# Doesn't dust or debris collect at the L4 and L5 Lagrangian points?

## Main Question or Discussion Point

referring to:

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

it would seem to me that over the billions of years that the Earth/Moon pair existed in a relatively stable orbit around their common center of mass, that at the two stable Lagrangian points (depicted as L4 and L5 in the Wikipedia article), a bunch of crap would congregate there and stay there unless some nasty comet or high speed asteroid happens to barrel right through it and disperses all of it.

has anyone pointed a decent telescope at these two L4 and L5 points and noticed stars disappearing behind it and reappearing as these two points pass over those stars in the background? i would think that it would have to visible.

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DaveC426913
Gold Member
There is some dust there, yes. The points are not all that stable. so stuff will drift in and out.

There is some dust there, yes. The points are not all that stable. so stuff will drift in and out.
if stuff drifts in, and if somehow the kinetic energy is dissapated (unelastic collisions with other stuff), then how can it drift out?

if it drifts in and the kinetic energy is not disapated, the object would either drift out right away or would somehow "orbit" the Lagrange point (as viewed by this accelerated frame of reference where you apply d'Alembert's principle to analyze the 2-body problem) until it did dissapate its kinetic energy or something else came along to nudge it out.

is that looking at it correctly? because otherwize i do not understand why anything would ever drift out, maybe get knocked out, but what action would cause it to drift out?

thnx.

D H
Staff Emeritus
Just because an object passes near the Earth doesn't mean it will be captured by the Earth. It's velocity has to be below the escape velocity. Similarly, just because some object passes near the Earth-Moon L4 or L5 point doesn't mean it will be captured by the point. The attractive force is fairly weak.

Solar radiation pressure will quickly get rid of any dust that does get captured by the L4 or L5 point. The odds of anything larger than dust passing near the L4 or L5 point and having a small enought relative velocity to enable capture are slight.

DaveC426913
Gold Member
Look at the Wiki example and examine the red and blue arrows showing forces acting on the points. The blue arrows are forces pulling objects out of the L points. The blue arrows are forces pulling objects into the L points.

The L points are simply areas where opposing forces cancel out.

In fact, the L points aren't gravity wells at all, they're the tops of hills! They're only stable as long as nothing nudges them away from that area.

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D H
Staff Emeritus
The blue arrows are wrong, in the sense that they rather strongly imply your interpretation. You are not alone (it's a Wiki article, after all):
From the discussion page:
Aren't L4 and L5 stable points in the Earth-Sun model? The blue arrows in de picture show them as instable, which I think is incorrect. --Mushlack 17:32, 2 August 2006 (UTC)
They are potential peaks, hence the blue downwards-slope arrows. They are, as you say, stable, even though the potential diagram does not show it. It requires slightly more sophisticated maths... 203.97.255.167 00:15, 3 September 2006 (UTC)

The Sun-Earth L4 and L5 are indeed weak gravity wells. Ignoring all other bodies in the system, all it takes is a mass ratio of about 25:1 or greater to make the L4 and L5 points stable.

DaveC426913
Gold Member
You're right. The text explains it better. The contour lines are are not gradients (i.e. that objects will cross prependicularly) - they are paths (i.e. that objects will follow). If you place an object on any line in the diagram, the object will trace out that line.

See diagram (please! I insist! It took me an hour to do that from scratch in Illustrator)

You can see that the Moon follows a line around the Earth. Objects placed near the Earth will actually follow a path that loops near Earth then far away (there are actually asteroids that do this). Objects at L4 and L5 will trace out very small lines around the L points.

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D H
Staff Emeritus
See diagram (please! I insist! It took me an hour to do that from scratch in Illustrator)

You can see that the Moon follows a line around the Earth. Objects placed near the Earth will actually follow a path that loops near Earth then far away (there are actually asteroids that do this). Objects at L4 and L5 will trace out very small lines around the L points.
If you insist ...

Very nice. Can you turn that into a rotating frame equipotential plot?

It's too bad the article describe the contents of the image, reference the image, or have anything to do with the image at all. Then again, it's a Wikipedia article. You get what you pay for.

DaveC426913
Gold Member
Can you turn that into a rotating frame equipotential plot?
I think I have a macro for that...

thanks for wasting an hour on a question i posted.

You're right. The text explains it better. The contour lines are are not gradients (i.e. that objects will cross prependicularly) - they are paths (i.e. that objects will follow). If you place an object on any line in the diagram, the object will trace out that line.
i didn't think it was either gradients or paths, but equi-potential curves (surfaces perpedicular to the gradient) .maybe that's the same as a path, but an object could have the counterpart to an "elliptical orbit" for its path and that would cross equi-potential lines, no?

Use Newton's 2nd law, and then you will be able to figure out that dust should not collect at any Lx point unless it has no motion and is somehow placed in those points.

Use Newton's 2nd law, and then you will be able to figure out that dust should not collect at any Lx point unless it has no motion and is somehow placed in those points.
that's like saying that a marble should not collect at the bottom of a hemispherical bowl when released on the side near the rim. if there was no turbulance or increase in entropy, then the ball would be rolling back and forth (or in a little elliptical orbit around the bottom of the bowl) forever. but, if you let it go around and around and come back the next day, might you find that marble at the bottom of the bowl? the other L1, L2, and L3 Lagrangian points are like saddle points so anthing left there is bound to drift away. but L4 and L5 are depressions although very slightly.

DaveC426913
Gold Member
From http://www.nineplanets.org/asteroids.html" [Broken]:

"Trojans: located near Jupiter's Lagrange points (60 degrees ahead and behind Jupiter in its orbit). Several hundred such asteroids are now known; it is estimated that there may be a thousand or more altogether. Curiously, there are many more in the leading Lagrange point (L4) than in the trailing one (L5). (There may also be a few small asteroids in the Lagrange points of Venus and Earth (see Earth's Second Moon) that are also sometimes known as Trojans; 5261 Eureka is a "Mars Trojan".)"

http://cfa-www.harvard.edu/cfa/ps/lists/Trojans.html" [Broken]

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tony873004
Gold Member
The Earth/Moon L4 and L5 points are not very stable at all. Even if an object is placed directly on the point, with no velocity relative to the point, it would quickly drift out of perfection, and ultimately be ejected from the Earth/Moon system, or collide with the Earth or Moon.

The solar gravitational tidal force is the reason for the instability. The size of the Earth/Moon system is large compared to the distance of the Earth/Moon system from the Sun. Its size is about 0.5% of Earth's solar distance. This creates a significant gravitational gradient across the Earth/Moon system by the Sun.

If the Earth/Moon system orbited much further from the Sun, then the Earth/Moon L4 & L5 points would be stable.

I've simulated, and the best I could do by placing an object directly on the L4 or L5 points was to have it last about 20 years. In one of the points, (I think L5) the object departs after only a few orbits. The other point (I think L4) does better, but usually can not hold onto the object for more than about 2 years (~24 lunar orbits), although my record is ~20 years.

Since such a simulation is time-reversable, it should imply that under the right circumstances, The Earth/Moon L5 point can temporarily capture objects.

There should be a formula that relates L4 & L5 stability to the Hill Sphere. For example, the Moon is about 1/3 the distance to Earth's Hill Sphere. At (1/3)*HS, the L4 & L5 points are not stable. At some point < 1/3, they will become stable. A series of simulations could probably expose the relationship.

tony873004