The Loop Width Distribution - PowerPoint Presentation

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The Loop Width Distribution – Are we Hitting Rock Bottom

Markus J. AschwandenSolar & Astrophysics Laboratory, Lockheed Martin Advanced Technology Center

8th Coronal Loops Workshop, Palermo Italy, 27-30 June 2017

Reference: Aschwanden M.J. and Hardi P. 2017, ApJ 840:4, 24pp “The Width Distribution of Loops and Strands in the Solar Corona – Are We Hitting Rock Bottom ?”

Monreale

Loops

Fabio

Reale Loops

Hardi Peter

Max Planck Institute for Solar System ResearchSlide2

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Previous Loop width measurements

Loop widths measured in optical, Ha, Lya, EUV, SXR, radioFrom w=20 Mm (radio wavelengths)down to w=70 km (0.1 arcsec

pixel size, Hi-C)52+ publications (1963-2017) with the instruments:Dunn/Sac Peak, Pic-du-Midi, Skylab, NRAO, VLA, CSIRO,Rockets, SXT/

Yohkoh, EIT/SOHO, TRACE, EIT/STEREO,VAULT, AIA/SDO, EIS/Hinode, CRIPS, Hi-C, IRISMost finest loop width measurements are 2-4 pixel sizesMost loop width measurements in EUV (171, 195 A)Higher instrumental resolution revealed progressivelysmaller loop widths.QUESTIONS:

Do we find finer loop widths with higher resolution ?Is there a fundamental physical limit ?Slide7

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Scale-Free Probability Conjecture

Nonlinear dissipative processes (avalanches) produce scale-free size distributions(i.e. no special size is preferred within a range).The scale-free range of a size distribution isnaturally be described by a power-law distribution, because the number of sizes N decreases by a constant factor with increasing sizes (w): w=1  N=1

w=1/2  N=4 w=1/4  N=16 w

=1/8  N=64 …. N(w) = w -2 For areas, the number of areas with length scale wdecreases by a power law index of a=-2Note that a power law distribution function iso

nly defined in a scale-free range [wmin < w < w

max]Slide11

Thresholded Power-Law Distribution,Pareto [type II] or Lomax Distribution

A lower threshold of the power-law distributionis produced by: - a physical threshold of an instability - incomplete sampling of smallest events - Contamination by an event-unrelated background

An upper cutoff is produced by - truncation effects at the largest events (due to finite system size)

N(w) = (w+wmin)-2 w < wmaxA threshold at

wmin produces a turn-over of the

power-law distribution to a const N

max at the lower end

Nmax = wmin

(-2) = const for w < wmin

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Loop Width Broadening Effects

A pixelized image has a spatial resolution of wmin ~ 2.5 pixel sizes

The observed loop width w is broadened bythe pixel size and loop background noise.The effective loop width can be added in quadrature:Slide13

Power-law with a Smooth Cutoff

Data noise, the instrumental point-spread function, and uncertainties in the background subtraction of loop profiles will smear out the theoretically predicted sharp peak of the size distribution at the lower end. A smooth power-law size distribution can be defined by a singularity at w/wmin=1

The size distribution has a smooth rollover from the

most likely value w

p to the absolute minimum wminand has 3 free parameters (

wp, wmin, a) and a

normalization constant (n0).Slide14

Power-Law Size Distribution with a Smooth CutoffFits the Observed

Distributions (AIA, Hi-C):

Simulation:

Hi-C:

AIA:

Hi-C rescaled to AIA:Slide15

Loop Width Measurements with OCCULT-2 Code:

AIA 171 A

OCCULT-2 code (Aschwanden, De

Pontieu, & Katrukha 2013, Entropy 15, 3007)”Oriented Coronal Curvature Loop Tracing”Slide16

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Monte-Carlo simulations of circular loops

with power-law distributions,Automated Tracing with OCCULT-2 codeSlide17

.Automated Tracing with OCCULT-2 CodeSlide18

.Size Distributions for pixel sizes ofdx=0.1”, 0.2”,…,2.0”

Normalized:Unresolved loops

have a peak at wp=2.5 pixels,resolved loopsat w

p > 2.5 pixelsSlide19

.Size Distributions for pixel sizes ofdx=0.1”, 0.2”,…,2.0”

Normalized:Unresolved loops

have a peak at wp=2.5 pixels,resolved loopsat w

p > 2.5 pixelsSlide20

.Hi-C, 2012 July 11, 18:54:16 UTSlide21

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fOriginal

dx=0.1”Filter:Nsm1=1Nsm2=3W=140 kmFilter:Nsm1=16Nsm2=18

W=1200 km

Filter:Nsm1=32Nsm2=342500 kmSlide22

.Hi-C data do not show

significant structures at the highest resolutionThe size distribution

shows a peak at wp/wmin=7.1,which indicates fully resolvedstructuresSlide23

.AIA data do showsignificant structures at the highest resolutionThe size distributions

hows a peak at wp/wmin

=3.1,which indicates marginally resolvedstructuresSlide24

.AIA shows a most frequentwidth at wp=1258 km,similar to Hi-C scaled to the

same resolutionThe most frequent loop

widths are seen at >2.5 pixelswhen unresolvedThe Hi-C data show a mostfrequent width at w=514 km,and similarly do the AIA datawhen scaled to the same

resolution of 0.1” The most frequent loop widths are seen at 500 km

when fully resolvedSlide25

.Hi-C shows a most frequentwidth at wp ~ 500 km,w

hich is fully resolvedsince wp >> 2.5 pixels (170 km)

The AIA data show a mostfrequent width at w ~ 500 km,which is marginally resolved,since wp ~ 3.0 pixels Slide26

Theoretical Interpretation :What sets the preferred (most frequently observed) value of w~500 km ? (granulation pattern of magneto-convection cells?)

Detection of small-scale granular structures in Quiet Sun with NST/Big Bear

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Abramenko et al. 2012, ApJL 756: L27)Diffraction limit of 0.0375”=77 kmTwo population of granules: regular granules and mini-granules

Mini-granular structure dominant at scales 100 km < L < 600 kmPower-law distribution with Kolmogorov spectrum -5/3 (obs

: -1.8)Regular granulation has Gaussian distribution with mean 1050 kmMini-granular structures are fragments of regular granules,

Subject to highly turbulent plasma flows in the intergranular lanes,Where the intensity of turbulence is enhanced (Nordlund

et al. 2009)Slide27

Conclusions :1) Largest statistics of loop width measurements (N~10

5) in Hi-C images with automated loop tracing code (OCCULT-2).2) Principal-Component Analysis produces differential occurrence rate size distribution of loop widths  power-law distribution3) Loop width distribution N(w) can be characterized by a thresholded power-law distribution (Pareto type II, Lomax distribution) with peak width

wp at

wp/wpixel ~ 2.54) Monte-Carlo simulations provide diagnostic on resolvability of finest loop/strand structures: Unresolved strands have wp/wpixel~2.5, while resolved strands have

wp/w

pixel >> 2.5.5) Hi-C (with a pixel size of 70 km) fully resolves loops, finds a most frequent value at ~500 km6) AIA/SDO (with a pixel size of 435 km) marginally resolves finest loops at ~500 km

7) Loop width ranges in agreement with Brooks et al. (2013), with 91 loops with a low cutoff at wmin

~ 200 km and a peak at wp ~ 640 km.8) Theoretical prediction: Instruments with higher spatial resolution will not show finer strands  no unresolved

nanoflare strands !!!9) Theoretical consequence: What sets the preferred (most frequently) value of w~500 km ? (magneto-convection or field line braiding at the granulation scale)