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Taking a break from the vicissitudes of trying to find evidence of
discrete self-similar order in exoplanet systems, which is a much more difficult
problem than I initially expected, I went to the library and chanced
to find the latest issue of Publications of the Astronomical Society
of the Pacific.

Therein, Fu et al have a paper [PASP, 121, 251-259, March 2009] on the
SX Phoenicis star DY Pegasi, and it struck me that this star might
have the right qualities for a possible test the discrete fractal physics of Discrete Scale Relativity. Below are the results of the test.

DY Pegasi has a mass of about 1.45 solar masses, and a high amplitude
radial oscillation frequency of about 13.713 cycles/day, or about
1.587 x 10^-4 sec^-1.

Discrete Scale Relativity predicts that one can identify a discrete
self-similar analogue on the Atomic Scale and test this by predicting
its oscillation frequency uniquely.

In DSR, 1 stellar mass unit (SMU), a scaled analogue to atomic mass
units (amu), is equal to 0.145 solar masses. Therefore 1.45 solar
masses corresponds to 10 SMU, and DSR predicts that we are dealing
with an analogue to a 10 amu atom, with the most likely candidate
being a Boron atom.

Given the P(n) = nP(0) relation that is known to apply to atoms, DSR
can be invoked to predict that the most likely value for n, given the
oscillation frequency of DY Pegasi and the scaling rules of DSR, is  n
= 4.

Since the oscillation of DY Pegasi is a fundamental radial
oscillation, DSR predicts that l ~ 0.

DSR predicts that if you divide the 13.713 d^-1 frequency of DY Pegasi
by 5.2 x 10^17, in order to scale it to the Atomic Scale, and
carefully convert the frequency into a wavenumber, you get 2752.869
cm^-1 and this should be the corresponding wavenumber of the discrete
self-similar Atomic Scale transition frequency.

So, we have a simple test: if you go to your Atomic Data Tables and
look up Boron will you find a likely self-similar analogue to the DY
Pegasi oscillation frequency? Luckily Boron is a fairly simple atom
with a 1s^2 2s^2 nl spectrum.

(1) Do we find a transition with a wavenumber of about 2752 cm^-1? We
do! We find a transition with a wavenumber of 2776.826 cm^-1, which
agrees at the 99.13% level. Note that ambient EM fields, temperatures,
pressures, etc. can shift the Stellar Scale periods and unfortunately
we cannot take stars into the lab to control these influences.

(2) Is the uniquely identified transition primarily an low l
transition? Yes! It is a p [l=1] to s [l=0] transition.

(3) Is the uniquely indentified transition associated with n = 4? Yes!
The uniquely identified transition is a 1s^2 2s^2 4p to 1s^2 2s^2 4s
transition.

(4) Do any other Boron transitions match our requirements? No! No
other transition provides an acceptable quantitative/qualitative fit.

So I suppose one could just say that by coincidence I just happened to
notice a particular Stellar Scale system and it just happened that by
coincidence it's properties matched up remarkably well with a uniquely
indentified Atomic Scale analogue. But I would suggest that there is a
simpler and more likely explanation, albeit one that implies that
nature has a mighty big surprise in store for us: nature has a discrete fractal organization with discrete Scales [i.e., subhierarchies like the Atomic Scale and the Stellar Scale] that are exactly self-similar to one another, as described by Discrete Scale Relativity.

Yours in science,
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
http://independent.academia.edu/RobertLOldershaw

The result seems antique although it rests on futuristic chaos theory. Gravity is supposed to be only attractive and passing through everything. Everything else acts against gravity. That's a hard thing to process. I think that it would be really cool if you analyzed gravitational resonance and explain geometric progression found in thickness of Saturn rings. It is similar to the Titius-Bode law.

The Solar System has a principal quantum number (n) of about 168 and an angular quantum number (l) of about 160.

Therefore it is a nearly "classical" system, and one that is very highly excited and exceedingly complex. The Titus-Bode Law tells us there are underlying "quantum" relationships involved in the planetary distances and moon distances, but nobody is ready to solve the Schroedinger equation for n=168, l=160 to see how well it predicts the Solar System's structure.

This would be a fascinating project for someone with the talent for mathematical physics that it will require. I am definitely not a candidate for this particular mission.

I am using far simpler systems with n=5 to n=10 to show the discrete self-similarity of variable stars and excited atoms undergoing energy level transitions.

I have now analyzed 3 SX Phoenicis stars, including 2 double-mode pusators that are a nice tool for demonstrating a unique fit between predicted and observed frequency spectra.

If you give me a high-amplitude radial-mode (or l = 0 or 1) variable star whose mass is known to 5% or better, then I can tell you its exact atomic analogue and the initial and final n values. Then I can uniquely and quantitatively predict that the identified atom will have transition frequency/energy [associated with the identified transition] that is related to the stars pulsation frequency by the discrete self-similar scaling of Discrete Scale Relativity.

Here's another simple example - this time with a RR Lyrae variable star.

DX Del has a mass of ~0.6 solar masses. DSR predicts that this is self-similar to 4 atomic mass units, so we are talking about a Helium analogue.

The period of 0.473 days can be used [as above with DY Peg] to determine that this is a n=9 to n=8 transition, and its nearly radial so l~0.

If you scale the transition frequency for a Helium atom undergoing a 9p to 8s (singlet) energy-level transition using Discrete Scale Relativity you get a predicted period of 0.4722 days. Or if you prefer, you can scale "down" the star's period to an Atomic Scale period by the inverse scaling transformation.

Rather good agreement, if I do say so myself. Any one can do this all day long with most of the classical types of variable stars. It is not a fluke.

Variable stars and excited atoms are exact self-similar analogues on neighboring Scales of nature's discrete fractal hierarchy.

But just try to get a physicist to even listen to this idea! Such ideas are not even "on their radar" at all, except as UFOs and quackery. Maybe in a couple of decades mankind will overcome its remarkable and totally unwarranted bias against stellar/atomic analogies.

This is probably a bit more of a response than you had hoped for. Sorry, I got a bit carried away.

Best,
RLO
www.amherst.edu/~rloldershaw

The system ate my message. Now I am sitting here being the lotus flower, Buddha, meditating how good it is to accept that the reply is lost and continue with the topic of Saturn. I am always carried away, I just have these gremlins around bragging about time and money.

Please go to
 http://www.youtube.com/watch?v=_1pdhdMR5p8&feature=PlayList&p=7000E2FA266DB3F7&index=13&playnext=13&playnext_from=PL (http://www.youtube.com/watch?v=_1pdhdMR5p8&feature=PlayList&p=7000E2FA266DB3F7&index=13&playnext=13&playnext_from=PL)

I just found out about this guy on wed 14 oct 2009.
 Enjoy and do not feel hopeless.