THE TITIUS-BODE LAW REVISITED BUT NOT REVIVED

Unbelievable!

I bet you're glad you didn't publish it, now...

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

NO WAY! Even as a hatchet job, the paper is lame! I particularly like this gem:

Now I'll go out on a limb here, and correct me if I'm wrong, but I'm pretty sure that both (1) and (2) are exponential equations.

Bottom line: you guys can save yourselves the trouble of reading the paper; believe me, it need not be taken seriously.

I think you may have misquoted the formulas, there...

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

Huh?

Formula (2) is missing an n on the right side.

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

The original Titius-Bode law employed a fudge factor for Mercury. 0, 3, 6, 12, 24, 48, 96… is not a proper doubling sequence. The first element should be 1.5.

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Curt Renz - "Centaur"
For monthly astronomical calendar visit:
www.CurtRenz.com/astronomical

Actually, the one planet that screws up the pattern is Earth itself. Take Earth away, and the TBL works great! It's almost as if somebody just inserted the planet into a preformed solar system. . . .

True, but maybe it's a translation issue.

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

Indeed! There could very well be a special distinction in Russian mathematics between equations of the form f(x) = a + bc^x and those having the form f(x) = ab^x

ha ha, here's a better fromula:

Let f(n) = -1049058.013341956+1049057.7556187088e^(.0011543770 084426366n) - 1213.8757604547038n + 3.9440957501468796 sin(0.19758114049325678+0.6014719275636937n)

then, f(n) gives the distance in A.U.s, reasonably accurately, for the nth planet (to include Ceres and Pluto):

1 = Mercury
2 = Venus
3 = Earth
4 = Mars
5 = Ceres
6 = Jupiter
7 = Saturn
8 = Uranus
9 = Neptune
10 = Pluto
11 = Planet x (55 AU)
12 = Planet y (70 AU)
13 = ....

and so on.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Nice formula Todd! It's no wonder you hang your hat in Maryland!


Just one question: what's the R-squared score?

(Don't worry about computing the expected random probability distribution of all R-squared scores--I've already done it!)

I haven't computed it, but it's probably rather terrible, given that this isn't a linear fitting.

Mathematica, incidentally, generated the coefficients (in a few seconds, after I told it what to use the coefficients on--that came from looking at a graph of the difference from one formula and the actual data, and saying things like, hey that looks like a sloppy sine curve, so I'll add a Sin term, etc.). If only Bode had Mathematica....for that matter, Newton and Einstein. Notice there are more digits in the approximating formula than in tha AU's of all the objects listed, even if the latter is taken a little beyond where the formula matches.

It does show that having a close-form formula for something is sometimes overrated (e.g. I once saw a closed-form formula for all prime numbers, but it's faster to run a Sieve of Eratosthenes--in fact, I think the formula encoded the Sieve of Eratosthenes in terms of complicated summations). A better goal is "simple expression for" where expression can be a formula, program, list, or whatever--and in the case of the planets, the simplest expression is probably just to list the A.U.s.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

I computed it the old fashioned way: by transforming the data first, and then trying to fit a linear model on to that. Then one computes the R-squared score.

A debatable procedure, since the quantities that actually interest you are not linearly related...

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

I'll be sure to write the authors of my statistics textbook to let them know...

I said it was a debatable process, not that it was wrong to do it. If it's a good book, they probably say the same.

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

Some statistics books are better than others. Hogg and Tannis is pretty good. The one I had in undergraduate school, I forgot what it was, is also good....for a paperweight.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

Biggest mistake I ever made.

When they say exponential function, I think they explicitly mean functions genre f(x)=a+be^xn.

I think you mean f(x) = a + be^cx. But such formulas can be changed into any base you want. Poveda & Lara use the natural logarithm base e, but most people express the formula as Kⁿ, where, ideally, K would be the ratio of successive orbits--and is thus more physically meaningful than the natural logarithm base.

Thus I think the most important finding, that Poveda and Lara pretty much overlook is that the TB scaling factor K for the Solar System (1.7) is quite different from 55 Cancri's (2.7).

What is the significance of the difference?

The only obvious physical difference between the Solar System and 55 Cnc is the metallicity.