presently don't know how.

There's a free version of Mathematica?

#### sdsds

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##### Re: EML2
« Reply #19 on: 05/10/2015 10:35 PM »
Responding to the last topic first:

There's a free version of Mathematica?

Yes. See Stephen Wolfram's blog post from November 21, 2013:

Putting the Wolfram Language (and Mathematica) on Every Raspberry Pi

This was only a little bit interesting back then, when the available Pi (model 1) had about as much computational power as a cell phone. But now that a quad-core Raspberry Pi 2 is available for \$35, and the "raspbian" operating system is easy to install and includes Mathematica...? For those comfortable using  the X Window System (and even "ssh -X") it is pretty enticing.

I've only been using it for a week or so, and have been meaning to start a, "Flying Mathematica to Cis-Lunar Destinations" thread, but you beat me to it with this thread!
« Last Edit: 05/10/2015 10:36 PM by sdsds »
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