1. Correlations and Scale in Circumstellar Dust Shells
PHYS 4811 – Statistical Mechanics
Spring 2009 Project
Kathy Geise
"Slice of the Universe" by de Lapparent et al 1986 (ApJLett 302, L1, courtesy AAS)

2. 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
Equilibrium correlation function of Ising model;
spins at different space and time

3. 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

4. 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.

5. Scaled image U Hya at 70m imaged by Spitzer space telescope
1 pixel = 4.92”
U Hya at 70m
imaged by Spitzer space telescope

6. Scaled image
1 pixel = 4.92”
U Hya

7. 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.

8. Analysis

9. Analysis

10. 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