The Loop Width Distribution – Are we Hitting Rock Bottom

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Presentation transcript:

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, 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 ?”

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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 (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, 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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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]

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)-2 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

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:

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).

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

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”

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

Automated Tracing with OCCULT-2 Code .

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 .

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 .

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

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

. 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 .

. 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 .

. 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 .

. 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 .

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 0.0375”=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)

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)