simple nonlinear models suggest variable star universality john f. lindner, wooster college...
TRANSCRIPT
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Simple Nonlinear ModelsSuggest Variable Star
UniversalityJohn F. Lindner, Wooster College
Presented by John G. LearnedUniversity of Hawai’i at Mānoa
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Collaboration
John F. LindnerThe College of Wooster
Vivek KoharNorth Carolina State University
Behnam KiaNorth Carolina State University
Michael HippkeInstitute for Data AnalysisGermany
John G. LearnedUniversity of Hawai’iat Mānoa
William L. DittoNorth Carolina State University
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Multi-Frequency Stars
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P. Moskalik, “Multi-mode oscillations in classical cepheids and RR Lyrae-type stars”,
Proceedings of the International Astronomical Union 9 (S301) 249 (2013).
Petersen Diagram
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Petersen Diagram Rescaled
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Spectral Distribution
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Strobe signal at primary period and plot its values versus time
modulo secondary periodto form the Poincaré section
If the function
represents the section,is it smooth?
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Expand function & its derivativesin Fourier series
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For smooth sections, expect Fourier coefficients to decay exponentially, so all the derivatives also decay
For nonsmooth sections, expect Fourier coefficients to decay slower, as a power law, so that some derivatives diverge
Invert to get
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Since an averaged spectrum decreases with mode number or frequency
reinterpret
to be the number of super threshold spectral peaks
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So rich, rough spectrahave power law spectral distributions
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Stellar Analysis
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KIC 5520878 Normalized Flux Sample
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Lomb-Scargle Periodogram
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Spectral Distribution
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Gutenberg-Richter law forvolcanic Canary Islands earthquake distribution
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Some Number Theory
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Golden Ratio
Slow convergence suggests maximally irrational
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Liouville Number
Example of nearly rational irrational
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Model 1:Finite Spring
Network
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Natural Frequencies
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Model 2:Hierarchical
Spring Network
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Natural Frequencies
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Model 3:Asymmetric Quartic
Oscillator
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Asymmetric Quartic Potential Energy
Sinusoidal Forcing
Drive Frequency a Golden Ratio Above Natural Frequency
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Model 4:Pressure vs. Gravity
Oscillator
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Adiabatic Simplification
Pressure & Volume
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Potential Energy
Force
Sinusoidal Forcing
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Model
Data
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Model 5:Autonomous Flow
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generalized Lorenz convection flow caused bythermal & gravity gradients plus vibration
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Adjust parameters so that
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Model 6:Twist Map
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Twist map is circles for vanishing push
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Push perturbation has vanishing mean
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Least resonant golden shift remains
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Insights from Helioseismology
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Helioseismology and asteroseismology have observed manyseismic spectral peaks in the sun and other nonvariable stars,
which correspond to thousands of normal modes
Yet, despite preliminary analysis, we have not discoveredpower law scaling in the solar oscillation spectrum
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Stochasticity and turbulence dominate the pressure waves inthe sun that produce its standing wave normal modes
In contrast, a varying opacity feedback mechanism inside avariable star creates its regular pulsations
In golden stars, interactions with this pulsating mode maydissipate all other modes except those a golden ratio away
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Discussion
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The simple nonlinear models suggest the importance of considering simple explanations
0: The golden ratio itself has unique and remarkableproperties; as the irrational number least well approximated by
rational numbers, it is the least “resonant” number
1: A finite network model of identical springs and masses hastwo normal modes whose frequency ratio is golden
2: An infinite network hierarchy can be mass terminated in twoways to naturally generate two modes whose frequency ratio is
golden, while a realistic truncation of the model generates aratio near golden, as observed in the golden stars
Summary
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3: A simple asymmetric nonlinear oscillator produces a richspectrum with a power-law spectral distribution
4: A more realistic oscillator model of pressure counteringgravity exhibits a recognizable but stylized golden
star attractor
5: An unforced Lorenz-like convection flow also produces asingular spectrum with a power-law spectral distribution,provided its parameters are tuned so that a golden ratio
characterizes its orbit
6: An ensemble of twist maps naturally evolve to a goldenstate, because golden shifts are least resonant with any
oscillatory perturbation
Summary
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The Feigenbaum constant delta ~4.67,which characterizes the period doubling route to chaos,
has been observed in many diverse experiments
Does the golden ratio ~1.62,or equivalently the inverse golden ratio ~0.62,
play a similar role?
Or does the mysterious factor of ~0.62, which characterizes many multifrequency stars,
merely result from nonradial stellar oscillation modes?
Simplicity vs. Complexity
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Some natural dynamical patterns result from universal features common to even simple models
Other patterns are peculiar to particular physical details
Is the frequency distribution of variable stars universal or particular?
Universality vs. Particularity
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Thanks for Listening