Lunastationary/Lunasynchronous orbits

Factual Questions
1

I have been looking at this page ( http://www.newton.dep.anl.gov/askasci/ast99/ast99159.htm ) but find myself dissatisfied. First, an explanation like:
“For an orbit far away from the moon our satellite would encounter the gravity of the earth and that would alter the orbit and make a stationary orbit unlikely. Because of the earth’s presence such an orbit is n ot possible”
is simply opinion, not demonstrated fact or convincing calculation.

Second, my verbal interpretation and math competency are completely inadequate for the following:
“If it were, you could use the value of the radius for earth and multiply it by the one-third power of the ratio of the moon to the earth masses and multiply by the two -thirds power of the number 28. That should give the value without the prese nce of the earth.”

So I put to you the question: Are lunastationary/lunasynchronous (shortened to “Luna_S”) orbits possible?

To some extent the answer is “Yes”. The 5 Lagrange points of the Earth-Moon system are Luna_S, and the Earth itself is Luna_S. Saying these things are in “lunar orbits” may not be the most accurate way to describe the situation, but from the surface of the moon all six of these spots are somewhat stable and somewhat unmoving, and are in Luna_S orbits sufficient to satisfy me.

I wonder about other orbits, though. The Earth has “Geo_S” orbits at about 35,800 kilometers; to what extent can the moon have a similar ring of Luna_S orbits? My best pass at figuring this (assuming no Earth in the calculations) is that the ring would be around 88,000 km above the moon’s equator.

Is the Newton site linked to above right in saying that such orbits are impossible? Because of the presence of the Earth I imagine the possible locations would not form a circular ring, and might not form a complete ring at all. But I do not see why stable Luna_S points (hereafter to be known as “Blade points” ( :slight_smile: )) cannot exist, say, 88,000 km preceding or trailing the moon in its orbit (like the L4 and L5 spots, but closer to the moon).

(If lunar satellite in a Blade point were big or bright enough to be visible from Earth, it would appear to be unmoving relative to the moon; that would be an interesting thing to see.)

So anyone want to help figure this out? Can there be “real” Luna_S orbits?

2

Okay, let’s get to the most important thing first.

A stable orbit around any astronomical body is only possible where that planet’s gravity well is largely undisturbed. About five times the distance of the moon from the earth, for instance, you could theoretically have a stable orbit around the earth (technically about the earth-moon system,) if those two bodies, and your tiny third body, your orbiter, were the only things that were involved. However, at that distance, IIRC, the earth’s gravity well is strongly disturbed by the influence of the sun, which would work slowly but surely to push or pull that little craft out of its earth orbit, sending it either deeper in towards the earth, or out into interplanetary space.

Things are even more extreme where it comes to lunar orbits. Because the earth is so much heavier than the moon, and so close to it relatively, the moon’s gravity well only exists at all quite close to it, and there really isn’t anywhere that it isn’t partially skewed by earth’s own gravity. I don’t remember the math to figure out where a theoretical L-s orbit would be, but it seems quite likely to be that it would be well out of the area where the moon can strongly define a gravity well and thus would not be anything like stable orbits.

So then, you’re left with figuring out where ‘stationary’ points would be in the Earth-moon system, especially since the moon happens to be tidally locked to the earth, and thus, as you pointed out, will keep the same face to any stationary point in the two-body system. L-1 and L-2 are particularly convenient as far as nearness to the moon, if I recall correctly, though any objects placed there would probably need stabilizing jets to keep them in place against tidal effects and the interference of solar gravity. (do objects in geostationary orbit need to correct their orbit so much?) One of those points will be hanging a little ways away from the near side of the moon, and the other above the far side of the moon, as it were.

Sorry if I didn’t explain things that well… it’s been too long since I’ve done any orbital dynamics.
If you were able to figure out that 88,000 km figure correctly, then try this: Work out how intense lunar gravity is, (in terms of acceleration in meters per second per second or some similar unit) at 88,000 km from the moon. Then figure out how intense gravity from earth would be at the distance of the moon from the earth minus 88,000 km.

3

I managed to struggle through a little of the math, using the old newtonian equation for force of gravity and using a 1 kg object in your 88,000 km orbit above the surface of the moon.

At closest approach to earth, that little 1 kilogram buoy is getting 0.63 milinewtons (0.00063 newtons) of gravitational force pulling it towards the moon, and 4.54 milinewtons of force pulling it towards earth. Whereas if you’re supposed to be in a stable orbit around the moon, you shouldn’t be getting gravitational forces from any other body 1/10th as strong as those from the moon. Lunasynchronous orbit just isn’t going to cut it. Does that make it easier for you to see??

:slight_smile:

4

I thank you for your responses, but maybe I still don’t understand properly. The Earth-Moon L4 and L5 points are much more strongly influenced by Earth’s gravity than by the Moon’s, but they exist and are stable. L4 and L5 are about 380,000 km from the moon, so I see no reason that points 88,000 km from the moon (leading and trailing the moon in its orbit) couldn’t be stable.

The physical layout of the Moon, the Earth, and (any of) the “stable” points I am considering remains constant, three points of a triangle that doesn’t change. Remember that while the satellite does orbit the Moon, it doesn’t appear to “move around the Moon”. If we see it at “15 degrees to the left of the Moon”, that is its fixed position relative to us and the Moon.

For any of the points in the proposed Luna_S “ring”, there will not be shifts in gravitational force or direction from either the Earth of the Moon, because each point in the ring has a fixed orientation with the Earth and the Moon at all times.

Do you see what I am trying to say? Does it make sense?

5

Umm… not really. I’m not quite sure how you’ve derived your S ring, but let’s try going over it again. For stable orbits of a small object (one not large enough to create a significant gravitational disturbance through its own mass,) in a system of two gravitational bodies, one more massive than the other, there are four general solutions.

1- a tight, close orbit around one body that is locked to a sufficient degree to ignore gravitational perturbations from the other body. This is what the space station and space shuttle orbit in, for instance, and commsats at a higher level, geostationary orbit. It is possible about the moon, but only at a very low orbital distance because the moon’s mass is so low compared to the earth’s. It does not allow for any sort of lunastationary orbit.

2- an orbit that alternates between the two bodies in a kind of figure-eight, loop-de-loop. This definitely doesn’t fit your requirements, because for large periods of the orbit the orbiting body would be far away from the moon. I only mention it for the sake of completeness.

3- A stable orbit that remains at exactly the same position relative to both primaries throughout the entire period. Now, I’m not entirely clear on this, but I strongly believe that the five mapped lagrange points are the only spots, (or regions since L4 and L5 are rather amorphous and don’t really behave like points in space) where this is feasible. Specifically, if you try to do it with an orbiting body as far out as the moon, but 80,000 km off to one side, it seems to me that there won’t be any reason that it won’t gradually be attracted by the moon and eventually be pulled into it and crash. the L4 and L5 points must manage to avoid this because they’re far enough out that their orbital speed exactly counters the efforts of the earth and moon to ‘pull them in’… or something like that, I’m not sure of the details. But I’m confident in Lagrange’s work, especially since so many other people have gone over it since then, and it’s pretty much accepted that there are no other spots that have the same properties. Sorry.

4- theoretically, you could have a stable orbit far enough out beyond both bodies to orbit their common center of gravity, but as I said earlier, this can’t be easily done with the earth-moon system because the sun’s influence interferes.

So… does this make anything clear?? If not, then it might be worth trying to work out the sum of the forces and vectors on your proposed luna-s ring, and see if they can be made to balance out to stability, or if there’s some driftage that can’t be gotten rid of. If you can get that far, either you’ll be able to see for yourself why it won’t work, or you might have actually made an incredible discovery that someone will name after you. :wink:

6

quick followup… I’ve done a little research, and the math to prove that L4 and L5 are stable is a little too much for me to follow… otherwise I’d have tried to post a similar proof that your points are unstable.

But I think an appeal to authority works to a certain degree here… considering that so many bright people have been looking for useful points of equilibria, the hypothesis that there are a whole ring of them right around the moon seems very hard to support. But I won’t say never. :slight_smile:

7

I know I’ve been posting a lot here, just one quick followup to clarify a point that might have been misunderstood here. What you’ve said above is true as far as it goes. However, for any such quote ‘stable point’, the problem is to prove that there is actually stability… specifically, three vectors… (gravity pull toward the moon, gravity pull towards the earth, and effect of inertia as the entire triangle rotates,) must balance out and equal zero. Otherwise, something put in that point, starting at rest with respect to the earth and the moon, will gradually drift in some direction or another, and not actually remain in the fixed theoretical position.

8

This is the scenario I am trying to describe. The satellite remains at exactly the same position relative to both the Moon and the Earth. It orbits the moon every 27.322 days (or whatever), thus remaining over exactly the same spot on the Moon’s surface. It also stays in exactly the same position compared to the Earth. The gravitational force on it from the Moon is unchanging, the gravitational force on it from the Earth is unchanging. There is nothing to unstabilize the orbit once it is achieved.

Admittedly the triangle (made by the Moon, Earth, and satellite) rotates around Earth every 27.322 days, but the geometry of the triangle is unchanging.

Believe me I have a hard time picturing it and evidently a harder time explaining it. The satellite is in a 27.332 day orbit around the Moon. The satellite seems simultaneously to be in a 27.322 day orbit around the Earth. The geometry and gravitational forces are the same from moment to moment and orbit to orbit.

I don’t doubt the Lagrange points–I just don’t see how they are more stable than the ones I am thinking of.

Again, thanks for your help in understanding this.

On preview I see you have posted twice more, but I will still submit this and then digest your new responses.

9

Ok, I see what you are saying, but I don’t have what it takes to figure out whether it would be properly stable or not. Maybe somebody with the “right stuff” will take pity on us and actually work out the numbers.

10

A stable lunasynchronous orbit is impossible. The same is true for a geosynchronous orbit. The gravitational influence of the moon (and, to a lesser extent the sun) will eventually knock any geosynchronous satellite out of that orbit unless the satellite does course corrections. This requires propellant. Eventually whatever propellant the satellite has will be exhausted. Should mankind annihilate itself totally in a nuclear holocaust tomorrow, and some aliens from another star system visit the Earth millions of years from today, they would find no satelites still in geosynchronous orbit.

11

That is the entire, um, point of the Lagrange points. They are the ONLY points in which you can remain stationary with respect to the Earth and the Moon. You just said, “The satellite remains at exactly the same position relative to both the Moon and the Earth.” You’re done, you have just defined “Lagrange Point”.

Now, you can be in orbit around some of the Lagrange Points, or be in an orbit that goes near multiple Lagrange Points, but that isn’t stationary to either the Earth or the Moon.

Some other interesting ideas, though:
Earth’s “second moon” Cruithne is in a horseshoe orbit
Lunar Space Elevator (to Lagrange Points)