Talk:Titius–Bode law

Updates

I'm updating the article to reflect the latest(!) work on the Titius-Bode Law. Please be patient, this is a substantial update. Note: this is thoroughly grounded in science; it is essential to correct long-held misconceptions about about the Law. The comprehensive authority for this is Michael Martin Nieto's book "The Titius-Bode Law of Planetary Distances: Its History and Theory" (1972). --Jekkara (talk) 05:47, 26 March 2021 (UTC)

Kepler's First Law

I have had to modify the first para of the text, which originally described a planet's distance from the sun as the "semi major axis" of its elliptical orbit. This is not true; it wrongly assumes the star to be at the center of the ellipse where the major and minor axes intersect. According to Kepler's First Law, a planet orbits a star in an elliptical path with the star at one focus of the ellipse and nothing at the other focus. In other words, the planet's distance from the star is the distance from the focus occupied by the star, to the opposite end of the major axis. User:Alan Marshall floratwokettles@hotmail.com —Preceding unsigned comment added by 198.163.53.10 (talk) 19:16, 16 January 2009 (UTC)

Here is why it makes sense to use "semi-major axis" for distance. In astronomy, the semi-major axis measurement refers to an ellipse with the primary (the Sun in our case) at one focus and the secondary (one of the planets in our case) travelling around the ellipse. See Ellipse for a diagram.

Call the semi-major axis a, and length from the center to a focus f. As the planet moves around the ellipse, its distance from the Sun will vary from a + f to a - f. The average of these extremes is a.

Aside: This particular average does not take into account how long the planet spends at different distances. See the "Average Distance" section in the Semi-major axis article for fuller discussion and other averages. —Preceding unsigned comment added by 68.145.187.67 (talk) 22:05, 15 April 2009 (UTC)

Yes.. This is why I reworded the para in such a way as to mention "average distance" only. However despite the flaws in the original definition, that definition is repeatedly reinstated in the text -- with increasing obviousness, this is more to do with territorial monkeys, who don't want their writing-territory violated, than with science. For them, sadly, status comes first and accuracy last.-- Alan Marshall —Preceding unsigned comment added by 198.163.53.10 (talk) 19:07, 16 June 2009 (UTC)

Uranus, Neptune, Pluto

"We" don't believe that anything in particular happened to Uranus, Neptune, and Pluto; edited accordingly. I've made some other related changes to the last few paragraphs, to try to make clearer than the whole thing may be pure chance, and thus not need an explanation, and to remove the "we" voice.Vicki Rosenzweig

The New Adjusted Bode Equation

Has been in the public for some time with absolutely no adverserial point of view. The error from actual (Not discussed here, considering new topic for it, is remarkable.) The difference between being popular and correct are different issues. Even though the popularity of the new derivative has gone up and down, it has never been discredited. —Preceding unsigned comment added by 71.215.54.11 (talk)

Applying Bodes law to the outer solar system

Recent arguments view the changes in the orbits of the outer planets in a favorable light in relation to the scattering of early planetismals. Hmmm... —Preceding unsigned comment added by 66.82.9.24 (talk)

Original work?

I have never seen the last section with the table and Mercury units and the 2:1 ratio elsewhere. Is this original work? Unless a reference gets tagged on it in the next few days, I recommend removing the section. The whole page needs references, too... --zandperl 14:57, 26 August 2005 (UTC)

I think almost a year waiting for the ref to appear is more than long enough, and have deleted this section. The whole entry has enough of a cranky feel to it as it is, without more of this stuff. Robma 14:18, 17 June 2006 (UTC)

Response

A quote at the top, "A weaker formulation with no geocentric point of view and less "ad-hocism" reads: The distance of one planet to the innermost one is about twice as much as that of the previous one." alludes to a web page with tables I have had on the internet for quite some time (the tipoff to me was use of the word "geocentric" which couldn't have come from anywhere else). So I added a table with a description, and a link. If this is not acceptable within Wikipedia guidelines, feel free to remove it, and remove the quote alluding to it as well. (If you do the math you will see the figures in the table are accurate, but the site has its guidelines, so go with what is necessary.) josephconklin 27 August 2005

Not verified

Does someone have a modern reference that treats the "Law" as other than trash? Melchoir 20:15, 8 June 2006 (UTC)

Do you have a modern reference that does treat the law as trash? Everything I've seen describes the Titius-Bode law as an interesting mathematical observation that may or may not have any deeper meaning, with a note that an insufficient number of solar systems are known to draw any conclusions. --Carnildo 21:47, 8 June 2006 (UTC)

Murray and Dermott's 1999 textbook Solar System Dynamics calls it a "simple mnemonic"... "...predicts a semi-major axis of 38.8 AU; the planet Neptune was ddiscovered in 1846, but it has a semi-major axis of 30.1 AU. The breakdown of the 'law' was complete with the discovery of Pluto in 1929..." (bolding mine) "...if every value of i is filled we should expect an infinite number of planets between Mercury and Venus!"

Their data table doesn't exclude Neptune as a "n/a" "violator", but includes it in the table: "Neptune 7 30.06 38.8/ Pluto 8 39.44 77.2". Then there's a statistical analysis of Uranus' moons...

"There is no compelling evidence that the uranian satellite system is obeying any relation similar to the Titius-Bode 'law', beyond what would be expected by chance. This leads us to suggest that the 'law' as applied to other systems, including the planets themselves, is also without significance. However, even though there are no grounds for belief in a Titius-Bode 'law', the various bodies of the solar system do exhibit some remarkable numerical relationships and these can be shown to be dynamically significant."

And then the the next section is an overview of actual mean-motion resonances, including: "The avalanche of planetary data in recent years has provided striking confirmation that our solar system is a highly structured assembly of orbiting bodies, but the structure is not as simple as Kepler's geometrical model nor as crude as that implied by the Titius-Bode 'law'."

So where is the evidence that the law has any meaning at all? Melchoir 22:45, 8 June 2006 (UTC)

I feel this whole entry needs a re-write, as it has a distinctly crank-like feel to it. I've begun the process by adding some references to serious attempts to explain the law, deleting the table of the Mercury-normalised "quasi-Titius-Bode" law, and fixing the first table to reflect the fact that Neptune has a perfectly well-defined TB value, and the fact that it doesn't fit it should be clearly represented in any NPOV entry. Robma 14:22, 17 June 2006 (UTC)

Since the "quasi-Titius-Bode" table has been deleted numerous times, I defer to popular opinion and will not add it in again. However, as someone quite some time back added the statement, "the orbit of each planet is approximately double the distance to the innermost one," the evidence of which is in the "quasi" table, I have deleted the statement as well. JosephConklin 17 June 2006

Sounds good - I think we're getting there....Robma 08:11, 18 June 2006 (UTC)

No one understands sarcasm. What I mean to say is that simple arithmetic is being ignored, and user "robma" celebrates this as "getting there." I guess this user has no calculator. But I can't keep copying the table back after it is deleted by countless clueless people who can't do simple calculations and see simple patterns. JosephConklin 22 June 2006

I'm sure Robma does not doubt your calculations, but under Wikipedia:No original research, they don't necessarily belong in a Wikipedia article just because they're true. Melchoir 04:14, 22 June 2006 (UTC)

It's not "original research," it is simply a recalculation of existing mathematics. How does one go about being "unoriginal?" This seems like the most bizarre dilemma. Plus, I think it answers your question, "So where is the evidence that the law has any meaning at all?" JosephConklin 22 June 2006

Ah, the fine art of being unoriginal! See WP:RS. A good resource for astronomy questions is [1]. Melchoir 04:32, 22 June 2006 (UTC)

Micheal sunanda 03:18, 30 August 2006 (UTC)== 2006 planetary redefinition by IAU ==

I removed the following text which was put in too hastily by someone. The IAU has made no decision yet, and it's premature to start rewriting Titus-Bode just because something is in the news:

Ironically, Ceres has come under consideration by the IAU to fit the revised definition of a planet after the discovery of 2003 UB313, which is larger than Pluto and calls into dispute definition of a planet.

The proposed definition reads, "A planet is a celestial body that (a) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (b) is in orbit around a star, and is neither a star nor a satellite of a planet."

Under the proposed definition, Pluto and Charon will be classified as plutons and thus planets.

Something like this could be put back in if the IAU makes an appropriate decision later. Please see 2006 redefinition of planet. Thanks.Derek Balsam 13:41, 17 August 2006 (UTC)

03:18, 30 August 2006 (UTC)03:18, 30 August 2006 (UTC)Micheal sunanda 03:18, 30 August 2006 (UTC)

Opinion: I am surprised reading the Planet definitions around Pluto exclusion by IAU, that no reference to Chiron between Saturn & Uranus was mentioned anywhere, & that 'Bodes law' was also excluded by IAU in its definition. I assume that means 'Bodes' is outmoded, considered an old mathmatical chance occurance with no practical function in our solar system. I think there is celestial function that Bodes law reflects for inner planets to Saturn. I wonder if Fibanacci formula series is related to the celestial harmonic attraction between orbiting bodies? If so, how does Bode, Fibanacci, Octaves or some formula reveal a pattern of hamonic attraction in our solar system? I call the celestial orbitng pattern - THORBs = 'Torus Harmonic Orbital Ring of Balance' that guides & holds a planet in its orbit according to the size & distance between the mother/centering body, planet or star & its orbiting body. I totally doubt the nebular planet formation theory held by astrophysics. I support using THORB model so explain where planets &* moons are captured comets into new orbit as described by James McCanney in his "Lunar & Planetary capture theory" of Moon captured by Planets & Planets captured by Stasr into orbit. If true, THORB is invisible harmonic zone in ecliptic planes of stars & planets. Micheal Sunanda.Micheal sunanda 03:18, 30 August 2006 (UTC)

If one modifies the rule so that planets form at half intervals in the outer reaches (presumably due to lesser density of available material), we can place Neptune and Pluto just where they should be. The question shifts to "What about the half-interval between Saturn and Uranus?" In the late 1970s, an unusual minor planet was detected in that range. Modifying the article's table, we have . . .

Planet k T-B Real

Mercury 0 0.4 0.39

Venus 1 0.7 0.72

Earth 2 1.0 1.00

Mars 4 1.6 1.52

Ceres 8 2.8 2.7

Jupiter 16 5.2 5.20

Saturn 32 10.0 9.54

Chiron 48* 14.8 13.7 (avg.)

Uranus 64 19.6 19.2

Neptune 96* 29.2 30.06

Pluto 128 38.8 39.44

*Half interval.

Chiron (10.0 + 19.6) / 2 = 14.8

Neptune (19.6 + 38.8) / 2 = 29.2

I am not advancing this seriously, but just to illustrate how, for a moment there it looked as if Chiron had rehabilitated the law. WHPratt (talk) 17:26, 1 June 2009 (UTC)

Periodicity

Hello anon user. While I admire your dedication to truth, there are several rules involved, particularly WP:NOT. I would imagine that 1 Ceres.com is your page. I'm not sure how I feel about your opinions from a mathematical sense, it doesn't seem right, but it might be true. There does appear to be some credence to your claims. Alas, Wikipedia isn't as interested in Truth, as it is in Verifiability. I'm not here to say that the claim is false, but I'm saying that it isn't verifiable. When 1ceres.com gets reviewed by a major journal or website, it will go in, even if unproven, because Verifiabiliy is more important to Wikipedia than truth.

On a seperate note, reverting a revert, especially without reasoning, is considered bad form. McKay 04:13, 18 August 2006 (UTC) Similar notes appear on his talk page and on Talk:Definition of planet and Talk:2006 redefinition of planet

I'm getting rid of the recent original research, as well as the current tag. Unless of course someone can produce a reliable source that claims that a redefinition of the word "planet" will shed new light on the Titius-Bode law. Melchoir 05:46, 18 August 2006 (UTC)

It seems that 71.215.54.11 and PlanetCeres are of like mind on this issue, as both are repeatedly inserting this material into numerous articles. The articles include Planet, Definition of planet, 2006 redefinition of planet, 1 Ceres, 2003 UB313, and Titius-Bode law. PlanetCeres also deleted Mckaysalisbury's comment here, which has now been restored. Here is the current (07:55 UT) text from the front page of the 1ceres.com site:

"Welcome to 1Ceres.com. You might be wondering what 1CERES.com is all about? Well, first off I am not a scientist by any recognition. I came across a periodicity that appears to be a wave across our solar system that was "proved wrong" in the 1800's and has not been revisited except to say "what a cute thing". This new periodicity defines 1CERES as the major mass lying within a planetary periodicity. So, I claim that by applying this periodicity 1CERES is properly defined as a planet. At this time I am claiming the Rediscovery of a planet by mathematical means. My name is Tyler Granger and I came across this periodicity while in prison. When I got to the internet I posted in various areas of this discovery. In the next hundred years I hope to be known as the "Discoverer of 1Ceres as a Planet" and not as the discoverer of 1Ceres as an asteroid."

And here is a relevant excerpt from Wikipedia:Verifiability:

"Self-published sources (online and paper)(See also Wikipedia:Reliable sources#Using online and self-published sources). Anyone can create a website or pay to have a book published, and then claim to be an expert in a certain field. For that reason, self-published books, personal websites, and blogs are largely not acceptable as sources. Exceptions may be when a well-known, professional researcher in a relevant field, or a well-known professional journalist has produced self-published material. In some cases, these may be acceptable as sources, so long as their work has been previously published by credible, third-party news organizations or publications. However, exercise caution: if the information on a professional researcher's blog is really worth reporting, someone else is likely to have done so."

--Ckatz^chat_spy 08:05, 18 August 2006 (UTC)

IAU Special Interests

Because the New Bodes Law competes with the IAU definitions, I will continue to support it. I have known of 1ceres.com for years. And, the previous works done on megspace.com/science/ceres first introduced the equation to myself years before this issue. Please refrain from accusations. The information has been verified. The article content has been edited by others. Your just playing god. Enjoy your toadstool.

Nice detective work there. I got a username just so we could discuss the matter. I can't believe that you used the "hello" portion of the web site while ignoring the information. Yah, and if you do a little archive search you'll see that information has been around for awhile. Thanks for communicating. But, deletion is all you seem to know. OK fine.

The offensive information that is not good enough: —Preceding unsigned comment added by PlanetCeres (talk • contribs)

Please indicate when you're done rewriting your comment so that I can reply. --Ckatz^chat_spy 08:32, 18 August 2006 (UTC)

OK - first off, please don't devolve this difference of opinion into a fight. I'll ignore the insults, and the fact that you've rewritten an older comment to focus it on me. By the way, I did read through the site (your site?), and not just the front page. It's an interesting theory. However, the point of all this hasn't been to debate the correctness of the theory. It is the fact that the citations you've presented - and the web site itself - do not provide sufficient reference to warrant adding your theory to Wikipedia's astronomy articles. That does not in any way mean that it shouldn't be published, or be on a web site, or discussed, or anything like that. It just means that it is not verifiable in a manner that meets the standards set by this project. --Ckatz^chat_spy 09:11, 18 August 2006 (UTC)

Please be aware that PlanetCeres is trolling multiple articles and talk pages with his periodicity theory. Check his contributions and those of anonymous IP 71.215.54.11 for details. Nick Mks 09:03, 18 August 2006 (UTC)

• Requesting Deletion of previous comment and nontopic information by a common contributor. Has no value to the topic. PlanetCeres 04:56, 18 August 2006

New Bode's Law

By changing k within the original equation to stop doubling and switch to a linear progression, the old Bode's law becomes substantially more "accurate". This new equation was discovered by Tyler Granger in the late 1990s prior to the discovery of 2003 UB313 (a.k.a. "Xena"). This new equation was shown to demonstrate periodicity within the solar systemTyler Granger (2002). "Discovery of Planetary Periodicity". 1Ceres. Retrieved 2006-08-17. and has been used to "demonstrate" that Ceres is a planet Tyler Granger (2003). "Further Exploaration of a New Bode Periodicity". 1Ceres. Retrieved 2006-08-17. 2003 UB₃₁₃ is found within the equation to a high degree of accuracy.

Old Titius-Bode where k=0, 1, 2, 4, 8, 16, 32, 64, 128 (geometrical progression)

New Adjusted Bode where k=0, 1, 2, 4, 8, 16, 32, 64, 96, 128, 160, 192, 224 (at 64, the geometrical progression becomes arithmetical)

Here are the distances of planets calculated from the new derivative and compared with the real ones:

Planet	k	T-B rule distance	Real distance
Mercury	0	0.4	0.39
Venus	1	0.7	0.72
Earth	2	1.0	1.00
Mars	4	1.6	1.52
Ceres	8	2.8	2.77
Jupiter	16	5.2	5.20
Saturn	32	10.0	9.54
Uranus	64	19.6	19.2
Neptune	96	28.2	30.06
Pluto	128	38.8	39.44
Unknown	160	48.4	0.0
Unknown	192	58.0	0.0
2003 UB313	224	67.6	67.67

What do you think? POV or Fact? Your chance to vote here. —Preceding unsigned comment added by PlanetCeres (talk • contribs)

• Please be aware that PlanetCeres is trolling multiple articles and talk pages with his periodicity theory. Check his contributions and those of anonymous IP 71.215.54.11 for details. Nick Mks 09:01, 18 August 2006 (UTC)
• Requesting Deletion of previous comment by a common contributor. Has no value to the topic. PlanetCeres 04:54, 18 August 2006

This remark terminates my attemps to engage in a constructive debate with PlanetCeres. Nick Mks 10:02, 18 August 2006 (UTC)

Sorry, just wanted to get the discussion area cleaned up a little. Not pointing a finger. Not counting the equation material, I have no problem with everything else I've done being deleted. Really, it's just got nonproductive stuff here. Don't take it personally, please. I just don't want to delete anything myself because everyone is so on edge it seems. And, if someone would clean this stuff up please include this. Thanks. PlanetCeres 05:39, 18 August 2006

• Damn, PlanetCeres has a step fuction in the law. That's nasty.

• When dealing with "waves" it is justified.

What_Wikipedia_is_not--91.34.222.170 (talk) 19:52, 17 October 2015 (UTC)

What's wrong with this?

² If we substitute 2003 UB313 in place of Pluto, the next planet out in progression matches better.

2003 UB313

256

77.2

67.67

I'd like to see how this qualifies as Original Research. We know 2003 UB313's orbit, we know the formula for the Titus-Bode law. Applying the formula gives 77.2. Xena is 67.67. And it has been stated in several places that Xena is bigger than Pluto, and some people have noted that it's a closer fit for Titus-Bode, so what's wrong with footnoting it? 132.205.93.88 03:30, 20 August 2006 (UTC)

It's easy to find the problems in Wikipedia:No original research:

• It introduces original ideas;
• It introduces an analysis or synthesis of established facts, ideas, opinions, or arguments in a way that builds a particular case favored by the editor, without attributing that analysis or synthesis to a reputable source;

...any facts, opinions, interpretations, definitions, and arguments published by Wikipedia must already have been published by a reliable publication in relation to the topic of the article.

The semi-major axis of 2003 UB313's orbit is not disputed, but we have no reliable source for the claim that it is relevant to this article. We certainly don't have a source that synthesizes this information with the Bode's Law prediction, much less claims that it is an acceptable match or even points out that it is "better" "substitute" for Pluto. Finally, including the claim in the article -- which is supposed to be a neutral reflection of the literature -- would give the reader the impression that scientists are continuing research on whether Bode's Law is a salvageable description of the Solar System; and there is no evidence of that, either. Melchoir 18:39, 21 August 2006 (UTC)

What_Wikipedia_is_not--91.34.222.170 (talk) 19:53, 17 October 2015 (UTC)

Asteroid Belt, Ceres

Shouldn't we just replace Asteroid Belt with Ceres? The data is for Ceres. And indeed Ceres has been used historically for that particular position on the table, so it is not original research (as a synthesis of existing data). 132.205.93.19 02:56, 25 August 2006 (UTC)

Go right ahead; I'm not sure why the asteroid belt is there in the first place. Melchoir 06:14, 25 August 2006 (UTC)

(I hope I am commenting correctly by editing the section.) The data may be for Ceres, but I believe it is simply the most accurate way of measuring the Asteroid Belt's significance in TB, being (the largest piece). — Preceding unsigned comment added by 69.143.164.189 (talk) 20:54, 5 December 2013 (UTC)

point-of-view tag

I have added a point-of-view tag to this article, as it seems insufficiently critical. In particular, the discussions of 'updating the law" worry me. Among astronomers and planetary scientists, the relationship is generally held to be a combination of equal parts resonance, coincidence, and shortage of degrees of freedom (at least three parameters and only eight or ten points). For example, Murray and Dermott, in the standard reference work 'Solar System Dynamics', show that the arrangment of the moons of Uranus can be fit very well by a Titius-Bode relationship, but that so could any arbitrary stable system. As an example of how lowly regarded Titius-Bode is, the planetary science journal Icarus specifically states in their instructions to authors that they will not accept submissions discussing it or similar relationships.Michaelbusch 05:24, 28 August 2006 (UTC)

That sounds about right. I didn't know about Icarus, but I'm not surprised! Melchoir 06:22, 30 August 2006 (UTC)

Having just put together a review of the Titius-Bode law controversy for a science magazine over here in the UK, I took another look at this entry, and I feel it needs a complete re-boot. Trouble is, who has the time to do this - esp. given the virtual guarantee of having the new version trashed by monomaniacal trolls in 24 hrs flat ? Robma 07:51, 30 August 2006 (UTC)

Actually, Icarus will not entertain a paper about "improved" versions of the erstwhile "law", unless it is "accompanied by some detailed physical/chemical arguments to explain why the new relation is to be preferred." One gets the impression this is the perpetual motion machine of dynamics: too many pseudo-numerological fruitcakes in the world, I guess. This needs to be mentioned in the article, as well as Murray and Dermott's statistical counter-argument given in Solar System Dynamics. mdf 13:26, 31 August 2006 (UTC)

Saturn's Moons?

What about the Saturnian satellites? How closely do they follow the rule? CFLeon 05:46, 27 April 2007 (UTC)

In some cases, planetary satellites fit similar rules—but not the very same rule! SharkD (talk) 04:26, 27 February 2008 (UTC)

Table of Bode's vs. orbital resonances

I added a table comparing the Bode's Law prediction of orbital periods between adjacent planet pairs to the suggested (perhaps coincidental) orbital resonances listed for planets in that article. The main reasons for this table are (1) to demonstrate that Bode's Law does not translate to the orbital resonance ratios commonly proposed, (2) to demonstrate that the proposed orbital resonances are not consistent but are picked out to match whatever the actual numbers are, (3) to remind people that Mercury's relationship in Bode's Law has been "fudged" - 0 is not a power of two!, (4) to hint that perhaps "around 2.3" is as good a power law as any (unless you can pull something informative out of The Well-Tempered Clavier). I do not believe I am committing any "original research" by dividing ratios or taking orbital distances to the 1.5th power, so don't even start with that - I'm just putting the models next to one another in the same language because until I thought about it I hadn't realized Bode's law predicts all irrational orbital resonances. 70.15.116.59 (talk) 21:08, 9 December 2007 (UTC)

While your statements are correct, they do start to tread into original research and I'm afraid I don't see what they add to the article. We already state that the apparent fit of the law is due to 'a combination of orbital resonance and a shortage of degrees of freedom'. I don't think a table is really needed - especially since, as you say, there is cherry-picking to fit the data points. Michaelbusch (talk) 00:29, 10 December 2007 (UTC)

The problem is, the table did not appear to support the idea that Bode's Law was due to orbital resonance, since it gives one value and orbital resonance gives the other. I never figured out what "shortage of degrees of freedom" meant, I'm afraid... Also, the orbital resonances provided were not my idea; they're from the orbital resonance article. 70.15.116.59 (talk) 05:03, 10 December 2007 (UTC)

Plotting the function

I am having trouble plotting the function as it appears in the article. I can only get the curve to "line up" with the first several planets if I change it to the following:

$a$ = 0.4 + 0.3 · 2 ^{(m - 1)}

See this image of where I've plotted both functions in some math software. The purple function f(x) is the original, as it appears in the article. The orange function g(x) is the modified function. Only the latter fits any of the plotted points. SharkD (talk) 07:39, 8 February 2008 (UTC)

Incomprehensible formulation

The formulation paragraph needs to be written in such a way that someone who doesn't already know the law can understand it: the current paragraph has two equations (the description of how they are related is either glib or simply wrong) and no clear algorithmic process for calculating the distance of a given planet. It's a simple algorithm, and the formulation paragraph ought to be able to explain it so any idiot (like me) can understand and apply it at once. 82.153.72.124 (talk) 16:38, 13 April 2008 (UTC)

But that's just it; it's not a simple algorithm. It's somewhat cobbled together, and really doesn't have an easy, universally applicable formulation. The best is that which is described at the top. Any algebraic formulation, such as that at the bottom, requires a power that yields zero, which, apparently (I wasn't aware of this) is minus infinity. Serendi^podous 19:13, 16 June 2009 (UTC)

Negative Bias

The article seems to have something of a negative bias intended to disprove the law before explaining it. The line where it states that the formula has been discredited and made moot in the eyes of the astronomers seems to display the contributors negative POV, It has been marked for citation for a while now and no relevant source has been sighted. The law was posited in the 18th century, for its time the law seems like a great achievement- predicting orbits of planets and bodies not even discovered then, I think someone should remove the questionable line already where the citation needed tag has been placed. —Preceding unsigned comment added by Theo10011 (talk • contribs) 12:39, 15 December 2009 (UTC)

But that's a fact. Astronomers do not accept Bode's Law as real. Many astronomical journals refuse to accept any papers on Bode's Law in principle. Given the number of cranks who've tried to add their own solutions to Bode's Law onto this page, I think such forthright language is appropriate. Serendi^podous 12:50, 15 December 2009 (UTC)

Again thats your opinion, or even if it was a generally held consensus among the astronomical community stating in the article without a reputable source makes it inappropriate. That line was marked for citation for that very reason I suppose, since there hasnt been any credible source sighted I think it should be removed. Again, Astronomers might or might not accept Bode's law, doesn't mean it doesn't exist, Newtonian Mechanics didn't lose their place in history with the advent of quantum mechanics, our understanding and the need to explain new phenomenon grew. I would like to point out the line about Icarus refusing to accpet reworked or improved versions of the law was half true pointed out by mdf above, they just require detailed physical/chemical arguments to accompany it. As for the "cranks" submitting there own solutions, well the law itself seems so simple and elegant even to a layman that people feel they can submit their own solutions, perhaps thats one of the attractions about the law. they just need to know that this isn't the place for original research.Theo10011 (talk) 21:17, 16 December 2009 (UTC)

I'm not sure it does exist; it only exists if you accept that the numerical value doesn't hold for Mercury (unless you accept a sequence of powers that goes from -infinity to 1) or Neptune (which it just skips over). As for the page as is, I wouldn't mind changing it but I don't want this article to give the impression that astronomers consider the law scientifically valid. Serendi^podous 21:57, 16 December 2009 (UTC)

Hi Serendipodous, I dont know what you mean by "it doesnt exist", if you're referring to the semantical argument between a rule or law or whether you believe that such a theory was never presented. Historically far more propostoruos scientfic theories have been proposed, the world believed at a point that the sun revolved around the earth or even that the world was flat in some point in history, do we purge the mention of these older laughable theories from history and encyclopedias today, do we ridicule them. I think you keep overlooking the time it was posited, In the 18th century for its time it was a remarkable achievement no matter what you think of it today. The law or rule isn't entirely useless either, It correctly picked orbits of the first 4 planet with a possible location for a fifth in 1801, you can not overlook the time it predicted those without any observational or empirical scientific evidence, even if it doesnt hold true for trans-jovian bodies it WAS relevant at one point and deserve to be mentioned without any reporter bias. The questionable tone of the article is present through out, the first para of the Theoretical Explanation seems consumed with questioning the validity of the law rather than providing what it should, a theoretical explanation just states alternative arguments for its apparent validity.Theo10011 (talk) 22:36, 16 December 2009 (UTC)

Charles Sanders Peirce

Bode's law is discussed as an example of fallacious reasoning in Lecture Five (pages 194--196) of his 1898 lectures:

Reasoning and the Logic of Things (RLT) (The 1898 Lectures in Cambridge, MA)

• Peirce, C. S., Reasoning and the Logic of Things, The Cambridge Conference Lectures of 1898, Kenneth Laine Ketner, ed., intro., and Hilary Putnam, intro., commentary, Harvard, 1992, 312 pages, hardcover (ISBN 978-0674749665, ISBN 0674749669), softcover (ISBN 978-0-674-74967-2, ISBN 0-674-74967-7) HUP catalog page. Text of the lectures that William James invited Peirce to give in Cambridge, MA. (Extracts variously from drafts and from delivered lectures were earlier published in CP 1.616-677, 6.1-5, 185-213, 214-221, 222-237, 7.468-517.)

Editorial Procedures, xi-xii Abbreviations, xiii-xiv Introduction: The Consequences of Mathematics, 1-54   (Kenneth Laine Ketner and Hilary Putman) Comment on the Lectures, 55-102   (Hilary Putman) Lecture One: Philosophy and the Conduct of Life, 105-122 Lecture Two: Types of Reasoning, 123-142 [Exordium for Lecture Three], 143-145

Lecture Three: The Logic of Relatives, 146-164 Lecture Four: First Rule of Logic, 165-180 Lecture Five: Training in Reasoning, 181-196 Lecture Six: Causation and Force, 197-217 Lecture Seven: Habit, 218-241 Lecture Eight: The Logic of Continuity, 242-270 Notes, 272-288 Index, 289-297

Thanks. Kiefer.Wolfowitz (talk) 18:44, 12 March 2010 (UTC)

About the formula

Excuse me people but I think that the formula a= 0.4 +0.3*2^m is WRONG. You cannot obtain the values in the table here with that law.

Planet	k	T-B rule distance	Real distance
Mercury	0	0.4	0.39
Venus	1	0.7	0.72
Earth	2	1.0	1.00

The correct law is a= 0.4 +0.3*m, this one correctly give the "T-B rule distance (AU)" for the right "m" (try to believe). For example for the Earth we have ("m" of Earth is "2")--> 0.4 +0.3*2=1. For Mercury ("m" of Mercury is ZERO) --> 0.4+ 0.3*0=0.4 (with the exponential formula you can never obtain 0.4) I did not change anything because I was already accused of vandalism (I did a mistake correcting a formula...). -- 22:00, 14 March 2010 85.218.11.204

Did you notice the line where it says

$\scriptstyle m=-\infty ,0,1,2\ldots$

? The first value in the series is negative infinity... AnonMoos (talk) 23:59, 14 March 2010 (UTC)

Error percentage

The table states the errors way too exact. For example, the error for Jupiter is said to be 0,00%. Yes, if Jupiter's real distance was exactly 5,20 AU it would be correct, but now it's not. Is it necessary to use 2 decimals for the error (as it's decimals of percents, it's actually 4 decimals)? Fomalhaut76 (talk) 12:07, 19 May 2010 (UTC)

Solar system HD 10180 offers new evidence

The five (possibly seven) planets in the solar system HD 10180 are positioned in a similar pattern as that predicted by Bode's law, according to the following article:

http://www.eso.org/public/news/eso1035/

Might be worth writing up as tentative

-Tristan (anonymous) —Preceding unsigned comment added by 173.248.194.155 (talk) 01:25, 25 August 2010 (UTC)

When I heard of the latest discovery of two more planets in Gliese 581 it occured to me that it might be an interesting new datapoint for Bode's law. I played around with some numbers and the distances seemed to fit reasonably well with a simple exponential curve d=0.0160129*EXP(0.539578*N) with a 'missing' planet number 6 at 0.41 AU. I suspected that someone else must have already thought of it and sure enough a quick search yielded this: http://www.univie.ac.at/adg/Conferences/ahw5/talks/TothZsuzsanna.pdf When it was written there were only 4 known planets in the system. I haven't spent as much time as I would like to on this but I thought fit to point it out just in case someone might be interested in looking further into it. — Preceding unsigned comment added by Grottlu (talk • contribs) 18:07, 21 December 2010 (UTC)

Newton to Bode

If Bode's Law is not coincidental, then it ought to be possible to begin with a mathematical model of the early solar system, with a flat protoplanetary disk surrounding a star, and then, by dividing the disk into hundreds of representative points (particles), which of course will all influence each other under the prevailing forces, use Newtonian gravity equations to mathematically derive a relation which encapsulates Bode, in terms of the distance apart of the accreted particle masses (planets), and also predicts alternative spacing patterns. This has never been done. However it would be immensely useful in the search for earth-type planets; for example if we find say 3 large planets around a star, then we could use this math to tell us, at once, if there will also be a planet in that star's habitable zone, without having to spend years or decades of telescope time searching for it. Any mathematicians up for it? —Preceding unsigned comment added by 142.161.207.211 (talk) 00:30, 22 January 2011 (UTC)

Question about a graph on the article:

On the article page about Bodes Law it has a graph with Bode's predicted positions and then the actual postions of the planets and the graph compares how accurate Bode was ... and there's the column labled "K" ... what does "K" stand for? In this discussion someone mentions the K factor but still doesn't say what it stands for. The article doesn't say so we the readers are left guessing as to what "K" stands for.

G2thef (talk) 14:13, 12 October 2011 (UTC)

Yes that's not very clear. Well if we sequence the planets, like, for Mercury N=1, for Venus N=2, Earth N=3..etc,.....where N means "number".... then k= 2 to the power of (N-2). Or k=2^(N-2). However this doesn't work for Mercury. Using N=1 for Mercury should give us k= 1/2. However this value works pretty well on the graph, so I don't know why they didn't use it for consistency's sake. (63.163.22.209 (talk) 05:37, 30 November 2011 (UTC))

Oh nevermind, I see "note 2" on the next column of that graph

173.238.43.211 (talk) 05:07, 22 July 2012 (UTC)

LEGITIMATE CONFUSION OVER YOUR CHART

Hi. The chart showing the actual values of the planets and the Titius-Bodes calculated value has a column for "% of error using real distance as the accepted value" and I have no idea what percentage calculator you used for that column because, for example, the difference of Uranus's Titius-Bodes number (19.6) from its actual distance number (19.2) is 0.4 and its "% of error" number is 2.08  % .... BUT THEN LOOK AT VENUS and its even closer with its Titius-Bodes number (0.7) from the actual distance number (0.72) so a difference of only 0.2 YET THE % OF ERROR ON THE CHART IS GREATER THAN URANUS'S WHICH IS FARTHER FROM IS PREDICTED VALUE? IF URANUS' ACTUAL DISTANCE IS FARTHER FROM THE PREDICTED VALUE THAN VENUS IS FROM ITS PREDICTED VALUE THEN SHOULDN'T THE % OF ERROR FOR VENUS BE SMALLER BECAUSE ITS CLOSER THAN URANUS IS TO ITS PREDICTED NUMBER? Your chart is giving a number the opposite of how the numbers are given or is there more to the procedure finding this "& of error" than is mentioned in the article? I don't know how you arrived at the numbers under the "% of error" column is what I'm asking sorry.

Thanks I appreciate understanding this graph!

173.238.43.211 (talk) 02:36, 3 August 2012 (UTC)

Dwarf Planets

Just a suggestion. We include the other Dwarf planets? Haumea, Makemake, and Eris Belong just as much on the list as Pluto and Ceres and they show the complete breakdown of bode's law.Donhoraldo (talk) 23:59, 31 March 2013 (UTC)

Pluto and Ceres were historically discussed in relation to the law, while Haumea, Makemake, and Eris weren't. Besides, they are probably not the only dwarf planets: Sedna, 2007 OR₁₀, Quaoar, and Orcus are very likely to be round as well. I think we should just stick to including all the bodies that were historically mentioned in discussions of the law, and no more. Double sharp (talk) 14:42, 14 June 2014 (UTC)

Perhaps it's what we count as qualifying for entry on the table that's the problem... noting that the law itself and most discussion of it, and indeed (given the recent speed of change in our understanding of the solar ecosystem) even some of the older pieces of debate on this very page happened prior to it being clear just how damn BUSY space actually is (suggesting Eris, under it's pre-name designation as it hadn't long been discovered, replace Pluto, for example). The model assumes mostly a complete void with a small number of large bodies floating around, only.

If we take a more holistic view instead, we could say:

Mercury - Venus - Terra-Luna - Mars (and all the smaller satellites, quasi satellites, and asteroids that cross or share each of their orbital zones),

then Ceres and the rest of the asteroid belt (it takes up a majority of the belt's mass, but there are a half dozen other asteroids that represent a reasonable fraction each, and then millions of smaller bodies spread out in a solar ring, whose presence and gravitational influence shouldn't be completely ignored),

maybe a distinction between that and the Hildas as well (or include them, to spread the belt out further?)

Jupiter (+ moons, + greeks/trojans, etc)

Saturn (+ moons, + rings...)

possibly the notable inter-Saturn/Uranus centaurs, though I feel it better that they either get clumped with whatever major planet their orbit most closely matches OR considered as secondary asteroid belt loosely centered on that zone

Uranus (+ moons, rings, etc)

and then it all gets difficult because of the considerable number of interactions between...

Neptune and the plutinos and all the other TNOs that come close enough to it to have gravitational interactions...

We could have Neptune (+ moons + other co-orbitals), *then* the Kuiper belt as a separate entity from it (as in everything from Plutinos to Twotinos), then other bands outwards from there - scattered disc etc. Considering everything that doesn't "clear its neighbourhood" as an anonymous member of a larger co-orbital group, rather than as a significant object in its own right when it comes to the model. Whether the more directly Neptune-interacting objects should be considered as part of a group that includes it - and given the sheer number of them, and their often eccentric orbits that have relatively large SMAs despite perihelions close to or just inside of Neptune - and therefore drags the *average* orbital position of their collective centre of mass (...barycentre, I suppose?) out towards what was originally predicted, or whether it's altered so that Neptune can be considered a discrete item and then the KBOs etc are considered as a discrete cluster-item at an even further radius, could stand some heavy calculation to figure out. Especially as we almost certainly still haven't detected everything that's out in that zone, let alone worked out their individual (or collective) masses, sizes, precise orbits, etc.

But even though the question of where you draw certain lines may be fuzzy, I definitely stand by my suggestion that it's silly, on that scale and with such an increasingly large known population of objects that share certain identificable SMAs or zones of them (many of the outermost ones in particular Neptune resonances), similar to the asteroid belt itself and indeed the Jovian system as well, plus the growing number of known centaurs and their fairly eccentric orbits, with the scattered disc, inner and outer Oort clouds beyond, and the possibility of a distant massive body perturbing them from a potentially circular original orbit into a cloud of elongated ovals around a distinctly extrasolar barycentre, to continue to base the predictions and attempts to fit things into them on an idea that each slot should only occupy a single object - as massive a one as we can find, but only one. We now know that they exist as a continual spread of material from inside of Mercury to way, way out into the interstellar zone, with the tracks of everything from Ceres outwards (and in a lesser way, the inner planets too) being more like belts of small objects being dragged around by larger ones, and, whether we like to represent it in our models or not, they are still there, still have mass and gravity, and still have influence on their myriad neighbours, their orbits and positions. And if you have a million 1-kilotonne lumps of rock clustered around a certain point in space, then averaged over a long enough time, it's reasonably similar to having a single 1-gigatonne lump. Finding a particular 300-megatonne one lurking in the midst of it all and claiming that as your representative is needlessly discriminatory. Particularly if its orbit within that belt is somewhat off the actual centre of orbiting mass, because that then gives you an incorrect figure.

After all, the idea itself isn't particularly daft. The system will have formed according to fairly normal physical laws, if there was a certain patterned or even just graduated distribution of gas and dust in the original nebula then a mathematically related sequence of clumping might have occurred. There's enough things in the system already that are in resonant orbital relationships and, where their own masses are small enough to not have a significant effect on individual periods around the primary body, that also implies a relationship (2/3s power vs period) of orbital radii. EG, all of the Plutinos should have semimajor axes approximately 1.31x that of Neptune, as that's 1.5^0.6666(...). Saturn and Jupiter have an almost but not quite 5:2 resonance, so that's a 1.842x multiplier. And so on and so forth. Gravitational interactions often encourage that sort of thing, though the actual relationships involved are not necessarily as neat as "double it every time".

And of course through the belt between the Plutinos and Twotinos (2:1 resonance, thus 1.587x Neptune radius), there's a not inconsiderable amount of mass. Maybe not equal to Neptune itself, or even a tenth, but given the positioning that might be enough to act as a virtual binary or very large moon that somehow always sits on the same side of the planet; together, might their overall mass and position drag the effective combined orbital radius out towards where Neptune was "expected" to be?

NB don't fall into the trap of considering the minor planets' *current* positions in space, btw. Always go with the average orbital radius, IE the SMA... eccentricity isn't really an issue for the main eight, as their orbits are damn near circular as far as the calculations for this chart are concerned, and so they're always at the same distance. But the minor bodies, as they're not gravitationally dominant in their orbital zone, are often rather eccentric thanks to the influence of larger inner (and outer?) planets, and their initial apparent position could easily vary twofold outwards OR INWARDS (or far more for the extreme examples, and that's before we get on to comets) from there over time depending on where it is along its track. And it's only the SMA that matters for these calcs 146.199.0.251 (talk) 19:53, 27 September 2017 (UTC)

A possibility

I will admit I am not in any possible way an expert on this, or deeply knowledgeable on the subject, or at all mathematically correct. I am only a Middle Schooler at the moment, so excuse my errors. I had a preliminary theory (very simple) that would explain a good many things about Ceres' (or so it seems) involvement in the Titius-Bode law. First of all, as many, I believe, I have assumed that the fifth number, 2.8, does not at all relate to Ceres, but instead to the planet located where the asteroid belt is today that never actually formed. My theory is this: Some time after planetary formation, most of the planets we know of today are fully formed, excepting the fifth, Theia (ring a bell?), which is disrupted by Jupiter's gravity and prevented from fully forming. I have calculated that Theia existed dangerously close to the 5:2 Kirkwood gap (Țp= 2.53) using the simple equation d³=p². Therefore, Theia's orbit becomes increasingly elliptical towards Jupiter, until, upon entering it's orbit, it is slingshotted towards the inner solar system, increasing it's velocity so that when it approaches earth orbit, Theia overtakes Earth from behind greatly reducing the force of impact and ensuring earth's survival. The rest is history. Additionally, I believe that the tenth Titius-Bode measurement, by some believed to be pluto, is in reality the Kuiper Belt, too sparse too form a planet in the first place. — Preceding unsigned comment added by 69.143.164.189 (talk) 22:09, 4 December 2013 (UTC)

Andrew Prentice and his modified Laplacian hypothesis DOES support the TB law.

Professor Andrew Prentice's work in the 1970s with his modifications to the Laplacian hypothesis of planetary creation by adding a component for turbulence, has provided a great deal of theoretical support for this theory. Amongst other things, he extended this theory to the system of Jupiter and its moons. His resultant predictions for the densities and compositions of the Galilean moons of Jupiter turned out to be far more accurate than anybody else's, as confirmed by the measurements taken by the Voyager and other Jupiter-reaching spacecraft. It resulted in NASA taking his theory very seriously. His theory also explains why the outer planets don't match; or more correctly, why there is a change to the formula that then makes it match.

Finally, my understanding is that Prentice's work IS indeed accepted by mainstream astronomers. I don't understand enough about it to add this to the article myself, which is why I'm only putting it here. 101.175.94.11 (talk) 11:55, 25 January 2014 (UTC)

A paper called Testing the Titius-Bode law predictions for Kepler multi-planet systems came out. This shows that this law does not apply to most exoplanetary systems as it found about 1 out of 28 predicted planets. However, this number could be significantly larger when taking a chance of the planet non transiting the star or the planet being too small into account. --Artman40 (talk) 23:09, 19 June 2014 (UTC)

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correct citation + dead link

"...supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety five."[1] not 96 as somebody wrote before-

new link for 95 is: http://dawn.jpl.nasa.gov/mission/background_02.asp

--91.34.222.170 (talk) 19:37, 17 October 2015 (UTC)

Time for a test: Planet Nine

Planet X is now officially back on the menu, with different packaging as Planet Nine. See [2] [3] The article guesses its semimajor axis is somewhere very roughly around 700 AU. Titius and Bode tell us it is within a few percent of 614.8 AU, or failing that, 1229.2 AU, or failing that, 307.6 AU.

None of this is sourceable or addable yet - don't even try - but it's bound to get mentioned sometime. (though one thing to watch for, while trying to drive a stake through this old chestnut until the ichor of mixed metaphors sprays across the stars, is whether it's possible for someone to say no, wait, it's the average distance, it's the perihelion, or the aphelion, or something) Wnt (talk) 16:47, 20 January 2016 (UTC)

Alright... I got tempted to add some of the weird distant bodies in the table to see what it looks like. [4] As a heuristic, this law actually seems pretty nice - I mean, with just a little effort you could use it to immediately understand in your mind which minor planets are in what general league in terms of distance. That said, I may well have bent the OR rules too far doing this... for example, while some sources refer to semimajor axis I have no idea if this particular near-pseudo-science actually limits itself to that particular measurement. I'm sort of on the fence about self reverting the whole thing at the moment. And yet ... my it's nice to just look and see things around Pluto, Eris, CR105, GB174, Sedna as five different classes of planets roughly as spaced out as the more familiar examples. Wnt (talk) 23:08, 12 March 2016 (UTC)

No mention of the Nice model

A mention of the Nice model in relation to Titius-Bode would be helpful to those who are familiar with it and who wonder about the change in orbits of Neptune and Uranus proposed by it. — al-Shimoni (talk) 10:54, 23 February 2017 (UTC)

OK, I'm having trouble getting my head around the maths here.

Now, naturally, we can handwave the original figures given, in tenths of AU, as just for the sake of convenience in calculating and writing things down in the 18th century; these days we could modify both the initial measurements and the assumed coefficient between them to be much more precise given that we have magic number slaves to do it all for us (ie computers, or even just pocket calculators).

But, even given that allowance, I, er... can't see how the progression works? There's supposed to be something meaningful there, but as far as I can see at the moment it's just a big old ass-pull. How are each of the numbers for successive planets arrived at, starting with Mercury as the baseline? There's mention of it doubling every time, and something called "k", but everything in the article is so convoluted that I'm having trouble picking it out. Is there any hope of having the concept laid out in a very straightforward manner? If all we're doing is coming up with very limited precision numbers between about 0.4 and 100 (or indeed, integers from 4 to 1000), then it can't be THAT complex.

After all if that's made plain, and there IS any kind of validity to the model which only becomes plain after e.g. integrating the data for dwarf and minor planets in certain bands as notional single items (maybe even combined with a nearby main planet) with an averaged-out orbital radius, correcting the Mercury baseline and adjusting other coefficients to make the datapoints fall more comfortably onto the trendline, we at least need to know what it is we're altering and how it was originally expected to work.

4, 7, 10, 14, 28, 52... etc. It doesn't speak of a particularly obvious relationship to me. Other than the doubling between Mars and Ceres? (I might have misremembered a number or two there?) 146.199.0.251 (talk) 20:01, 27 September 2017 (UTC)

It's pretty simple: take Mercury's orbit radius as one constant ("a") and the Venus orbit radius minus the Mercury orbit radius as a second constant ("b"), and then the radiuses of the planet orbits beyond Mercury are predicted as a+(b*2^n), where n=0 for Venus, n=1 for Earth, and so on. (Actually, "b" is usually chosen so that a+2b equals one astronomical unit, but that's a relatively minor difference.) The formula has problems, but it did basically predict Uranus and the asteroid belt... AnonMoos (talk) 14:38, 29 September 2017 (UTC)

Superseded by What?

The introduction to this page says that the law has been superseded. Superseded by what? What was the next theory accepted by scientists, and what is the current theory accepted by scientists? There are links for further reading at the bottom, but I don't see any indication about why the links are relevant to this page, except that they must be about a similar topic (that is without having read the articles).

I feel like this article leaves me at a cliffhanger. 209.133.79.7 (talk) 00:48, 7 January 2018 (UTC)

It's been superseded by an understanding that real world orbits don't usually obey simple numerical laws of this type. -- AnonMoos (talk) 10:45, 17 January 2018 (UTC)

According to the article, the law has been discredited... because of Pluto and Eris... which aren't even planets. Go figure! DAVilla (talk) 05:10, 22 December 2021 (UTC)

programmed the Titius_Bode equation into TCL

• • • • • • • • • • • • • • • • • • • 11Jul2021*************************************

Under Titius_Bode , Wikipedia page. I have programmed the Titius_Bode equation into a Tool Control language TCL gui program and very interested in modeling Titius_Bode-like calculations also. I have run the JPL Keplar data with automatic curve fitter. Maybe there is a contact interested in working or reviewing this item? The autofitter found a = 5.14E-01 and b = 1.55E+00 for the Modified Power function <y=a*C**x>, which was close to the Armellini formula for planetary distances. In 1921, G. Armellini suggested a planetary distance formula, equivalent to <y = 0.2792 x 1.53**n> . Other authors have suggested 1.73 or, 1.842, for the C term in a planetary distance power function <y=a*b**x>. The Modified Power function error is at 5 percent of the Y axis, which is at the non-significant borderline level generally cited. The Weibull growth curve should fit the Y axis with a 1 percent error. In these growth functions, the Levenberg-Marquardt (L-M) algorithm was used with double precision C's.Thank you. Dewi7 Ps. The TCL WIKI editor is locking, so I have more material than evident. (talk) 15:42, 11 July 2021 (UTC) Dewi7 (talk) 19:48, 13 July 2021 (UTC)

Perhaps you might like to write a section about Armellini's formulation? However, I found this almost immediately: "A supposed new law for planetary distances", Badolati, E., The Moon and the Planets, Volume 26, Issue 3, pp.339-341, DOI 10.1007/BF00928016. The letter points out several flaws in this formulation, and suggests that the existing Blagg/Richardson formulation is still the most accurate, and general. Jekkara (talk) 06:37, 14 July 2021 (UTC) ___ posted this item in TCL WIKI___

• • • The Wave-Particle theory of the Solar System***

The Soltani paper presents the theory that the Titius-Bode Law is derived from the existence of a standing and cosine wave wave packet in solar system. There are successive solar wave nodes in a Schrodinger standing wave originating from the Sun in the Soltani position. Planets and objects may reside at successive solar wave nodes of a stationary or standing wave, but some nodes are not filled. The solar nodes start from the origin line, with the first solar node which is 0.1 AU. The first solar node is empty of a planet. After the first solar node, each successive solar node is a length of 0.3 AU. The planet Mercury resides on the second solar node at pi/6 ==>> 0.4 AU. The planet Venus resides on the third solar node at 5*pi/2 ==>> 0.7 AU. The Earth resides on the fourth node at 7*pi/2 ==>> 1.0 AU. The fifth solar node is empty, but the location is at 9*pi/2 ==>> 1.3 AU. The planet Mars resides on the sixth solar node at 11*pi/2 ==>> 1.6 AU. For previous mainstream use in physics, Quantum mechanics, the Schrodinger equation, and the de Broglie wavelength relation were thought only for subatomic scale and subatomic objects. Small TCL routines can be written added to the TCL gui to study the implications and graph the possible interrelations to the Titius-Bode Law .

1. 1. 1. 1. Here is a summary of Distance formulas and planet assignment numbers for starting the adventure to infinity and beyond. The Titius-Bode formula set < Earth = 1 ... > as number assignment scheme for planets. The Asteroid Ceres is included on most number assignment schemes. Ceres represents the Asteroid belt between Mars and Jupiter. The planet Neptune and dwarf planet Pluto are included on some number assignment schemes. Other planet number schemes are for power laws are available here. The Wurm_formula_1803 was expr {(0.3 + (3.*2.** $nn))*0.1} with the setting < Mercury = -1 ...> But the setting < Mercury = 1 ...> was used by Gaussin (1880) and Armellini (1921) for their power laws. Very interesting, Nicolini transformed planet number scheme in Titius_Bode to < Mercury = 1 ...>, refer to Nicolini. The nicolini_formula_1957 was expr {0.4 + 0.075 * (2. ** $nn)} , $nn=l for Mercury,2,... The Basano_Hughes_formula_1979 was expr {0.285 * (1.523 ** $nn)}, $nn=l for Mercury, 2,... Sven-Ingmar Ragnarsson, 1994, modeled Jupiter and the outer planets to Uranus, separately. See the Ragnarsson paper. For the TCL code, keeping track and comparing these planet assignment numbers is tricky. The Ragnarsson paper used $nn=0 for Jupiter, 2 = Saturn, 3 = Uranus ..... The most successful players separate the non-linear distances into two or three piecewise sections, rather than having a single formula try to cover all the planets and objects.

1. 1. 1. The set of measured planet distances is a non-linear function, which possibly should be modeled in three sections. From the log plots, there appears to be a sine wave or sigmoidal growth function component in the measured planet distances, meaning non-linear S-curve in slang. Again from the log plots, one can see how the power laws or doubling rules like the Titius-Bode formula are breaking down beyond Jupiter and Uranus. The Ragnarsson paper in 1994 showed fair results in modeling Jupiter and the outer planets as a separate section for the non-linear function.
         1. 1. I plotted the Armellini (1921) output. I think Armellini was trying to optimize the Asteriod belt. The outer planets in Armellini are not a reasonable math to the Kepler measured data set, in opinion. I think it would tricky to endorse the Armellini fit, other than to point out the Armellini formula is breaking down for outer planets. I think Badolati is giving some sound advice.

• • • • • Armellini (1921) output ****

1 0.43 mercury 2 0.65 venus 3 0.99 earth 4 1.51 mars 5 2.30 Ceres?

---

• • • • Testcase 2, Planet distances in AU, Armellini formula ****

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%| Table Armellini formula | | I | | printed in TCL format |% %| | formula d= 1.53**$n, $n=-2,-1,0,1,2,3... | | | avg error 0.937 percent | |% %| Planet Name | assigned number | Armellini formula | Kepler measured data | percent error | comments, if any |% &| Mercury | -2 | 0.427 | 0.39 | 9.5349038448 | |& &| Venus | -1 | 0.654 | 0.723 | -9.5996167024 | |& &| Earth | 0 | 1.000 | 1 | 0 | Earth is set to 1 AU |& &| Mars | 1 | 1.530 | 1.524 | 0.3937007874 | |& &| Vesta Asteroid | 2 | 2.341 | 2.36 | -0.8093220339 | Asteroid, Armellini mentioned |& &| Cammila Asteroid | 3 | 3.582 | 3.48 | 2.9188793103 | Asteroid, Armellini mentioned, but Italian spelling used |& &| Jupiter | 4 | 5.480 | 5.203 | 5.320253892 | |& &| Saturn | 5 | 8.384 | 10 | -16.158864007 | |& &| gap? | 6 | 12.828 | | | Armellini mentioned gap in formula sequence |& &| Uranus | 7 | 19.626 | 19.18 | 2.3272759364 | |& &| Neptune | 8 | 30.028 | 30.06 | -0.1052946353 | Armellini made fair prediction for Neptune |& &| Pluto | 9 | 45.943 | 39.44 | 16.4892827128 | dwarf planet, Pluto is not considered a planet. |& &| Note | Pluto is not considered a planet. | | | | |& &| Note | Vesta representative of Asteroid belt here | | | | |& &| Note | Bode-Like Laws etc usually break down for Neptune and Pluto, confusing references. | | | | |&

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The original Armellini planetary distance power function <y=C**x>, C=1.53, nn = -2 for Mercury, -1, 0=Earth, 1, 2, 3.... The Armellini average error was 0.937 percent here. Later authors modified the Armellini formula with different planet number assignments and assumed different gaps, and got some slightly different results. The calculation of average error is tricky because some authors interpreted planets and gaps differently. The original Armellini over 9 planets and objects was about half the relative error of the Titius-Bode results.

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Dewi7

P.S.The paper from Abolfazl Soltani looks impressive on Titius_Bode relations(to me). Solar System Wave Function and its Achievements by Abolfazl Soltani Department of Physics, University of Birjand, Birjand, Iran Email provided in paper

Dewi7

So, Blagg knew about dwarf planet Eris in 1913...

Whoever wrote, or created the final form of, the paper that is supposedly Blagg's 1913 paper, apparently knew about dwarf planet Eris. So, my conclusion is that it is not a 1913 paper.

I am talking about the paper available here, which, at least as displayed on my computer: contains Blagg's formula, Distance=A * (1.7275)ⁿ * {B+f($\alpha$+n$\beta$)} ; it also contains a long 7-term expression (starting with 0.4594) for function f; and it also contains the coefficients A=0.4162, B=2.025, $\alpha$=-0.8°, $\beta$=56.6°.

So:

1. The numbers don't add up. As in, if you try to perform calculations based on the formula, you won't get the expected result. But that can be easily corrected. It might have been that author of the paper had two formulations of the "Blagg's law", and mixed them up.

First formulation: A=0.1395, counting planets starts from n=0 for Mercury.

Second formulation: A=0.4162 (which is 0.1395*1.7275²), counting planets stars from n=-2 for Mercury, and you need to adjust the formula by either:

- in the formula, where there is f($\alpha$+n$\beta$), change it to f($\alpha$+(n+2)$\beta$)

- or just increase $\alpha$ by a value equal to 2$\beta$, which has the same effect

- or just modify the function f by shifting its graph by 2$\beta$ to the left, which has the same effect

I chose to use the second formulation, increase $\alpha$ by 2$\beta$, just like in Roy's and Nieto's papers, so $\alpha$ is now 112.4°. And for the function f, I use the long 7-term expression provided in the paper. Now the formula works. Still weird that there was an error in the original paper.

2. The formula predicts Eris (n=8) at about 67 AU. Awesome! And the value of function f for Eris is 0. That function f (continuous) takes values from 0 to 1 (continuously), but its value for Eris seems to be almost exactly 0 (well, 0.0067).

Just as if someone knew Eris was there, and adjusted the function accordingly.

3. The formula looks like someone took a simpler equation and masked it my making it more complicated. Why didn't anyone notice that $\beta$=56.6° is close to 1 (in radians)? Combine that with the fact that $\alpha$ was almost 0 in the original paper, and $\alpha$ and $\beta$ disappear:

Distance = A * (1.7275)ⁿ * {B+f($\alpha$+n$\beta$)}

becomes simply:

Distance = A * (1.7275)ⁿ * {B+f(n)}

And I think that's the original form of the formula: for each n (for each planet), pick an arbitrary value of f(n) that makes the formula fit the data.

By complicating the formula further, by adding additional parameters like $\alpha$ and $\beta$, maybe you can make the formula work for satellite systems (not only for the Solar System) - but my guess is that is not discovering a genuine physical law, my guess it's just a trick.

4. The formula also predicts the Kuiper Belt (n=7) at 42 AU. Not about 42 AU, exactly 42 AU, with accuracy to two digits after the dot (so, 42.00 AU). Another great prediction as the Kuiper Belt wasn't known yet. But the formula should not give an integer value. Looks like someone tried to make the function match all the values exactly, including a weird integer value of 42 AU for the Kuiper Belt (it's an integer as it's not a real value, but an approximation), and it shows.

5. As a continuation of point number 2, I'll try to show how the person who created the formula knew that Eris was there.

Let's consider how someone would create the formula knowing only the planets up to Neptune; and then, how someone would create the formula knowing the planets up to Eris.

The process is the same: you have a list of objects, numbered starting with n=-2, with distances from the Sun (in AU) for each object, and you are trying to make the distances follow the rule: distance~1.7275ⁿ. Of course, there are deviations from that rule:

Name	n	Actual distance	Distance according to the rule (1.7275n)	Fraction (Actual distance as fraction of what the rule says)
Mercury	-2	0.387	0.3351	1.1549
Venus	-1	0.723	0.5789	1.2490
Earth	0	1	1	1
Mars	1	1.524	1.7275	0.8822 (minimum)
Ceres	2	2.77	2.9843	0.9282
Jupiter	3	5.2	5.1553	1.0087
Saturn	4	9.58	8.9058	1.0757
Uranus	5	19.22	15.3847	1.2493 (maximum)
Neptune	6	30.07	26.5771	1.1314
Kuiper Belt	7	42	45.9120	0.9148
Eris	8	67*, (67.8)	79.3130	0.8448, (0.8548) minimum, but not known in 1913

* 67 is not the actual value, but is very close to what Blagg's formula "predicts", so let's assume someone used that value in the calculations

You will want to use a "corrective" function f, with values from 0 to 1. Specifically, 0 for objects with the lowest value in the Fraction column and 1 for objects with the highest value in the Fraction column.

Back in the year 1913, you would have mapped f(n)=0 to Mars (0.8822) and f(n)=1 to Uranus (1.2493), stretching the difference of 0.36709 into 1, because these were the extremes (Eris, the object with an even lower value of 0.8448 or 0.8548, hadn't been discovered yet). You would have ended up with a formula like:

Distance = 0.36709 * 1.7275ⁿ * {2.4032+f(n)}

A=0.36709, B=2.4032. Not similar to the values in the paper.

But after Eris was discovered, you would have mapped f(n)=0 to Eris (0.8448) and f(n)=1 to Uranus (1.2493), stretching the difference of 0.4045 into 1. You would have ended up with a formula like:

Distance = 0.4045 * 1.7275ⁿ * {2.0882+f(n)}

A=0.4045, B=2.0882 - you can see that this is much closer to the formula in the paper. The paper was written/edited by someone knowing to map Eris to f(n)=0.

Or, using the actual distance to Eris, 67.8 AU and not 67 AU, you would have mapped f(n)=0 to Eris (0.8548) and f(n)=1 to Uranus (1.2493), stretching the difference of 0.3944 into 1. You would have ended up with a formula like: distance = 0.3944 * 1.7275ⁿ * {2.1672+f(n)}

Left as exercise: how was a supposedly 1913 paper written/edited by someone knowing about the Kuiper Belt AND Eris?

Periwinklewrinkles (talk) 06:30, 19 April 2022 (UTC)

As reference, please see this paper, dated December 30, 1973, from the Irish Astronomical Journal, “THE DISTANCE AND PERIOD OF A TRANSPLUTONIAN PLANET AS BASED ON THE TITIUS-BODE LAW,” in which it is stated that “a new object should occupy a position at a distance of 67.7 a.u.” and that it will thus have “a period of revolution of 560 years.”

https://articles.adsabs.harvard.edu//full/1973IrAJ...11..138B/0000138.000.html OculusCapricorn (talk) 03:20, 20 January 2023 (UTC)