Lagrangian Points

[Chaos]

I have some questions about the Lagrangian points. I´ve read the Wikipedia article on them, but unfortunately the math present in it makes my head hurt just looking at it, and it doesn´t adress two of the question questions at all.

So...

1) The article mention that the L4 and L5 points, the Trojans, are stable if the mass ratio, M1/M2, is at least 24,96. What happens if the ratio falls below that? Do they not exist at all, or are they just unstable like the other points?

2) Intuitively, I would say that, the smaller the difference between M1 and M2, the more L1, the point between both bodies, would be to the midpoint between these bodies - to the point where, if M1 = M2, both the L1 point and the barycenter would be exactly at the midpoint between the two bodies. Is that correct?

3) In which direction do L2 and L3 "move" if the difference between M1 and M2 is smaller than in the Earth/Moon, Sun/Earth and Sun/Jupiter systems discussed in the Wikipedia article - say if we´re talking about a close binary consisting of two similar stars? Where would they be if M1 = M2?


Thanks in advance for taking the time to answer these questions.

[antoniseb]

I'm not sure what you use to conceptualize things, but basically, we're just looking for orbits that keep everything in the same orientation.
So, If M1 is many orders of magnitude larger than M2, L1 & L2 are practically on M2. If M1 = M2, L1 is midway between them, and L2 is somewhere giving it a one (year) orbit around the center of mass (coincidentally L1). L3 for small M2 is the same orbit as M2. For M1=M2 is a bit further out, being the same distance from M1 as L2 is from M2.

[frankuitaalst]

I have some questions about the Lagrangian points. I´ve read the Wikipedia article on them, but unfortunately the math present in it makes my head hurt just looking at it, and it doesn´t adress two of the question questions at all.

So...

1) The article mention that the L4 and L5 points, the Trojans, are stable if the mass ratio, M1/M2, is at least 24,96. What happens if the ratio falls below that? Do they not exist at all, or are they just unstable like the other points?

2) Intuitively, I would say that, the smaller the difference between M1 and M2, the more L1, the point between both bodies, would be to the midpoint between these bodies - to the point where, if M1 = M2, both the L1 point and the barycenter would be exactly at the midpoint between the two bodies. Is that correct?

3) In which direction do L2 and L3 "move" if the difference between M1 and M2 is smaller than in the Earth/Moon, Sun/Earth and Sun/Jupiter systems discussed in the Wikipedia article - say if we´re talking about a close binary consisting of two similar stars? Where would they be if M1 = M2?


Thanks in advance for taking the time to answer these questions.

I think the link herunder shows you a graph which may answer your questions 2 and 3.
http://www.orbitsimulator.com/cgi-bi...13908048/10#10

[grapes]

I have some questions about the Lagrangian points. I´ve read the Wikipedia article on them, but unfortunately the math present in it makes my head hurt just looking at it, and it doesn´t adress two of the question questions at all.

This one?
http://en.wikipedia.org/wiki/Lagrangian_point

So...

1) The article mention that the L4 and L5 points, the Trojans, are stable if the mass ratio, M1/M2, is at least 24,96. What happens if the ratio falls below that? Do they not exist at all, or are they just unstable like the other points?

They exist, but they are unstable, according to the wiki article.

2) Intuitively, I would say that, the smaller the difference between M1 and M2, the more L1, the point between both bodies, would be to the midpoint between these bodies - to the point where, if M1 = M2, both the L1 point and the barycenter would be exactly at the midpoint between the two bodies. Is that correct?

Yes, of course.

3) In which direction do L2 and L3 "move" if the difference between M1 and M2 is smaller than in the Earth/Moon, Sun/Earth and Sun/Jupiter systems discussed in the Wikipedia article - say if we´re talking about a close binary consisting of two similar stars? Where would they be if M1 = M2?

They would be symmetric, with respect to the barycenter, of course.

If the distance between M1 and M2 is 2R (distance to the barycenter, the center of the orbit, is R), then I get L2 and L3 at a distance of .7789R, where 1.7789 is the solution to x = 1/(x+1)^2 + 1/(x-1)^2, or x^5 - 2x^3 - 2x^2 + x +2 = 0

Wolframalpha.com
x^5 - 2*x^3 - 2*x^2 + x -2 = 0

Thanks in advance for taking the time to answer these questions.

De nada