Download PHYS 4811 Stat Mech

Correlations
and Scale in
Circumstellar
Dust Shells
PHYS 4811 – Statistical Mechanics
Spring 2009 Project
Kathy Geise
http://www.newscientist.com/article/dn14200-galaxymap-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
r
r
r r
C( r ,  )  S( x,t)S( x  r ,t   )
Equilibrium correlation function of Ising model;
spins at different space and time
Sierpinski triangle & Mandelbrot set
www.mathworld.wolfram.com
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
1
H II regions of the interstellar medium
(2)P  n 212 1 (12)
The probability of finding two galaxies in the solid
angles δΩ1δΩ2 at angular separation θ,
where n is the angular mean galaxy density.

 g ( )  Ag
  1.77  0.04
The angular 2-point function is
well represented by a power law
across many scales.
Scaled image
1 pixel = 4.92”
U Hya at 70m
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
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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