I can safely report that that is not true. It is not necessarily bound to happen.

Bahcall revived the field with a series of papers in 1984.

The one which (re-)introduces the Bolzmann equation (and the Poisson equation), and thus the 6D (phase) space ('ordinary' space and velocity space) is The distribution of stars perpendicular to galactic disk:It is also of interest to note that, in a series of papers starting in 1989, Kuijken showed how important a careful accounting of systematic errors was (and still is): when analysed using the combined Boltzmann and Poisson equations (as developed by Bahcall), rather than the more limited Jeans equations, the evidence for dark matter in the local part of the galactic disk disappears* (overturning Oort's conclusions, which had stood for several decades).

Of course, much more, and much better, observational data helped, but the role of proper treatment of systematic errors was the breakthrough.

A curious 'cycle' may thus be observed, in modern astronomy: a similar (as we now see) improper treatment of systematic errors lead to a blind alley in extra-galactic astronomy, one that still features prominently in some popular literature (not to mention on many crank and crackpot websites). Curiouser (to quote Alice) is that Bahcall played an important role in both.

*This is, of course, a simplification; possibly an over-simplification.

Thanks.

So this 'cycles' stuff is, indeed, just a (quite small?) subset of applications of transforms like Fourier transforms.

Image processing and analysis (and deconvolution) includes some pretty heavy duty use of FFTs and similar tools; the estimation of the CMB angular power spectrum is just one recent (and pretty spectacular) example. Another is in 'imaging' starspots (the equivalent of sunspots) on stars whose disks are far, far below the angular resolution limit.

I don't get the bit about 'waves'; a point in the graph of the Fourier transform* of some input is, by definition, the power (in an appropriate sense - 'y-axis value') of the 'wave' of that frequency ('x-axis value'), and so all such 'waves' 'wrap around' the space defined by the inputs.

What am I missing?

*I know, I know; this is too terse.

Fourier transforms are a way of expanding any function in terms of a basis of periodic functions. But one can expand any function in terms of a basis of lots of different kinds of functions, like nonperiodic "wavelets" for example. The reason Fourier transforms are so prevalent is expressly because they bring out the deeper connection physics often has with periodic phenomena.

For example, Fourier transforms have convenient properties like the fact that taking the spatial derivative of a function in "real" space is tantamount to simply dividing each Fourier component by the wavelength of that component. But that convenience stems from the fact that derivatives when applied to periodic functions conveniently map into a known rate of change of phase. The importance of the convenience is that physics often involves derivatives, but that's also the reason it involves cycles-- these are two halves of the same coin. The structure of reality seems to involve derivatives, the simplest derivatives are either a constant rate of change of relative magnitude or a constant rate of change of phase (Gould's two fundamental types of time evolution, or exponentials of real and imaginary parameters), so one of the simplest entities in reality that connect to its fundamental structure are cycles, the other being decay. (The type of transform that connects to the exponential in magnitude type of behavior is called the LaPlace transform, and is kind of the younger brother to the Fourier transform because of the importance of these two types of simple evolution.)

So why then is the CMB analyzed in terms of a spatial Fourier power spectrum? Because fundamentally what they are looking for is something spatially periodic, that connects to what are essentially sound waves in the early universe. If they were looking for stochastic variations instead of periodic ones, I suspect they'd use autocorrelation functions instead of Fourier transforms, but I don't do that stuff so I'm guessing a bit.

Ken G's post just before this is an excellent statement of relationships between waves, derivatives and Fourier analysis. I did refer to some of this in the ATM thread on Harmonics Theory regarding finding a cycle in a redshift histogram. By taking the derivative it removed the selection effects which are low frequency (not many inflexion points) and enhance the periodicity which is a higher frequency. There is a page on my web site that has a worked example of this sort of thing, showing the effects of taking logs and differences on a FFT. These are useful things to know as ways of dealing with various problems when looking for periodicity.

Well I would say that it is not a subset nor a superset, but an overlapping domain. I probably have a wider interest in all the applications of FFT than some cycles researchers. As I mentioned, FFT is just one tool. Ken G also mentions autocorrelation functions. These can also be useful for cycles where there is a clear cycle but not a stable period.

Analysis of sea waves and many other phenomena show that at short range (several cycles away) the autocorrelation is strong and then gets weaker. In the case of sea waves the autocorrelation then does a series of additional increases and decreases in the wave shape. These can be interpreted in multiple ways which are all more or less equivalent.

Firstly, the autocorrelation wave can be seen as coming and going on a longer cycle. Secondly the autocorrelation can be seen as beats (two nearby frequencies interacting - actually three for waves). Thirdly it can be seen as amplitude modulation of a single frequency.
Yes. The great compression of JPG is because periodicity is everywhere. We do not notice it clearly when multiple similar but unrelated periods are present. We do notice it when a single period, related periods or periods differing by large ratios are present. Of course doing a lot of cycle analysis does tune the mind for seeing this better.
Squeezing the most out of data is something that appeals to me a lot. Actually FFT is not as good at MESA at doing this when the time periods are short. But with long time periods and lots of data it is so much faster to run.
I presume that you refer to the diagram at the bottom of my post. If you forget what the meaning of that diagram is to begin with, and simply look at the heights of the peaks, you will see that at about every 12th step in the histogram (ignoring the very left part also for now) there occurs one or more peaks and half way between these there are troughs. This is exactly the same sort of thing that you get when you analyze the waveform of a musical instrument, be it piano, guitar or mouth organ.

The meaning of the repeating waves in the spectrum is that frequencies near multiples of a certain frequency are stronger than others. In other words harmonics of some base frequency is present. The base frequency in this case is 1/(80 minutes). In a musical instrument this behaviour results from the fact that sound is restrained within a particular sized cavity, and so only frequencies that fit a multiple number of times non-destructive.

This suggest a similar restraint in the case of the Sun, that it produces harmonics of 80 minutes. The question is what represents the cavity? Or it may be looked at as a strongly non-linear process with a single driving period of 80 minutes. What is that process?

Now return to the bit on the left of the graph that shows that sub-harmonics of 80 minutes are also present, namely 1/2 and 1/3. These are not generally created as harmonics in systems of any type, but can arise in some extreme non-linear cases.

In almost every solar variable that I analyze (or where I see other people's analysis) I see these periods: 88, 44, 22, 11, 5.5 years; 0.33 years; 155, 78, 52, 26 days; 160 and 80 minutes; 5-6 minutes. Only some of these are adequately understood as natural resonances of the Sun. Even where the natural resonance such as the 5-6 minute ones (and weaker ones further about that vicinity) are understood, existing theory says that they should die out, but they do not. There is a lot more that is to be learned and cycles analysis is a useful tool there.

If you find a common cycle in several variables but the causal connection is unclear you know to look deeper. If the cycle has a slightly different period then in terms of frequency you look at the difference. As an example, if you didn't know what caused tides, and you find that the tides are nearly once and nearly twice per day you look at the difference between the frequencies of the tides (there are multiple) and the frequencies of days, full moons and seasons, you will see the cause as being primarily the moon assisted by the sun. When you look a bit deeper, you will discover additional components such as the elliptical nature of the earth and moon orbits, the inclination of the lunar orbit and the precession of the lunar orbit. All this can be done without telescopes!

If you look at atomic spectra in this way, you find that all spectral lines have fine structure. The beats etc present point to longer period modulations which are taken to be behaviours of the atom. The fine structure has still finer structure which indicates ever longer periods of modulation and explains the connection between things such as the 21 cm H wave as equal to beats between the the hyper-fine structure lines - or the other way around if you want.

These same structures are found in everything! Stock markets show cycles with multiple levels of modulation too. When this is seen clearly then they are not seen as some chaotic thing that is only human in origin, but that human nature is driven by external conditions. Chizhevsky did much to show this with such things as his war index that shows wars are increasingly likely at sunspot maxima, especially on alternate cycles (with the same polarity). Dewey's reanalysis of Chizhevsky's 2000+ years of war data.

Ah, I think I have more or less got it. I was thinking on something related to this recently - the motion of stars as they oscillate above and below the galactic plane. I know that that oscillation is very different to a normal orbit because of the matter distibution, and I was trying to figure the "attraction law" of a plane. It seems it must be more like 1/r rather than 1/r^2, but clearly with no infinity at r=0, is that right? This is of interest in the 62 million year species diversity cycle, which is being attributed (in peer review articles) to the increased radiation on one side of the galactic plane. Additionally the passage through the plane with resulting increased meteorite and comet impacts is another theory, but this is to explain the ~30 MY cycle that is accurately measured at 26.6 MY. So both cannot be right. Another astronomy - paleontology connection using cycles though. Some references below.

But my interest is in calculating stellar orbits to see whether moderately stable grids are possible. They clearly aren't with inverse square with a central mass like the solar system except very special cases like libration points, but I suspect that they are in the galaxy.

I might come and ask more about this some time. From tomorrow I will be away for about 10 days, so will rejoin the discussion after that.

http://www.lbl.gov/Science-Articles/...diversity.html

These following info not linked properly. Springer actually gave them both the same URL, so it is obviously wrong to quote.

Episodes of terrestrial geologic activity during the past 260 million years: A quantitative approach
JournalCelestial Mechanics and Dynamical Astronomy PublisherSpringer Netherlands ISSN0923-2958 (Print) 1572-9478 (Online) IssueVolume 54, Numbers 1-3 / March, 1992 CategoryInvited Papers DOI10.1007/BF00049549 Pages143-159

The “Shiva Hypothesis”: Impacts, mass extinctions, and the galaxy
JournalEarth, Moon, and Planets PublisherSpringer Netherlands ISSN0167-9295 (Print) 1573-0794 (Online) IssueVolume 71, Number 3 / December, 1995 DOI10.1007/BF00117548 Pages441-460

Perhaps you are unaware that correlations with sunspot cycles is an especially unfortunate example to use-- it has essentially become an inside joke whenever someone wants to underscore the difference between correlation and causality.

I'm not sure I'm following you here, KenG.

If I may rephrase, in my own words:

Continuous stuff is either (quasi-)periodic or monotonic.

If the former, then you can nearly always find a way to transform it into something for which you can infer a ~1/r potential. Thus 'cycles' can be said to be 'natural'.

If the latter, Fourier transforms (and so 'cycles') don't tell you much.

For stuff that's not continuous, Fourier transforms (and so 'cycles') give no insight into the underlying physics.

I'm sure I've missed something ... in any case, I can't see how 'cycles' tells you anything other than (maybe) where to look for a 1/r potential. But that surely isn't very helpful ... I mean, what would the derived 1/r potential for Canadian Lynx tell you about population biology?

To give a direct counter-example: you can (if I remember correctly) relate hydrogen-like line spectra to electrons as standing waves. Sure it's cute, but gives you no insight into the underlying physics (quantum mechanics).There's a Q&A thread on this, it contains everything you could possibly want to know ... but mostly in the references, not the thread itself!

I think you'll find the reasons for looking at the CMB angular power spectrum, rather than anything else, are pretty straight-forward - that spectrum provides the cleanest, quickest set of tests of alternative cosmological models (or, if you prefer, a means to estimate the few parameters in those models, with the fewest degeneracies) ... unless you take a very general interpretation of 'spatially periodic'. In fact, causally connected regions of the universe at the time of matter-radiation decoupling (surface of last scattering) are only ~1^o in size, in the CMB we see today (so you wouldn't need WMAP, if all you were interested in was "essentially sound waves in the early universe").

Yes, and another way to look at it is if the phenomena has an effectively infinite (periodic) or finite (monotonic) span, which is often the deciding factor in using Fourier or LaPlace transforms to bring out certain conveniences.More often the potential is ~x^2, like a ball rolling around in the bottom of a rounded bowl. That's typically what happens when you perturb a system a little from a stable configuration.
Potentials are a special case of periodicity, I'm not sure the mathematics of lynx populations supports that kind of description. More fundamentally, I would say that cycles stem from the phase relationship between the thing and it's own rate of change. If the rate of change of something is either in phase or opposite phase with the thing itself, then you have monotonic behavior, whereas if it is 90 degrees out of phase, then you have cyclical behavior. Combinations are of course also possible. So when lynx populations are in exponential growth of rapid decay, we would expect to look for something that is driving the population change in phase with the present population. But when you see cyclic behavior, you expect the first derivative to be 90 degrees out of phase with the population, which is most easily achieved either by having the second derivative of the lynx population be 180 degrees out of phase with that population, or by having the first derivative depend on the value of another population (perhaps a food source) which in turn has a first derivative that depends on the negative of the lynx population (the predator/prey relationship). These are the linear responses-- in reality populations tend to have nonlinear terms so produce stochastic behavior when we leave the linear regimes.

Orthogonal phase relationships are nice ways to yield long-term stability, like an orbit or an atom. Synchronized phase relationships are nice ways to produce growth, like blowing a whistle or jumping on a trampoline. Ultimately, dissipation truncates the stable cycle or saturates the growth-- and then there's the nonlinear behavior to really throw a wrench in the works!
I don't see that, it seems that there is great insight into quantum mechanics obtainable from the standing wave picture. The basic energies in a quantum well, for one thing. Note the shape of the potential is not the key issue there, it's the size of the well, because the phase relation inside the well stems from the potential energy being a positive multiple of the amplitude there, and the kinetic energy has to do with the second spatial derivative of the amplitude, so that's our cyclical situation. What constrains the energy is that the well is bounded, so has regions where the kinetic energy goes negative and that means the second spatial derivative is a positive multiple of the amplitude, so the first spatial derivative is 180 degrees out of phase, and that's decay. So we have a standing wave that must bounce back and forth in the well, and that constrains the possible energies.

WMAP uses a multipole expansion in spherical harmonic basis functions, so it's not really a "Fourier transform" in the formal sense, but they are periodic basis functions and you may be right that they're just looking for any convenient way to characterize the scales at which structure appears. The "first bump" is very much related to the causality horizon, so subsequent bumps are on smaller scales and are sensitive to those "sound waves". The first bump could probably be achieved with autocorrelation functions too, the choice of spherical harmonics might relate to the sound waves, or it might just be a convenient set of basis functions that many people know, I don't know which.

[edited: potential energy was intended in place of kinetic energy.]

I am aware of this. However the fact that people make such jokes is not evidence that such relationships do not happen. The first person recorded as making observations relating to sunspots and events on earth was Sir William Herschel who said that wheat (or corn) prices were related to the Sunspot cycle. I did an analysis of over a century of wheat prices and found a strong cycle of 5.54 years present, exactly half the average sunspot cycle. Chizhevsky's correlation between sunspot peaks and war has continued in the 5 sunspot cycles since then.

In general many systems, be they physical, economic, social, biological etc, have the character that there are restorative forces when an equilibrium is disturbed. If the restorative force is proportional to the disturbance then that is the necessary condition for a simple harmonic oscillator. If it is not, then the period may not be stable but depend on the amplitude of the disturbances. However in many cases there are secondary periodic forcing disturbances that, though weak, cause a cycle to stay periodic or at least return to a common phase even after wandering away.

Examples like the lynx and many other species following the 9.6 year cycle, with no obvious connection between most of them, shows that quite small disturbances of even unknown origin can cause dramatic widespread behaviour. In the lynx case the population variation is by 1000%.

It is evidence of what hideous results occur when people don't understand that correlation does not imply causation.
I rest my case. Make a prediction, test it, suggest a mechanism, and then test that. That's science. In the absence of any plausible mechanism, coincidence, or the kind of "active interpretation" one finds in astrology, is still the likeliest candidate. We know what causes sunspots, and we know what causes war. If we are wrong about either, how do you know the wars aren't causing the sunspots? I'm sorry, the idea that war is causally connected to sunspots is simply laughable.

My point is that it generally is proportional to distance for small perturbations, that's a ubiquitous feature of stable systems that takes effort to avoid.

Yes, the field of "nonlinear dynamics" is very much in its formative stages, and has far more questions than answers, which is why most of physics is linear. But one should be careful to ascertain that a system is expressly nonlinear before expecting sensitivity to small perturbations.

I disagree that the sunspot cycle is fully understood. There are serious unsolved problems in solar cycles as presently being discussed in another thread.

If the causes of war were fully understood, then why are wars still happening?

Here are some things that make a possible connection. Some of these are little known knowledge.

1. Solar activity affects terrestrial e/m activity including the strength of Schumann resonances and related ELF/ULF activity.

2. Human reaction times are affected by the presence of ELF/ULF waves near the Schumann resonance (normal brainwave frequencies) with slightly slower waves causing slower reaction times and slightly faster waves causing faster reaction times.

3. Accidents are more frequent when ~3 Hz waves are stronger.

These results show that solar activity can affect human brains and behaviour and possibly cause entrainment. It is not difficult to understand how aggression could also be affected.