It's difficult to know if Warren Platts will be pleased or irritated to learn that a variant of his ATM idea has been submitted and accepted for publication in Revista Mexicana de Astronomía y Astrofísica by Poveda & Lara: The Exo-planetary System of 55 Cancri and the Titius-Bode Law.

Grant Hutchison

How fun. I'm sure dark matter can be arranged in an appropriate fashion to dismiss this.

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jwj

If you always believe what you already know, you can't learn anything - Liz

Hmmm, the Titus-Bode law seems to be similar to the "Gas Giants only exist in the ice belt" way of thinking that all solar systems will be similar to ours. The article is trying to make 55 Can., fit into our mode of thinking of planet formation, which is looking to be incorrect due to our system bias.

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Fields of Space

LOGIC, n.
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In the Year 2525.

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If an astronaut doesn't need good grammar, niether does you.

Host of Seraphim

The authors are aware of that problem for their hypothesis, even though they considerably over-egg the significance of their curve-fitting efforts:Grant Hutchison

Yes, how do we understand the persistence of something that doesn't actually persist?

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WOW!! And a Mexican journal no less!

But first let me say it's great to hear from you again Dr. Hutchison! How goes it in the land of Hume?

I knew if I sat around, I was going to get scooped. It only took three months. We should have jumped on it Tony! There goes the Nobel. Oh well. . . .

It's a nice paper, (skimmed so far--detailed comments soon to follow) but I don't think they plagiarized me because they left out the most important part: the Monte Carlo simulations. They haven't shown that the 55 Cancri distribution is statistically significant.

But hey, I must say that the best thing of all is that it's nice to know that I'm not the only completely crazy person on the planet!

Edit: Dr. Hutchison, I couldn't help noticing you started this thread in the mainstream astronomy section. Thanks for the compliment!

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

I will say this now. If this Titius-Bode pattern turns out to be fairly common for well-behaved systems with multiple planets with circular orbits confined to one plane, it will have significant implications for planetary formation theory because it will evetually have to be admitted that gas giants can form much closer to their parent stars than was previously thought. My theory is that the reason the Sol system inner planets are rocky is because, paradoxically, there never was much rocky material within the inner solar system. The metallicity is much higher at 55 Cancri; this allowed big, rocky cores to form near the star capable of holding onto massive atmospheres of light elements in situ.

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

These figures are exactly what I found.

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

After reading the paper, I still fail to see the significance after the papers by Graner and Dubrulle. Although a nice little result, it is a paper that tells 2 + 2 = 4, but why this is is left untouched.

Secondly, the reason for not assuming Mercury as one of the planets to make the new exponential TB law seems totally superfluous, no reason is given for it, apart from the original TB law also did not use this planet. Does not sound like a real reason to me. And if you have to do this, why not leave out the first planet in the 55 Cancri system?

I think maybe reading Lynch and Neslusan will give some more insight into real or chance. From the latter abstract:

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Optimism does not change the laws of physics. (T'Pol)
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Martin ( http://www.geocities.com/DrMartinV )

Hey tsenfem

Glad to hear from you!
I saw that too.

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

Just a couple of things:

• A Google search on "Titius-Bode" and "55 Cancri" reveals that other thread on the front page.
• They report an r² score like that's a normal thing to do, yet in the literature I've been scouring lately, the only other reference where I found something similar is in an obscure letter to the editor of the The American Statistician from 1969, (Vol. 23, No. 5 (Dec., 1969), pp. 52-53), penned by a C. Mitchell Dayton where he calculates a correlation coefficent for the Solar System applied to a linear TBL model. (BTW Poveda & Lara mistakenly refer to their R² as a "correlation coefficient" which is just plain R; R² is the "coefficient of determiniation".)
• The forumula they use is mathematically virtually identical to the one I constructed, yet it's like they changed it around to make it look different, sort of like using the capitalized R².

Well, as they say, imitation is the most sincere form of flattery; or was it that great minds are to be found in the same gutter? At least, they've given the idea the publicity it deserves. Indeed, the internet's practically on fire!

In any case, they didn't get into significance testing, so thus the meaning of their R² is open to interpretation, to say the least. Dayton, in his letter goes on to say that Remember, Dayton was writing in 1969, when computing power was slow and expensive. For example, Ephron (1971, Journal of the American Statistical Association Vol. 66: 552-559--it's in JSTOR) attempted a Monte Carlo simulation of sorts, but he could only do a measly 2,000 attempts, and his histogram is pathetically jerky-looking.

(BTW, there was an interesting flurry of interest in the Titius-Bode law among certain American statisticians in the very late 1960's and very early 1970's. I'm trying to put together a bibliography.)

But nowadays, a typical desktop computer could do 1 billion or more simulations in a single day if it wanted to. So I think I was on the right track by calculating a probability distribution for each specific scenario. However, reading through the statistical literature on the TBL has made me realize that I perhaps made too much of the actual distributions that I constructed. This is because the question of whether any proposed hypothesis is statistically significant is relative to an alternative null hypothesis that is rejected. Here is what Ephron has to say:
So in that other thread, I chose an uninteresting alternative null hypothesis, all right--just take the random planetary distances from a totally uniform distribution. The only problem is that it was too uninteresting to be interesting. It was a straw man, really, I must admit, guaranteed to produce the desired result. As Ephron's rival, I. J. Good wrote (1969, American Statistical Association Journal 64: 23-34, page 30):
BAM! Cut down to size by a single sentence by a man writing 40 years ago!

Quoting Good, Ephron added
So, noting that as one progresses out of the Solar System, the spacing between successive planets always increases, Ephron proposed his own uninteresting null hypothesis: the law of increasing differences.

Ephron's "law" of increasing differences may or may not seem absurd to you, but it has taken on some historical significance because it was addressed in some detail in Hayes and Tremaine's (1988) Icarus paper (preprint available at the ArXiv; the original at ScienceDirect). Now, Hayes and Tremaine's paper is one of the more opaque papers I've read in a long time (though they provide a complete discussion of possible radius exclusion principles). But their bottom line with regard to the law of increasing distances is that any system "that satisfies the law of increasing differences will often satisfy all but the most stringent exclusion laws." (p. 555) In other words, "the law of increasing differences is a much more restrictive assumption than radius-exclusion laws, and in the absence of any physical justification, it does not form a sound basis from which to judge the validity of Bode's law." (p. 555-556). That is, the law of increasing differences is an unfair null hypothesis because it's too strong.

On the other hand, if it could be shown that the law of increasing differences as a comparatively uninteresting, alternative, null hypothesis to the Titius-Bode law should be rejected, that would indeed warrant the assertion that the Titius-Bode spacing pattern of 55 Cancri is a real pattern.

So, naturally, I had to try it.

I used a simple broken stick algorithm to generate random systems that satisfy the law of increasing differences.

The first step is to break a stick of length 1 into 7 pieces--2 end pieces and 5 interior pieces. This is accomplished by drawing six numbers from the uniform distribution ([U[0, 1]). These numbers are then sorted in a nondescending order.

Then I calculated the length of the interior sticks by taking the absolute value of the differences between adjacent random numbers, and then the interior sticks themselves were sorted.

The next step to generating the random system was to reconstruct the semimajor axes: a₂, a₃, a₄, a₅. Note that the one exception to the law of increasing differences in our Solar System is Mercury, in that it is 0.4 AU from the Sun, whereas the distance to Venus is 0.7 AU, thus the difference is only 0.3 AU. Thus, a₁ and a₆ are already given because the two end pieces were never sorted.

The final step was to throw out the 5th planet, because of the empty orbital that Poveda and Lara's model assumes.

Once the random system is generated, then the r² can be computed in the ordinary matter, and stored into the histogram to construct the probability distribution necessary to conduct a significance test.

Once the histogram was constructed out of 100 million random solar systems (see attached chart), it was determined that the 95th percentile of r² fell out at right about 0.9967. Thus, for a 5% level of significance, 0.9967 would be the critical value. The r² reported by Poveda and Lara of 0.997 exceeds the critical value. (I reported an r² of 0.9975; but that was rounded up from 0.997488; if I had chosen three significant figures, I also would have had 0.997.) Therefore, we can say with 95% confidence that we can reject the null hypothesis of the law of increasing differences as a possible explanation for the spacing pattern an 55 Cancri.



Granted, a 95% confidence level is not nearly as high as the 99.9% confidence level I reported in that other thread, but there's no glory to be gained in showing that a straw man is false with a confidence level of 99.9%; it is much more satisfying to be 95% sure that one is warranted in rejecting the strongest null hypothesis proposed in the literature so far.

(Caveat: these are preliminary results, I need to double check everything and increase the number of random systems to 1 billion to smooth out the curve even more before I send the manuscript into Icarus!)

Attached Images

PROBDIST.png (142.7 KB, 7 views)

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

REM *** Increasing differences model: Solar System
REM *** Use broken stick technique to generate distances,
REM *** then sort the distances to get random system with
REM *** increasing distances. Then figure out new A(I)'s.
REM *** Then calculate R-squared and add to histogram.

REM *** input the number of orbitals (planets)
LET ORBITALS = 9

REM *** input the semimajor axis of the innermost planet
REM *** note that INNERMOST = mercury's semimajor axis
REM *** less A(min) (which is 0.15 AU)
LET INNERMOST = 0.23710

REM *** input the semimajor axis of the outermost planet
LET OUTERMOST = 29.91107

REM *** input average x for least squares now, because it
REM *** never changes
LET AVGX = 5

REM *** number of trials between PRNG seed changing
LET MAXTRIALS = 100000

REM *** number of PRNG seed changes
REM *** MAXTRIALS x MAXCYCLES = total # systems generated
LET MAXCYCLES = 10000

REM *** A(i) is for the semimajor axes. A(1) and A(outermost)
REM *** never change, so we can set their values now
DIM A(ORBITALS)
LET A(1) = INNERMOST
LET A(ORBITALS) = OUTERMOST

REM *** we transform the semimajor axes into logs to compute
REM *** the R-squared goodness of fit statistic
DIM LOGA(ORBITALS)
LET LOGA(1) = LOG(INNERMOST)
LET LOGA(ORBITALS) = LOG(OUTERMOST)

REM *** D(i) the differences between consecutive orbitals
DIM D(ORBITALS - 1)

REM *** X(i) is for the x-axis in the least squares zone
REM *** X(i) aka the ordinal of the orbitals w/ planets

DIM X(ORBITALS)

REM *** load the x-axis that keeps track of 9 planets
REM *** note it goes {1,2,3,4,5,6,7,8,9} because the
REM *** scenario calls for 9 planets, no skipped orbitals,
FOR I = 1 TO ORBITALS
LET X(I) = I
NEXT I

REM *** HIST(i) is the histogram that defines the
REM *** probability distribution which is the final goal
DIM HIST(10001)

FOR CYCLE = 1 TO MAXCYCLES
REM *** we print the cycle # to keep track of progress of program
PRINT CYCLE
RANDOMIZE
FOR TRIAL = 1 TO MAXTRIALS
REM *** We are normalizing to the size of the whole system.
REM *** We generate random numbers from within the innermost
REM *** to the outermost orbitals
FOR I = 2 TO (ORBITALS - 1)
LET A(I) = INNERMOST + (OUTERMOST - INNERMOST) * RND
NEXT I

REM *** now sort in ascending order
LET J = 3
DO
LET I = J
DO
IF A(I - 1) > A(I) THEN
LET BUFF = A(I)
LET A(I) = A(I - 1)
LET A(I - 1) = BUFF
END IF
LET I = I - 1
LOOP UNTIL I = 2
LET J = J + 1
LOOP UNTIL J = ORBITALS

REM *** calculate the length of the sticks
FOR I = 1 TO ORBITALS - 1
LET D(I) = A(I+1) - A(I)
NEXT I

REM *** now sort the sticks
LET J = 2
DO
LET I = J
DO
IF D(I - 1) > D(I) THEN
LET BUFF = D(I)
LET D(I) = D(I - 1)
LET D(I - 1) = BUFF
END IF
LET I = I - 1
LOOP UNTIL I = 1
LET J = J + 1
LOOP UNTIL J = ORBITALS

REM *** now reconstruct the semimajor axes
REM *** you already know A(1) and A(ORBITALS)
FOR I = 2 TO ORBITALS - 1
LET A(I) = A(I - 1) + D(I - 1)
NEXT I

REM *** the following statements are for debugging purposes
REM FOR I = 1 TO ORBITALS
REM PRINT A(I)
REM NEXT I
REM PRINT "R-SQUARED = ";

REM *** time to figure out the r-squared score
REM *** start by tranforming data by taking logarithm
REM *** of the semimajor axes, A(i)
REM *** we already know the logs of the first and last A(i)
LET SUMY = LOGA(1) + LOGA(ORBITALS)
FOR I = 2 TO ORBITALS - 1
LET LOGA(I) = LOG(A(I))
LET SUMY = SUMY + LOGA(I)
NEXT I
LET AVGY = SUMY / ORBITALS

REM *** now figure the sums of the squares of the
REM *** deviations (x - avg(x)) and (y - avg(y))
REM *** note I don't do X^2 because X*X is faster
LET SUMXSQ = 0
LET SUMYSQ = 0
LET SUMXY = 0
FOR I = 1 TO ORBITALS - 1
LET SUMXSQ = SUMXSQ + ((X(I) - AVGX) * (X(I) - AVGX))
LET SUMYSQ = SUMYSQ + ((LOGA(I) - AVGY) * (LOGA(I) - AVGY))
LET SUMXY = SUMXY + ((X(I) - AVGX) * (LOGA(I) - AVGY))
NEXT I

REM *** B is the slope of the best-fit least squared
REM *** line of the form y = a + bx
LET B = SUMXY / SUMXSQ

REM *** SSRESID is the so-called residual sum of squares
LET SSRESID = SUMYSQ - (B * SUMXY)

REM *** now we calculate the R-squared score!
LET RSQ = 1 - (SSRESID / SUMYSQ)

REM *** next statement for debugging purposes
REM PRINT RSQ

REM *** this adds the R-squared to the histogram that will
REM *** eventually form the probability distribution for
REM *** determining the critical value that the observed
REM *** R-squared of 55Cnc must exceed in order to reject
REM *** the law of increasing differences null hypothesis
LET HIST(INT(RSQ * 10000)) = HIST(INT(RSQ * 10000)) + 1
NEXT TRIAL
NEXT CYCLE

REM *** time to output the probability distribution
REM *** the left hand tail is of no interest, so we skip that part
FOR I = 1 TO 4999
LET COUNTER = COUNTER + HIST(I)
NEXT I
FOR I = 5000 TO 9999
REM *** prints out the bins corresponding an R-squared score
PRINT USING "%.####": (I / 10000);

REM *** print out the number of systems for that bin
PRINT USING "##########": HIST(I);

REM *** the running total of bins from lowest to highest
LET COUNTER = COUNTER + HIST(I)
PRINT USING "###########": COUNTER
NEXT I

LET TOTALTRIALS = (TRIAL -1) * (CYCLE - 1)
PRINT "TOTAL TRIALS = "; TOTALTRIALS
PRINT "FOR TOTAL PLANETS = "; ORBITALS

PRINT "TOTAL LENGTH IS "; OUTERMOST - INNERMOST; "FROM "; INNERMOST; "TO "; OUTERMOST
END

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

Hello fellow amateurs,

After 17 hours of cranking, my computer generated

1,000,000,000

random solar systems resembling 55 Cancri in that there are 5 planets, 6 orbitals, with the 5th orbital apparently empty--as our near-sighted telescopes report for the real 55 Cancri system--and, best of all, all

1,000,000,000

random solar systems satisfy the law of increasing differences.

I also rechecked every single mathematical and statistical step against the textbooks. The only thing that changed, is that the critical value, once interpolated, edged up ever so slightly from about 0.99673 to 0.99678.

The attached chart only shows the very right-handed portion of the probability distribution (the left-handed tail extends all the way back to 0.5003, so it's useless showing all that). But you can see the critical value, and how the R-squared exceeds the critical value. Therefore, we must reject the null hypothesis--the law of increasing differences--in favor of our old friend--the Titius-Bode law--for 55 Cancri.

I challenge anybody to find anything wrong with the present analysis. . . .

Next stop:

The Solar System

Attached Images

CritVal.png (55.7 KB, 8 views)

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

Well, I tryed out the new technique on the Solar System. At first I was dismayed and bummed out that the Solar System didn't even come close to satisfying the 95% confidence level. Now, I'm merely perplexed. It turns out that the probability distributions based on the law of increasing differences behave oppositely compared to the distributions based on the pure random spacing null hypothesis. That is, as you add planets, the former more closely resemble exponential progressions and the median R-squared scores actually improve, whereas for the latter, the median and 95% critical values decline as one adds planets.

Thus, the R-squared for the Solar System, 0.9933--figure includes Mercury and Ceres, but excludes Pluto for the a priori reasons (1) that we know enough about Pluto to know that it is Kuiper Belt object that has somehow made its way into the inner Solar System, and (2) Pluto is in any case locked in an orbital resonance with Neptune, and therefore does not constitute an independent sample--is less than that for 55 Cancri (0.9975); but on the purely random null hypothesis, they had the same level of significance (0.1%). I had expected that under the law of increasing differences null hypothesis, the situation would be similar.

However, the Solar System's R-squared fell out at about the 87th percentile according to the probability distribution I generated based on 9 planets, no empty slots, rather than close to the expected 95th percentile. But as you can see from the attached charts, the first clearly shows how sensitive the median R-squared is to increasing planets. The second chart shows that the 95% critical value is not as sensitive, but the important point is that it doesn't decline when more planets are added. (The arrow at the far left is the observed R-squared of the Solar System. The number of trials for each distribution was 10 million.)

So now I'm coming around to Hayes and Tremaine's (1998) way of thinking: that the law of increasing differences is an unfair null hypothesis to test against Titius-Bode laws.

I'm open to suggestions at this point.

Meanwhile, back to the drawing board. . . .

Attached Images

	image001.gif (21.4 KB, 5 views)
	image003.gif (20.7 KB, 6 views)

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Fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings one to the far edge of the known world. -- Bradley Ephron

I've been reading, and I found this paper by Neslusan (2004), and he says that it's not Mercury or Neptune that are the odd planets out--Earth is the odd planet.

So I tryed in my own way to replicate Neslusan's results--and it's true: Earth is by far the planet that doesn't belong!

Consider the table below (in all scenarios Ceres is included, but Pluto is excluded because of the a priori reasons that we know enough about Pluto to say it's obviously a Kuiper Belt object, and since it's locked in an orbital resonance with Neptune, Pluto wouldn't constitute an independent sample in any case). The numbers are the corresponding r² scores: