Zero energy trajectories - a thread to discuss this topic

Bill White · 2004-03-25 14:47:36

Here are some links on zero/low energy trajectories:

[http://en.wikipedia.org/wiki/Interplane … perhighway]Wikipedia, to start

[http://www.gg.caltech.edu/~mwl/publicat … ateway.pdf]AIAA Paper

And quotes from a member of the NASA Genesis team:

J.B. What have you found to be the most fascinating thing about the Genesis mission? What new science understanding will Genesis provide?
D.S. The most fascinating thing about Genesis is the trajectory… it is like we are flying around the edge of an enormous potato chip… I am excited that Genesis will provide a new understanding about the composition of our Sun from the solar wind samples we collect, and how that will help us understand why our Earth is different from the Sun and different from the other planets in our solar system.
J.B. What do you see as the riskiest part of the Genesis mission?
D.S. The two riskiest parts are right at the start and right at the end. At the start we must do a big midcourse correction just 48 hours after launch to make sure the spacecraft gets to the Earth-Sun libration point, L1. At the end we must get the spacecraft to return to a tiny spot in the Utah desert and then snag it with a helicopter before it hits the ground.

Snag it with a helicopter? Now that is cool!

The question I have is, can we routinely transit from the Earth - Sun libration point (L1) to the Mars - Sun libration point (M1?) while spending very little delta V? If true, cargo trips to Mars have become very much easier since M1 to Mars needs very very little delta V, right?

A NERVA class tug could throw hundreds of tons to Mars as fast as HLLV can throw the stuff up, right?

Please, if this is DEAD WRONG, someone please tell me!

SBird · 2004-03-25 15:24:33

From everything that I know about the subject (the links you gave above pretty much encompass all I know), you are correct. The hard part is getting to lunar L1 if you are going to Mars. Coming back, the hard part is getting to Martian L1. Everything else, including going from Martian L1 to Mars/LMO and from lunar L1 to Earth or LEO is basically free.

IMO, the most interesting thing about Genesis is the trajectory. Honestly the composition of the solar wind is sort of interesting but I think that we've gota pretty good idea what is is already. The demonstration of the low energy trajectories is invaluable, though.

I love how the 48 hour course correction is the ONLY major thrusting they have to do aside from station keeping and a couple of minor mid-course corrections. Basically, one thrust is enough to go from LEO to Earth L1, sit there for a 2.5 years, loop back past Earth L2 and land in Utah.

With this technology, getting from any lunar or Earth L point is a matter of delta Vs in terms of m/s, not km/s.

I or someone here needs to contact MArtin Lo and find out from him if any serious work has been done to look at Earth Mars trajectoreis and if we might be able to help out by donating computing power.

Bill White · 2004-03-25 15:48:27

This stuff might not be all that [http://www.marsinstitute.info/rd/facult … /in04.html]new.

SBird · 2004-03-25 16:00:05

Check [http://groups.msn.com/DaveDietzler/thel1gateway.msnw]it out! A nice summary page.

Heres some info from there about getting from LEO to lunar L1:
Delta V from 200 km LEO to lunar L1 - 3116 m/s + 50m/s stabilization burn at L1.
Delta V from 500 km LEO to lunar L1 - 3042 m/s + 50-100 m/s burn.
travel time: 4 days.
travel time with NEP: 6-7 days?! (I'd like to confirm this number, it seems very low)
Delta V from lunar L1 to earth L1 or L2: 50 m/s

LEO to Mars could therefore require a total delta V of something like 3.5 km/s of delta V. That's a full 1km/s we don't have to spend. Furthermore, the fuel/mass ratio is exponential so that going from a 4.5 km/s to 3.5 km/s Mars transit and a 450 Isp H2O2 engine would bring our non-fuel fraction of the spacecraft from 36.1% to 45.2% or a 25% increase in usable cargo. Alternately, you can view it as effective increase in engine Isp from 450 to 580

SBird · 2004-03-25 16:02:14

I think that overall idea is old but that the computational means to actual realize them is very recent.

Bill White · 2004-03-25 17:04:33

I googled "Sun-Mars libration point" and seemed to have hit some good stuff.

One possible article:

[http://www.stk.com/pdf/white_papers/070 … vities.pdf]http://www.stk.com/pdf/white_papers/070 … vities.pdf

Another:

[http://www.stk.com/pdf/white_papers/020 … ib_pts.pdf]http://www.stk.com/pdf/white_papers/020 … ib_pts.pdf

Here is a book that is said to include a section on "Sun-Mars Libration Points and Mars Mission Simulations"

[http://www.amazings.com/sbb/reviews/review0485.html]http://www.amazings.com/sbb/reviews/review0485.html

$330.00 at amazon.com (free shipping!) yet it may be at university libraries.

Another:

[http://www.marsinstitute.info/rd/facult … /in05.html]http://www.marsinstitute.info/rd/facult … /in05.html

An ISV slowly orbiting the Earth-Sun L-1 point, 1.5 million kilometers in toward the Sun, could fire its rockets for a short time to leave the L-1 point, then perform gravity-assist swingbys of the moon and Earth to place itself on course for Mars, thus saving large amounts of propellant.

SBird · 2004-03-25 17:31:10

[I read through the second paper and although I don't understand all the terminology (what's C3?) I think I got the general gist. This isn't the low energy trajectory in use. They're using Mars' L1 and L2 to hold communication satellites. They use a low Mars flyby in combo with a braking maneuver to lower the fuel consumption. The principles they use for this braking maneuver are the same ones used on the low energy trajectory and demonstrate its usefulness. However, they are getting there through standard orbital mechanics.

John Creighton · 2004-03-26 10:05:30

I noticed one of the papers was titled invariant [http://mathworld.wolfram.com/Manifold.html]manifold [http://www.control.lth.se/~funonlin/200 … slides.pdf](1) [http://www.math.vt.edu/people/renardym/ … ode18.html](2). I understand a center manifold to be a subspace for which trajectories either approach or diverge from. One special case of a manifold is a [http://mathworld.wolfram.com/LimitCycle.html]limit cycle. A limit cycle is a closed trajectory for which other trajectories are either attracted two or diverge from. Approximate solutions to limit cycles can be found using [http://www.ee.unb.ca/jtaylor/Publicatio … _final.pdf]describing function analysis. I wonder if I can find approximate solutions to some of the zero energy trajectories using describing functions.

RobS · 2004-03-28 09:26:41

I have just read the Wikipedia article (vague and doesn't seem very reliable) and the IAAA paper (meatier). I don't see much that's new. A Hohmann trajectory to Mars is 3.8 km/sec, and 3.2 km/sec of that is for escaping the Earth. You have to do that to get to L1 anyway. With cargo, though, one could slowly lift it to lunar-Earth L1, then transfer it by ion tug to Earth-solar L1, then send it on a gravity assist past the moon and Earth, and MAYBE that will be enough to send the cargo vehicle to Mars by, essentially, a Hohmann trajectory. That saves a lot of fuel but adds a lot of time.

If you want to send people this raises the transit time from six months to maybe a year, so it isn't practical. You could save almost as much energy and send the Mars ship to Venus, then gravity assist to Mars.

I have yet to see anything that says Earth L1 to Mars L1 is "free." That's why I called the idea an "interplanetary wormhole." It still looks like that to me. Once someone "shows me the beef" (the lower delta-v) I'll change my tune.

-- RobS

SBird · 2004-03-28 14:36:01

RobS - your concerns are well founded. I've been looking for some sort of numbers for the Moon-Mars L1 shunt and haven't been able to get any. One big advantage is that very small Delta V allows you to do a low speed Mars surface landing without having to expend delta V on Mars orbital insertion or aerobraking. the 3.8 km/s assumes an aerobraking maneuver which is dangerous in the Martian atmosphere with the variable density of the atmosphere and even more dangerous with large loads. A direct insertion burn is much preferrable and would add considerably to the total mission delta V. the 0-energy trajectory should be safer and allow you to stay < 3.5 km/s. OTOH, the orbital trajectories are going to be much more stringent, relying upon chaos effects.

Also, if we set up some sort of lunar L1 space station, we can refuel spacecraft there and have the ability to send mass to both Mars and the Moon with the same starting location. The moon is particularly accesible from L1. Every point on the moon is accessible with minimal delta V in under 12 hours from L1. That would allow the use f space tugs to carry freight to L1, lowering the fuel requirements for the spacecraft and greatly lightening the HLV cargo capacity requirements.

I've been too busy to E-mail the originating professor. I think that that's the way to go, though. It looks as if he's at least run some rough numbers on the subject.

SBird · 2004-03-28 14:46:32

To add, I believe that there's an Earth-Mars trajectory that takes off straight from lunar L1. (there's an interesting if information poor animation on the professor's Marton Lo's website that can be found about halfway down [http://www.gg.caltech.edu/~mwl/communic … tions2.htm]this page.)

I don't know how accurate this anim is but I'm assuming that since they went to the trouble of animating it, they've run a simulation demonstrating the trajectory.

Of course, this wouldn't be used for moving personnel but it's got a lot of utility for moving heavy cargo to the rest of the solar system. The L1 departure point is still a fairly good location for departing astronauts, however, as it allows creew modules to geta refuel before departing on a fast Hohmann trajectory.

RobS · 2004-03-28 14:56:04

Here are a few numbers that might make my concerns clearer. They come from Arthur C. Clarke's *The Promise of Space,* published in 1968, page 219.

Let us say we have a spacecraft orbiting the sun in an orbit identical to the Earth's, but not near the Earth (outside its gravity well) and we want to fly it to the orbit of Mars. The spacecraft is orbiting the sun at 66,000 miles per hour (29.5 km/sec). To go to Mars we have to raise the aphelion by about 50 million miles or 80 million kilometers. A delta-v of 7,000 mph (3.1 km/sec) raises our speed around the sun to 73,000 mph (32.6 km/sec) and the spacecraft will fly away from the sun until the gravity slows it enough to pull it back. At that velocity, our aphelion will be the orbit of Mars. When we get there we will be moving at 48,000 mph (21.5 km/sec) and will then fall back to the orbit of Earth. If we want to stay in the orbit of Mars we need a circularization delta-v of 6,000 mph (2.7 km/sec) to raise our orbital speed around the sun to 54,000 mph (24.2 km/sec).

These are Clarke's numbers. The situation is different if we are departing from a deep gravity well and arriving into another gravity well. Leaving low Earth orbit, a delta-v of 3.8 km/sec will take us away from the Earth AND give us a hyperbolic excess velocity of 3.1 km/sec, so we combine the escape velocity and the delta-v in solar orbit into one manuever. Arriving at Mars, aerobraking accomplishes both for free.

Maybe you see why I am skeptical about "free" orbits between planets. The sun's gravity dominates everything outside the lagrange points. Earth's and Mars's gravity together will not neutralize the sun's gravity in the space between those worlds. A spacecraft going from the Earth to Mars HAS TO rise AGAINST the sun's gravity. The lagrange points are not magical entry points into tunnels where the sun's gravity is canceled out for millions of kilometers.

But this is not to say the lagrange points are of no use. On the contrary. You may recall that my Mars-24 plan called for departure for Mars from the earth/moon L1.

-- RobS

John Creighton · 2004-03-28 15:58:43

I hope I understand this right.

“The Genesis trajectory (see Figure 6.1(a)) was designed using
dynamical systems theory. The three year mission, from launch all the way to Earth return, requires only
a single small deterministic maneuver (less than 6 m/s) when injecting onto the halo orbit.” (from section 6 of [http://www.gg.caltech.edu/~mwl/publicat … eeBody.pdf](1))

Thus to get to earth L1 from LEO only requires a delta v of 6 m/s. However because systems exhibiting chaos are very sensitive to initial conditions some fuel is required to make corrections.

“ The most important error in the launch
of Genesis is the launch velocity error. The one sigma expected error is 7 m/s for a boost of nearly 3200
m/s from a circular 200 km altitude Earth orbit. Such an error is large because halo orbit missions are
extremely sensitive to launch errors. Typical planetary launches can correct launch vehicle errors 7 to 14
days after the launch. This correction maneuver is called TCM1, being the first TCM of any mission. In
contrast, for orbits such as the Genesis transfer trajectory, the correction maneuver ?V grows sharply in
inverse proportion to the time from launch. For a large launch vehicle error, which is possible in Genesis’
case, the TCM1 can quickly growb eyond the capability of the spacecraft’s propulsion
“(from section 6 (The Trajectory Correction Maneuver (TCM) Problem) 0f [http://www.gg.caltech.edu/~mwl/publicat … eeBody.pdf](1))

Thus more fuell is needed to correct for errors when going from LEO to earth L1 then is needed to go from LEO to L1.

“Moreover, for all cases that we investigated, the optimal costs are well within the
?V budget allocated for trajectory correction maneuvers (450 m/s for the Genesis mission). The cost
function is very close to being linear with respect to both TCM1min time and launch velocity error. Also,
the halo orbit insertion time is always close enough to that of the nominal trajectory as not to affect
either the collection of the solar wind or the rest of the mission.”(from section 6 (Halo Orbit Insertion (HOI) Problem.) 0f [http://www.gg.caltech.edu/~mwl/publicat … eeBody.pdf](1))

Thus 450 m/s is more then sufficient fuel to go from LEO to earth L1.

John Creighton · 2004-03-28 16:26:41

As I understand it one a space ship reaches earth L1 it can go to earth L2 for practically free. I am not sure what velocity it has when it reaches earth L2 but I think if the initial conditions are right it can probably continue all the way to mars L1. The space craft must pick up some velocity when traveling by earth (I think?). A space craft can also go from earth L1 to lanner orbit with practically know fuel. This is called a ballistic capture. There was a Japanese probe called Hiten, that used some of these principles. (see [http://www.gg.caltech.edu/~mwl/publicat … Energy.pdf]Low energy transfer to the moon). I am curious how much delta V it takes to land on the moon from lunar orbit.

John Creighton · 2004-03-28 16:33:06

Martin Lo uses a planner three body model ((earth, sun, spacecraft) or (earth,moon, spacecraft) when developing the theory behind the trajectories. Thus the theory for going from earth to mars should not depend on the position of the moons. Thus there should be at least one possible trajectory every year. I suspect if these manifold trajectories are too slow the space craft could start on one and due some kind of elliptical trajectory once the craft was past earth L2.

RobS · 2004-03-28 17:39:08

The quotations you give say it "requires only
a single small deterministic maneuver (less than 6 m/s) when injecting onto the halo orbit." Note the WHEN INJECTING ONTO THE HALO ORBIT."

The second quote says :The most important error in the launch of Genesis is the launch velocity error. The one sigma expected error is 7 m/s for a boost of nearly 3200
m/s from a circular 200 km altitude Earth orbit."

Note that FIRST you boost the spacecraft from low Earth orbit to 3200 meters per second (escape velocity!) then you need a 7 /sec correction. You still need 3200 m/s to get fropm low earth orbit to the halo orbit, but the only minor corrections to keep it there.

There ain't no such thing as a free lunch.

-- RobS

RobS · 2004-03-28 19:13:14

Here's a very useful article by the expert, Martin Lo, titled "Lunar Sample Return via the Interplanetary Superhighway." It was on his website. He gives these figures for Low Earth Orbit (200 km, 23.5 degree inclination) to Aitken Basin (180 degrees longitude, 57 degrees south latitude):

Direct "traditional" route:

Earth escape: 3100 m/s
Lunar orbit insertion: 979 m/s
Apoapsis burn: 23 m/s
Landing burn: 1703 m/s

The last three total 2705 m/s

To return to the Earth directly is tricky because you're taking off from the back side; he says its 3200 m/sec

Travel time to moon is 4 days.

On the other hand, this is better:

Fly from LEO to the Earth/sun L1 point: delta-v, 3193 m/sec
Ride the earth/moon/sun gravity "ridge" to the earth/moon L2 point (behind the moon): 60 m/sec
Injectin into a "halo orbit" around Earth-moon L2: 13 m/sec
Leave a communications satellite there to maintain permanent communications with far side
Land on moon: delta-v, 2325 m/sec
Flight time: 1 year

Lunar surface to L2: 2424 m/sec. From there to the Earth is 50 m/sec, but it seems to take about six months to get back to the earth. This is a big improvement over 3200 m/sec, though, and has two advantages: it involves a single burn rather than lunar orbit insertion and then trans-Earth injection, and the communications satellite provides continuous communications between the entire far side and Earth.

Lo also says that from Earth to Earth-Lunar L1 or L2 is 3100 m/sec of earth escape plus about 600 m/sec to enter into a halo orbit around the point (you never go "to" it, only around it). I think that's high; somewhere I have seen that one can use the moon to provide a gravity assist to get to L1 for 200 or 300 m/sec.

-- RobS

Bill White · 2004-03-28 20:57:08

RobS - another potential benefit of using L1 for Mars (even if the moon proves unable to yield meaningful quantities of H2 or O2) is to use solar ion to slowly spiral a cargo module to L1 and then push the cargo for a lunar fly-by and Earth fly-by followed by chemical rocket ignition as you describe in your novel.

At a minimum, this could be used for the pre-positioned ERV/Sabatier portion of Mars Direct, IF a solar ion tug were already deployed.

The solar ion tug would follow a different lunar fly-by trajectory with the goal of returning to LEO and another cargo. If there is NO useable H2 on Luna, we would only need a small handful of tugs (maybe just one) yet we could still leverage far more mass to Mars that way.

Its not free - - yet to use a solar ion tug that never leaves Earth orbit to ferry cargo to L1 would seem to greatly enhance the deliverable payload from a single Ares launch. After all, the solar ion tug collects sunlight and expends xenon rather than rocket propellant, right?

Could a 120 ton Mars vessel meet an ultra-light fast CEV style crew taxi at L1 and start a MarsDirect style mission there?

= = =

Also, look at Martin Los work concerning a mini-grand tour of the Jovian moons. Its all just real careful use of large numbers of fly-bys yet he has a plan that allows careful study of 8 Jovian moons with essentially ZERO propellant.

= = =

One NERVA tug could push payloads to L1 for lunar / Earth fly-bys and chemical ignition and the net payload increase to Mars would seem significant, right? Dozens or NERVAs of VASIMIR or gas core nuclear would be nice, yet given proliferation concerns, one and only one NASA owned and Pentagon operated NERVA tug could still prove really useful.

SBird · 2004-03-28 23:51:18

RobS - Again, I share your concerns. Of course, there isn't a free lunch here, just cheap ones. My biggest concern is, of course the process of getting to Mars L1 without having to spend all the energy fighting the Sun's gravitational field.

It's roughly analagous to a chemical reaction where you have two similar energy states seperated by a high energy activation barrier. This system seems to act as a catalyst that lowers that energy barrier, making it eaiser to get between the 2 equivalent energy states but I'm a bit unclear on HOW. I can easily understand how lunar L1 can easily feed to Earth L1 but Lunar L1 to MArs L1 just seems almost too good to be true.

I guess that I'm just relying on faith here - Dr. Lo has managed to successfully design a NASA mission that's almost finished and these 0-energy trajectories seem to be hot stuff these days. I assume that if they didn't work, someone would have raised an objection by now.

A qualitative analysis of the Genesis orbit seems to suggest that the L points basically let you hit any point in the solar system for nearly free. After all, after the inital orbital injection for Genesis was enough to send the probe to L1 and back to Earth.

Of course, getting to lunar L1 is expensive, I've maintained that all along. However, the advantage is that the equivalent to the OCS thrusters is enough for the rest of the mission. Even a simple refuelable chemical thrust tug could ferry stuff to L1 and then detach. The remaining engines on your cargo can be tiny along with smaller fuel tanks, etc. The overall weight savings will be substantial. All those heavy, expensive high delta V engines and tanks can go back to LEO for reuse. You therefore limit your high delta V mission requirements to the Earth-L1 corridor. I think that there will be significant infrastructural savings from this approach. Also, you can share booster tug elements with the faster crew and cargo missions. They can go to the lunar L1 with similar sorts of tugs (fast and slow) where they can depart on standard Hohmann trajectories.

(I am aware that the L1 requires a halo orbit for stability, I was leaving this out for succintness. I think that even L4 and 5 have a sort of halo orbit that forms from the Coriolis stabilization)

I'm not too concerned about Mars to Mars L1 since I don't anticipate large cargo loads going this way for the time being.

Rxke · 2004-03-29 00:40:51

Several tugs are in the making by private companies, like Orbital recovery: [http://www.orbitalrecovery.com/]http://www.orbitalrecovery.com/

I took them as an example because they have a particular nice design: the tug fits snugly in the docking collar of a launcher, taking up no additional space... Estimated first launch: 2007...

SBird · 2004-03-29 00:51:24

That's a really cool idea - using the apogee kick motor as a docking point. Of course the eventual tugs will be orders of magnitude bigger but this sort of experience will be invaluable. (NASA should have made one of these things 10 years ago)

Now that O'Keefe is pushing to have Hubble be rescued by a robotic mission (WTF?!), it might provide some impetus to finally create a decent telerobotics presence in LEO. There's no reason to send astronauts to do work that someone can do from the ground.

Rxke · 2004-03-29 00:53:32

I can easily understand how lunar L1 can easily feed to Earth L1 but Lunar L1 to MArs L1 just seems almost too good to be true.

Isn't it 'just' a case of falling towards the center of the gravity well (sun) at an oblique angle, and in that way winning momentum, afterwards you 'coast' towards your goal, (indirectly, maybe with additional gravity assists) losing that momentum again, because you 'climb'... This would explain the slow arrival speed at Mars? The whole idea is that you can precicely direct your original push from any given libration with very little propellant towards where you want to 'dive'?

So it's 'just' some kind of fancy juggling with gravity assists if you want to climb, but from libration point to libration point...

Rxke · 2004-03-29 00:56:27

Now that O'Keefe is pushing to have Hubble be rescued by a robotic mission (WTF?!)

Guess what... [http://www.spaceref.com/news/viewsr.html?pid=10083]http://www.spaceref.com/news/viewsr.html?pid=10083

SBird · 2004-03-29 01:30:12

Re: 0-energy trajectories:
I don't think so, based on what I've been able to understand in the papers. Lo talks about the manifolds that seem to be sperical shells that surround the sun with radii the same as each planet's orbit. These orbits can't pass through these shells except where the planet actually sits and givea 0-energy trajectory through. What puzzles me is that this means that an Earth -Mars mission just leaves here and heads to Mars. Obviously, you can't get an Earth gravity boost from a spacecraft that's still in its gravity well. Perhaps there's enough ofa gravity boost that one gets from swapping between the Earth's and the Moon's gravity wells sequentially but that strikes me as fishy.

Re: Hubble:
Well I'm glad that someone is thinking along these lines. However, this is a temporary solution. The Hubble's gyros are about to go belly up. I doubt that this tug has the fine pointing capacity to keep Hubble operational if the gyros fail. At least, it won't become a 'ballistically implanted reef' as the company so finely puts it. Also, there's no way to install the upgraded camers which have already been built.

I'd love to see some sort of actual tug that has a few robotic arms that can reproduce a human's movements reliably. I mean, most of our cars are built by robots, what's so hard about making a three-joint manipulator arm and putting in on a spacecraft for cryin' out loud.

Rxke · 2004-03-29 01:54:10

Thanks for the insight.

'bout the tug, IIRC NASA had one in a fairly developed stage, but abandoned it... because cost-overruns. Sounds faamiliar?
Somewhere i saw pics of it, looked like a ballet with arms, if i'm not mistaken, one of the websites discussing a DIY approach on settling the Moon have more info on it... No, wait... it's on [http://www.spaceislandgroup.com/omv.html]The SIG siteLooks like a manned version, launched from the shuttle, a bit of over-kill, but you could scrap the cockpit and install teleoperation hardware... (and retrofit it with ion-engines, of course)

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