# The Naked Scientists Forum

### Author Topic: How do equilibrium points measure up?  (Read 1223 times)

#### jeffreyH

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##### How do equilibrium points measure up?
« on: 14/02/2015 17:00:02 »
Using contours of equal gravitational potential Lagrange found equilibrium points. These should occur within different field strengths. So if we find an equilibrium point within the sun earth moon system how would this differ to a point between two large dense masses in a binary system? Do the forces produce different results and what are those differences?

Details of what I am talking about can be found here:

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/lagpt.html
« Last Edit: 14/02/2015 17:17:46 by jeffreyH »

#### evan_au

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##### Re: How do equilibrium points measure up?
« Reply #1 on: 15/02/2015 02:53:32 »
The location of the L1, L2 & L3 points depends on the ratio of masses of the two larger objects.

The L4 & L5 points (the "Trojan" positions) are positioned 60° ahead and behind the position of the Moon. (This may be different if there were two large objects of equal mass).

Lagrange's 5 solutions were found by locating orbits which are in a fixed position relative to the two more massive objects. Astronomers have found one asteroid at the Earth's L4 point. (if I were an alien observer, this may be where I would choose to put my telescope...)

More recently, astronomers have found some objects orbiting the Sun/Earth system which are not at the Lagrangian points. 3753 Cruithne is an asteroid which has the same average distance from the Sun as the Earth. It follows a "C" shaped orbit relative to the Earth. When it gets close to the Earth, it slows down or speeds up to move away from the Earth, and then travels around the other side of the Sun to approach the Earth from the other direction, 770 years later.

Lagrange's 5 points assumed that there were only 3 objects in the system. However, the Solar System has 9 planets and multiple moons, so asteroids at Earth's Trojan points are continually tugged by other objects (especially Jupiter), and so these L4 & L5 positions may not be stable over astronomical timescales.

#### jeffreyH

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##### Re: How do equilibrium points measure up?
« Reply #2 on: 15/02/2015 23:00:06 »
These Lagrangiam points are like dimple in spacetime. The gradient around them in three dimensions is not uniform. The best way to restate my question is by taking cavities inside masses centred around their centres of gravity. Say between the Earth and Saturn. The centre of gravity would then be the equilibrium point. My question is would every such equilibrium point have the same profile? Would moving away from equilibrium require the same force in every case? The forces cancel but the gradient of the gravity well differ? I could sit and figure this out but at the moment I am looking at other things.

BTW I mean like for like motion. So that if we consider moving one metre away from equilibrium in the Earth cavity we consider the same one metre movement in the Saturn cavity.
« Last Edit: 15/02/2015 23:02:02 by jeffreyH »

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##### Re: How do equilibrium points measure up?
« Reply #2 on: 15/02/2015 23:00:06 »

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