That Bode's Law is an ATM concept is evidenced by the fact that the main planetary science journal Icarus does not accept manuscripts that discuss Bode's Law.

I take it that the essence of Bode's Law is that planetary spacing is approximately logarithmic in scale.

One problem is that there's not a lot of data sets to test it on. However, five exoplanets have been discovered around 55 Cancri A. Amazingly, the semimajor axes are well predicted by the following simple relation:

a = 0.039e^n-1

where a is the semimajor axis, e is the natural logarithm constant (2.7 ...), n is the number of the planet starting from it's sun, and the 0.039 is the semimajor axis of the closest planet in AU's.

The tables show the data for 55 Cancri and the Sol system (1st column = observed value; 2nd column = predicted value; 3rd column = the square of the difference between the first two columns; data from the Wikipedia):

55 Cancri

1. 0.038 --- 0.039 --- 0.0000010
2. 0.115 --- 0.106 --- 0.0000808
3. 0.240 --- 0.288 --- 0.0023207
4. 0.781 --- 0.783 --- 0.0000055
5. ????? --- 2.129 --- ??????????
6. 5.770 --- 5.788 --- 0.0003281
7. ????? --- 15.73 --- ??????????

Solar system (planet #5 is Ceres)

1. 00.400 --- 0.390 --- 0.0001
2. 00.700 --- 0.720 --- 0.0004
3. 01.000 --- 1.000 --- 0.0000
4. 01.600 --- 1.520 --- 0.0064
5. 02.800 --- 2.770 --- 0.0009
6. 05.200 --- 5.200 --- 0.0000
7. 10.000 --- 9.540 --- 0.2116
8. 19.600 --- 19.20 --- 0.1600

Thus the average for the squared differences for 55 Cancri is 0.0005472, whereas that for the Sol system is 0.0474250. Even if we just take the average of the squared differences from Mercury through Jupiter, the average is 0.00130--still higher than that predicted for 55 Cancri.

It is striking that the very first extrasolar test of Bode's Law-like relations agrees with the predictions better than our own solar system does.

If Bode's Law like spacings are ubiquitous, that suggests that logarithmic spacings are not in fact merely coincidental, but result from a deep, underlying physical explanation.

The best reference so far I've found is Graner and Dubulle (1994).

Edit: this thread is a continuation of a discussion started in the Is there a pattern to how our solar system is laid out? thread in the Astronomy section.

Attached Images

	55CancriLogGraph.gif (15.5 KB, 12 views)
	55CancriPlanetV.gif (8.3 KB, 12 views)

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This is interesting: earlier I wrote:

Then this from Graner and Dubrulle (1994):

Yet another example of teleological reasoning in astronomy.

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"Ignorance more frequently begets confidence than does knowledge" -- Charles Darwin

You're on a roll, Mr. Platts -- why spoil it by dragging in that teleology nonsense?

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The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

Seeing the TB relation in 55 Cancri is pretty cool. I agree that it will be interesting to see if this holds up for broad classes of stars.

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I'm not trying to make too big of a deal out of it. Of course we all know that mere physical systems don't have souls or spirits, and are not alive. Nevertheless, it is a fact of the history of science that physical scientists occasionally employ biological language and reasoning when trying to come up with a working hypothesis that can explain a given phenomenon.

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It will be interesting to see if it holds up for 55 Cancri when we actually get images of the planets in question.

That's right, the values I used aren't written in stone; therefore, any conclusions reached on those numbers must be regarded as provisional.

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Heh. I found out about Bode's law about a month ago, and immediately started wondering if it would be found to apply in other planetary systems.

Icarus maybe, but as I wrote in the other thread in "Astronomy" papers have been published, at least in 1999 by Graner & Dubrelle, who investigated Titus-Bode like laws and where they come from in great detail and they published it in "Astronomy & Astrophysics". I do not know where you get the information that Icarus would not publish a well written scientific paper on the appearance of TB-laws. Naturally, only fitting stuff will not get you published, you will have to give a reasonable explanation why this "law" exists, just like Graner & Dubrelle did.

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Although I've been very critical of Warren's idea, I can't help agreeing. It is true that many natural phenomena, in a wide variety of fields, seem to follow power laws* at least approximately (for an old example, google "Zipf's Law"; for a slightly mind-boggling one, google "Benford's Law"). Unfortunately, I don't think anyone has ever managed to clearly explain why this happens in general, though there are plenty of partial attempts in the mathematical and probabilistic literature.

*I should add that the Titius-Bode law is not a power law. It's a logarithmic/exponential law. But it still made me think of power laws.

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And thanks for the Graner & Dubrulle link, but I beat you to it in the first post of this thread. Graner and Dubrulle were able to get away with their paper because they framed it as a paper about papers about Titius-Bode laws. I found the following quote to be insightful regarding the historical discussion:

Skeptical theories can be divided into two camps: (1) those that say the observed pattern is real but that it's a mere coincidence--there's no underlying physical explanation for the pattern; and (2) those that say the pattern is actually indistinguishable from random expectations and that anybody can "curve-fit" some sort of relation to any set of points.

Theory (1) is plausible, and one can always say that a sample-size of one solar system is too small to go off half-cocked inventing physical explanations for a pattern that may not be general. On the other hand, there are the solar system's gas giants, and now we have tantalizing results from 55 Cancri, and soon data will be flooding in from many other systems, so (1) is increasingly harder to maintain. Theory (2), however, is false on the face of it. All you have to do is look at the pattern in this solar system--the observed pattern is not similar to that generated by a blindfolded dart thrower. So proponents of (2) have to make assumptions like the planets cannot be too close together, and so the distribution is random within those constraints. But this is just sour grapes. By imposing excluded volume constraints they're invoking a physical explanation in order to deny the physical explanation. Moreover, even the true believers don't deny that there's going to be random variation within fairly wide error bars for any real world solar system.

Of the physical explanations for Titius-Bode relations, Graner and Dubrulle identify two broad classes: (1) those that are the result of celestial mechanics operating after the planets form; and (2) those that say the observed pattern is a fossil relict of conditions at the time of planetary formation. I don't see how theories of type (1) can work, just based on my own fooling around with tony873004's GravitySimulator computer program: when orbits get in a stable configuration, they just stay there pretty much, and there are lots of stable patterns that aren't necessarily Bode law-type patterns.

So that leaves the "dynamical" theories having to do with the formation of planets and moons out of primordial disks. Graner and Dubrulle discuss several. The two I find most plausible are (1) the pattern is based on the average eccentricity of particles making up the primordial disk; and (2) the observed pattern is the result the scale of vorticity in the primordial disk that's assumed to be fully turbulent. Although, my original thinking was more along the lines of (1), I'm now leaning towards (2) mainly because a large vortex is a better design for matter-eating than are single planitesimals, and it's clear that the gas giants formed out of minidisks within the main primoridial disk.

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What's the difference between a power law and a logarithmic/exponential law?

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"Ignorance more frequently begets confidence than does knowledge" -- Charles Darwin

In a power law, the base is variable and the exponent (i.e. the power) is a constant. In a logarithmic/exponential law, the base is constant and the exponent is variable. Their graphs are different curves.

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I have a confession to make: in the first post I said that the planets for 55 Cancri obey Bode's Law better than the planets of our own solar system. That's not exactly true. I took the straight errors, and then squared them, and because most of the planets at 55 Cancri are very close in compared to our solar system, the absolute errors are smaller, but if the errors are expressed percentage-wise, the errors at 55 Cancri are larger than the classic formulation of Bode's Law for our solar system. To wit: under the formulation I proposed, there average of the absolute value of the percentage errors is about 6% for 55 Cancri and about 2% for the classic formulation for the Sol system (excluding Neptune). The biggest source of error is for 55 Cancri III (to use the old Star Trek classification), which is 20%.

Graner and Dubrulle list several forms of mathematical structures a Titius-Bode law can take, but there are two that are in common currency: (1) what I call the classic formulation; and (2) what I call the "natural" formulation. The general form of the classic formulation is:

a_n = a₁ + (a₂ - a₁)K^m

where m is a number from the following series: (-∞, 0, 1, 2, 3, ...). The "-∞" makes the second term go to zero (whoever said 1/0 wasn't useful! ) so you get a₁, and when m = 0, K^m = 1, so you get a₂. In other words, you get your first two planets for free, so the classic formulation of Bode's law is more open to the charge of "curve-fitting" than the natural formulation. In our solar system, the values usually used are a₁ = 0.4 and a₂ = 0.3, with K set to 2. So the formula becomes:

a_n = 0.4 + (0.3)2^m

One can attempt a classic formulation for 55 Cancri, but one has to set K to equal 3 instead of 2:

a_n = 0.039 + (0.077)3^m

in which case the average percentage error can be reduced to about 5%.

The natural formulation, according to Graner and Dubrulle is:

a_n = a₁K^n-1

which is the formulation I first used for 55 Cancri. Which also apparently entails that my value for K (i.e., e) must be a total coincidence. But what an utterly, freaky coincidence!!! The only number that works better than 2.71828182845904 is 2.7180. So, naturally, I went with e for K. But apparently that's a pure fluke since the K for our solar system, according to Graner and Dubrulle (which I subsequently confirmed myself) is 1.7. On this model, the average error for our solar system is 10% (but it has the virtue of bringing Neptune back into the fold at least). So, on the natural formulation of Bode's Law, 55 Cancri does in fact do better than our own solar system. If, however, you divide our solar system into inner and outer planets, and assume a K of 1.57 for the inner and a K of 1.855 for the outer, the errors can be reduced to 4%.

Now, the main point of the Graner and Dubrulle paper is that any attempt to physically model the layout of a solar system that assumes initial conditions are radially independent (i.e., there is no variation that depends on the distance r from the center of the system), a Titius-Bode's like relation will fall out, no matter which of the 15 underlying physical theories you use. They then use this result of theirs to trivialize attempts to more deeply understand the Titius-Bode relation: hence the subtitle of their paper: "Scale indepence explains [away] everything".

But I don't think that's fair. If it's really the case that a system obeys the Titius-Bode relation, that says something important about that system: namely, that the process that caused the pattern was radially independent, and that that is going to place huge contraints on what a theory to explain the layout of the solar system must be like. Conversely, if a solar system does not obey the Bode relation (as our solar system apparently does not under the natural formulation), that also says something: that conditions changed as one progressed toward the outer edge of the primoridial disk.

Also, the scale factor K says something important. I suggest that the K factor says something about the rate of formation of planets and solar systems: the higher the K factor, the faster the rate of formation (you heard it here first folks! ).

On both the classic and natural formulations for 55 Cancri and the Sol system, 55 Cancri has a larger K factor. So why would I say that the 55 Cancri system formed faster than our own system?

So what happened is that at 55 Cancri--because of the superabundance of iron and other heavy elements--heavy, iron-rich, powerful, hungry cores capable of holding mass-sucking atmospheres were able to rapidly form at the center of large vortices; and so planetary and orbital evolution at 55 Cancri was accelerated, compared to the Earth's solar system. Hence, the wider spacing of planets at 55 Cancri--which, paradoxically, allowed planets to form much closer to the sun than at our solar system.

Meanwhile, back here, there was less iron, core's formed less readily, evolution was slower, and so the primordial disk had time to become more organized (less eccentric), and that furthermore, since K, the Bode scale factor, for the outer planets is 1.86, but only 1.57 for the inner planets, then the evolution must have progressed inward starting with the outer planets first, with the inner planets forming somewhat later.

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Thank you. But then what do you call it when both the base and the exponent are variables?

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"Ignorance more frequently begets confidence than does knowledge" -- Charles Darwin

If it has a name, I don't know what it is. The simplest case is y=x^x, but this is just one curve. There's no free shape parameter there we can adjust to define a family of similar curves (like your K).

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Let's be very clear on one important point. The Titus-Bode (or Titius-Bode) relationship is not a law in the scientific sense. At best, it is an hyposthesis. I say "at best" because, while it claims a relationship of secondary to orbital distance from its primary, it offers no explanation for why this might hold.

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