Get a Straight Answer

    I am currently studying astronomy, and found your article on Lagrangian points thoughtful and very useful in helping me understand. I do have one question for you if you don't mind, however. You mention that were it not for other influences, the Lagrangian points would be stable. How can this be? If would appear to me that as an object starts to move away from one of the points, the change in the gravitational pull from the sun would cause its orbital velocity to change, which in turn would cause it to move farther away from the L point. A closely related question is this: how can an object orbit a L point without having some mass to which the object is attracted to?
    I am sure the answer is simple, but my brain is hurting trying to figure this out. Your answer will be most appreciated.
    Regards, Larry

Reply

  I wrote in "Stargazers" that if it were not for other attractions, L4 and L5 would be stable--but one should add that L1 and L2 are unstable. (Still, I am not sure about some "halo orbits" near them--see "The Art of the Orbit" by Gary Taubes, p. 620-622 Science, vol 283. 29 January 1999, section after the subhead "Three-body perfection.").

  If you are studying astronomy at the college level, you might find a relevant derivation in Symon's text "Mechanics." For objects that keep fixed positions in a ROTATING frame, the equilibrium can be studied in that frame by adding a centrifugal force, and then you can obtain a potential function and draw its contours. The problem then resembles that of a small ball rolling with no friction on a curved surface: if the L4 point is the center of a pit, small displacements would cause the ball to roll back, so the equilibrium is stable. Or else it could circle the pit, like a marble in a bowl: it needs no attraction from the middle.

  If instead it is on top of a dimple, a small displacement will cause the ball to roll even further away, never to return, which signifies an unstable equilibrium.

    If a meteor of significant mass hit the earth wouldn't this cause the earth in turn to move. Would its orbit be disrupted?

Reply

    To give a short answer to your long question--not likely. Asteroids are far too small. An asteroid with a 10 km radius would have a volume less than one part in 200 million of the Earth, and if its mass were similarly scaled, the impact on the Earth would negligibly affect its orbit. Anything large enough to shift our orbit would have to be larger than any known asteroid, and the collision would be violent enough to wipe out all life.

    Personally, I would think so. The logic being that if I assumed that the mass of the upper crust were zero, the closer the object moves towards the core the greater the gravitational pull (till the object penetrates the core).

Ron

Reply

    That is some neat question you have asked, and your qualitative argument is absolutely right. A short calculation (using some elementary calculus) makes it more precise.

    Suppose we are at a distance R from the center, the local density is D(R), and we move a test mass m downward by a small distance dR. If G is the constant of gravity and M the attracting mass, does gravitational attraction increase or decrease?

    In a spherically symmetric mass, any mass closer to the center than the attracted one acts as if they were concentrated at the center, while any which is more distant has no effect. That result is part of the theory of the potential, although Newton cleverly derived it from elementary considerations even before such a theory existed.

Therefore, as our test mass advances a distance dR towards the center, the mass that is attracting it diminishes by dM = 4 p R2 D(R) dR, and the attracting force decreases by

where K = 4p Gm. On the other hand, the closer approach to the center adds to the force

    Let us ignore signs and just recognize the contributions are in opposite directions (the fact R is positive upwards while the force of gravity points downwards can confuse). If the average density of the mass M below is , then

Substituting in the equation, canceling the cube power and introducing K gives

Thus if D(R) is smaller than (2/3), gravity increases, if larger it decreases, which includes the case of constant density, D(R) = . A nice problem!             David

    When I saw those movies, there was always something that confused me so much. What's the differences between LIGHTSPEED, HYPERSPACE, and WORMHOLE?

I can understand about lightspeed, but I don't know if a wormhole could be used in space travel. As far as I know, quantum theory was just used to prove other dimensions of our world (parallel worlds), so is there any connections here between this wormhole and space travelling?

Well, Mr. Stern, I think these are the questions to which I'd like to know the answers. Can you please help me?

Reply

    The stories of science fiction movies come from professional writers, not from scientists. About 100 years ago Einstein found (something confirmed since then in many ways) that no material object can move faster than light, 300,000 kilometers per second. (If YOU moved that fast, time would pass at a different rate, so TO YOU the speed might seem greater--but not to someone in the outside world).

    Writers of fantasy stories, and later of fantasy movies, felt restricted by that fact, which suggested that back-and-forth travel or communication with civilizations on planets outside the solar system was impossible on the short time scale of travel and communications between countries on Earth. As seen now, a projected trip to another world (even using technology we do not have yet!) might take many thousands of years.

    So writers picked up some scientific terms, suggesting some day in the future the limitation of light speed may be overcome, by using hyperspace or wormholes. However, these are just ways for literature and films to imagine things which physics says (at least right now) cannot be done. I am not sure about wormholes, which have to do with general relativity: the added dimensions proposed by some theories extend only a very short distance into our universe, and are not likely to help us navigate the three principal dimensions of our universe (or 4--though time is a different kind of dimension)

Reply

My almost 8 year young son Adam and I have a question about the revolution of the sun. We know that the planets revolve around the sun, and all have rotational periods also. We see that the sun aside from having a rotational period, also has a revolution of some 250 million years. We are curious what it is that the sun is revolving around?

Reply