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- Thread starter mufcdiver
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Considering two equal masses and a small test mass midway between these, then yes, there will be a point of unstable-equlibrium. In a real situation, any minor deviation from this point would cause the test mass to move to the nearest large mass.

There will be a plane between the two large masses whereby theoretically the test mass could make an orbit in this plane at right angles to the axis joining the two large masses. I.e It would orbit around 'nothing'. It would be unstable and would work mathematically. In reality any minor disturbance would make the test mass move to either of the large masses.

This orbit can be imagined as if you put the test mass midway between the two large masses and to one side of the central axis, it wolud experience a resultant restoring force directed towards the mid-point. Hence , given a push tangentially, it should orbit in this plane.

For two unequal large masses, there would be an unstable equilibrium point between them that is nearer to the smaller mass, but off-hand I don't think the test mass could make an orbit as above as there would be no plane at right angles in this situation.

For more than two large masses, then there can still be one equilirium point for the test mass. As long as one of the masses is physically so large as to make the equilibrium point inside the large mass or indeed the arrangement makes it so the equilibrium point is inside any particular large mass.

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DaveC426913

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It should still work fine. I think the plane might be more accurately described as a very shallow hyperbolic plane.For two unequal large masses, there would be an unstable equilibrium point between them that is nearer to the smaller mass, but off-hand I don't think the test mass could make an orbit as above as there would be no plane at right angles in this situation.

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All of you are treating this problem statically, and that is something you should not be doing. The two massive objects will attract each other gravitational as well as attracting the test mass. Each of the massive objects will be in some orbit about each other (and about their center of mass). Ignoring this motion leads to an incorrect identification of the equilibrium point. For example, the point at which space is gravitationally flat is not an equilibrium point unless the two massive bodies have the same mass. An unstable equilibrium point between the two massive bodies does exist, but it is not at the point where space is gravitationally flat.

This problem in general is called the three body problem. The problem becomes more tractable if the gravitational attraction induced by the test mass can be ignored (i.e., a non-massive test mass); this is the restricted three body problem. The special case of the massive bodies being in a circular orbit is called the restricted circular three body problem. Five equilibrium points exist for the restricted circular three body problem, one of which lies between the two massive bodies. This is the L1 equilibrium point. An inertial observer will see the L1 point as being in a circular orbit about the system center of mass. An observer in a rotating frame based on the two massive bodies will see all five LaGrange points as having fixed locations.

A satellite at the Sun-Earth L1 point would have unrestricted viewing of the Sun. However, the L1 point is an unstable equilibrium point. A spacecraft positioned at the Sun-Earth L1 point would have to expend a lot of fuel to maintain that position. A better alternative is to "orbit" the L1 point. Such orbits are still unstable, but not nearly so bad as trying to stay exactly at the L1 point. This is exactly the strategy used for the SOHO satellite, which is in a halo orbit about the Sun-Earth L1 point.

This problem in general is called the three body problem. The problem becomes more tractable if the gravitational attraction induced by the test mass can be ignored (i.e., a non-massive test mass); this is the restricted three body problem. The special case of the massive bodies being in a circular orbit is called the restricted circular three body problem. Five equilibrium points exist for the restricted circular three body problem, one of which lies between the two massive bodies. This is the L1 equilibrium point. An inertial observer will see the L1 point as being in a circular orbit about the system center of mass. An observer in a rotating frame based on the two massive bodies will see all five LaGrange points as having fixed locations.

A satellite at the Sun-Earth L1 point would have unrestricted viewing of the Sun. However, the L1 point is an unstable equilibrium point. A spacecraft positioned at the Sun-Earth L1 point would have to expend a lot of fuel to maintain that position. A better alternative is to "orbit" the L1 point. Such orbits are still unstable, but not nearly so bad as trying to stay exactly at the L1 point. This is exactly the strategy used for the SOHO satellite, which is in a halo orbit about the Sun-Earth L1 point.

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Is there one of these points at the [gravitational] center of the Milky Way?

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There would be corresponding LaGrange points for the Milky Way and Sun

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The simple answer is yes.

In a two body system a lateral thrust (say at right angles to the two bodies in question) that is not sufficient to reach escape velocity will eventually after the oscillation calms down translate into torque on the small body that would result in it returning to the balance point with spin that accounts for the conservation of angular and linear momentum. In a three, or multibody system if the thrust is just right (planar is easier to visulize, but not a requirement) it is possible if the masses create a locus of equipotential gradient that a small central mass is perturbed it will settle into some orbit around the centre of potential along the gradient well proporortional to its mass relative to the base masses and the distance from the theoretical centre and the velocity of orbit and the amount of energy translated into spin of the small object. See the previous posters links and some research on momentum.

Nothing is as odd as the truth.

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