PDA

View Full Version : How do Astrophysicists locate the position of Lagrangian Points?



arslan
2011-Mar-11, 11:43 PM
Is there like a mathematical formula or equation? If so, can I know what it is? Because Im really curious about how these points are located, especially L4 and L5.

Grashtel
2011-Mar-11, 11:51 PM
The Wikipedia page on Lagrange points (http://en.wikipedia.org/wiki/Lagrange_point) has information about them and explains how to work out their locations.

Hornblower
2011-Mar-11, 11:54 PM
Is there like a mathematical formula or equation? If so, can I know what it is? Because Im really curious about how these points are located, especially L4 and L5.

L4 and L5 are in the same orbit, 60 degrees from the planet, assuming a circular orbit.

swampyankee
2011-Mar-12, 02:05 PM
If you don't like Wikipedia, follow this link (http://www.physics.montana.edu/faculty/cornish/lagrange.html) and read the linked pdf (http://www.physics.montana.edu/faculty/cornish/lagrange.pdf).

grapes
2011-Mar-12, 02:56 PM
L4 and L5 are in the same orbit, 60 degrees from the planet, assuming a circular orbit.Not quite the same orbit. Since the centers of mass of the three bodies form an equilateral triangle, but the orbits are about the barycenter of the first/second body, that means the third body has a slightly larger orbit.

korjik
2011-Mar-12, 04:00 PM
Not quite the same orbit. Since the centers of mass of the three bodies form an equilateral triangle, but the orbits are about the barycenter of the first/second body, that means the third body has a slightly larger orbit.

two concentric circles cannot have different radii. Two circles of different radii cant be concentric.

Nowhere Man
2011-Mar-12, 05:04 PM
What? Concentric means "having the same center."

Fred

korjik
2011-Mar-12, 06:13 PM
What? Concentric means "having the same center."

Fred

yeah, and being in the same orbit generally implys concentric-ness. It is really difficult to do otherwise

Shaula
2011-Mar-12, 07:03 PM
It was your comment that two circles with different radii couldn't be concentric that was causing the question, I think. They can. Orbits, trickier. But circles can be concentric and different radii very easily!

korjik
2011-Mar-12, 07:51 PM
It was your comment that two circles with different radii couldn't be concentric that was causing the question, I think. They can. Orbits, trickier. But circles can be concentric and different radii very easily!

yeah, thats right, I didnt mean what I said :)


Two identical orbits certainly cant have different centers tho.

grapes
2011-Mar-12, 08:43 PM
yeah, thats right, I didnt mean what I said :)



Two identical orbits certainly cant have different centers tho.If they're identical, they're ... identical. :)

But in this case, they have the same (moving!) center, but different radius.

loglo
2011-Mar-12, 09:24 PM
If you don't like Wikipedia, follow this link (http://www.physics.montana.edu/faculty/cornish/lagrange.html) and read the linked pdf (http://www.physics.montana.edu/faculty/cornish/lagrange.pdf).

I am surprised that there is no use of the Lagrangian in this derivation of the Lagrange points. The wiki article implies that Lagrange developed the Lagrangian for this specific problem so is it hiding in a form I don't recognise or did Lagrange do it differently?

korjik
2011-Mar-12, 09:28 PM
If they're identical, they're ... identical. :)

But in this case, they have the same (moving!) center, but different radius.

You cant have the same center and a different radius and still be on the same orbit. From the barycenter frame of reference, the primary and secondary each trace ellipses that are 180 degrees out of phase, with a relative size of (ms/ms+mp)as=ap with the barycenter on one of the focii of each ellipse. An object orbiting at either the L4 or L5 point is constrained to the same orbit as the secondary. this means it orbits on an ellipse with the barycenter on one of the focii, and it has to be the same focus as the secondary. With elliptical orbits, the instantanous radius will not be the same, but the barycenter, and both axis will be identical to the secondary.

You can see this in the equilateral triangle situation in the case of the circular orbit. In that case, the primary isnt orbiting, the secondary is in a circular orbit, and there is an equilateral triangle between the L-point the primary and the secondary. By definition, the distance from the primary to both the secondary and the L-point must be identical. Since the triangle applies to all points of the orbit, then at all points of the orbit the l-point and the secondary must have the same distance, which means they are on the same circle.

So, you cant have the same circle and have two different centers or two different radii.

chornedsnorkack
2011-Mar-12, 10:17 PM
An object orbiting at either the L4 or L5 point is constrained to the same orbit as the secondary. this means it orbits on an ellipse with the barycenter on one of the focii, and it has to be the same focus as the secondary. With elliptical orbits, the instantanous radius will not be the same, but the barycenter, and both axis will be identical to the secondary.

You can see this in the equilateral triangle situation in the case of the circular orbit. In that case, the primary isnt orbiting, the secondary is in a circular orbit, and there is an equilateral triangle between the L-point the primary and the secondary. By definition, the distance from the primary to both the secondary and the L-point must be identical. Since the triangle applies to all points of the orbit, then at all points of the orbit the l-point and the secondary must have the same distance, which means they are on the same circle.

So, you cant have the same circle and have two different centers or two different radii.

If the mass of the secondary is not zero then the primary is orbiting.

Primary, secondary and tertiary body are on an equilateral triangle, but the centre of mass of the three is near the corner where the primary is. If secondary and tertiary are of equal mass (which is possible) then secondary and tertiary do indeed have the same orbital radius. If not then the orbital radii of secondary and tertiary are different.

arslan
2011-Mar-14, 04:09 AM
Grapes is right, the orbit of L4 and L5 points in the Earth-sun system is slightly larger than the orbit of the Earth. This is because the L4 and L5 points orbit the barycenter of the Earth-sun system, which is a little farther out from the center of the sun, though still being inside the sun.

grapes
2011-Mar-14, 11:27 PM
So, you cant have the same circle and have two different centers or two different radii.I'll grant you that! :)

However, the distance between the three centers of mass is the same. Since all orbit the barycenter, the distance between the barycenter and the two smaller bodies has to be different, geometrically, even in circular orbits.
If secondary and tertiary are of equal mass (which is possible) then secondary and tertiary do indeed have the same orbital radius. My head hurts! 'course part of it is because I'm on jury duty...