Celestia Forums

http://fr.arxiv.org/abs/0803.2240

55 Cancri

The Exo-planetary !

Using the same Titus-Bode relation that they used for 55 Cancri, Chaos Syndrome at Extrasolar Visions (Ajtribeck here?) got results for other solar systems.

http://www.extrasolar.net/forums/viewtopic.php?t=1759

...on the other hand, there is no guarantee that planets predicted by the Titius-Bode "law" exist... e.g. in our solar system, one of the "slots" is occupied by the asteroid belt. Furthermore, it does not predict which "slot" corresponds to the outermost planet.

The interesting question is why such "laws" exist - particularly in situations where planetary migration is thought to have occurred.

ajtribick wrote:The interesting question is why such "laws" exist - particularly in situations where planetary migration is thought to have occurred.

I, too, have pondered this. Let us not forget planetary ejections. I can't think of any reason why the TB relation should work.

It is an exponential relationship among the orbits.

Consider the large moons of Jupiter. Io has an average period 1/2 that of Europa, Europa's period is half that of Ganymede, and Ganymede's is about 3/7 that of Callisto. A good fit to the ratios is 3:6:12:28. Or, as logs:
0.477, 0.778, 1.079, 1.447.

So the periods of the first three moons of Jupiter as a logarithm differ by the logarithm of 2 (0.301). Ganymede to Callisto does not fit the pattern, but the 7/3 ratio is close enough to 2 (2+1/3) that the logarithm is also close to 0.301 (0.368).

Time for a thought experiment.

Consider a pair of bodies of similar mass, A and B with A orbiting faster and both orbiting close to a parent body. If their orbital periods differ by a constant K, the orbits tend to settle down into a stable resonance if K is roughly the same as the period of the faster body, A (in other words, the periods are roughly A=K, and B=2K). When the planets pass each other, they do not spend much time close together so they do not perturb each other much. I won't explain why it happens, just that it does.

Now imagine that these same planets are orbiting 4 times farther from the parent body, but the orbital separation remains the same. Now imagine these planets are passing each other in their orbits. Now that they are orbiting farther out from the parent body, they are orbiting more slowly. To be precise, they would orbit 8 times more slowly, as a consequence of the Keplerian relation D cubed being proportional to T squared. And this means that the time available for the bodies to perturb each other is 8 times what it was before, so they pull on each other for 8 times as long on each pass, thus the perturbation on each pass must therefore be 8 times greater. The passes also happen only 1/8 as often, but with one big push instead of 8 little ones, there is more chance of a damaging perturbation. It would be like pushing a child on a swing: 8 little pushes would get the child swinging nicely, but if instead there was one big push the child is likely to be pushed out of the swing because there was too much force applied at once. So too would a pair of close bodies be more subject to disruptive perturbations.

It is thus not surprising that given enough time bodies tend to settle down into orbits that are spaced according to a logarithmic relation because closely-separated bodies tend to perturb each other.