A sort of L-1 primer

by: Ken

Some folks that stop by out here in the Selenian Boondocks might be wondering just what exactly this whole L-1 thing is all about.

Simply put, L-1 is the point on the line connecting the center of mass of the Earth and the Moon where the pulls from either side are balanced out. It’s mostly a function of the Earth’s gravitational pull, the Moon’s gravitational pull, and centrifugal force. An object, like a space station, placed at this point will remain there, absent any outside displacement. Since both the Sun and Jupiter are outside displacements, objects put there aren’t going to stay there. This also means that debris will not accumulate there.

There are ways to solve the problem, like halo orbits and a modest (on the order of 10s of m/s) expenditure of station-keeping propellant. Other orbits, like a lissajous orbit, are more complex but offer different benefits. What is important, though, is that the L-1 point is the gravitational ‘high ground’ between here and the Moon. This makes it the lowest delta-V launch point to a number of other destinations of interest.

It’s exact distance from the Earth or Moon varies throughout the month proportional to the variance of the distance to the Moon in its orbit around the Earth, but generally the average is taken to be around 58,500km from the Moon, or 57,660km, or 66,000km, or somewhere in between depending on which source you use. And of course, most sources don’t specify whether thatโ€™s to center of mass or to the surface of mass (the more important consideration, since that’s where you’re landing).

Its orbit doesn’t follow the regular rules of orbital mechanics, since its period has to match that of the Moon. Even more bizarrely, objects that move ahead of the L-1 point in its plane of orbit will be pushed back towards the L-1 point. Same thing if it falls behind. This is because the gravipotential warp starts climbing ‘uphill’ towards the L-4 and L-5 points in front of and behind the Moon. It helps to look at a topographical type image to picture it.

The thing to remember is that it is closest at the Moon’s perigee and farthest at apogee. Because its location is defined by the gravitational fields, it lies in a network of gravity boundaries. Objects sent along these curves are in effect ‘surfing’ the gravity boundaries between all the planets. They use very little fuel, at the expense of time, to get where they’re going. Much more importantly, they can be made to return to Earth, which opens up a completely new realm of possibility in how we study our Solar system.

There are a few papers available on the Internet to explore this fascinating topic.

“The Lunar L1 Gateway: Portal to the Stars and Beyond” by Martin Lo and Shane Ross. These are the guys that helped to pioneer the concept of the ‘Inter-Planetary Superhighways’ or IPS that connects all of the planetary Lagrange points in our Solar system (though its use dates back to the early sixties). It includes lots of graphics to illustrate the points, and notes that the Genesis spacecraft return trajectory was very similar to the one taken by comet Shoemaker-Levy-9 in it’s date with fate on Jupiter, and quite possibly also the asteroid that wiped out the dinosaurs. This kind of knowledge only serves to highlight how important it is that we make a concerted effort to catalogue the other objects in orbit around the Sun, in part so that we can develop a gravitometric map of near-Sun space and look for those objects that might sneak in through the back door. Such a map would serve a lot of other useful purposes as well.

Kind of an interesting paper is “CEV Architectures – Cost Effective Transportation System to the Moon and Mars” by Leisman, Joslyn and Siegenthaler. These are folks that teach in the Department of Astronautics at the USAF Academy and work in flight research at the USAF Test Pilot School. They effectively advocate an EELV launch architecture and station at L-1. Some of the assertions are nothing short of astounding, like “A 70% reduction in launch costs could be realized if EELV has block buys over 30” (sourced from AIAA 2002-4314). From which follows the same wisdom I advocate (though from a vastly different analysis perspective):

“Not only would economies of scale be realized in the American launch industry for the first time in years, but the demand would enable a healthy launch market to maintain both EELV contractors ensuring NASA, the DoD, and commercial users assured access to space”
(Under the heading ‘NASA Should Not Develop an Ultra Heavy Lift Vehicle’)


“If NASA launches three Heavy EELV per year along with the military and commercial requirements for Medium and Heavy EELV, you have just built economies of a scale to make the EELV family a low cost system for all three customers (win/win/win)”.

Gotta love that. The paper also advocates use of LH and LOX, since both are also used in fuel cells and water needs.

So what happens when a group of Aerospace Engineering undergraduates at UMCP takes a look at the problem? You end up with a nice project paper like “Clarke Station: An Artificial Gravity Space Station at the Earth-Moon L1 Point”. This is a nice design project that ventures a little more into the station concept than the prior references.

“Strategic Considerations for Cislunar Space Infrastructure” by two well-respected Lunar scientists, Wendell Mendell and Steven Hoffman, also looks at EML-1 and strongly advocates its use based on meeting such criteria as:
-It should be near the edge of the Earth’s gravity well so that reusable interplanetary vehicles can come and go with a minimum of propellant.
-It should be accessible from the Earth and the Moon with a minimum of constraints on launch windows.
-There should be no hazard from artificial debris.
-The region of space in which it sits should support colocated unmanned scientific platforms which can be reached from the space station by small delta-V transfers.
-The fuel requirements for station-keeping should be small.

Sounds to me like they stacked the deck a bit, but those are all important considerations for a permanent space-based space exploration infrastructure. There are a couple of very useful tables, including mass surcharge comparisons.

These should be helpful in understanding why the L-point between here and the Moon is really the next logical destination beyond LEO. First to install instruments, later facilities. An investment now in doing so can have enormous value benefits not just in going back to the Moon, but also for operations in GEO, traveling to NEOs of interest, and launching for Mars so that half the planet can watch and cheer on those brave souls.

Since our space efforts are about human culture as well, below are the lyrics that I sing to ZZ Top’s Lagrange. I always envision a crew vehicle coming sweeping in towards the station, with the Moon further off in the distance, and then everyone disembarking to stretch their legs, breathe some new air, and hit the bar to socialize. And of course a shot of the lovely and competent young lady working comm during the ‘nice girls’ line, coquettishly winking at the camera.

Well you must’ve been down
in that station town,
in that shack outside Lagrange

You know what I’m talking about.

Just let me know
if you want to go
to that home out on the range

They got a lot of nice girls, heh. ๐Ÿ˜‰

Well the air is fine
if you’ve got the time
and the grand to get your ship in.

A hmm, hmm.

And I hear it’s tight
most ev’ry night,
but now I might be mistaken.

hmm, hmm, hmm.

(with my sincere apologies to ZZ Top)

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14 Responses to A sort of L-1 primer

  1. Paul Dietz says:

    You last link there is messed up.

  2. Ken says:

    L1 seems like a nice place to wave to as you sail by at top speed, not a place to waste fuel stopping at. Putting stations in orbits means you don’t have to come to a full stop any time and you already have partial speed when you’re ready to leave.

  3. murphydyne says:

    ken, the L-1 point is moving, tracking the Moon in its orbit around the Earth. One paper (not noted here, I’ll have to look for it), notes the advantages for doing swing-by assist trajectories.

    As for wasting fuel, it depends on what you want to do. If you’re going to the Moon it doesn’t add a whole lot to the equation. It’s on the return from the Moon, wherein most people assume a direct-return trajectory to the Earth’s surface, versus a backwards trip through L-1 and LEO.

    Now some folks may be okay with exploring the Solar system entirely from the LEO, and that’s okay. But that’s not going to lead to a permanent space infrastructure and regular access to many destinations of interest.

    Did you actualy read the papers, ken?

  4. Ben Reytblat says:

    L1 might provide a very good reason to stop by. At least at first blush, it seems to be a great place to put a Lunar tether through.

    It would seem that the delta-v necessary to stop at L1 from after trans-lunar injection from LEO (well under 1km/s, according to some recent threads here) and drop off/pickup cargo is less then cost of landing on Luna.

    Given Luna’s 1/6g, it might even be possible to build the tether with more-or-less current materials.

    On the other side of the equation is the question of whether a tether can be built with Lunar materials, and how much (mass/$$$) would it cost to do so. Granted, capital costs of such construction might eventually be paid back via operational savings. But when?

    It would be interesting to figure out if it makes sense to build the L1-Luna tether before attempting GEO-Earth. Does anyone know if this analysis has already been done?

  5. Billy Gibbons says:

    Well, LaGrange, Texas rhymes with ‘range’ and the LaGrange points in space sound like ‘Grand’ pronounced with a nasally ‘a’.

  6. Anonymous says:

    Re: Ben Reytblat

    yes, a delta v of less than 1km/s is well within range of commercially available tether materials such as spectra and zylon fiber.

    In fact, even with some redundancy and a healthy safety factor current materials allow for capture velocities of up to 4km/s without excessive tether mass ratios.

    See the publications at http://www.tethers.com for more info.

    It is a mystery to me why everybody ignores space tethers (except in the currently impractical form of a space elevator).

    A tether infrastructure would allow for propellantless travel from LEO to the surface of the moon and back (assuming that the mass flow is balanced).

  7. murphydyne says:

    Hi ben and anonymous,

    Personally, I think it’s a wee bit early to be worrying about tethers. We should keep looking at and studying them, but really the focus right now should be on actually getting further out into space than LEO, and building up our capabilities between here and the Moon.

    Plus, as noted in the post the L-1 point is not a ‘fixed’ distance from the Moon’s surface, so where are you going to put the station? And the counterweight out at the end is going to be sliding up and down the gravity well, stretching and shrinking the length of the tether.

    My personal feeling is that we would have to capture a carbonaceous chondrite asteroid and park it at L-4 or L-5 (because from there the supply chain is all ‘downhill’) to harvest the carbon to use. L-1 is a good place to get started on that.

    Tethers are the long term solution, but we have to be sure we don’t build our crystal castles on foundations of sand. Let’s not put our cart too far beyond the horse.

    Please read the papers.

  8. Anonymous says:

    Re: murpyhdyne

    > Personally, I think it’s a wee bit
    > early to be worrying about tethers.
    I think you overestimate the technical complexity of small delta v tethers.

    The gravity gradient forces at the L-1 are so small as to be negible.

    And you do not need lots of material to build a low delta-v tether. For a 10 ton payload and a delta-v of 1000m/s the tether weight would be at most 5 tons, and that is with a lot of margin.

    So there is really no need to mine any asteroids for this.

  9. murphydyne says:

    I don’t disagree that tethers are really the permanent transportation solution, but usually we have a bit of traffic before we start building bridges. That’s usually the way it works.

    Besides, I don’t want to have to be the one that has to inform world leaders that oops, our tether broke, and there’s a big counterweight sliding down the Earth’s gravity well.

    As I said, I think we need to keep studying tethers, and I think they are ultimately the ‘permanent’ soution. But looking near term of 20, 50 75 years I don’t pragmatically see a tether being the kind of ‘magic bullet’ that a lot of people think. Eventually yes, but I think we have to evolve into it versus imposing it as a critical element in the design from the get go.

    By the same token, we don’t need tethers for L-1 to be useful.

  10. Anonymous says:

    re: murphydyne

    Even for a major tether system such as the one proposed for LEO to GTO transfer in this paper, the weight of the counterweight is just 10 tons. And the maximum delta-v in the unlikely case of a tether break would be on the order of the tip velocity, so in case of a L-1 tether the counterweight would still be in orbit.

    I hope that we will see tethers much earlier than you imagine. Low cost launch vehicles such as the falcon will allow companies like tethers unlimited to do on-orbit experiments.

    But I agree with you that we do not need tethers for L1 to be useful.

  11. Anonymous says:

    The L1 point is the minimum kinetic energy point between the earth and moon. Think of it as the mountain pass between two towns.

    Ken, If you wave, It would probably be at the slowest crawl managable.

    It’s also a great place to store propellants and potential energy without any aerodynamic losses.

    (We’ll skip the whole thing about Rotating frames of reference transformations, Changes in total energy and Zero velocity curves).


  12. Anonymous says:

    Using Dnepr, tethers unlimited can start their practice today.

    Using Falcon, we gotta wait for a little while longer.

  13. Ken says:

    Let me clarify for those that don’t think I can read. The benefit of emL1 is that it is a stable orbit. Under no circumstance, to any location, does it save fuel. Go ahead, argue the point.

    Read the chart and understand, when I say stop, I mean stop. All movement is relative. Since L1 is the gravitational point between two orbiting bodies, yes it is moving, but from LEO to emL1 the chart says a deltaV of 3.71 and emL1 to LLO of 0.72 where direct LEO to LLO is 4.22.

    Stop is what a ship does when it docks with a station as L1, we don’t need to consider that L1 is in orbit of two bodies. (We say the Moon orbits the Earth but that’s an unqualified statement and I’m not going into all the permutations it would take to qualify it.)

    Starting from emL1 may seem cheaper than from LEO but only if you discount getting logistics to L1 in the first place.

    Having said all that, why not L1? The cost isn’t that much greater and at least where not likely to misplace whatever we put there.

  14. Ken says:

    It turns out emL1 is unstable as you suggested. So I guess you could lose stuff ๐Ÿ˜‰ Now I really don’t know why it would be of advantage to put anything there. For your other point, I think using the Moon for gravity assist to other destinations outweighs any other factors.

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