I believe I've been admirably restrained, occasionally verging on the solicitous.

I don't bet, Warren.
But now I know you're viewing this as some sort of competition, perhaps I understand the thread a little better.

Grant Hutchison

BTW: Just in case anyone else wants to jump in and take up Warren's bet, please note that his choice of wording means he wins no matter what the reported semimajor axis is.

Grant Hutchison

GARGH! They really do teach good logic up there in Scotland. But believe me, I've had my share of logic here. Once I even developed a completely expressible propositional caluculus (PC) based solely on unary truth-functions. Which I'm sure you'd put into the same class as the Titius-Bode relation.


Oh, by the way, "Titius-Bode" is kind of reminiscent of a certain parts of the female anatomy in certain foreign languages. Sorry if I offended anyone by starting this thread.

As for this thread being a competition, you should remember this is a science forum, and as the cover of this philosophy of science book attached demonstrates, science consists of a bunch of naked men battling with swords, Braveheart-style.

I wish you'd reconsider the bet; really it's a sucker's bet, because if 55 Cancri is like our solar system, there's an asteroid belt in slot V. Something about slot V. . . . Probably the solar system bifurcated more than once. So the asteroid belt is a sort of phase transition zone, where original symmetry was broken. The outer protoplanetary vortices started doing their own thing, while the central zone still thought it was all one big empire. After a while, though, the close quarters became untenable, so the parent retracted to enjoy its dotage, leaving the younger kids a meagre inheritance.

So, perhaps the same thing happened on 55 Cancri--which is billions of years old.

So what do you say. I know you're not a gambling man. But hopefully that's because of your principles and not bitter experience. So why not make an exception. It's really not a bet, it's more of a duel. But dueling with guns and swords is outlawed these days, and even a good old-fashioned fist-fight is liable to wind yourself up in jail no matter what country your in these days, so forcing the other to buy a superexpensive bottle of booze is the closest either one of us can hope to inflicting any real pain. So what do you say there, mate? (I win if 55 Cangri g semimajor axis is greater than or equal to 1.665 AU AND less than or equal to 2.572 AU.)

Attached Thumbnails

 

I don't bet, because a bet on the outcome either glorifies the trivial or demeans the significant.
In this case, the former applies, since it won't matter a whit whether or not there's a planet in the appointed slot: you will still have speculated beyond the data available to you today.
There could be a planet in the slot, but no general exponential relationship to be found when 55 Cnc is included in the analysis of good data from multiple systems. There could be no planet in the slot, but widespread exponential scaling detectable across multiple systems, sufficient to require explanation.

We don't know. You don't know. And that's the point people have been trying to make.

Grant Hutchison

Wagering is frowned upon around here, so if would be best for you to discontinue this and simply present your evidence.

No, the point of this thread, as stated in the OP is that 55 Cancri is already a data point to be chalked up on the TBL side of the board, as we build up that database of 50 systems that Van Rijn said would satisfy him. Right? 55 Cancri certainly can't be counted as evidence against the TBL. Can it? Sure, it's only one data point; but data points have to be added together. Don't they?

55 Cancri is Iowa, baby, and she's voting for TBL!!!

That cannot be denied.

That just cannot be denied.

I did the statistics myself.

To deny that--as I'm sure none of you can--though you will anyway--is the very essence of pure

FUD.

And that's a nice line about gambling demeaning the admirable--I'll have to remember that one. But like I said, this isn't about gambling for the sake of gambling anymore. This is between you and me buddy. I'm supposed to pack up my tent and leave?!? How bout we just put our money where our mouths are instead?

R.A.F., don't be such a control freak. You know as well as I do that there's been a tradition among scientists and philosophers to place long term bets on the outcome of some future crucial experiment or observation. Wasn't Fermi taking bets on whether the atmosphere would ignite when they lit off the first atomic bomb?

And I did present my evidence as simply as standard statistics allows. It's there. In the post above with the two charts attached. Read it. Read it again. The post where it says that the planets at 55 Cancri have a logarithmic spacing pattern. The one where the regression line had an r² of 99.7%.

FUD will not work anymore.

Hey...just friendly advice.

NOT ON THIS BOARD.

Done and done, yet the "evidence" you present is simply not convincing.

Bodes-Titus doesn't work...not without "twisting" the data all to hell.

Maybe they should get around to finally passing rule # 37.1.412.III(a): "No bets on future scientific observations allowed."

What? 99.75% of the variance explained by the logarithmic model is not good enough for you?

Just where did I twist the data? Was it when I computed the logarithms of the semimajor axes, or when I summed the squares of the residuals; was it when I looked up the t-value in the table at the back of the book, or was it when I plotted that suspicious-looking error bar on that one chart?

That pretty much sums it up. Fitting equations to the data, then yelling "FUD" at anyone that points out you can't leap to conclusions, is pretty sad behavior. I'm open to the possibility that a common, simple, relationship may fall out with more data, but at this point, it's speculation. I don't have any problem discussing speculation, but I do have a problem when someone insists it is more than that.

__________________
I say there is an invisible elf in my backyard. How do you prove that I am wrong?

Disclaimer: Avatar is not an official NASA image and does not imply any specific interplanetary or interstellar capability.

The Leif Ericson Cruiser

What makes you think that I would have the slightest interest in proving you wrong?

You're the proponent of this idea...prove yourself right.

...and repeating "this can not be denied" is not proof, it's handwaving.

Could someone shed some light on Graner & Dubrulle for me, please. My understanding was, that if you start out with a gas cloud with axial symmetry and scale invariance and let it collapse to a star with planetary system, you're going to get some sort of "regular relation" in the spacing of the final bodies.

That relation can be anything, but once the collpase gets started and the chaotic processes fall a certain way, you will get "some sort of relation".

-Richard

You might just pause, during your little dance, to recall that these data are necessarily correlated, just because of the way you've generated them. You numbered the semimajor axes in ascending order of magnitude. You've sorted them: it's pretty much impossible for them to show no correlation.
This sort of thing is a common enough error in data analysis, when people accidentally put in "pre-correlated" data (because of an unnoticed mathematical linkage between the dependent and independent variable, or because the two variables turn out to be measuring the same thing in two different ways), and are then impressed by the results of a correlation test. But it's unusual, I think, for it to be quite so flagrant as correlating a dataset with a simple measure of the relative magnitude of its data points.

And then you inserted a gap in order to meet your specific expectation of an exponential.

Grant Hutchison

Yes, I think that's what they're saying.

Grant Hutchison

No, what they're saying is that if whatever physical mechanism responsible for planetary spacing is scale invariant with respect to distance from the primary, then a logarithmic planetary spacing pattern will result.

You're suggesting that a "logarithmic planetary spacing" is not a "'regular relation' in the spacing of the final bodies"?

Grant Hutchison

Here's a good one on Arxiv:

http://arxiv.org/abs/astro-ph/9710116

IOW, generate a random solar system, apply stability criteria, a voila!, you get regular spacing that follows a pattern, roughly, especially if you allow yourself some "peremptory challenges" so to speak of data points you don't like.

-Richard

Hmm...ignoring data based solely on a predetermined outcome sounds like bad science to me.

I'm not sure what they did was as simple as generating a "random" solar system. By assigning a uniform distribution to log(r), aren't they basically assuming Bode's Law?

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

It certainly seems so, and I don't see any discussion or defence of that particular random distribution in the paper, unless I'm missing something. They also use a Hill radius exclusion, which will help to weed out systems that deviate strongly, by chance, from the log distribution they've created. And then they get to "weed" the result! Very odd.

Grant Hutchison

It still holds. I just did a Monte Carlo simulation where I generated 31 random 55 Cancri systems with 5 planets each. I then looked at the log plot for each one, and gave myself one unknown planet to fool around with. I manipulated the arrangement by hand, to give the best r². The r²'s ranged from .7722 to .99479. The mean was 0.91840. So, that's pretty darn close to being in the 95% confidence area.

I'd like to do more simulations, but I need to automate the process, and I don't have programming capability right now.

But even with the sample size I did generate, 55 Cancri is obviously exceptional.

There can be no doubt about it.

That's true -- random in log(r) vs r just didn't hit me. But here's what they say about it:

So their argument seems to be that available stable slots have to follow such a rule anyway, and so let's fill that in at random for some number of planets, letting no two planets be in each other's exclusion zone.


-Richard

Were your randomly generated systems demonstrably stable (and demonstrably detectible)?
You have three problems I can think of with your claim that 55 Cnc is exceptional. (Ummm. At the risk of sounding like a Monty Python sketch, you have four problems.)
1) The data are already strongly correlated (as previously described).
2) The population from which the data are drawn is not random. Considerations of stability require some minimal spacing, which will depend on the masses of the planets and the possibility of resonant interactions. So each successive "random" member of a planetary system constrains the probability distribution for the next, in a way which will depend upon mass and eccentricity.
3) The population is also not random by reason of detectibility: close and massive sends a more detectible signal than far and low-mass.
4) Titius-Bode allows, indeed encourages, the "adjustment" of data that don't match the required relationship. This further "de-randomizes" the dataset on which a correlation is performed.

So any sensible hypothesis test on the 55 Cnc data would require us to allow for the strong correlation going in, the non-random spacing considerations arising from stability (and detectibility), and the rules by which we decide to "adjust" poorly matched data. Only then could we decide whether there was any statistical significance to your correlation. I suspect the prospect of framing a sampling distribution for that lot would make a strong statistician blench, but Disinfo Agent may be able to cast more light.

So "obviously exceptional" seems just a tad overstated.

In the absence of a sampling distribution, doesn't just looking at the data give you the tiniest frisson of misgivings? You've got a rough line of four objects and then a clear distant outlier, the location of which you have adjusted to catch the upstroke of a fitted exponential. Looks to me as if your exponential fit pivots on a single datum, which happens to be the one you've messed with.

Grant Hutchison

I didn't run GravitySimulator on them--yet! But see below.

True, and I was surprised with my simulation just how strongly correlated an exponential plot was with several of the random systems.

True again. For my first attempt at a Monte Carlo simulation, I kept it as simple as possible. But taking into account the Hill radius, for example, will mainly affect the left-ended, lower-valued outliers. Just by looking at them, the worst-performing simulations were the ones with one planet relatively close-in with four other planets planets that were farther out that were clustered together.

A planetary spacing system for a half dozen (and we should always be very cautious about extending TBL type relations much beyond that) planets that perfectly followed a logarithmic pattern would be spaced apart far enough so that Hill radii wouldn't be a causal factor. The Hill radii would simply spread those four clustered planets, increasing the r² for the best linear model of the log plot for that trial. But such trials would not affect the right handed tail of the distribution curve for r², which is what we're interested in.


I must admit that as I started through the random trials, I was thinking those guys at BAUT will say that I would just add unknown planets to fudge the model. So I started allowing two unknown planets, but then I got an additional two r²'s in the 99%+ category.

However, the highest 7-slot trial [99.6%] was still lower than the 6-slot model I proposed for 55 Cancri (r² = 0.9975). I then redid, and expanded the data set to only include the 6-slot trials. Nevertheless, the exercise clearly demonstrates your point that adding slots for sure makes it easier to fit a curve to.


Which is only ~ 0.9

Which are too small to affect TB models
We do that by minimizing the number of missing planets.

ALL RIGHT!!!

It wouldn't be that hard. Just run 10,000 random trials incorporating your hill radius or whatever, and then see if the observed, apparent TBL pattern still falls in the right-handed tail.

Is anyone aware of a good, simple, free BASIC programming language out there?

Not to me!!!

I must say I did feel a little shiver as I started making random models. But that's the thing, the line of four objects are not so rough, and they point right to the outlier. It's almost like a road sign to the missing planet.

And one which, by pointing at the planet, seems to say "We're not part of an exponential relationship!"

I edited in a fourth consideration for your delectation, presumably while you were posting. Any sampling distribution must also simulate what we can detect, not what's actually out there.

With reference to your suggestion of "just" knocking off 10,000 simulations of the dataset, i have to wonder how you might do that, since no-one on Earth knows how planetary systems are actually constructed, or how much of them we currently detect. This, once again, refers back to our tediously repetitive suggestion that you are speculating beyond the data. And publius's paper is a rather timely and lovely illustration of the whole principle that you get out exactly what you put in, when you build simulations.

Anyway. You seem to be rather more chipper and less confrontational today. Have you banished all thoughts of duels and combat and inflicting pain? Or is there more of that stuff to come?

Grant Hutchison

But you have to agree that 5 points is very little. Given that we know they'll fall along some increasing curve, it's almost fatal that an exponential model will provide a good fit. The thing is, so would literally infinitinely many other curves.

Grant raised another caveat which I thought was pertinent: correlation. It's quite possible that the data for the five planets are not independent (remember that it all started out as one estimated planet). If not, this invalidades all the t-tests and confidence intervals you made. The oddly large correlation coefficients you got between x and y (though, as Grant noted, a positive correlation was predictable) could also be due to correlations between different data points.

__________________
"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

Here's a graph to illustrate what I was talking about regarding Hill radii.

The four planets are way too close to be realistic, but imposing minimum spacing won't improve the r² enough to rival the correlation at 55 Cancri.

Attached Thumbnails

 

Oh yes; isn't that a good point!
We're not looking at individual little sparkly points in the sky, after all, which we can guarantee are unique entities. (Warren, this links back to our previous exchange about "following an exoplanet for a complete orbit to be sure it's real" as compared to "following Pluto for a just bit of an orbit to refine the data".) The original datset for 55 Cnc is just a single, wobbly line connecting individual Doppler observations and their accompanying error bars, which has been deconvoluted into discrete frequencies to produce the best fit. Tweak just one of those frequencies, and the others will shift to compensate.
So none of these "planets" (temporary scare quotes) is actually independent of the data describing all the others. I'm remote from my filing cabinet, but I think I correctly recall the paper from Macy's team, announcing the fifth planet. They took the two best-represented frequencies, across all observations, as "givens", and then fitted the residuals. Three additional frequencies did the trick, four did no better.

The mutual dependence of the data is pretty clear.

Grant Hutchison

How do you know? It's an impossible system. Eliminate and roll the dice again; It's not acceptable just to nudge excluded data towards reality. (Although medical research would work so much better if that were the case. )

Grant Hutchison

Some models fit better than others. The proportion of variation in spacing explained by the linear version of the logarithmic model (r²) for randomly generated models is about 0.92 (1.0 is perfect)--so there's a grain of truth to your charge.

However, the r² for 55 Cancri is 0.9975; I handcrafted 31 models for describing 31 randomly generated solar systems, and not one of them had a higher r².

No matter how you slice it, the data for 55 Cancri are exceptional.

Assuming the astronomers are correct for their empirical estimates for the semimajor axes of the alledged planets at 55 Cancri, they are independent variables. It is trivially true that as one progresses from the center to the outer edge, the absolute distance must necessarily increase.

However, the spacing between planets need not always increase. It's the spacing we're after, and that is an independent variable.