Placing an Asteroid in Moon Orbit: Is It Possible?

[Plastic Photon]

So had an idea the other day, and...Too make a gob of statements that started out incoherently seem coherent this is what I was thinking of doing:
Take an asteroid (roughly 2 million pounds in weight) from the asteroid belt and place it in moon orbit.
Please don't ask me about how I would remove the asteroid, how wide it would be, how fast it is going, unless you feel it as a necessary contribution to the information that I am looking for, which is: (I have been searching the web with no avail) for information about moon satellites, orbits or anything that would lead me to understand whether it is possible to place such an object in moon orbit for more than 20 years without having to make adjustments.
Aside from a hs physics class I know next to nothing about physics.
The problem I am imagining with placing such an object like this so close to the Earth is that it will eventually make its way to Earth even if it is orbiting the moon. I think the idea is complete bunk, but I want to give it a shot until I am for certain of this.
My school library is closed for another week and a half and I have until the second week of Feburary to complete this concept and submit it if it is up to par.
So, thank you in advance for any information.

 

[Danger]

You might have while to wait, PP. I asked a similar question about the possibility of a naturally-occurring satellite-of-a-satellite several months ago and still haven't heard anything.

 

[pervect]

Read up on the "Hill sphere"

http://en.wikipedia.org/wiki/Hill_sphere

I think there are some posts about it in the forum.

https://www.physicsforums.com/showthread.php?t=42668&highlight=Hill+sphere

Basically, if the asteroid is inside the Hill sphere of the moon, it won't be able to escape from the moon's orbit.

Making sure that the asteroid doesn't hit the moon is a different problem, and is trickier. Low lunar orbits have a problem with lunar mass concentrations (masscons) which IIRC tend to cause them to become more elliptical and eventually hit the moon. I don't have a lot of hard data on this, but I think it would usually take longer than 20 years unless the orbit was already very low.

 

[russ_watters]

I'm not sure what you are asking: are you asking if it were possible or what we would gain by it? You didn't say why you would want to put an asteroid in orbit of the moon...

 

[Tide]

If the moon had a satellite it would have to be in a very tight (i.e. nearly grazing the surface!) and almost perfectly circular orbit in order to prevent Earth's tidal forces from breaking up the orbit.

 

[Danger]

Thanks, Tide and Pervect.

 

[KingNothing]

I'm not sure what you are asking: are you asking if it were possible or what we would gain by it? You didn't say why you would want to put an asteroid in orbit of the moon...

I think it was a hypothetical question. Hypothetical questions aren't always practical but they can still help us better understand something for when we do put it to use.

Anyhoo, why is it that it'd be so hard with the moon?

Danger...the possibility of a naturally-occurring 'satellite of a satellite', huh? What do you think our moon is?

As far as I know, there are lots of 'satellites of satellites' etc, even our moon does have debris that orbits it, much like Earth has some debris. Even the sun itself revovles around the center of the milky way.

 

[Vixus]

Danger...the possibility of a naturally-occurring 'satellite of a satellite', huh? What do you think our moon is?

Brilliant! :D

Yes, I think if you had the space you could have a long 'series' of satellites. For example, the entire solar system revolves around the centre of our galaxy, in a way, the planets orbit the sun and the planets have moons. Then, if we want pictures of those moons, we usually put a satellite in orbit around them, don't we?

 

[Danger]

Danger...the possibility of a naturally-occurring 'satellite of a satellite', huh? What do you think our moon is?

:tongue:
In my original question, I specified a 2nd generation satellite. This time I didn't bother because it seemed to be defined by Plastic's post. In fact, I had asked about a huge satellite such as Titan capturing and maintaining a moon similar in size to those in our own system such as Phobos, Charon, or even our Luna.

we usually put a satellite in orbit around them, don't we?

Which is why, once again, I specified a naturally occurring one. It just seems to me, intuitively, that it would be almost impossible for something to just fall into a stable orbit around something like that without interference from the parent body.

 

[pervect]

If the moon had a satellite it would have to be in a very tight (i.e. nearly grazing the surface!) and almost perfectly circular orbit in order to prevent Earth's tidal forces from breaking up the orbit.

Not all that tight - nearly grazing the surface would in fact probably cause the object to impact the moon in the near future due to the mascon problem.

To keep the Earth's tidal forces from causing the object to escape, the orbit basically has to be inside the Earth-moon L1 and L2 Lagrange points. This would be somewhere around 60,000 km away from the surface of the moon.

(There are two parts to the problem - preventing escape from the moon is one part, avoiding impact with the moon is the other part).

The detailed mathematical basis of the Hill sphere, as described in the wikipedia article previously mentioned

http://en.wikipedia.org/wiki/Hill_sphere

is based on the conservation of the Jacobi intergal function.

http://scienceworld.wolfram.com/physics/JacobiIntegral.html

This is a conserved quantity for the restricted three body problem, very similar to an energy. (It's actually the Hamiltonian of the system in the co-rotating coordinate system, so it has units of energy.)

As can be seen by examining plots of this function

http://www.geocities.com/syzygy303/

if an object is close enough to the moon, the conservation of the Jacobi intergal function prevents the escape of the object, keeping it in a region close to the moon.

If the object can pass close to L1, it can esape from the moon into a low Earth orbit.

If the object has slightly more "energy" it can pass close to L2, and escape the earth-moon system entirely.

 

[Tide]

pervect,

I think the more restrictive condition would be the cumulative elongation of the orbit due to Earth's tidal forces which would lead to the satellite either escaping or colliding. This would happen even if the orbit started well within the Lagrange points.

I'm not up to solving the three-body problem tonight, however. :)

 

[pervect]

The Wikipedia article doesn't go into the math, unfortunately, but the Hill sphere formula given there comes from a three-body analysis. It will be impossible for a body to escape the moon's orbit if it is within the Moon's Hill sphere due only to the Earth's tidal pertubations.

I'd be happy to go into the math - oh, you're not in the mood :-)

Well, if anyone wants to see the math, I can post more. (The traditional approach uses the Hamiltonian formulation of the problem). I think I've also posted on this topic here in the past, I'm not sure how much I covered.

One might also try some of the gravity simulator programs for a "hands-on" approach. I've never used it myself, but one of the posters here (Tony) has a gravity simulator freeware program at http://www.orbitsimulator.com/.

Caveats:

The Hill analysis is a three body analysis, so it includes the Earth's tidal pertubations. It doesn't include the sun's tidal pertubations, however.

The region of stability is only approximatly sphereical. If you look at the exact equation, the region of stability becomes more egg-shapped than spherical. The important thing is really that the orbit stays away from L1 and L2.

The analysis only considers escape, not collision. (But low orbits are bad for collision, only a slight pertubation and the satellite will collide).

The analysis assumes an exact 1/r^2 gravity force (which would actually be the case if the planets were ideal point masses or exactly spherical) - but they aren't quite sphereical, and for the moon especially mass concentrations can be a problem for low orbits.

The simulation route will also assume point masses, BTW.

 

[Plastic Photon]

Well, this is not good. I do not know Langragian math, but thanks for the information. I will have to work with what I know.

 

[tony873004]

Well, this is not good. I do not know Langragian math, but thanks for the information. I will have to work with what I know.

The math for the Lagrange points is pretty simple. Here's a calculator I wrote that will do it for you.

http://orbitsimulator.com/cmc/HillSphere.html

 

[pervect]

I fired up Tony's gravity simulator, and I have a few more notes.

If you want the orbit to be nearly circular, you want to keep the orbital radius around the moon a lot smaller than the Hill radius - maybe 20-25,000 kilometers (with the Hill sphere radius being about 60,000).

Here is a .qsim file for a "barely bound" case, which can be run using Tony's simulator. This was made by putting the moon in a perfectly circular orbit around the Earth (which isn't quite right, but makes the display easier), and creating the body "barely" with coordiantes and velocities of .85 x the coordinates of the moon, then letting the simulator run for a bit. A frame co-rotating with the moon is chosen for the display mode. Letting this file run in Tony's gravity simulator will trace out almost all of the egg-shaped "Hill sphere".

Where did the factor of .85 come from? It came from
$$1 - \sqrt[3]{\frac{m_{moon}}{3 m_{earth}}}$$

plus a very small safety margin. (m_moon/m_earth = .0123, by the way).

You can see that the resulting orbit for the barely-bound case is highly non-circular, and has undesirable "close passages" to the moon which could result in collisions.

<GrAVITY>
<Distance Boxes>

0
850
Earth
Moon
1
2
</Distance Boxes>
<Orbit Boxes>
</Orbit Boxes>
<SData>
0.00000050625
341866413
64
279.164327753673
-2135.50393588358
1
3
False
False
0
-10000
D:\GAMES\Gravity Simulator\barely.gsim
True
2357659.45456642
True
0
2
-60
-60
15480
11160
True
False
1
10
True
False
True
True
D:\GAMES\Gravity Simulator
D:\GAMES\Gravity Simulator\barely.gsim
False
0
0
0
0
</SData>
<Command Buttons>
-1
-1
525
2280
-1
-1
-1
-1
-1
-1
</Command Buttons>
<Captions>

MS Sans Serif
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</captions>
<Thrust Box>
</Thrust Box>
<Objects>

Floating
0

0
0
26108872225.9702
-25968776769.555
1882507645.28996
905.504918677805
-986.286317905421
50.7050402169655
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Earth
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Earth
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Moon
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Earth
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-3891383013.37169
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Barely
6.85683674835943E-19
Earth
4194432
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</Objects>
<Auto Pilot>
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</Auto Pilot>
</GrAVITY>

[add]
I ran this file for a while, and "barely" actually escaped. I'm not sure if this is due to round-off errors in the simulator, or whether my initial conditions were not quite in the stable region.

(I'm not sure what integration algorithm the current version of the program is using.)

 

[tony873004]

Try making another "Barely" in a retrograde orbit (inclination = 180). You'll see that retrograde orbits are a lot more stable towards the edge of the Hill Sphere. Another fun thing to try is to place objects in Earth orbit external to the Moon's orbit. You should find that no stable prograde orbits exist. The Moon ejects or collides with anything inbetween the Moon's orbit and the edge of the Hill Sphere (of course there is no Earth Hill Sphere without the Sun) in only a couple of orbits. But retrograde objects can stick around a while if not indefinately, with only a small spacing between their orbit and the Moon's.
Another thing to try is to introduce the Sun into the system. Start from scratch as it's difficult to add the Sun after the fact. The solar gravitational tide through the Earth / Moon system reduces the stable region even further.

I'm not sure if this is due to round-off errors in the simulator, or whether my initial conditions were not quite in the stable region

I'd guess math error, since once they're introduced they grow. But here's how you can find out:
Take it out of rotating frame mode (mnu View>Rotating Frame, uncheck) and Focus on the Moon. (Press F8 or F9 to display the Focus Object window). Then zoom in on the Moon until it comes close to filling the screen. As "Barely" makes its closest approaches to the Moon, it is jumping about 1.5 "Barely" diameters per step rather than tracing a continuous line. This is a crude method of determining that your time step may be a little too fast and math errors may contribute to your results. Although 1.5 diameters is not that bad. You definitely want to avoid "Barely" jumping significant portions of a Moon diameter, or "Barely" may survive what should have been a Moon collision.
Two workarounds to this problem:
1. Reduce your timestep by half and see if you get the same results. Different results for different time steps implies round-off error or truncation error.
2. Reduce your time step by a lot, perhaps down to 1. Then make up for the difference by running in the "Don't Plot" mode, which is the P button on the Graphics Options window. In the Don't Plot mode, the graphics are not refreshed, and the program appears frozen. But behind the scenes, it's crunching the numbers 25x faster. Approximate how long you need to do this to exeed your ejection date, then turn the Plot mode back on and see if "Barely" still exists.

I'm not sure what integration algorithm the current version of the program is using

Euler

double tM;
double tM2;
double dx;
double dy;
double dz;
double D;
double f;
double fx;
double fy;
double fz;
for (int k=1;k<=NumObjects;k++) {
tM = ObjMass[k] * 398600440000000;
for (int j=k+1;j<=NumObjects;j++) {
dx = objx[j] - objx[k];
dy = objy[j] - objy[k];
dz = objz[j] - objz[k];
D = sqrt(dx*dx+dy*dy+dz*dz);
if (tM > 0) {
f = (1/D) * (1/D) * tM;
fx = (dx / D) * f;
fy = (dy / D) * f;
fz = (dz / D) * f;
objvx[j] = objvx[j] - fx;
objvy[j] = objvy[j] - fy;
objvz[j] = objvz[j] - fz;
}
tM2 = ObjMass[j] * 398600440000000;
if (tM2 > 0) {
f = (1/D) * (1/D) * tM2;
fx = (-dx / D) * f;
fy = (-dy / D) * f;
fz = (-dz / D) * f;
objvx[k] = objvx[k] - fx;
objvy[k] = objvy[k] - fy;
objvz[k] = objvz[k] - fz;
}
}
}
for (int h=0;h<=NumObjects;h++) {
objx[h] = objx[h] + objvx[h];
objy[h] = objy[h] + objvy[h];
objz[h] = objz[h] + objvz[h];
}

 

[Plastic Photon]

This all very relevant information, and I must say after examining it to the best of my ability; I will not be pursuing my original plan. I think the margin for sucess is to slim when dealing with such an object so close to Earth and I have only begun to research what would happen if the asteroid were to possibley crash into the moon break off and head to earth. I think my idea is either hopeless or for another day.