Lagrange Points, Maximum mass and their effects

[Chand-Ashoka]

So, I'm working through some ideas dealing with Lagrange points.

I understand that, the rotation and mass of 2 objects in space create stable areas where an object of "insignificant Mass" compared to the objects it's balancing against, allows for the placement of an object in a stable area.

My first question deals with the idea of what exactly is "insignificant mass" on a planetary term. For the example, let's use the Earth-Moon system for placement of objects within the Lagrange points.

I assume (big mistake I know) - that the mass would be balanced against the mass of the moon, as it's the smaller of the two objects. But how much mass could be safely "stored" in a particular Lagrange point? Honnestly, even if someone could ball park a percentage of the Moons mass (IE no greater than 1%) - I could wrap my head around this a little easier.

Second would be, is the total mass that is able to be placed in stable positions a combined total, or does each Lagrange point have it's own maximum total separate from the other 4 (Or, as would make sense a combination of both L1 can only have a maximum of Xkg of mass at it while the entire Lagrange system cannot exceed Ykg of mass).

Third, and I guess, the most interesting to me, what would happen if this mass was exceeded. I would assume (again, bad) - that if the mass was exceeded by a small ammount, the object in question would be caught up in one or the others (Moon or Earth, depending on placement) - and get dragged into that object. Or... would the centrifugal force cause them to break orbit and be flung off?

In addition, if we can let our imagination wander for half a second - what if something was to be introduced at an L point that was greatly outside the Maximum mass for that point IE 2-3 times as much mass? Would the same effect happen (either flung into space or attracted to the moon/earth and all the consiquences there in) - or would it affect the orbit of either of the two anchoring object? Again, my assumption would be that it would have a greater chance of affecting the moon than the earth, in the Earth-Moon Lagrange system...


Sorry if some of these are really easy questions - but, some of this is well beyond my area of knowledge - and as I looked into it, these are the questions that cropped up

 

[zhermes]

These are all good questions, the main thing you're missing is why they say 'insignificant mass' in the first place. "insignificant mass" is required just so that the two bodies (e.g. Earth and moon) aren't themselves effected by the mass at the lagrange points.

The lagrange points arise out of a static situation (i.e. a nice smooth, periodic orbit), if you added a third object that had a comparable mass, the force of gravity from that object on the two primaries would upset that static situation, making the lagrange points unstable.

My first question deals with the idea of what exactly is "insignificant mass" on a planetary term. For the example, let's use the Earth-Moon system for placement of objects within the Lagrange points... if someone could ball park a percentage of the Moons mass (IE no greater than 1%) - I could wrap my head around this a little easier.

I suppose the answer comes down to what type of time-frame you're looking at. The actual lagrange points for the earth-moon system aren't perfectly stable because the earth-moon system isn't perfectly stable (the moon is very slowly moving away, and its orbit is precessing, etc etc). Now, if you added a large third mass, the bigger the mass the faster the system would destabilize.

Roughly, I'd guess that 0.1% moon mass would be pretty much negligible compared to the existing causes of instability. 1% mass would destabilize the system in the long term; 10% would be immediately unstable.

Second would be, is the total mass that is able to be placed in stable positions a combined total, or does each Lagrange point have it's own maximum total separate from the other 4

Again, its just a matter of how much additional gravitational force you're exerting on the initial two planets.

Third, and I guess, the most interesting to me, what would happen if this mass was exceeded.

This is just a "three-body" problem, and the results vary tremendously depending on the initial conditions.

 

[Chand-Ashoka]

These are all good questions, the main thing you're missing is why they say 'insignificant mass' in the first place. "insignificant mass" is required just so that the two bodies (e.g. Earth and moon) aren't themselves effected by the mass at the lagrange points.

The lagrange points arise out of a static situation (i.e. a nice smooth, periodic orbit), if you added a third object that had a comparable mass, the force of gravity from that object on the two primaries would upset that static situation, making the lagrange points unstable.

That's actually why I was asking about the insignificant mass - I figured, if you overloaded one or more Lagrange points, that it wouldn't just destablize the object within the point, but have an (undesirable) effect on the other two objects.

But, for me anyway, the size and scope of what we're dealing with makes me wonder what exactly "insignificant" is when dealing with object that are, at least, in the Quintillions of megatons

I suppose the answer comes down to what type of time-frame you're looking at. The actual lagrange points for the earth-moon system aren't perfectly stable because the earth-moon system isn't perfectly stable (the moon is very slowly moving away, and its orbit is precessing, etc etc). Now, if you added a large third mass, the bigger the mass the faster the system would destabilize.

Granted, while L1, L2 and L3 are relatively stable, L4 & L5 are strange, with objects in them moving around in a... what kidney bean shape area?
So, does that make them less stable in the over all, meaning that less mass can be placed in those spots due to instability, before they're disrupted by the gravity of the other two objects?

Roughly, I'd guess that 0.1% moon mass would be pretty much negligible compared to the existing causes of instability. 1% mass would destabilize the system in the long term; 10% would be immediately unstable.

Again, its just a matter of how much additional gravitational force you're exerting on the initial two planets.

This is just a "three-body" problem, and the results vary tremendously depending on the initial conditions.

What if it's taken beyond a 3 body problem? What if the gravitational forces are exerted upon the moon, say from the 4 surrounding Lagrange points? (L1, 2, 4 and 5). From what I understand, the forces would exert equally on the moon from all for sides, potentially destroying it... which... well, would be bad. (and all the issues that come with it).

Please forgive, I haven't exactly done physics in over a decade. I'm currently doing a little writing and want to make sure the concepts are correct and I'm not just going off on an implausable tangent...

The good news is, for being a complete noob at this... I'm on the right track :D


Edit: and again, thank you for your help :)

 

[zhermes]

Granted, while L1, L2 and L3 are relatively stable, L4 & L5 are strange, with objects in them moving around in a... what kidney bean shape area?
So, does that make them less stable in the over all, meaning that less mass can be placed in those spots due to instability, before they're disrupted by the gravity of the other two objects?

Lol, yeah, kidney beans. Horseshoes in the extreme case ("horseshoe orbits").
As for stability, the L4 and L5 (I think) are actually the most stable per se, because of those sub-orbits. In regards to masses in the L points perturbing the primary binary, I'd say L1 and L2 are the least stable and the rest about equivalent.

What if it's taken beyond a 3 body problem? What if the gravitational forces are exerted upon the moon, say from the 4 surrounding Lagrange points? (L1, 2, 4 and 5). From what I understand, the forces would exert equally on the moon from all for sides, potentially destroying it... which... well, would be bad. (and all the issues that come with it).

It would be pretty hard to 'destroy' the moon that way. If you distributed a given mass to all 4 of those points it would increase the stability, for sure. The detailed dynamics of it would be very complex, however.

Happy to help; hopefully I'm not leading you astray.

And don't worry too much about the implausible tangents, those are fun; its the impossible ones you have to watch out for.

 

[Chand-Ashoka]

Lol, yeah, kidney beans. Horseshoes in the extreme case ("horseshoe orbits").
As for stability, the L4 and L5 (I think) are actually the most stable per se, because of those sub-orbits. In regards to masses in the L points perturbing the primary binary, I'd say L1 and L2 are the least stable and the rest about equivalent.

Ah ok, so out of them, it's L1 and 2 you need to worry about overloading a bit more.
What about L3, being way the hell on the other side of the planet. Now, I'm just thinking out loud.

It would be pretty hard to 'destroy' the moon that way. If you distributed a given mass to all 4 of those points it would increase the stability, for sure. The detailed dynamics of it would be very complex, however.

Happy to help; hopefully I'm not leading you astray.

And don't worry too much about the implausible tangents, those are fun; its the impossible ones you have to watch out for.

Not at all, you're pretty much leading me to where my mind was headed anyway - but as I said, I've got no back ground in this, beyound curiosity, so I wanted some back up :D.

But, yes, I imagine distrubting 1% of the mass over the 4 Lagrange points there would cause it to be a bit more stable, than having the full 1% at anyone specific point. The issue I'm tackling was, what if you put 1% at each of those points?

On a basic level, what would happen if you over loaded the L1, L2, L4 and L5 Lagrange points with a signifigant amount of mass. Something less than the over all 10% CRAP THE WORLD IS GOING TO END weight, but more than the "this is slightly bad" weight.

I know these are all off-the-cuff numbers, but that's all I'm looking for. As well as the theoretical end results.

 

[qraal]

Hi Guys
The Moon wouldn't be destroyed by "overloading" the L points - putting equal masses in L 4 & 5 would stabilize the system since the perturbations from the two extra masses would cancel each other out. But there is a maximum mass ratio which is stable between the Secondary body and the body in the L point of 1/24. Else the two end up on walking orbits and eventually collide. That's probably how Theia - the Mars mass object that smashed into proto-Earth to make the Moon - ended up with a similar composition to Earth, because it formed in an L point (L 4 or 5 I'm unsure) then got too big and began wandering back and forth, eventually smacking into Earth.

 

[qraal]

Hi Guys
The Moon wouldn't be destroyed by "overloading" the L points - putting equal masses in L 4 & 5 would stabilize the system since the perturbations from the two extra masses would cancel each other out. But there is a maximum mass ratio which is stable between the Secondary body and the body in the L point of 1/24. Else the two end up on walking orbits and eventually collide. That's probably how Theia - the Mars mass object that smashed into proto-Earth to make the Moon - ended up with a similar composition to Earth, because it formed in an L point (L 4 or 5 I'm unsure) then got too big and began wandering back and forth, eventually smacking into Earth.

Actually that stability limit is for the primary and secondary bodies, not the L point occupant/s. I'm not altogether sure there's a stable mass-ratio that can be found mathematically, as most studies I've read on the formation of Theia use Many-Body codes to model the system over several million years, thus it's AFAIK not a derived result. Perhaps someone out there in Physics Forums Land knows a derivation?

 

[Chand-Ashoka]

So, forgive

Overloading the Lagrange points wouldn't cause damage to the moon, per se, due to gravity stress, but would, instead, drag the objects of our their stable position and towards one of the two other objects.

Not quite as dramatic as doing damage to the moon, but still... not something you'd want to have happen, especially if they wander to the Earth.

 

[qraal]

So, forgive

Overloading the Lagrange points wouldn't cause damage to the moon, per se, due to gravity stress, but would, instead, drag the objects of our their stable position and towards one of the two other objects.

Not quite as dramatic as doing damage to the moon, but still... not something you'd want to have happen, especially if they wander to the Earth.

Wandering objects sharing an orbit are more likely to collide with each other or be ejected from the system than collide with the central body - but, being a chaotic three-body system, I wouldn't rule that out of the possible outcomes either. One theory of the asteroid belt involves a fifth planet which was eventually thrown into the Sun because of chaotic interactions with Mars and Jupiter, and in the next 5 billion years there's a non-zero chance of Mercury being sent diving into the Sun by interactions with Earth & Venus. A low chance, but not impossible. It's unlikely because to detach a body from a distant orbit means adding only an extra ~41.4% velocity, whereas as dropping a body into the central object means subtracting 100% of its velocity. Thus the former is more likely than the latter.