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#11
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How big is L5?
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#13
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How big is L5?
(Karl Hallowell) writes:
(Gordon D. Pusch) wrote in message ... (Karl Hallowell) writes: Erik Max Francis wrote in message ... snip You're not alone; it's an extremely common misunderstanding, simply because of the weirdness of the stability of the Trojan points (as you say, due to Coriolis pseudoforces, not the usual mechanism). Plenty of informative Web sites have it wrong, as well. It "seems" obvious it's a valley, first perceptions can be wrong. I'm still not getting this. So what everybody is saying is that L4 and L5 are normal stable points (ie, is a "valley") with respect to the usual differential equations describe the dynamics of the orbits, No, we are saying L4 and L5 are =NOT= in "valleys" --- they are both sitting at the summits of "hills" !!! (Or, at least they are when viewed from the co-rotating frame.) Hence, if it weren't for the coriolis pseudo-force terms, the L4 and L5 points would be UNSTABLE, =NOT= stable. Ok, I see what's going on. I think we are making this unnecessarily complicated though. If I were piloting spacecraft in the neighborhood of L4 or L5, I would treat the coriolis pseudo-force (which is just an artifact of my co-rotating frame of reference) as a real force. That's because in my frame of reference, it looks like a force. Then I am moving in a "valley". No, I'm afraid that is =NOT= the case. Example: Suppose you bring yourself to rest relative to L4 or L5 as viewed from the co-rotating frame. Since your velocity vanishes (relative to the co-rotating frame), the coriolis pseudo-force =ALSO= vanishes, since it is given by 2 M V x \Omega. However, the combined gravitational force plus the "centrifugal" pseudo-force does =NOT= vanish --- and it points =AWAY= from L4 or L5 !!! Therefore, an object placed "at rest" relative to L4 or L5 will initially "fall" _AWAY_ from L4 or L5, =NOT= toward them !!! No matter how you try to twist it, L4 and L5 are _THE TOPS OF HILLS_ --- =NOT= the "bottoms" of "valleys" !!! In other words, people expect to find valleys with stable points and saddles and hills with unstable points. That is a natural and effective analogy that IMHO we should work with rather than against. Sadly, REAL physical phenomena do =NOT= always conform to our naive expectations --- which more nearly correspond to Aristotle's incorrect physics than to Newton's correct physics. This is one example where intuition breaks down. L4 and L5 are =NOT= "valleys," in ANY sense of the word !!! It seems better to me to point out that these stable points are in valleys with the sides partly consisting of pseudo-forces rather than to state in a confusing manner that the stable point is not a valley, but that it is when you add pseudo-forces, I'm sorry, but stable points are =NOT= necessarily "valleys" in 3-space, even when you "add the pseudo-force." If you begin from an incorrect premise, you will reach incorrect conclusions. I just don't see gain from this mental pretzel. The "gain" is that you are using a PHYSICALLY CORRECT model, instead of a PHYSICALLY WRONG model. If people are going to live and work in space, they are going to have to learn to understand REAL Newtonian physics on a intuitive level, not the quasi-Aristotelean pseudo-physics that living at the bottom of a gravity well has incorrectly conditioned them to expect. When intuition disagrees with reality, it is the INTUITION that must change, not the reality. People who live and work in space are going to need to become INSTINCTIVELY familiar with "An object in motion remains in motion, Every action has an equal and opposite reaction, and Force equals mass times acceleration," and the orbital dynamics mantra, "In takes you East, East takes you Out, Out takes you West, West takes you In. North and South bring you back" on the same gut intuitive level that we groundhogs FALSELY expect that an object at rest remains at rest, an object in motion comes to rest, and velocity is proportional to force. We will have no choice but to UN-learn our FALSE intuitions, and learn the ones that are correct for the space environment --- because those who =DON'T= un-learn their groundhog Aristotelean expectations WILL DIE IN SPECTACULARLY MESSY WAYS. It's that simple. -- Gordon D. Pusch perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;' |
#14
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How big is L5?
(Gordon D. Pusch) wrote in message ...
(Karl Hallowell) writes: snip Ok, I see what's going on. I think we are making this unnecessarily complicated though. If I were piloting spacecraft in the neighborhood of L4 or L5, I would treat the coriolis pseudo-force (which is just an artifact of my co-rotating frame of reference) as a real force. That's because in my frame of reference, it looks like a force. Then I am moving in a "valley". No, I'm afraid that is =NOT= the case. Example: Suppose you bring yourself to rest relative to L4 or L5 as viewed from the co-rotating frame. Since your velocity vanishes (relative to the co-rotating frame), the coriolis pseudo-force =ALSO= vanishes, since it is given by 2 M V x \Omega. However, the combined gravitational force plus the "centrifugal" pseudo-force does =NOT= vanish --- and it points =AWAY= from L4 or L5 !!! Therefore, an object placed "at rest" relative to L4 or L5 will initially "fall" _AWAY_ from L4 or L5, =NOT= toward them !!! No matter how you try to twist it, L4 and L5 are _THE TOPS OF HILLS_ --- =NOT= the "bottoms" of "valleys" !!! snip Ok, I read through the discussion of the physics of the stability of these points. One beef I have is that a number of them stop short of proving stability. Ie, it's common to see linearization of the equations to yield an evolution equation (ie, dx/dt = Ax, where A is a matrix and x a vector function of time t). However, this yields completely imaginary eigenvalues which indication rotation occurs around the equilibrium point but imply neither indicate stability nor instability in themselves. Ie, the stability is nonlinear. See for example: http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html http://map.gsfc.nasa.gov/ContentMedia/lagrange.ps (a detailed analysis) http://scienceworld.wolfram.com/phys...ngePoints.html The last one (from the World of Physics) refers to the following text, "Fundamentals of Celestial Mechanics, 2nd ed., rev. ed.", by J. M. Danby, where a discussion of this nonlinear stability occurs. Does anyone have an opinion on this book? In other words, people expect to find valleys with stable points and saddles and hills with unstable points. That is a natural and effective analogy that IMHO we should work with rather than against. Sadly, REAL physical phenomena do =NOT= always conform to our naive expectations --- which more nearly correspond to Aristotle's incorrect physics than to Newton's correct physics. This is one example where intuition breaks down. L4 and L5 are =NOT= "valleys," in ANY sense of the word !!! snip I agree with your statements now. We have complex rotating orbits around this point. One would need to be careful when piloting through such a region. Also, here's a answer of sorts to the original question. It's an arXiv.org article which estimates the size of the stability region of the Sun-Jupiter L4 point. This is probably just the tip of the iceberg, but what I found with some casual searching. http://www.arxiv.org/abs/astro-ph/0012225 Abstract: "We study the spatial circular restricted problem of three bodies in the light of Nekhoroshev theory of stability over large time intervals. We consider in particular the Sun-Jupiter model and the Trojan asteroids in the neighborhood of the Lagrangian point $L_4$. We find a region of effective stability around the point $L_4$ such that if the initial point of an orbit is inside this region the orbit is confined in a slightly larger neighborhood of the equilibrium (in phase space) for a very long time interval. By combining analytical methods and numerical approximations we are able to prove that stability over the age of the universe is guaranteed on a realistic region, big enough to include one real asteroid. By comparing this result with the one obtained for the planar problem we see that the regions of stability in the two cases are of the same magnitude." ----- Karl Hallowell |
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