Potential valleys in our solar system

[dianaj]

Hi all,

Our sun is approximately a spherical body and accordingly it has a spherical potential. I have just learned that this potential - accoring to general relativity - has a peak and a valley before smoothing out around zero at large distances. One can have an unstable circular orbit at the position of the peak and a stable orbit at the position of the valley. But what is more normal: one can have an elliptical orbit with a mean radius being that of the vally. This is the kind of orbits our planets move in. So far so good. But the model with the spherical body only produces a single valley while we clearly have several stable elliptical orbits in our solar system. How can this be? The only answer I can come up with is that the masses of the bigger planets change the overall potential and induce several potential wells. Is this correct?

/Diana

[Nabeshin]

I assume you're talking about the effective potential as calculated from GR. In this case, elliptical orbits do not correspond to a minimum in potential, but rather the difference in two points around this minimum. (Which is why the radius of orbit is not constant: the turning points correspond to different radii)

Refer to this picture:
http://upload.wikimedia.org/wikipedia/commons/2/2c/EffektivesPotential.jpg

Note: This is actually a newtonian potential, but for what you're considering it doesn't matter.

So, the circular orbit corresponds to the minimum in potential. Elliptical orbits correspond to drawing a line of constant E greater than that minimum, and then you get r+ and r- as the two turning points of the function it oscillates between. Clearly, for different values of energy you get different radii, and thus, our elliptical orbits.

Hope this clears it up at least a little.

[D H]

dianaj, Newtonian mechanics does a very good job of explaining the behavior of the solar system. Newtonian mechanics cannot fully explain the anomalistic recession of the planets. The error is greatest for Mercury, for which Newtonian mechanics misses the mark by all of 43 seconds of arc per century.

I think you might be talking about what is now called the "interplanetary transportation network (http://en.wikipedia.org/wiki/Interplanetary_Transport_Network)". Read the link. If that hits the mark, you can google that phrase. You can also google the names Martin Lo, Jerrold E. Marsden, Wang Sang Koon, and Shane Ross, some of the key people behind this concept.