Correlations and Scale in Circumstellar Dust Shells PHYS 4811 – Statistical Mechanics Spring 2009 Project Kathy Geise http://www.newscientist.com/article/dn14200-galaxy-map-hints-at-fractal-universe.html?full=true "Slice of the Universe" by de Lapparent et al 1986 (ApJLett 302, L1, courtesy AAS)
Correlation functions Correlation functions describe how systems evolve in space and time (Sethna pg. 216) (eg: relaxation time tr = 4.5 (T-Tc)-1.85) Example: Ising model Equal-time correlation function C(r, 0) Above Tc order is short range; correlation function goes to zero at long range Below Tc order is long range; at long distances, the correlation function goes to m2 (net magnetization, m, per spin) At Tc the correlation function decays as a power law (Equal-position) spin-spin correlation function C(0, τ) Goes to zero for T > Tc Goes to m2 for T < Tc Decays as a power law for T = Tc Sierpinski triangle & Mandelbrot set www.mathworld.wolfram.com Equilibrium correlation function of Ising model; spins at different space and time
Scale invariance Similar structures at different scales Fractal-looking, rough, with irregularities Scale invariant systems are symmetrical under changes of length scale Coarse-graining Characterized by scaling functions multiplied by power laws Example: Critical point of the Ising model; The Ising model at Tc is statistically self-similar Fluctuations occur at all length scales Phase transitions The Ising model near critical point on four different length scales www.ibiblio.org/e-notes/Perc/ising.html
Correlations and Scale in Astronomy Cosmic microwave background radiation (WMAP) Scale invariant initial velocity flelds (Sethna pg. 232) Galaxy distribution and Large Scale Structure (LSS) of the Universe Angular 2-point correlation function (Peebles) Spatial 2-point correlation function using redshifted data Higher order correlations Cluster correlations Fourier space analysis of the power law spectrum Turbulence in molecular clouds H II regions of the interstellar medium The probability of finding two galaxies in the solid angles δΩ1δΩ2 at angular separation θ, where n is the angular mean galaxy density. The angular 2-point function is well represented by a power law across many scales.
Scaled image U Hya at 70m imaged by Spitzer space telescope 1 pixel = 4.92” U Hya at 70m imaged by Spitzer space telescope
Scaled image 1 pixel = 4.92” U Hya
Analysis A scaled image of U Hya with the inner and outer ring indicated in green. The star center was estimated using sky2xy coordinates for the larger (212 x 294 pixel) image. This image is 106 x 147 pixels.
Analysis
Analysis
Next Steps Brightness is an estimate of density of dust at this temperature (40K) Radial distribution analysis in one dimension around the ring Density fluctuations are approximated by a random Gaussian process