1. The Loop Width Distribution – Are we Hitting Rock Bottom
Markus J. Aschwanden
Solar & Astrophysics Laboratory,
Lockheed Martin Advanced Technology Center
Hardi Peter
Max Planck Institute for Solar System Research
Monreale Loops
Fabio Reale Loops
8th Coronal Loops Workshop, Palermo Italy, 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 ?”

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6. Previous Loop width measurements
Loop widths measured in optical, Ha, Lya, EUV, SXR, radio
From w=20 Mm (radio wavelengths)
down to w=70 km (0.1 arcsec pixel size, Hi-C)
52+ publications ( ) 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, IRIS
Most finest loop width measurements are 2-4 pixel sizes
Most loop width measurements in EUV (171, 195 A)
Higher instrumental resolution revealed progressively
smaller loop widths.
QUESTIONS:
Do we find finer loop widths with higher resolution ?
Is there a fundamental physical limit ?

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10. 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 is
naturally 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 w
decreases by a power law index of a=-2
Note that a power law distribution function is
only defined in a scale-free range [wmin < w < wmax]

11. Thresholded Power-Law Distribution, Pareto [type II] or Lomax Distribution
A lower threshold of the power-law distribution
is 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) w < wmax
A threshold at wmin produces a turn-over of the
power-law distribution to a const Nmax at the lower end
Nmax = wmin(-2) = const for w < wmin

12. Loop Width Broadening Effects
A pixelized image has a spatial resolution of
wmin ~ 2.5 pixel sizes
The observed loop width w is broadened by
the pixel size and loop background noise.
The effective loop width can be added in quadrature:

13. 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 wp to the absolute minimum wmin
and has 3 free parameters (wp, wmin, a) and a
normalization constant (n0).

14. Power-Law Size Distribution with a Smooth Cutoff Fits the Observed Distributions (AIA, Hi-C):
Simulation:
AIA:
Hi-C:
Hi-C rescaled to AIA:

15. 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”

16. Monte-Carlo simulations of circular loops
with power-law distributions,
Automated Tracing with OCCULT-2 code .

17. Automated
Tracing with
OCCULT-2 Code .

18. Size Distributions for pixel sizes of dx=0.1”, 0.2”,…,2.0” Normalized:
Unresolved loops
have a peak at
wp=2.5 pixels,
resolved loops
at wp > 2.5 pixels .

19. Size Distributions for pixel sizes of dx=0.1”, 0.2”,…,2.0” Normalized:
Unresolved loops
have a peak at
wp=2.5 pixels,
resolved loops
at wp > 2.5 pixels .

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

21. . Filter: Nsm1=16 Nsm2=18 W=1200 km fOriginal dx=0.1” Filter: Nsm1=1

22. . Hi-C data do not show significant structures
at the highest resolution
The size distribution
shows a peak at wp/wmin=7.1,
which indicates fully resolved
structures .

23. . AIA data do show significant structures at the highest resolution
The size distribution
shows a peak at wp/wmin=3.1,
which indicates marginally resolved
structures .

24. . AIA shows a most frequent width at wp=1258 km,
similar to Hi-C scaled to the
same resolution
The most frequent loop
widths are seen at >2.5 pixels
when unresolved
The Hi-C data show a most
frequent width at w=514 km,
and similarly do the AIA data
when scaled to the same
resolution of 0.1”
 The most frequent loop
widths are seen at 500 km
when fully resolved .

25. . Hi-C shows a most frequent width at wp ~ 500 km,
which is fully resolved
since wp >> 2.5 pixels (170 km)
The AIA data show a most
frequent width at w ~ 500 km,
which is marginally resolved,
since wp ~ 3.0 pixels .

26. 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
(Abramenko et al. 2012, ApJL 756: L27)
Diffraction limit of ”=77 km
Two population of granules: regular granules and mini-granules
Mini-granular structure dominant at scales 100 km < L < 600 km
Power-law distribution with Kolmogorov spectrum -5/3 (obs: -1.8)
Regular granulation has Gaussian distribution with mean 1050 km
Mini-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)

27. Conclusions :
1) Largest statistics of loop width measurements (N~105) 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 distribution
3) 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 ~ 2.5
4) Monte-Carlo simulations provide diagnostic on resolvability of finest loop/strand structures: Unresolved strands have wp/wpixel~2.5, while resolved strands have wp/wpixel >> 2.5.
5) Hi-C (with a pixel size of 70 km) fully resolves loops, finds a most frequent value at ~500 km
6) 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)