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I was reading on up Lagrangian points and the restricted three-body problem.

From what I was able to tell, the Lagrangian points are 5 points in a two-body system such that a third body would be relatively at rest. The first three are unstable, and the last two are stable.

How is this possible? Because we know from Kepler's laws that the orbits are in the shape of ellipses with the sun at a focus, and we also know the planes sweep out equal areas in equal time and so the speed varies. And so, how can the third body have a constant distance when there is an ever-changing speed gap between the orbital speeds.

And could anyone provide (visually) an explanation for the Lagrangian points, how to deduce them, and what assumptions were made in order to deduce them?

LE: So, in the three body problem, the orbits more closely resemble circles than ellipses? And if so, is the speed relatively constant so that the difference in distance between the third and second body is negligible?

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As mentioned in that same article, L4 and L5 are only stable if the ratio of the masses of the objects exceeds $(25+3\sqrt{69})/2$, which happens to be the case for both the Sun-Earth system and the Earth-Moon system. –  Chris White Aug 14 '12 at 3:27

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The apparent contradiction with the second Kepler law which you refer to is due to incompatible assumptions of the models in question.

Both Kepler laws and Lagrangian points can be derived from Newtonian laws of motion and the law of universal gravitation under very specific assumptions about which objects can be neglected and which must be taken into account. In order to derive Kepler laws one assumes the universe with only two bodies orbiting around their barycenter. In order to derive Lagrangian points one assumes the universe with three bodies with one of them having mass negligible compared to the other two.

For example, calculating SOHO's orbit around the Sun using Kepler laws amounts to assuming Earth's influence on SOHO is negligible. This would give SOHO higher orbital velocity, but since Earth's influence is in fact very significant it would disagree with observations completely. On the other hand, employing Lagrange solutions to the restricted three body problem assumes that the Sun's and Earth's influence is significant. In both cases the influence of the Moon, Jupiter and other planets and objects of the Solar system is assumed to be negligible. Thus both solutions produce approximations to the real world with the first being very inaccurate and entirely useless while the second having some useful relation to the observed reality.

As for the derivation of Lagrange points, you will find it here and in most books on orbital mechanics, e.g. A.E.Roy's Orbital Motion.

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So, in the three body problem, the orbits more closely resemble circles than ellipses? And if so, is the speed relatively constant so that the difference in distance between the third and second body is negligible? –  andreas.vitikan Aug 13 '12 at 13:57
    
Circular orbit is not the consequence, but part of the assumptions in the derivation of the Lagrange points (see the wikipedia article you linked to in the question). The distance between the second and third body isn't negligible, but remains constant due to zero relative velocity. –  Adam Zalcman Aug 13 '12 at 16:48

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