Orbiting around Lagrange points

[gregweymann]

Physics gurus: I understood from Newton's Law that a 2 bodies would rotate around their common center of mass. Should one body disappear (Harry Potter invoked here), the other would go flying off at a tangent... like a 'David's Sling" releasing a missile. The mass of the bodies was crucial to figuring the orbits; without 2 or more bodies possessing mass, there was no curvilinear motion.

Einstein proposed that mass 'deformed' space-time so that the track of a satellite orbiting a massive second body was the most natural path. Again with the MASS.

Could someone please attempt explaining the orbits of ARTEMIS-P1, Planck, and Herschel around Lagrange points which are not massy at all? There's no MASS there.

Thank you.

Greg Weymann

 

[DrFurious]

Basically the triangular Lagrange points L4 and L5 are stable - there is a minimum of potential energy in these places. When perturbed, an object will orbit around in a kind of a "kidney bean" shape.

The mass is provided by the two bodies. The orbit is described by how these bodies warp spacetime.

 

[gregweymann]

DrFurious: Thanks for the quick response. May I assume that you're saying "... the two bodies..." are either Earth-Moon or Earth-Sun whose configuration produces local areas of minimum potential energy?

In a non-inertial frame of reference, how does the mass of the 2 bodies project a 'virtual mass' at a third location? Doesn't this third location (a Lagrange point) still need to simulate a mass around which another body may orbit?

I can visualize an object held in such a potential energy 'well'; but, why do the satellites orbit that 'well' without recourse to constant thrust?

 

[DrFurious]

Yeah, so in the Earth-Moon system the mass of the Earth and the Moon warp spacetime. The curvature of spacetime is this potential energy "well". Satellites orbit the Earth because of this curvature. It doesn't matter that there's no mass at L4 and L5, only that spacetime is warped in a certain way (due to the system). Objects are perfectly happy orbiting away as long as there's a potential energy minimum.

In a non-inertial reference frame, you would just feel the effects of this potential energy well. It doesn't make sense to say that there's "virtual mass" there because the curvature is described perfectly by the objects with mass.

I'm having a hard time explaining this very well. My best advice would be to work out the gravitational potential energy of such a system yourself. Then you can decide what the mathematics are saying for yourself.

tl;dr The curvature of spacetime is all that matters.

 

[Bill_K]

This has nothing to do with Einstein or the warping of space, just Newtonian mechanics and the gravitational field. A particle near one of the Lagrange points L₄ and L₅ is orbiting the Sun in the Earth's orbit, either 60 degrees ahead or behind. Its orbit wobbles slightly, giving the appearance that it is orbiting the point rather than the Sun.

 

[DrFurious]

This has nothing to do with Einstein or the warping of space, just Newtonian mechanics and the gravitational field. A particle near one of the Lagrange points L₄ and L₅ is orbiting the Sun in the Earth's orbit, either 60 degrees ahead or behind. Its orbit wobbles slightly, giving the appearance that it is orbiting the point rather than the Sun.

I didn't say Einstein. Matter warps spacetime... and the gravitational potential energy well is just a good example of that.

Why do you say that it only has the "appearance" of orbiting the point? ₄ and L₅ are potential energy wells... can't think of a reason why they wouldn't be orbiting them in any reference frame.

 

[Bill_K]

Why do you say that it only has the "appearance" of orbiting the point? L₄ and L₅ are potential energy wells... can't think of a reason why they wouldn't be orbiting them in any reference frame.

Because the "well" you're referring to only appears in a rotating frame. It includes the centrifugal effect of a once-per-year rotation about the sun. In an inertial frame the particle orbits the sun.

Matter warps spacetime... and the gravitational potential energy well is just a good example of that.

Relativistic effects at a distance of 1 AU from the sun are entirely negligible and have never been detected.

 

[DrFurious]

Relativistic effects at a distance of 1 AU from the sun are entirely negligible and have never been detected.

Actually, and I didn't know this until a couple of weeks ago, but the first detection of relativistic effects were done HERE on Earth using two clocks in a tower. One clock on the top of the tower and one clock at the bottom. The difference in the curvature of spacetime caused by the Earth at the top of the tower and the bottom are enough to be measured due to the different (in one reference frame) rates the clocks tick.

 

[sophiecentaur]

I thought the Mossbauer effect was detectable (when I was at Uni in the 60s) on different levels of a building. Isn't that a relativistic thing?

 

[tankefugl]

To re-iterate Bill_K, the five Lagrange points have nothing to do with the warping of spacetime: The points can be found with Newtonian mechanics.

The exact derivations can be found by googling for it, so I'll just provide an example:

L1 lies between the Earth and the Sun. Due to the gravitational pull of the Earth, any object placed in L1 will have the same time of orbit as the Earth. Because of this it will seem to be stationary, from our frame of reference moving around the Sun.

There is no mass there, it's just a pont where the gravitational pull of the Earth and Sun together make it move in such a way that it is stationary with regards to the Earth's orbit.

The "orbits" around some of these points are due to some of them being the local minimas of potential energy, thus allowing objects to oscillate around them.

 

[gregweymann]

So, conflating the replies, may I infer that the "orbits" of bodies around Lagrange points aren't.
They are not "orbits" in the true sense of the word, but the paths are better described as 'wobbles'?

I return to a basic conundrum: the whole assemblage (Earth, Moon, & associated Lagrange points) does orbit around the Sun. Bill_K points out that the Lagrange point does not describe a smooth curve, but has a wobble... and a small-period oscillation of a low-mass object near that point might be construed as an "orbit" about that point.

So, is the satellite continually falling into the space where all the gravitational forces approach a minimum (thanks, tankefugl)? Like a kitten chasing a string?

 

[sophiecentaur]

I think they are just as real as other 'orbits'. They rely on more than one dominant mass so, in that way they are a bit different. But - look at the orbit of the Moon around the Sun. That's pretty complex. And what about the horseshoe orbits that satellites of planets can follow?
One man's wobble is another man's orbit. Why the preoccupation with semantics?

 

[tankefugl]

I return to a basic conundrum: the whole assemblage (Earth, Moon, & associated Lagrange points) does orbit around the Sun. Bill_K points out that the Lagrange point does not describe a smooth curve, but has a wobble... and a small-period oscillation of a low-mass object near that point might be construed as an "orbit" about that point.

So, is the satellite continually falling into the space where all the gravitational forces approach a minimum (thanks, tankefugl)? Like a kitten chasing a string?

Lagrange points are stationary and do not wobble or osccilate. Objects may orbit around the Lagrange points and it's these objects who "wobble".

The gravitational forces do not approach a minimum in these points: They are points of equilibrium in the potential energy. This means that the change in potential energy as you move them around these points is small.

Note that equilibrium can be minimum, maximum or neither! Sadle points, for example. L1, L2 and L3 are such sadle points: To stay at any of these points requires frequent minute adjustments. L4 and L5 are minimums.

I do like the image of a kitten chasing a string though =)

 

[sophiecentaur]

Why do you say the points don't wobble? Their position, relative to Earth, (although dominated by Sun and Earth) must also be affected by the position of the Moon wrt Earth and that is always changing so they must change too.

 

[tankefugl]

Ah, yes, in that respect you are right. I must admit I primarily regarded the Earth-Sun system above.

 

[sophiecentaur]

OK. We're speaking the same language then.

I have just one comment about this though. The potential minimum of an object at the LPs, presumably assumes that it is already in (or near to) the appropriate orbit around the Sun. An object 'passing through' on a different trajectory would, presumably, just experience the sum of the two G fields from Earth and Sun and be 'unaware' that it was passing through a 'special place' (unless it noticed the large amount of space debris that it would pass near to).