Earth-Moon L1

[eveningsky339]

This is a simple question, but I haven't been able to find the answer on Almighty Google or Orbiter Forum. "Where" is the Earth-moon lagrangian point 1? Is there a certain altitude and inclination to place, say, an L1 depot?

 

[jwcornish]

Excellent question.

The L1 point is not a fixed point in space, that is to say a location a specific distance from the Earth or from the Moon.

The L1 point (as you no doubt have read) is where the Earth's gravitational field and the Moon's gravitational field are of equal strength

If you picture the Earth and the Moon's gravity fields as dents/warps in space/time, L1 is simply the point where those to fields meet and there is a little flat spot, like the top of a hill.

And, essentially, it is. If you put an object just a bit Earthward or a bit Moonward of the L1 point it would fall towards that body.

This is, of course, imagining the Earth/Moon system as being fixed, or more techically, viewing it in a co-rotating frame.

It's unfortunately much more complicated.

1) the Moon's Orbit is elliptical. That means, in a corotating frame the Moon doesn't just sit still. It moves towards the Earth and away.
Also when the moon is at it's perigee, it's moving faster, so on our corotating picture, it would be a bit upspin. At apogee, it's moving slower, so it'd be a bit downspin.

So, in a corotating frame the Moon doesn't sit still, it does a little bean shaped orbit around the location it'd be in if it's orbit were a perfect circle.

1a) the Earth isn't exactly still either. The Earth Moon system rotates on it's barycenter, located a couple thousand kilometers below the Earth's surface, but NOT at it's center.

2) F = G(m1*m2/r^2)

This means that nice L1 point (and L2, L3, L4, L5) moves around, at the balance point between the Earth and Moon's gravity fields.

3) It's the top of a hill, essentially, and with no friction things roll off of it.

4) It just isn't just the Earth and Moon. The Sun( and Jupiter and Venus to a much smaller degree) also have gravity fields that affect the Earth Moon system.

Look up '3 body problem'--its very very complex.

So, you pretty much cannot put an object at L1, L2, L3 as they are hilltops which move around at the whims of the

but that doesn't mean you cannot put an object around L1. Look up the subject 'L1 Halo Orbit'
http://en.wikipedia.org/wiki/Halo_orbit
http://en.wikipedia.org/wiki/Lissajous_orbit

These are stable, kind of. Both require a lot of computation to keep station.

 

[tblaxland]

Use this: http://www.orbitsimulator.com/formulas/LagrangePointFinder.html, made by the same author of Gravity Simulator.

Note that it is an unstable position since it lies on a "saddle" in the gravity potential, so it is better to try and orbit the L1 point rather than try and sit on it. Either way, you will need corrections to maintain you position there.

You can also use this to help visualise the position: [ame="http://www.orbithangar.com/searchid.php?ID=3644"]Lagrange MFD 0.7[/ame]

 

[HopDavid]

Excellent question.

The L1 point is not a fixed point in space, that is to say a location a specific distance from the Earth or from the Moon.

The L1 point (as you no doubt have read) is where the Earth's gravitational field and the Moon's gravitational field are of equal strength

Not quite true. There are three forces, the earth's gravity, the moon's gravity and so called centrifugal force. Centrifugal force is actually just inertia but it seems like a force when you're in a rotating frame, the outward tug you feel when you're riding on a merry-go-round, for example. At L1 (as well as at the other Lagrange points) these three all cancel out.

If you draw a line from moon's center to earth's center, the L1 is about 15.09% of the line from the moon and 84.91% of the line from earth's center. The moon's semi major axis is 384400, so the L1 is usually around 58030 km from the moon's center.

During the moon's perigee it's 54815 km from the moon's center and during apogee it's 61245 km from the moon's center.

 

[eveningsky339]

but that doesn't mean you cannot put an object around L1. Look up the subject 'L1 Halo Orbit'
http://en.wikipedia.org/wiki/Halo_orbit
http://en.wikipedia.org/wiki/Lissajous_orbit

These are stable, kind of. Both require a lot of computation to keep station.

Ah, now I understand why NASA was unhappy with the L1 scenarios put forth during the development of the CEV.

I was intending to put a depot at L1, but perhaps I can put a depot into a Halo orbit around L1 instead.

Is there any way to figure out suitable L1 launch windows? All the historical L1 missions have been done at the Sun-Earth L1 point, which obviously doesn't help me very much.

 

[HopDavid]

Ah, now I understand why NASA was unhappy with the L1 scenarios put forth during the development of the CEV.

I was intending to put a depot at L1, but perhaps I can put a depot into a Halo orbit around L1 instead.

Is there any way to figure out suitable L1 launch windows? All the historical L1 missions have been done at the Sun-Earth L1 point, which obviously doesn't help me very much.

I believe an ordinary Hohmann from LEO (300 km altitude) to EML1 would have a period of 7.8 days. To get major axis of this Hohmann I added altitude, radius of earth and distance from earth's center to EML1. This major axis is .4332 times the moon's major axis. So the period is .4332^(3/2) times the lunar period or .2851 * 27.321.

A Hohmann trip is half of a period so the trip would be 3.9 days.

During the Hohmann trip the moon would advance 51.32 degrees while the spaceship advances 180 degrees. So the spaceship should be launched from a window where the moon has a 128.7 degree lead.

Towards the end of the trip, the moon's gravity will bend the path from elliptical. So a trip to EML1 will deviate slightly from a straightforward Hohmann ellipse.

 

[Flightoffancy]

This sounds such a solution that is needed for Dark side bases on the moon to stay in communication with Earth: A satellite orbiting the L2 point. (with tracking from ground stations of course or omni directional antennas)

I haven't the skills to come up with the orbital elements to make this happen in a scenario (translating Hopdavid's theory), but if someone could post them, that would be in their debt.