What's your point? As has already been said in this thread, TB has problems in the solar system, we certainly don't have enough information on exoplanets to draw a conclusion, and there are rather huge questions on the mechanism. Now, when we have detailed maps of, say, fifty systems and if then we see a consistent pattern, then it might rise above the level of hypothesis.

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The Leif Ericson Cruiser

My point is that I'm sorry I didn't include a poll in the first thread: it would have saved you a few keystrokes to register your vote. Thanks for voting!

What's your point here? That it's pointless to speculate that extrasolar planetary spacing is often logarithmic before we have a sample size of 50 solar systems with at least 5 planets each, or is it that BT is so fouled-up that it cannot predict where some of those planets will be found before they are found as we build the sample size of 50 systems?

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

The point is that it would need a lot more supporting data before it could go beyond a hypothesis. There are problems with the idea already, based on our own solar system, and we don't have enough data for any other solar system, including 55 Cancri, to take it very far.

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The Leif Ericson Cruiser

In the interests of brevity, what Van Rijn said (in his last several posts).

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I have a quick question for you guys: What would be the curve that better fits the data: (1) a curve that minimized the average of the percentage errors; or (2) a curve that minimized the standard deviation of the average of the percentage errors?

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

We were having enough philosophical trouble with 'law'; now you want to bring in 'hypothesis'. . . . Very well then. What do you mean by 'hypothesis' and what would it be like to "go beyond" a hypothesis?

And which hypothesis are you referring to?

1. H1: Major planets that formed out of a primordial disk under normal conditions have an approximately logarithmic spacing; or
2. H2: The semimajor axis of 55 Cancri g is greater than 1.9 AU and less than 2.3 AU.

How much more supporting data will it take for H2 to go beyond a hypothesis?

I seriously hope you aren't referring to asteroids and Kuiper Belt objects. . . .

And what about the fact that the major satellites of the gas giants in this solar system obey the TBL?

How can you possibly say that being able to predict the location of a new planet doesn't count as "taking it very far"? Is it that extrasolar planetary science is an utterly trivial and boring pursuit?

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

Here's a chart showing the position of where Planet V is predicted to be found.

Also, I revised the TBL formula for 55 Cancri by minimizing the sum of the square of the (percentage) errors:

a_i = 0.0374(2.722)^n-1

To still insist we don't have enough data to make useful predictions for 55 Cancri is just pure FUD.

Attached Images

55CancriChartofPlanets.gif (10.3 KB, 6 views)

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

Eh? The TBL formula is:

a = 0.4 + 0.3 · 2^m

where "a" is the distance from the star in AU, and "m" starts with negative infinity (that to fit Mercury), then 0, 1, 2, 3 etc. Your equation no longer is "the TBL formula" but rather a formula of your own devising, designed to fit different data.

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The Leif Ericson Cruiser

So, to be clear, you're proposing to make a prediction of another planet for 55 Cancri based on your custom calculation, but this calculation would not be applicable to any other star system?

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The Leif Ericson Cruiser

I thought that both Jim and I had covered this. The "philosophical trouble" is because it does not conform to what is usually referred to as a physical law. It is an observation based on limited data, and there is a hypothesis that there is an underlying, simple rule to the observation. I didn't "bring in" the hypothesis aspect: That's always been there.

Definition of hypothesis:

A hypothesis (from Greek ὑπόθεσις) consists either of a suggested explanation for a phenomenon or of a reasoned proposal suggesting a possible correlation between multiple phenomena.

Normally, you go beyond a hypothesis by showing validity through supporting data. That generally requires a decent sample size.

I was actually picturing something like:

Spacing of planets follow the pattern described by the Titius-Bode formula:

a = 0.4 + 0.3 · 2^m

where "a" is the distance from the star in AU and "m" starts with negative infinity, then 0, 1, 2, 3, etc.


Those, the Mercury "fix" and especially Neptune. Why would I ignore that data? Mercury is a special case in the equation. Neptune doesn't follow the rule. Pluto seemed to follow the rule, except that it turns out it is just one rather low mass object among many, so that's meaningless. There was supposed to be a planet between Mars and Jupiter according to the rule, but again, we just find a large number of small objects (and Ceres is like Pluto). In fact, of course, Jupiter would prevent formation of a large planet. To me, that suggests that the situation is too complex for a blanket rule, and a generally applicable method would need to consider (among other things) the mass of the individual planets.

I wasn't aware that was a "fact." I am aware that apparent patterns show up in systems of satellites, but I'm not aware that systems of satellites follow the TB formula. References, please?

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The Leif Ericson Cruiser

This isn't your average "Titus [sic]-Bode Law--ooh shiny!" thread. I'm trying to take it to the next level--to "go beyond", as some might say. Before, I merely suspected that you did not read thread--now I know for sure. So allow me to bring you up to speed.

In post #1, I first posted this formula:
Then in post #14, I basically parrot Graner and Dubrulle's (1994) paper, where they identify two basic forms of the TBL (and there are more):

Actually, the classical formulation is easily shown to be a special case of the natural formulation where a solar system is broken into two zones, each with its own scaling factor K.

And to say that TBL doesn't apply to 55 Cancri because the parameters of the model for 55 Cancri are different from our solar system is such a total red herring: you might as well say that someone who applies Newton's Laws to an extrasolar system is using a formula "of their own devising designed to fit different data" because they adjusted the mass parameter of the primary to something different from the Sun.

Re-read the above.

That's just false on the face of it. The TBL is usually referred to as a law (and I guess it would have to be a physical law, because it sure isn't a biological or economic law) at least 5 to 1 (as I point out in post #24) compared to other descriptions like 'rule' or 'relation' (you're the first to speak of the TB "hypothesis").

BTW, the wiki article on scientific laws you cite is junk. If you want an understanding of the issues involved, read the Stanford Encyclopedia of Philosophy article instead. But it's going to take more than a couple of minutes. . . .

I've got 5 data points for 55 Cancri. I've done the least squares statistical reduction. The standard deviation for the results is rather wide (about 6%), but that's still not bad, and it generates good results, and at least one concrete prediction, as you can see from the attached graph. If you're going to say that's statistically insignificant, you'd better be prepared to back up that assertion with something more than IIRC handwaving about sample sizes.

First of all, you contradict yourself within one paragraph when you say that we shouldn't ignore asteroids and KBO's, and then you say that Pluto fits the rule but doesn't count because it's a KBO.

Secondly, the TBL is, and never was, intended to be a "blanket rule" covering every object in the night sky. The proper domain of the TBL is major planets in normal solar systems. TBL--like Newton's Law of Gravity--incorporates a common-sense ceteris paribus proviso. Thus, in situations where other things are not equal--as when Jupiter prevents the formation of a planet--you're not going to find a planet there. Nevertheless, the presence of the asteroid belt where TBL says a planet should be is usually considered a confirmation of TBL--you're the first to suggest otherwise.

Thirdly, as for Neptune and Mercury--they both do fit the TBL. The formula for the inner solar system is:

a_i = 0.413(1.56)^n-1

The standard deviation for the average of the (percent) errors above (using the s.d. for a sample of a population [=STDEVA(...) in Excel]) is 4.4%. The formula for the outer system (Jupiter through Neptune) is (s.d. 3.6%):

a_i = 5.32(1.81)^n-1

The above formula predicts Neputune's orbit to within 5%. Now before you go off saying I am engaging in ad hoc curve-fitting, reread what I said before about ceteris paribus clauses. In other words, by giving two formulas, I am saying that conditions during the origin of the solar system were not the same for the inner and outer zones. My evidence? As an old professor of mine used to say, just look at it! The inner planets are all rocky, and the outer planets are all gas giants--so other things are obviously not equal. If there's any conclusion to be drawn from the ~256+ extrasolar planets discovered so far, it is that the situation here in our solar is not normal--gas giants regularly form elsewhere at distances much less than 2 AU. Therefore, it's not surprising that the scaling factor K for the two zones is different. In post #14, I speculate on the physical significance for the K factor for planetary formation:

And your suggestion that planetary mass has something to do with planetary spacing is belied by the evidence.

This fact has been noted several times in other threads on the TBL in this forum and Graner and Dubrulle (1994) list the K factors for Jupiter, Saturn, and Uranus as 1.6, 1.5, and 1.4 respectively (which according to my theory implies that the Jupiter system formed first).

Attached Images

55CancriChartofPlanets.gif (10.3 KB, 6 views)

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

There is a tendency towards increased spacing with increased orbital radius for the regular moons of the gas giants, which is of course what allows exponential fits to be carried out.
Good fits in this case need ad hoc rejections of uncomfortable data points.
Here is a log plot of the orbital radii of all the regular moons of the four gas giants. I've suppressed only the Saturnian ring shepherds, and have treated each coorbital family around Saturn as a single data point. If an exponential fit were going to be good, we would expect to see a family of straight lines.

Grant Hutchison

I don't know what you mean by "regular moon". You're falling into the same fallacy that says that the BTL has to apply to every asteroid and KBO out there. If you restrict your analysis to the five largest moons in each, I think you'll find a good correlation.

The physical basis of the BTL has to do with the formation of planets. Celestial mechanical events that happen subsequently to formation would be expected to cause deviations from the primitive logarithmic spacing.

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

The inner moons of the gas giants have near circular prograde orbits in approximately the equatorial planes of the their parent planets. The are called "regular" for that reason.

Please let us know which logical fallacy we are commiting by asking you to not to select only those data that best fit your curves.

In which case, in what way and to what extent can we expect your "law" to be predictive of orbital spacings?

Grant Hutchison

Let's take H1 (if that is what you are proposing).

Define "major planet."
Define "normal conditions."
Define "approximately" logarithmic.

Oh, and since you mentioned it,
Define "major satellites."
Explain why this applies only to the gas giants.

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