It seems that Bode published Titius' discovery without crediting Titius, so I will call it Titius' Rule. Titius' Rule, in it's modern adaption, says there is a geometric progression in the distance the planets circle the Sun. It's not a perfect rule, and Neptune doesn't fit at all. However, the astroid belt serves in place of planet 5, and Pluto serves for planet number 9. The pendulum of controversy has swung back and forth a bit. Is it meaningful, or happenstance? Apparently this topic hasn't been discussed on physicsforum since 2004 [edit: 2008], and in the mean time we now have data from observations of other solar systems. This article has a nice graph of data made available 2 months ago. http://www.economist.com/blogs/babbage/2010/08/planet_hunting Titius' Rule in modern form is [tex] ln(a) = mN+b \ .[/tex] 'a' is the semimajor axis of planet number 'N'. 'm' and 'b' are constants chosen for best fit curve. I tend to imagine that this orbital spacing is the most stable. For orbits to be stable we can't have orbital periods in simple ratios or perturbations would accumulate with each realignment of two planets until the arrangement blew itself up. I suspect this geometric relationship must somehow minimized gravitational harmonic coupling. With C, a constant, the orbital period is [tex]T = C a^{3/2} \ .[/tex] This leads to the same sort of equation, but in T. [tex]ln(T) = cN+d[/tex] Cleaned up a little, we can write [tex]e^N = k(T/T_0) \ .[/tex] Is this planetary spacing the most stable?
Three of Jupiter's moons have orbital periods with simple ratios. The other sixty do not. So not quite an apt example. Regarding our solar system: That there isn't a planet between Mars and Jupiter somewhat belies the Titus-Bode law. Saying that Pluto rather than Neptune fits the ninth slot (counting Ceres as the fifth) is a real stretch. The "law" is at most suggestive rather than a true law.
If I remember correctly, there has been numerical studies into solar system formation that has found that power laws like the Titius-Bode law is a normal occurrence in solar system formation. Without having studied the mechanics behind it, I would guess that such laws ties into, and can be explained more or less from, the phenomenon of orbital resonance.
Calling things laws is an historical artifact of over confidence physicist has sometime post-Newton. Which is why it's better to call it a suggestive rule. We don't have to be for or agin'it, but see how this plays-out as the periods of more planetary systems are measured. Personally, I suspect it will be found that it is statistically unlikely that planetary systems not follow this rule, although the linear fit is not exact for any system so far discovered, so there will be some method to determine what is statistically unlikely.
http://en.wikipedia.org/wiki/Io_(moon)#Orbit_and_rotation http://en.wikipedia.org/wiki/Orbital_resonance
Granpa, what about the other sixty satellites of Jupiter? Saying that Jupiter's moons obey Bode's law because 3 out of 63 are in orbital resonance is a bit disingenuous. It is cherry picking. Phrak, the fit was maybe OK for 18th century scientists. Add in Neptune and the law pretty much falls apart. Throwing out Neptune, Jupiter:Saturn:Uranus are 1:1.8:3.7. That is not a 1:2:4 ratio. We physicists like just a bit more precision than that.
most people I know would interpret 'the moons of jupiter' to mean the main moons. if you wish to twist my words to mean something stupid so you can argue then all I can say is 'be that way then'. I am not going to be drawn into a pointless argument with you are anybody else. I have better things to do with my time.
I don't think planetary motion should be exact, but something akin to thermodynamics, so we shouldn't be looking for an exact fit but a statistically significant fit. It's an evolving system. Ignoring interaction between planets, Each time a comet designs to drop in it may perturb orbits, and on the average over large numbers, reduce the kinetic energy of the system. Also, in the modern version, the ratios are not 1:2:4 but something close. We would be looking instead for the best fit curve to [tex] e^N = k(T/T_0) \ . [/tex] As you've pointed in the case of Neptune, and perhaps the asteroid belt as well, we can't always expect all the slots (i.e.: the N values) to be filled. Hmm... But of course, Neptune doesn't doesn't fit anywhere and we can't just ignore it entirely.
DH, These are the ratios as given by wikipedia. As you can see, the errors between Jupiter, Saturn, and Uranus as less than your own, but they are using some sort of modified version I haven't examined well. It seems to be Titius' original rule, with extra numerology to make a better fit. Also notice that 'k' for Mercury is not 1/2 but 0. Code (Text): Planet k T-B(AU) AU % error Mercury 0 0.4 0.39 2.56 % Venus 1 0.7 0.72 2.78 % Earth 2 1.0 1.00 0.00 % Mars 4 1.6 1.52 5.26 % Ceres1 8 2.8 2.77 1.08 % Jupiter 16 5.2 5.20 0.00 % Saturn 32 10.0 9.54 4.82 % Uranus 64 19.6 19.2 2.08 % Neptune 128 38.8 30.06 29.08 % Pluto1 256 77.22 39.44 95.75 % Edit: Yes, on further examination, it is a lot of numerology, tweaking and adding parameters to make the equations fit the data without further thought. I should delete the post, but on the off chance someone has read it between the time of posting and now, I'll just leave it here as an excellent example of science gone bad.
Numerology is the keyword here. This is one of those topics that might have some limited basis in reality due to orbital resonances and dynamical systems theory. This is also one of those topics that draws in the cranky crowd, bigtime.
I knew you'd appreciate it's mentioned. But it's fairly irrelevant. Titius advanced the numerical relationship N={0,3,6,12,24,...}. Obviously zero doesn't fit the doubling rule. There are plenty of crude approximations in physics, including Newtonian mechanics, thermodynamics, and I would include all the various versions of quantum mechanics from Bohr to string theory. Empirical, as it is, I still find Titius' Rule interesting, and I stress, in modern form, because 1) it appears to fit multiple planetary systems with statistical significance, and 2) it is a simplification of Titius' original rule rather than an over-parametrically adjusted equation as one would find in chemistry, fluid dynamics or medicine.