The Width Distribution of Loops and Strands in the Solar Corona—Are We Hitting Rock Bottom?

As mentioned in the introduction, the statistics of nonlinear energy dissipation events often exhibit a power-law-like function in their size distribution. The most general testable prediction that can be made for scale-free nonlinear energy dissipation processes is the so-called scale-free probability conjecture (Aschwanden 2012), which universally applies to SOC systems (Bak et al. 1987) and predicts a power-law distribution N(L) for length scales L,

$$\begin{eqnarray}&&N(L)\ {dL}\propto {L}^{-D}\ {dL},\end{eqnarray} \tag{ 1 }$$

where D is the Euclidean dimension of the system. If we use the widths w of coronal loops as the length scale of the cross-sectional area over which an "avalanching" mechanism dissipates energy and heats a particular coronal loop, the Euclidean dimension of cross-sectional areas is D = 2, and thus a power-law distribution of $N(w)\propto {w}^{-2}$ is expected. The diagram shown in Figure 1 depicts three sets of loops (with widths of w = 1, 1/2, and 1/4), which are packed into 2D areas with identical sizes (L = 1), yielding bundles of n = 1, 4, and 16 strands, as expected from the size distribution specified in Equation (1). Even when only a fraction of the packing area is used (e.g., q = 0.25 as illustrated by the dark-gray loop areas in Figure 1), the relative probability or size distribution follows the same statistical power-law relationship as specified in Equation (1).

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The scale-free range of a size distribution (for instance of widths x) exhibits an ideal power-law function for the differential occurrence frequency distribution, which is characterized by a power-law slope a,

$$\begin{eqnarray}&&N(x)\ {dx}\propto {x}^{-a}\ {dx}.\end{eqnarray} \tag{ 2 }$$

However, there are at least four natural effects that produce a deviation from an ideal power-law size distribution, which includes (1) a physical threshold of an instability, (2) incomplete sampling of the smallest events below a theshold, (3) contamination by an event-unrelated background, and (4) truncation effects at the largest events due to a finite system size. These can all be modeled with a so-called "thresholded power-law" distribution function, also called a Pareto [type II] or Lomax distribution (Lomax 1954),

$$\begin{eqnarray}&&N{(x){dx}\propto (x+{x}_{0})}^{-a}\ {dx},\end{eqnarray} \tag{ 3 }$$

which was found to fit the flux distributions of solar and stellar flare data well, in the scale-free range $x\gtrsim {x}_{0}$ (Aschwanden 2015). The additive constant x₀ turns a power-law function into a constant for small values $x\ll {x}_{0}$ at the left side of a size distribution (Figure 2(a); dashed curve), a feature that fits some data, but not all. Examples for different phenomena are shown in Figure 8 of Aschwanden (2015), based on data presented in Clauset et al. (2009). The functional shapes of the size distributions are shown in Figure 2, which shows the ideal power-law slope at $x\gg {x}_{0}$ (Equation (2); thin solid line in Figure 2(a)), and the thresholded power-law function (Equation (3); dashed line in Figure 2(a)), where the threshold is set at ${x}_{0}/{w}_{\min }=2.5$, all shown on a log–log scale in Figure 2.

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Since we are going to analyze the contributions of the thinnest loop strands to the size distribution of loop widths, we have to include a number of effects that set a lower limit on the measurement of loop widths, such as the finite instrumental spatial resolution and loop background noise, which we discuss in more detail with a quantitative example in Section 2.5. At this point we combine these effects into the variable w_min, which represents an absolute lower limit in the theoretical model distribution of loop widths w. The effectively detected loop width (or observed loop width) w can be defined in terms of adding the true loop width w_true and the combined broadening effects expressed in w_min in quadrature,

$$\begin{eqnarray}&&w=\sqrt{{w}_{\mathrm{true}}^{2}+{w}_{\min }^{2}},\end{eqnarray} \tag{ 4 }$$

so that the true width w_true can be obtained from the observables w and w_min,

$$\begin{eqnarray}&&{w}_{\mathrm{true}}=\sqrt{{w}^{2}-{w}_{\min }^{2}}.\end{eqnarray} \tag{ 5 }$$

Substituting the variable x(w) into the thresholded distribution function $N(w){dw}=N(x[w])$ $| {dx}/{dw}| \ {dw}$ (Equation (3)), where we denote $x={w}_{\mathrm{true}}$ (in Equation (5)) and ${w}_{\min }={x}_{0}$ (in Equation (3)), and inserting its derivative, ${dx}/{dw}\,=w/\sqrt{{w}^{2}-{w}_{\min }^{2}}$, we obtain the theoretical size distribution N(w) of apparent (or modeled observed) loop widths w,

$$\begin{eqnarray}&&N(w)\ {dw}\\ &&\quad \propto \,\left\{\begin{array}{l}0\\ \mathrm{for}\ w\leqslant {w}_{\min }\\ w\ {({w}^{2}-{w}_{\min }^{2})}^{-1/2}{(\sqrt{({w}^{2}-{w}_{\min }^{2})}+{w}_{\min })}^{-a}\ {dw}\\ \mathrm{for}\ w\gt {w}_{\min }\end{array}\right.,\end{eqnarray} \tag{ 6 }$$

which is shown in Figure 2(a) (dashed–dotted line) as a function of the normalized width $w/{w}_{\min }$. Thus, the spatial resolution causes an upturn with an excess above the thresholded power-law distribution right near $w/{w}_{\min }\gtrsim 1$, while no events are detected below the limit $w/{w}_{\min }=1$. However, the sharp peak at $w/{w}_{\min }=1$ may not be observable, because the loop width measurements inherently have some uncertainties that smooth out such sharp peaks in the observed size distributions and show the most frequent detections at a peak value of ${w}_{p}/{w}_{\min }\approx 2.5$, rather than at the absolute limit $w/{w}_{\min }=1$ (see numerical simulations in Section 4 and point-spread functions of extreme ultraviolet (EUV) imaging instruments, compiled in Table 3 and references therein).

In our data analysis, the width w of a loop cross-section is measured from the equivalent width of a Gaussian-like loop flux profile, where a local background is subtracted with a low-pass filter and the high-frequency data noise is filtered out with a high-pass filter. This method, as well as many others, produces uncertainties in the loop width measurement of order of one pixel for the thinnest loop strands, mainly because the true background flux profile is unknown and perturbed by data noise. As a consequence, the size distribution of the loop widths exhibits a smooth drop-off toward the minimum value at $w\gtrsim {w}_{\min }$.

In order to provide a realistic analytical model of a size distribution that can reproduce such an absolute cutoff value, we have to replace the threshold parameter ${w}_{\min }$ with a reciprocal function $1/(w-{w}_{\min })$ that has a singularity at $w={w}_{\min }$,

$$\begin{eqnarray}&&N(w)\ {dw}={n}_{0}{\left(w+\displaystyle \frac{{w}_{0}^{2}}{(w-{w}_{\min })}\right)}^{-a}\ {dw},\end{eqnarray} \tag{ 7 }$$

where we introduced an arbitrary constant w₀, to be determined. This type of distribution contains the desired singularity at $w={w}_{\min ,}$ and setting it to zero for smaller values $w\lt {w}_{\min }$ enforces an absolute lower cutoff at $w={w}_{\min .}$

We explore the maximum of this distribution function by setting the derivative to zero, i.e., ${dN}/{dw}{| }_{w={w}_{p}}=0$, and find that the peak value w_p of the distribution function (Equation (7)) is related to the constant w₀ by

$$\begin{eqnarray}&&{w}_{0}=({w}_{p}-{w}_{\min }),\end{eqnarray} \tag{ 8 }$$

while the maximum value N_p of the distribution function at w_p amounts to

$$\begin{eqnarray}&&{N}_{p}=N{(w={w}_{p})={n}_{0}(2{w}_{p}-{w}_{\min })}^{-a}.\end{eqnarray} \tag{ 9 }$$

Inserting the constant w₀ (Equation (8)) into the distribution function N(w) (Equation (7)) yields then

$$\begin{eqnarray}N(w)\ {dw}=\left\{\begin{array}{ll}0 & \mathrm{for}\ w\leqslant {w}_{\min }\\ {n}_{0}{\left(w+\displaystyle \frac{{({w}_{p}-{w}_{\min })}^{2}}{(w-{w}_{\min })}\right)}^{-a}\ {dw} & \mathrm{for}\ w\gt {w}_{\min }\end{array}\right..\end{eqnarray} \tag{ 10 }$$

This definition of a size distribution has the following properties, which are illustrated in Figure 2: (1) the distribution has two regimes, a scale-free range at large values $({w}_{p}\lt w\lt {w}_{\max })$ that follows a power-law function and a smooth cutoff range at small values $({w}_{\min }\lt w\lt {w}_{p});$ (2) a peak of the distribution ${N}_{p}=n(w={w}_{p})$ at the value $w={w}_{p};$ (3) the size distribution is completely defined in the entire range of $({w}_{\min }\lt w\lt {w}_{\max })$ with an absolute lower cutoff at $w\approx {w}_{\min };$ and (4) the distribution function can be represented with four parameters (${n}_{0},{w}_{0},{w}_{\min },a)$, or more conveniently expressed in terms of the peak parameters, $({N}_{p},{w}_{p},{w}_{\min },a)$, with ${N}_{p}={n}_{0}{(2{w}_{p}-{w}_{\min })}^{-a}$ and ${w}_{p}={w}_{\min }+{w}_{0}$. Since the parameter w_min is a constant for a given instrument, only the three parameters $({n}_{0},a,{w}_{p})$ need to be optimized in a fitting procedure for each data set. A family of such size distributions with various peak values w_p is depicted in Figure 2(b). We will see that this type of power-law distribution with a smooth cutoff range $[{w}_{\min },{w}_{p}]$ suits the observed size distributions much better (Figures 6(h), 10(h), 14(h)) than the thresholded power-law function with the threshold constant ${x}_{0}={w}_{\min }$ (Equation (3)), or the sharply peaked power-law function with a lower cutoff at the spatial resolution w_min (Equation (6)).

Let us discuss the physical meaning of the minimum loop width w_min that we introduced in the previously defined theoretical distributions. The major contribution to the minimum detected loop width is the instrumental point-spread function w_psf. Typically, the instrumental point-spread function of EUV or SXR telescopes amounts to ${w}_{\mathrm{psf}}\approx 2.0\mbox{--}2.5\,\mathrm{pixels}$ of the CCD detectors, mostly dictated by the electronic charge spreading. For instance, the azimuthally averaged point-spread function of the Transition Region and Coronal Explorer (TRACE) has been determined with a blind iterative deconvolution technique to be $\approx 2.5\,\mathrm{pixels}$, or 125 (Golub et al. 1999).

In addition, the measured loop widths are affected by data noise in the subtracted background counts in every pixel, which is most severe for faint loops on top of a strongly varying background, especially since we apply a high-pass filtering technique (rather than a Gaussian fitting of loop cross-section profiles). From numerous tests, we evaluated that this loop width broadening component has a standard deviation of ${w}_{\mathrm{noise}}\approx 1$ pixel. A more detailed discussion of this quantity including Quiet-Sun background counts, Poissonian photon noise, readout noise, digitization noise, lossless compression, pedestal dark current, integer subtraction, and spike residuals is given for the TRACE instrument in Aschwanden et al. (2000c).

Combining these loop width broadening effects by summing in quadrature, we specify an observed loop width w as

$$\begin{eqnarray}&&{w}^{2}={w}_{\mathrm{true}}^{2}+{w}_{\mathrm{psf}}^{2}+{w}_{\mathrm{noise}}^{2},\end{eqnarray} \tag{ 11 }$$

where w_true represents the unknown true width for a given loop segment. A quantitative example is illustrated in Figure 4(e), where a mean FWHM width of $w=2.9\pm 1.1\,\mathrm{pixels}$ is measured from 75 loop cross-sections. If we estimate a point-spread function width of ${w}_{\mathrm{psf}}=2.5\,\mathrm{pixels}$ and a data noise component of ${w}_{\mathrm{noise}}\approx 1.0$ pixel, we obtain with Equation (11) a true loop width of ${w}_{\mathrm{true}}={[{w}^{2}-{w}_{\mathrm{psf}}^{2}-{w}_{\mathrm{noise}}^{2}]}^{1/2}$ $\approx \,{[{2.9}^{2}\mbox{--}{2.5}^{2}\mbox{--}{1.0}^{2}]}^{1/2}$ $\approx \,1.1\,\mathrm{pixels}$. Thus, the expected minimum observed loop width for unresolved loop strands (with ${w}_{\mathrm{true}}\ll 1$ pixel) is ${w}_{\min }\,={[{w}_{\mathrm{psf}}^{2}+{w}_{\mathrm{noise}}^{2}]}^{1/2}\approx 2.7\,\mathrm{pixels}$.

In summary, the following symbols are used for loop widths: w is generally used for the observed (or modeled observed) loop width, w_true is the true (fully resolved) loop width, w_min is the minimum width in the theoretical model distribution, w_sim is the simulated loop width, w_psf is the point-spread function, and w_noise is the broadening due to background subtraction data noise. In the numerical simulations, an instrumental point-spread function with a width of ${w}_{\mathrm{psf}}=2.5\,\mathrm{pixels}$ is applied.

The automated detection of features with a large range of spatial scales is not trivial, especially when small and large structures coexist and blend into each other or overlap with each other, as is the case for a solar corona image with many different loops along any given line of sight.

We make use of an already existing automated loop detection code, the so-called Oriented Coronal CUrved Loop Tracing code OCCULT-2 (Aschwanden et al. 2008c, 2013a; Aschwanden 2010), which is customized for automated detection of curvilinear coronal (or chromospheric) loops (or fibrils) with relatively large curvature radii, compared with their cross-sectional width. The numerical code, written in the Interactive Data Language, is publicly available in the SolarSoftware library, with a tutorial given at http://www.lmsal.com/~aschwand/software/tracing/tracing_tutorial1.html. The OCCULT-2 code can be applied to any 2D image with an intensity (or flux) distribution $F(x,y)$, from which the Cartesian coordinates $[x(s),y(s)]$ of a set of 2D curvilinear structures as a function of their length coordinate s are determined. Applications range from the solar EUV images of TRACE and AIA/SDO, Hα images of the Swedish Solar Telescope, to microscopic images of microtubule filaments in live cells in biophysics (Aschwanden et al. 2013a). We provide in the following a brief description of the OCCULT-2 code.

The original image is first filtered with a high-pass filter (with a smoothing 2D box car with a length of n_sm1 pixels) and with a low-pass filter (with a smoothing 2D box car with a length of n_sm2 pixels), which together act as a high-pass filter for curvilinear structures with a (transverse) length scale range of $[{n}_{{sm}1},{n}_{{sm}2}]$, where the bipass-filtered image is defined as

$$\begin{eqnarray}{F}_{\mathrm{bipass}}(x,y) & = & \mathrm{smooth}[F(x,y),{n}_{{sm}1}]\\ & & -\mathrm{smooth}[F(x,y),{n}_{{sm}2}].\end{eqnarray} \tag{ 12 }$$

The high-pass filtering of an image is also known as unsharp masking.

The numerical algorithm of automated curvilinear pattern detection works as follows. First, the image location $[{x}_{1},{y}_{1}]$ of the maximum flux ${F}_{\max }({x}_{1},{y}_{1})=\max [({F}_{\mathrm{bipass}}(x,y)]$ is detected, from where the search of the first structure (with the highest contrast) starts, by detecting the directional angle $\alpha ([{x}_{1},{y}_{1}])$ of the local ridge (Figure 3),

$$\begin{eqnarray}&&\alpha ([{x}_{1}(s),{y}_{1}(s)]=\arctan \left(\displaystyle \frac{{dy}/{ds}}{{dx}/{ds}}\right).\end{eqnarray} \tag{ 13 }$$

In addition, the curvature radius ${r}_{\mathrm{curv}}({x}_{0},{y}_{0})$ of the local ridge is also determined in order to obtain a second-order polynomial approximation of the traced loop structure. A geometric diagram of the parameterized first- and second-order elements measured in a single point of a curvilinear ridge is shown in Figure 3. Then, the same direction is extrapolated in second order as an approximate prediction for the continuation of the loop direction. The exact direction of the traced loop segment at the next loop coordinate ${s}_{i+1}={s}_{i}+{\rm{\Delta }}s$ is then evaluated by maximizing the cross-correlation coefficient between the (generally Gaussian-like) subsequent loop cross-sectional profile $F[r,\alpha ({s}_{i+1})]$ compared with the previous profile $F[r,\alpha ({s}_{i})]$. If the direction of the loop tangent is ${\boldsymbol{s}}=[{dx}/{ds},{dy}/{ds}]$, the perpendicular direction is defined by ${\boldsymbol{r}}=[-{dy}/{ds},{dx}/{ds}]$, along which the cross-sectional loop profile F(r) is defined. The automated loop detection continues from s_i to ${s}_{i+1}={s}_{i}+{\rm{\Delta }}x$ in the first loop segment step, and then from ${s}_{i+1}$ to ${s}_{i+2}$ in the second loop segment step, and so forth. The end of the first half-loop segment occurs when the bipass-filtered flux values drop below a selectable threshold (typically set at one standard deviation of the flux of the background variation). The code then traces the second half of the loop segment in the same way, except in the reverse direction, starting from the initial maximum value at $[{x}_{1},{y}_{1}]$. Once the end of the second half-loop segment is reached, both the first and second half segments are joined together in the same direction, which constitutes the coordinates $[x({s}_{i}),y({s}_{i})],i\,=0,\ldots ,{n}_{p}$ with n_p points of the full loop segment. The image area that covers the coordinates of the first loop, with a margin of $\pm {n}_{{sm}1}\,\mathrm{pixels}$ in the x- and y-directions, is then erased in the bipass-filtered image, so that an already traced loop segment is not used multiple times in the tracing of a new structure. This procedure of detecting the first loop is then continued at the location of the next image maximum $[{x}_{2},{y}_{2}]$ and repeated for the structure with the second-highest contrast, and so forth, until the zero floor or a preset threshold value in the bipass-filtered image is reached. In the end, the code produces a list of $[{x}_{j}({s}_{i}),{y}_{j}({s}_{i})]$ loop coordinates for $j=1,\ldots ,{n}_{j}$ loop segments, each one containing $i=1,\ldots .,{n}_{i}$ coordinate points.

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There are a few control parameters that can be set in the OCCULT-2 code, such as the maximum number of traced structures n_struc per image or wavelength, the high-pass filter n_sm1, the low-pass filter n_sm2, the minimum accepted loop length l_min, the minimum allowed curvature radius r_min, the field line step ${\rm{\Delta }}s$ along the (projected) loop coordinate, the flux threshold q_thresh1 (in units of the median value of positive fluxes in the original image), the filter flux threshold q_thresh2 (in units of the median value in the positive bipass-filtered fluxes), and the maximum allowed gap n_gap with zero or negative fluxes along a traced ridge. In our analysis, we choose the following default control parameters: ${n}_{{sm}1}=1,\ldots ,\,64\,\mathrm{pixels}$, ${n}_{{sm}2}={n}_{{sm}1}+2\,\mathrm{pixels}$, ${l}_{\min }=2\,\mathrm{pixels}$, ${r}_{\min }=100\,\mathrm{pixels}$, ${\rm{\Delta }}s=1$ pixel, ${q}_{\mathrm{thresh},1}=0$, ${q}_{\mathrm{thresh},2}=2$, and ${n}_{\mathrm{gap}}=0$. We will present some parametric studies in Section 4 in order to investigate systematic trends of the algorithm.

The OCCULT-2 code applies a bipass filter to an image $F(x,y)$, consisting of a high-pass filter with a selectable smoothing box-car length n_sm1 and a low-pass filter with a box-car length n_sm2. Such a bipass filter selects only structures within the width range of $[{n}_{{sm}1},{n}_{{sm}2}]$ in the automated pattern recognition algorithm. The narrowest possible width for a bipass filter is ${n}_{{sm}2}={n}_{{sm}1}+2$, which yields the highest sensitivity to detect structures with a width of $w\approx ({n}_{{sm}1}+{n}_{{sm}2})/2$. For instance, if we set a low-pass filter of ${n}_{{sm}1}=2$, the high-pass filter is ${n}_{{nsm}2}={n}_{{sm}1}+2=4$, and the selected loop widths are $w\approx ({n}_{{sm}1}+{n}_{{sm}2})/2=(2+4)/2=3\pm 1$. An example of an automatically traced loop with extraction of the loop width profiles in the direction perpendicular to the loop axis is shown in Figure 4. The actual loop width w at a given loop location s_i along a loop is measured from the equivalent width w of the loop cross-sectional flux profile F(r),

$$\begin{eqnarray}&&w=\displaystyle \frac{\int F(r){dr}}{\max [F(r)]},\end{eqnarray} \tag{ 14 }$$

where the coordinate r is defined in the direction perpendicular to the local loop axis s and the value ${F}_{{ij}}=F({s}_{i},{r}_{j})$ at a particular location $({s}_{i},{r}_{j})$ is obtained from bilinear extrapolation in the 2D image $F(x,y)$. In the example shown in Figure 4(e), the mean loop width is found to be $w=2.9\pm 1.1\,\mathrm{pixels}$, where the uncertainty mostly results from the point-spread function, the noisy background, and multiple overlapping loop strands. The automated measurement of loop widths is performed in steps of ${\rm{\Delta }}s=1\,\mathrm{pixels}$ along each loop segment, which typically yields 10–100 times more width measurements than loop detections.

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For the detection of structures with cross-sectional widths covering a range of two decades, we can select a series of independent filter width components that are increasing by powers of two,

$$\begin{eqnarray}&&{n}_{{sm}1,k}={2}^{k},\qquad k=0,1,...6,\end{eqnarray} \tag{ 15 }$$

which essentially corresponds to the orthogonal basis functions in Fourier transforms or in PCA techniques (e.g., Jackson 2003). PCA techniques have been applied to solar physics data sets in various ways (e.g., Lawrence et al. 2004; Casini et al. 2005; Cadavid et al. 2008; Zharkova et al. 2012). Hence, we can run the automated loop detection for every image multiple times with independent (orthogonal) bipass filters ${n}_{{sm}1,k}$ in order to obtain a completely sampled range of width measurements. The smallest smoothing box-car width (${n}_{{sm}\mathrm{1,0}}=1$ pixel) approximately corresponds to the instrumental resolution, while an upper limit of ${n}_{{sm}\mathrm{1,6}}={2}^{6}=64\,\mathrm{pixels}$ covers about two orders of magnitude in the range of width measurements, with ${w}_{\max }={2}^{6}=64\,\mathrm{pixels}$.

The choice of filters by powers of 2 yields a uniform distribution of widths ${w}_{i},i=0,\ldots ,6$ on a logarithmic scale, log (w). The OCCULT-2 code then yields a number of detected loop structures ${N}_{i},i=0,\ldots ,6$ for each selected width w_i. This yields directly a frequency occurrence distribution (or size distribution) N(w) of loop structures, which is generally displayed on a $\mathrm{log}(N)\mbox{--}\mathrm{log}(S)$ (number versus size) plot. In other words, the option of width filters in the OCCULT-2 code can be directly used for the measurement of size distributions N(w). Of course, structures that are detected on relatively large scales of $w\gg 1\,\mathrm{Mm}$ are likely to consist of more complex structures (e.g., arcades of loops) rather than traditional field-aligned monolithic loops. The size distributions can have different functional shapes, each one being a characteristic of the underlying physical process. For instance, random measurements of linear processes yield a normal Gaussian distribution, while nonlinear dissipative processes yield a power-law distribution. By measuring the size distribution, we thus obtain a diagnostic of the underlying (linear or nonlinear) physical generation mechanism.

We generate a thresholded power-law size distribution of loop widths, with a threshold at w₀ (according to Equation (3)), where the symbol w_sim stands here for the simulated fully resolved loop widths,

$$\begin{eqnarray}&&N{({w}_{\mathrm{sim}}){{dw}}_{\mathrm{sim}}\propto ({w}_{\mathrm{sim}}+{w}_{0})}^{-a}\ {{dw}}_{\mathrm{sim}}\quad \mathrm{for}\ {w}_{\mathrm{sim}}\geqslant {w}_{\min },\end{eqnarray} \tag{ 16 }$$

with a power-law index of a = 2, bound by the range $[{w}_{\min },{w}_{\max }]$. Such a distribution can be produced by using the transformation (e.g., Section 7.1 in Aschwanden 2011; Equation (15) in Aschwanden 2015)

$$\begin{eqnarray}{w}_{\mathrm{sim},i} & = & [(1-{\rho }_{i}){({w}_{\min }+{w}_{0})}^{(1-a)}\\ & & {+{\rho }_{i}{({w}_{\max }+{w}_{0})}^{(1-a)}]}^{(1/(1-a)}-{w}_{0},\end{eqnarray} \tag{ 17 }$$

where ${\rho }_{i}$ are random numbers uniformly distributed in the interval $[0\lt {\rho }_{i}\lt 1]$. Such a set of values ${w}_{\mathrm{sim},i}$ then forms a size distribution $N({w}_{\mathrm{sim}})$ of widths in the form of a power-law function. The peak value of the cross-sectional loop profiles F(r) are chosen according to the emission measure definition in an optically thin plasma,

$$\begin{eqnarray}&&{F}_{i}\propto \int {n}_{e}^{2}\ {dz}\propto \langle {n}_{e}^{2}\rangle {w}_{\mathrm{sim},i}\propto {w}_{\mathrm{sim},i},\end{eqnarray} \tag{ 18 }$$

where the mean electron density $\langle {n}_{e}\rangle$ is assumed to be independent of the loop width, which means that the flux increases proportionally to the column depth $w=\int {dz}$, being equal to the loop width ${w}_{\mathrm{sim},i}$ for circular loop cross-sections. Note that the simulated loop widths ${w}_{\mathrm{sim},i}$ defined in Equations (16)–(18) are fully resolved, because each simulated loop strand is made up of numerical substrands with a width of 1 pixel.

A spatial 2D image $F(x,y)$ is then composed by superposing the brightness distributions of semi-circular loops (as shown in Figure 5), each one consisting of a space-filling bundle of loop strands with a total Gaussian cross-section of width ${w}_{\mathrm{sim},i}$ (Equation (17)) and flux F_i (Equation (18)). The loop positions are randomly distributed in 2D space with a fixed height of the curvature center (of the semi-circular loops) in photospheric height. In this setup, we simulate a total of n = 128 loop bundles, where the strands have a cutoff or minimal width of ${w}_{0}=1$ pixel, projected onto a 2D image with a size of 1000 × 1000 pixels. In addition, the simulated images are convolved with a 2D Gaussian kernel that has an FWHM of ${w}_{\mathrm{psf}}=2.5\,\mathrm{pixels}$ in order to mimic a realistic instrumental point-spread function. The peak count rate is 10⁵ photons per pixel, and a background with random noise of 10% is added. An example of such a simulated image is shown in Figure 5(a), along with a bipass-filtered rendering in Figure 5(b). The added random noise is visible in the filtered image in Figure 5(b).

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We now perform a validation test of the capability to retrieve the underlying power-law distribution function of loop widths as follows. We apply seven bipass filters with ${n}_{{sm}1}={2}^{i}=1,2,4,8,16,32,64$ and ${n}_{{sm}2}={n}_{{sm}1}+2$ to produce seven bipass-filtered images with different filter widths. This corresponds to the PCA described in Section 3.2. We run the automated loop detection code OCCULT-2 on each of the seven images. Using a threshold of q₂ = 2.0 standard deviations (in the bipass-filtered image), we require a minimum length of ${l}_{\min }=2\,\mathrm{pixels}$ for the selected loop length segments, and show the detected loop tracings in Figure 6 (red curves), which appear to be fairly complete in all filters, as judged by visual detection. Comparing the detected structures with the theoretically simulated image (Figure 5(a)) on a pixel-by-pixel basis, we find that 80% of the simulated image pixels are retrieved with the correct values, using the automated OCCULT-2 code. We then fit the histograms of the loop widths, which includes 152,564 loop width measurements in 1254 loop segments detected in seven different filter images. For the power-law slopes of the resulting width distributions, we find the values of $a=2.82\pm 0.07$ for the differential frequency distribution (Figure 6(h)) and a similar value from the cumulative frequency distribution (Figure 6(i)).

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The specific value of the power-law slope a cannot easily be retrieved with our method, since most loops show up in multiple width filters, and thus are counted multiple times with different widths. We tested input values of the power-law slope in the range of $a=1.5\mbox{--}4.0$ in the simulations, but invariably obtained values of $a\approx 2.7\pm 0.2$ after auto-detection with our multiscale sampling method. A method for the power-law slope retrieval can in principle be designed by eliminating multiple countings of loops, but this requires a unique definition of the mathematical function of loop cross-section profiles, which is an ambiguous task and is not necessary for the measurement of the size distribution of the smallest loop strands here.

Inspecting the peak of the size distribution of widths $N({w}_{\mathrm{sim}})$, we find a peak width of ${w}_{p}/{w}_{\min }=3.0$ (Figures 6(h) and (i)). The reduced chi-square value of $\chi =1.9$ (Figure 6(h)) is satisfactory for the fit of the differential size distribution, given the simple analytical model used here (Equation (10)).

Thus, this test demonstrates that we can (1) measure a complete size distribution of loop widths within a range of about two orders of magnitude above the instrumental resolution limit, (2) retrieve the functional shape of a power-law function (although not the exact value of the power-law slope), and (3) detect the rollover of a smooth cutoff in the range of $1\leqslant {w}_{\mathrm{sim}}/{w}_{\min }\lesssim 3\,\mathrm{pixels}$ near the resolution limit w_min. Comparing simulated loop images (Figure 6) with observed images (Figures 13–14), however, we notice that the numerical simulations contain idealized loops only, without taking temperature inhomogeneities into account, such as "moss structures" at the footpoints of hot SXR emitting loops (e.g., Berger et al. 1999). The present study therefore does not discriminate between "classical coronal loops" and "loop-like transition region phenomena," nor does it disentangle the 3D topology of nested loop structures. Nevertheless, the sampling is complete in the 2D image plane (thanks to the automated loop detection algorithm), and thus no ad hoc assumptions of loop selection criteria are applied, which makes it more suitable to compare with theoretical models of statistical distributions rather than a hand-picked sample of selected (loop) structures.

The detection of a lower cutoff for the widths of the smallest loop strands depends crucially on the spatial resolution or pixel size of the image. In a Monte Carlo simulation, we can vary the spatial resolution and study the effect of the spatial resolution limit on the detection of the finest loop strands. In the basic example shown in Figure 6, we have chosen a minimum loop strand width of ${w}_{\min }=1$ image pixel, and sampled the size distribution in the range between ${w}_{\min }=1$ and ${w}_{\max }={10}^{2}\,\mathrm{pixels}$ (Figures 6(h) and (i)). We note that the resulting differential occurrence frequency distribution shown in Figure 6(h) exhibits a peak at a width ratio of ${w}_{p}/{w}_{\min }\approx 3.0$. It turns out that this ratio of the peak width to the pixel size is a persistent feature in our simulations of power-law distributions $N({w}_{\mathrm{sim}})$ of loop widths, for samples with unresolved structures down to the pixel size ${\rm{\Delta }}x$.

In order to investigate the universality of this ratio of the peak value w_p of a size distribution $N({w}_{\mathrm{sim}})$ to the pixel size ${\rm{\Delta }}x$, we perform 20 numerical simulations of identical 2D images (the same as shown in Figure 5(a)), but with 20 different pixel sizes, from ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 1$ to ${\rm{\Delta }}x=2\buildrel{\prime\prime}\over{.} 0$, while the scaling of the standard simulation shown in Figure 6 corresponds to an AIA pixel with ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 6$. For conversion into units of Mm, we use the scaling $1^{\prime\prime}$ (arcsec) = 0.725 Mm. We overlay the histograms and power-law fits of the 20 size distributions with different pixel sizes in Figure 7(a), which shows a systematic shift of the peak with pixel size. However, when we normalize the size distributions $N({w}_{\mathrm{sim}})$ to the pixel size ${\rm{\Delta }}x$ on the x-axis and to the peak value ${N}_{p}=N({w}_{\mathrm{sim}}={w}_{p}$) on the y-axis, we see that the peaks line up at a peak value of ${w}_{p}\approx 3.0{\rm{\Delta }}x$ (Figure 7(b)), and thus the normalized size distribution of loop widths exhibits a universal shape. However, this is only true for distributions that do not resolve the finest loop strands (red curves in Figure 7), when the smallest loop width is smaller than the pixel size, while the distributions with pixel sizes smaller than the finest loop widths exhibit a larger ratio ${w}_{p}/{\rm{\Delta }}x\gtrsim 3.0$ of the peak width to the pixel size, because there is a relative scarcity of detected loops at small widths w_sim. Thus, we can use the ratio ${w}_{p}/{\rm{\Delta }}x$ as a diagnostic for discriminating which size distributions resolve, or do not resolve, the smallest loop strands.

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In order to quantify this new diagnostic criterion, we perform 4 × 20 Monte Carlo simulations for four different cutoffs w_min of the loop width distributions, ${w}_{\min }=0\buildrel{\prime\prime}\over{.} 3$, 06, 12, and 18, and for 20 different spatial resolutions ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 1,0\buildrel{\prime\prime}\over{.} 2,...,2\buildrel{\prime\prime}\over{.} 0$. For each of the 80 simulated images, we repeat the automated loop detection, produce the size distributions of loop widths, and measure the ratio of the peak width to the pixel size, ${w}_{p}/{\rm{\Delta }}x$. The results of these ratios as a function of the pixel size are plotted in Figure 8 (black diamonds). From the four plots in Figure 8, we see that each function ${w}_{p}/{\rm{\Delta }}x$ exhibits two regimes, a fully resolved regime at $w\lt {w}_{\min }$ on the left and an unresolved regime at $w\gt {w}_{\min }$ on the right side. We can model this dependence of the peak width ${q}_{w}={w}_{p}/{\rm{\Delta }}x$ on the spatial resolution ${\rm{\Delta }}x$ with the function ${q}_{w}({\rm{\Delta }}x)$,

$$\begin{eqnarray}&&{q}_{w}=\sqrt{{\left(\displaystyle \frac{{w}_{\min }}{{\rm{\Delta }}x}\right)}^{2}+{({w}_{\mathrm{psf}})}^{2}},\end{eqnarray} \tag{ 19 }$$

for which best fits are shown in Figure (8) (red curves), yielding approximate values for the point-spread function (${w}_{\mathrm{psf}}\approx 2.5\,\mathrm{pixels}$) and the minimum width w_min for each of the four simulations. A more accurate value for the minimum width w_min is obtained by interpolating the fitted function at the critical value q_w = 3.0 pixels, which is the separator between the fully resolved and unresolved regimes. For the four cases simulated in Figure 8 we used minimum loop widths of ${w}_{\min }=0\buildrel{\prime\prime}\over{.} 3$, $0\buildrel{\prime\prime}\over{.} 6$, $1\buildrel{\prime\prime}\over{.} 2$, and $1\buildrel{\prime\prime}\over{.} 8$, while the predicted values inferred from the critical value q_w = 3.0 are ${w}_{\min }^{\mathrm{pred}}=0\buildrel{\prime\prime}\over{.} 35$, $0\buildrel{\prime\prime}\over{.} 51$, $0\buildrel{\prime\prime}\over{.} 72$, and $2\buildrel{\prime\prime}\over{.} 07$, and thus agree with the values used in the numerical simulation within $\lesssim 30 \%$.

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In summary, our numerical simulations demonstrate that rebinning an image with different pixel sizes and sampling of the resulting size distributions of loop widths are capable of predicting the minimum width w_min at the cutoff of the finest loop strands contained in the image, using the diagnostic of the peak width ratio to the pixel size, ${w}_{p}/{\rm{\Delta }}x$. The applied range of rebinned pixel sizes should not extend below the instrumental pixel size of the image, because there is no information on finer scales in the image. If the finest structures are all unresolved at full resolution, the peak width ratio has a constant value of ${w}_{p}/{\rm{\Delta }}x\lesssim 3.0$. If the peak width ratio is larger, this indicates that the finest loop strands are resolved and the critical limit ${\rm{\Delta }}x$ or minimum loop width w_min can be determined from the inversion of ${q}_{w}({\rm{\Delta }}x)$ (Equation (19)). This diagnostic gives us a reliable tool to determine the size w_min of the smallest loop strands in the corona in the case of resolved structures, or an upper limit in the case of unresolved structures. If this cutoff is detected on a macroscopic scale, we can conclude that we hit "the rock bottom of the smallest loop strands."

We analyze AIA images from 2011 February 14, 00:00 UT, in the six coronal wavelengths 94, 131, 171, 193, 211, and 335 Å. The AIA images have a full image size of 4096 × 4096 pixels, with a pixel size of 06, corresponding to a spatial resolution of ${w}_{\mathrm{res}}=0\buildrel{\prime\prime}\over{.} 6\times 2.5\,\mathrm{pixels}\times 0.725\,\mathrm{Mm}\approx 1.1\,\mathrm{Mm}$ on the solar surface). The time cadence of AIA is 12 s. For the automated pattern recognition, we cut out a subimage with a field of view of 0.3 solar radii (or 487 × 487 pixels), centered on the active region NOAA 11158, which has a heliographic position of S20 and E27 to W38 during the six observing days. One example is shown for 2011 Feb 14 in Figures 9 and 10. The same active region is the subject of over 40 published studies (for a list of references, see Section 5.1.1. in Aschwanden et al. 2014). Descriptions of the SDO spacecraft and the AIA instrument can be found in Pesnell et al. (2011) and Lemen et al. (2012).

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In each AIA image (at six wavelengths) we run the automated loop recognition code OCCULT-2 separately, sampling the number of loops (n_loop) and detected loop widths (n_wid), and performed power-law fits for both the differential and cumulative size distributions. The results of the total number of detected loops n_loop, the power-law slopes a_diff and a_cum, and the goodness-of-fit ${\chi }^{2}$-values are listed in Table 1.

Table 1.  Measurement of the Power-law Slope of the Loop Width Distribution Functions Obtained in Different Wavelengths of the AIA Image of 2011 February 14, 00:00 UT

Wave	Number	Number	Power-law	Power-law	Goodness	Goodness
Length	of Loops	of Widths	Slope	Slope	of Fit	of Fit
(Å)	nloop	nwid	adiff	acum	${\chi }_{\mathrm{diff}}$	${\chi }_{\mathrm{cum}}$
94	80	2653	2.5 ± 0.3	1.8 ± 0.2	0.8	1.0
131	199	5796	2.7 ± 0.2	2.6 ± 0.2	1.0	2.2
171	463	15017	2.7 ± 0.1	2.8 ± 0.1	1.1	1.9
193	401	11589	3.1 ± 0.2	2.7 ± 0.1	1.1	1.0
211	310	9232	2.7 ± 0.2	2.2 ± 0.1	0.8	2.7
335	125	3871	2.4 ± 0.2	1.9 ± 0.2	0.8	1.1
Mean	⋯	⋯	2.7 ± 0.3	2.3 ± 0.4	⋯	⋯

Notes. The columns are the number of detected loops n_loop, measured widths n_wid, the power-law slope of the differential frequency distribution a_diff and of the cumulative frequency distribution a_cum, and the goodness of fit values χ_diff  and χ_cum.

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When we compare the results from AIA (Figure 10) with those of our numerical simulation (Figure 6), we find similar values for the power-law slope, $a\approx 3.1$, and the peak width, ${w}_{p}\approx 2.9\,\mathrm{pixels}$, which confirms that our Monte Carlo simulation designs closely reproduce the functional shape of the observed coronal loop width distributions. A new insight of this study is how the spatial resolution affects the occurrence frequency distribution N(w) in the range from w = 1 pixel to the peak of the size distribution of ${w}_{p}\lesssim 2.9\,\mathrm{pixels}$, which can be diagnosed from the normalized peak width value ${w}_{p}/{\rm{\Delta }}x$.

For the power-law slopes of the loop width distributions, averaged over all AIA wavelengths, we found mean values of ${a}_{\mathrm{diff}}=2.7\pm 0.3$ and ${a}_{\mathrm{cum}}=2.3\pm 0.4$ (Table 1), which corroborates the consistency between the differential and cumulative power-law fitting method. The variation of the power-law slope among different wavelengths is of order 10% (Table 1). The measured power-law slopes a can be understood as the fractal dimension of the geometric volume of an active region, because the simulated loop bundles are not space-filling, although the strands inside a loop bundle are space-filling. If an active region would be solidly filled with loops, we would expect a fractal dimension that corresponds to an Euclidean dimension of $a=D=3$. On the other side, if all detected loops are located in a 2D layer with a fixed width in the third dimension, for instance caused by gravitational stratification, the expectation would be a Euclidean dimension of $a=D=2$. The fact that we find a power-law index with a mean value $a\approx 2.7\pm 0.3$ indicates that the spatial distribution of the measured loop cross-sections is fractal. In comparison, a fractal dimension of ${D}_{3}=2.0\pm 0.5$ has been inferred from 3D modeling of 20 solar flare events (Aschwanden & Aschwanden 2008a, 2008b).

We turn now to coronal images with the highest spatial resolution ever recorded, which were obtained during a rocket flight by the High-resolution Coronal Imager (Hi-C) on 2012 July 11. Instrumental descriptions are available from Kobayashi et al. (2014) and Cirtain et al. (2013). We use an image recorded at 18:54:16 UT, in the wavelength band of 193 Å, which is dominated by the Fe XII line originating around $T\approx 1.5$ MK. The image has a size of 3880 × 4096 pixels, with a pixel size of 01 (corresponding to 73 km/pixel), covering a field of view of 300 Mm² ($\approx 0.37{R}_{\odot }$), and was sampled with an exposure time of 2 s. The same image was analyzed previously by Peter et al. (2013).

We present the analyzed image in Figure 11(a), which contains a sunspot (in the upper half of the image), moss regions (in the upper-left quadrant), as well as coronal loops in the periphery of an active region (bottom-right quadrant), which are analyzed in Peter et al. (2013).

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The structures hidden in the image can be enhanced by bipass filtering (Equation (12)), as revealed in Figure 12(a) with a bipass filter of ${n}_{{sm}1}=16$ and ${n}_{{sm}2}=18$, which is most sensitive to structures with a width of $w=17\times 0\buildrel{\prime\prime}\over{.} 1\,\times 725\,\mathrm{km}=1200\,\mathrm{km}$, or in Figure 12(b) with a bipass filter of ${n}_{{sm}1}=32$ and ${n}_{{sm}2}=34$, which is most sensitive to structures with a width of $w=33\times 0\buildrel{\prime\prime}\over{.} 1\times 725\,\mathrm{km}\,=2400\mathrm{km.}$ However, filtering at the highest resolution, as shown in Figure 11(a) with a bipass filter of ${n}_{{sm}1}=1$ and ${n}_{{sm}2}=3$, being most sensitive to structures with a width of $w=2\,\times 0\buildrel{\prime\prime}\over{.} 1\times 725\,\mathrm{km}$ = 145 km, reveals no significant structures above the noise level anywhere (Figure 11(b)), which is a surprising result that we will scrutinize in the following analysis in more detail.

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In Figure 13, we show the automated loop detection applied to the 3880 × 4096 pixel full-resolution Hi-C image, using the seven bipass filters from ${n}_{{sm}1}=2$ to ${n}_{{sm}2}=128$. Almost no significant structure is seen at the highest resolution (Figures 13(a) and (b)), while a maximum of ${N}_{\det }=661$ structures is detected at the filter ${n}_{{sm}1}=32$ (Figure 13(e)). The resulting size distribution of widths shows a peak width of ${w}_{p}/{\rm{\Delta }}x=7.1\,\mathrm{pixels}$ (Figure 13(h)), which according to our numerical simulations is indicative of fully resolved width structures at the smallest scale of ${w}_{\min }=1$ pixel or 01, because we would expect a peak width of ${w}_{p}/{\rm{\Delta }}x\approx 2.5$ for unresolved structures at full resolution.

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A comparison of the Hi-C image with the simultaneous and cospatial AIA image at the identical wavelength of 193 Å has been studied in Peter et al. (2013), from which we use the same images, which have been coaligned and corrected for a rotation of 19. The only difference between these two images is the pixel size, with 01 for Hi-C and 06 for AIA, being a factor of 5.8 different. AIA images have been taken 3 s before and 6 s after the Hi-C image. The results of our inter-comparison are tabulated in Table 2.

Table 2.  Measurement of the Power-law Slope of the Loop Width Distribution Functions Obtained from Simultaneous Hi-C and AIA 193 Å Images Scaled to Each Other's Pixel Size

Instrument	Pixel	Number	Number	Peak	Power law	Power law	Goodness	Goodness
	size	of loops	of widths	width	slope	slope	of fit	of fit
	(arcsec)	nloop	nwid	wp	adiff	acum	${\chi }_{\mathrm{diff}}$	${\chi }_{\mathrm{cum}}$
AIA	0.6	784	20,927	3.1	2.6 ± 0.1	2.2 ± 0.1	1.7	1.3
Hi-C & AIA pixels	0.6	770	20,629	2.9	2.6 ± 0.1	2.3 ± 0.1	1.7	1.7
Hi-C	0.1	2459	138,965	7.1	1.4 ± 0.1	2.0 ± 0.1	1.5	4.8
AIA & Hi-C pixels	0.1	3205	188,926	8.3	2.7 ± 0.1	2.5 ± 0.1	...	...
Mean	⋯	⋯	⋯	⋯	2.3 ± 0.6	2.3 ± 0.2	⋯	⋯

Notes. The columns are the number of detected loops n_loop, measured, the normalized peak width w_p/Δx, the power-law slope of the differential frequency distribution a_diff and of the cumulative frequency distribution a_cum, and the goodness of fit values χ_diff and χ_cum.

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We perform an identical analysis of the AIA image (Figure 14) as we did for the Hi-C image (Figure 13). In the AIA image, significant structures are already visible at the full resolution (Figure 14(a)), which are comparable with those seen in the Hi-C image in the ${n}_{{sm}1}=17$ filter (Figure 13(d)). Inspecting the size distribution of loop widths obtained in the AIA image, we find also a different peak width of ${w}_{p}/{\rm{\Delta }}x=3.1$ (Figure 14(h)), which indicates, according to our numerical simulations, that the finest structures are barely resolved in the AIA image at its full resolution (06), while they appear to be fully resolved in the Hi-C image at its full resolution (01).

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In Figure 15 we show a juxtaposition of the size distributions of the AIA image with an original pixel size of 06 (Figure 15(a)), the AIA image degraded to the Hi-C resolution of 01 (Figure 15(d)), the Hi-C image with the original pixel size of 01 (Figure 15(c)), and the Hi-C image rescaled to the AIA resolution of 06 (Figure 15(b)). We find that the size distributions of widths are almost identical for equal pixel sizes, such as AIA and the rescaled Hi-C image with $0\buildrel{\prime\prime}\over{.} 6\,\mathrm{pixels}$ (Figures 15(a) and (b)), or the Hi-C and the rescaled AIA image with $0\buildrel{\prime\prime}\over{.} 1\,\mathrm{pixels}$ (Figures 15(c) and (d)). This means that there is essentially the same information stored for equal pixel sizes if all structures are resolved at this pixel size, which however would not be the case for unresolved structures. Further we find that rescaling of the same instrument does not preserve the peak width w_p or the normalized peak width ${w}_{p}/{\rm{\Delta }}x$, as can be seen in the comparison of AIA (Figure 15(a)) and the rescaled AIA (Figure 15(d)), or in the comparison of Hi-C (Figure 15(c)) and the rescaled Hi-C image (Figure 15(b)). The results of this inter-comparison are summarized in Table 2. This is all consistent with our method that the peak width ratio ${w}_{p}/{\rm{\Delta }}x$ can be used in the rescaling of the same image as a diagnostic for discriminating whether the finest structures in an image are resolved or not.

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In the results of the numerical simulations presented in Figure 8, we demonstrated that the peak width ratio has a constant value of ${w}_{p}/{\rm{\Delta }}x\approx 2.5$ for unresolved structures when the image is rescaled to different pixel sizes ${\rm{\Delta }}x$, while the ratio increases to higher values at the highest resolution (of one pixel size) for images with fully resolved structures. For this experiment, we rebin the Hi-C image from a pixel size of ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 1$ to ${\rm{\Delta }}x=2\buildrel{\prime\prime}\over{.} 0$ in 20 equidistant steps of ${\rm{\Delta }}{x}_{i}=0.1\times i,i=1,\ldots ,20$. Thus, the rescaled image varies in the number of pixels from 3880 × 4096 pixels down to 194 × 205, but covers the same field of view and thus the same structures. We repeat for each of the 20 rescaled Hi-C images the automated pattern recognition algorithm and the sampling of width histograms, from which we extract the peak width ratio ${w}_{p}/{\rm{\Delta }}x$ and plot it as a function of the pixel size ${\rm{\Delta }}x$ (Figure 16(a)). We find that the ratio is near constant for large pixel sizes, while it increases with smaller pixel sizes. The separation point between fully resolved and unresolved loop widths can be read from the critical value ${w}_{p}/{\rm{\Delta }}x=3.0$ on the y-axis, which yields a value of ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 77$ on the horizontal axis, corresponding to ${w}_{p}\approx 550\,\mathrm{km}$. Thus, we conclude that the structures detected with Hi-C have the most frequent width value of ${w}_{p}\approx 550\,\mathrm{km}$, while thinner loop strands become rapidly sparser below this limit.

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We repeat the same experiment with the AIA image for full resolution (down to 1 pixel) and show the results in Figure 16(b). If we employ the same critial value of ${w}_{p}/{\rm{\Delta }}x=3.0$ on the y-axis, we can read off a value of ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 58$, which corresponds to ${\rm{\Delta }}x=420\,\mathrm{km}$, agreeing with the value from Hi-C of ${\rm{\Delta }}x=550\,\mathrm{km}$ within $\approx 25 \%$. Thus, the AIA data indicate that all structures are mostly unresolved but become marginally resolved at the full resolution of 1 AIA pixel, i.e., ${\rm{\Delta }}x\approx 0\buildrel{\prime\prime}\over{.} 6$. Only with the additional information of Hi-C can we conclude that most structures are resolved below the AIA resolution.

The ratio of the detected (or observed) loop width w to the true (or simulated) loop width w_true follows from Equation (4), i.e., $w=\sqrt{{w}_{\mathrm{true}}^{2}+{w}_{\min }^{2}}$, as visualized in Figure 17. From Figure 17 we can read off the ratio of the peak widths w_p to the minimum width w_min, which is found to be ${w}_{p}/{w}_{\min }=6.4$ for AIA and ${w}_{p}/{w}_{\min }=19.4$ for Hi-C, while we would expect a ratio of ${w}_{p}/{w}_{\min }\approx 2.5$ for instruments that do not resolve the loop strands. Thus, the diagram shown in Figure 17 tells us that Hi-C fully resolves the bulk of the loop strands, while AIA resolves most of them, too, but only by a small margin compared with a non-resolving instrument.

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We provide a comprehensive compilation of 52 studies that dealt with coronal loop width measurements in Table 3, with a graphical visualization in Figure 18. Summaries of loop width measurements can also be found in Bray et al. (1991; Chapters 2 and 3) and Aschwanden (1995, 2004; Chapter 5.4.4). The graphical representation in Figure 18 divides the loop width measurements in four wavelength regimes by color (optical + Hα + Lyα, EUV, SXRs, and radio) and ranks the width ranges by the smallest detected width in ascending order. There are several trends visible in this overview. Minimum loop widths have been measured from $w\approx 20\,\mathrm{Mm}$ down to ${w}_{\min }\approx 0.1\,\mathrm{Mm}$. The finest loop widths have been detected preferentially in EUV, while the loop widths measured in SXRs and optical wavelengths tend to be significantly larger and are found to be the largest in radio wavelengths. This can be explained because coronal loops in EUV and SXRs are produced by optically thin bremsstrahlung, which yields a better contrast than the partially optically thick free–free and gyroresonance emission in radio wavelengths.

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Table 3.  Compilation of Coronal Loop Width Measurements in Chronological Order

Object	Pixel	Spatial	Loop	Wavelength	Instrument	Ref.
	Size	Resolution	Width
	wpix (Mm)	wres (Mm)	wloop (Mm)	λ (Å)
Green corona loops	...	...	3–8	5303	Dunn/SacPeak	1
Large green corona loops	...	...	8–12	5303	Dunn/SacPeak	2
Green corona loops	...	3.6	11–22	5303	Pic-du-Midi	3
Red corona structures	...	3.6	8–20	6374	Pic-du-Midi	3
Cool loops	...	3.6	2–5	1032 O VI	Skylab	4
Cool loops	...	3.6	<2	465 Ne VII	Skylab	4
Hot loops	...	3.6	3–12	1032 Mg X	Skylab	4
Hot loops	...	3.6	3–12	465 Si XII	Skylab	4
Loop prominence	...	1.5	$6.5-7.2$	160-630	Skylab	5
Active region loop	...	1.5	7–11	Fe XVI	Skylab	6
Radio loop structure	...	5.3	22	11 cm	NRAO	7
Radio loop structure	...	1.7	9	3.7 cm	NRAO	7
Active region loops	...	3.1	20	20 cm	VLA	8
Active region core loops	2.2	...	<2.2–5	SXR	ASE rocket	9
Compact volume loops	2.2	...	5–15	SXR	ASE rocket	9
Outward extending loops	2.2	...	10–30	SXR	ASE rocket	9
Hot coronal loops	...	1.8	8–14	170-630	Skylab	10
Coronal loops	...	3.1	16–20	20 cm	VLA	11
Active region loops	...	0.7	0.93–2.1	Hα	CSIRO	12,13
Very thin loops (chrom)	...	0.7	0.44–0.58	Hα	CSIRO	14
Very thin loops (photo)	...	0.7	<0.13–0.32	Hα	CSIRO	14
Coronal loops	0.6	0.7	2–3.5	Lyα	LMSAL rocket	15
X-ray loops	...	1.4	6–13	8–65	ASE rocket	16
Soft X-ray loops	3.5	...	7–9	SXR	SXT/Yohkoh	17
Soft X-ray flare loops	...	1.8	4.2–18.4	SXR	SXT/Yohkoh	18
Active region loop	1.9	...	5.8	171, 195	EIT/SOHO	19
Active region loops	1.9	...	7.1 ± 0.8	171	EIT/SOHO	20
Active region loops	1.9	...	7.8 ± 0.8	195	EIT/SOHO	21
Active region loops	1.9	...	7.9 ± 1.4	284	EIT/SOHO	21N
Nanoflare loops	0.35	0.82	1.8–6.1	171, 195	TRACE	22
Postflare arcade loops	0.35	0.82	0.9	171, 195	TRACE	23
Soft X-ray loop	3.5	...	21	SXR	SXT/Yohkoh	24
Oscillating loops	0.35	0.82	8.7 ± 2.8	171, 195	TRACE	25
Nanoflare loops	0.35	0.82	3–10	171, 195	TRACE	26
Nanoflare loops	...	1.8	7–50	SXR	SXT/YOHKOH	26
Oscillating loops	0.35	0.82	5.1 ± 3.9	171, 195	TRACE	27
Cooling loops	0.35	0.82	1.8–12.0	171, 195	TRACE	28
Heated loop	0.35	0.82	3–6	171, 195	TRACE	29
Elementary loops	0.35	0.82	1.4 ± 0.3	171, 195, 284	TRACE	30,31
Cool (<0.1 MK) loops	0.09	0.22	0.7–2.2	1216	VAULT	32
Cooling loops	0.35	0.82	<0.35–8.0	171, 195	TRACE	33
3D-reconstructed loops	1.15	2.30	2.6 ± 0.1	171,195,284	EIT/STEREO	34,35
Coronal loops	0.35	0.82	0.7–2.2	171,195	TRACE	36
3D-reconstructed loops	1.15	2.30	<1.0 ± 1.0	171,195,284	EIT/STEREO	37
Coronal loops	0.44	0.98	4.7–9.7	94–335	AIA/SDO	38
Oscillating loop	0.44	0.98	4.9 ± 0.6	94–335	AIA/SDO	39
Cooling loops	0.35	0.82	1.29–1.50	171, 195	TRACE	40
Coronal loops	0.73	...	0.7–1.9	195	EIS/Hinode	41
Coronal rain	0.06	0.14	0.31	6563	CRISP	42
Auto-detected loops	0.44	0.98	3–8	94-335	AIA/SDO	43
Coronal loops	0.07	0.22	0.2–1.5	193	Hi-C	44
Coronal loops	0.07	0.22	0.117–0.667	193	Hi-C	45
Inter-moss loops	0.07	0.22	0.675–0.803	193	Hi-C	46
Oscillating loops	0.07	0.22	0.15–0.31	193	Hi-C	47
Filament threads	0.07	0.22	0.58 ± 0.07	193	Hi-C	48
Finely structured corona	0.07	0.22	>0.22	193	Hi-C	49
Coronal rain	0.06	0.14	0.15–0.28	6563	CRISP	50
Coronal rain	0.08	0.15	0.32–0.57	3968	SOT/Hinode	50
Coronal rain	0.12	0.30	0.44–0.72	2796	IRIS	50
Coronal rain	0.12	0.30	0.40–0.62	1330	IRIS	50
Coronal rain	0.12	0.30	0.46–0.70	1400	IRIS	50
Coronal rain	0.44	0.98	1.02–1.24	304	AIA/SDO	50
Coronal rain	0.44	0.98	0.51–1.52	304	AIA/SDO	50
Coronal rain	0.44	0.98	1.19–2.36	171	AIA/SDO	50
Coronal rain	0.44	0.98	1.17–2.73	193	AIA/SDO	50
Fine structure loops	0.12	0.30	0.12–0.15	1400	IRIS	51
Penumbral jets	0.07	0.22	<0.6	193	Hi-C	52

References. (1) Kleczek (1963); (2) Dunn (1971); (3) Picat et al. (1973); (4) Foukal (1975); (5) Cheng (1980); (6) Cheng et al. (1980); (7) Kundu et al. (1980); (8) Kundu & Velusamy (1980); (9) Davis & Krieger (1982); (10) Dere (1982); (11) Lang et al. (1982); (12) Loughhead & Bray (1984); (13) Loughhead et al. (1985); (14) Bray & Loughhead (1985); (15) Tsiropoula et al. (1986), Transition Region Camera (TRC); (16) Webb et al. (1987); (17) Klimchuk et al. (1992); (18) Aschwanden & Benz (1997); (19) Aschwanden et al. (1998); (20) Aschwanden et al. (1999); (21) Aschwanden et al. (2000a); (22) Aschwanden et al. (2000b); (23) Aschwanden & Alexander (2001); (24) Aschwanden (2002); (25) Aschwanden et al. (2002); (26) Aschwanden & Parnell (2002); (27) Aschwanden et al. (2003a); (28) Aschwanden et al. (2003b); (29) Petrie et al. (2003); (30) Aschwanden & Nightingale (2005); (31) Aschwanden et al. (2007a); (32) Patsourakos et al. (2007); (33) Aschwanden & Terradas (2008); (34) Aschwanden et al. (2008a); (35) Aschwanden et al. (2008b); (36) Lopez-Fuentes et al. (2008); (37) Aschwanden & Wülser (2011); (38) Aschwanden & Boerner (2011); (39) Aschwanden & Schrijver (2011); (40) Mulu-Moore et al. (2011); (41) Brooks et al. (2012); (42) Antolin & Rouppe van der Voort (2012); (43) Aschwanden et al. (2013b); (44) Peter et al. (2013); (45) Brooks et al. (2013); (46) Winebarger et al. (2013); (47) Morton & McLaughlin (2013); (48) Alexander et al. (2013); (49) Winebarger et al. (2014); (50) Antolin et al. (2015); (51) Brooks et al. (2016); (52) Tiwari et al. (2016). For summaries, see Bray et al. (1991), Aschwanden (1995), and Aschwanden (2004; Chapter 5.4.4).

Download table as:  ASCIITypeset images: 1 2

In Table 3 we also list the pixel size w_pix and the spatial resolution w_res of the instruments. In Figure 18 we can clearly see at one glance that the smallest loop widths are almost always measured a significant factor above the instrumental pixel sizes, which is partially explained by the point-spread function that typically amounts to 2.5 pixels in current EUV imagers (TRACE, AIA/SDO, STEREO, IRIS, Hi-C). Interestingly, the same ratio of ${w}_{p}/{\rm{\Delta }}x\approx 2.5$ is also found in our present work, which corresponds to the optimum detection efficiency of our automated loop-tracing algorithm (using bipass filters) in the case of the Monte Carlo simulations, besides the effects of the point-spread function in the analyzed AIA and Hi-C data. Since every loop width measurement is subject to noise in the background, which causes some scatter on the order of a pixel size, for instance $w\approx 2.9\pm 1.1$ (pixels) in the example shown in Figure 4, the lowest width measurements can be as low as $w\gtrsim 1$ pixel using the equivalent-width method (Equation (14)). Some measurements shown in Figure 18 exhibit even smaller widths than one pixel, because the authors attempted to determine the true width after deconvolution of the point-spread function, i.e., ${w}_{\mathrm{true}}\approx \sqrt{{w}^{2}-{w}_{\mathrm{res}}^{2}}$ (Equation (4) and Figure 17).

Virtually all studies that report loop width measurements are based on a single loop or a small number of visually selected (hand-picked) loops, which may contain the smallest features visible in an image, but they are not representative samples that would allow us to derive the entire width distribution of all coronal loops seen in an image. Our work thus represents a unique method that completely samples the entire size distribution function $N(w){dw}$ over a range of about two orders of magnitude, characterized by the peak value (w_p), a smooth cutoff range $[{w}_{\min },{w}_{p}]$ down to the minimum width value $({w}_{\min })$, and by a power-law function in the upper part $[{w}_{p},{w}_{\max }]$ up to the maximum size $({w}_{\max })$. Consequently, the range of loop widths measured in the present study ($w\approx 0.1\mbox{--}10\,\mathrm{Mm}$) entails all previous EUV measurements of loop widths, as can be seen at one glance in Figure 18.

However, we may ask whether the distribution of loop sizes continues at the low end if a future instrument faciliates a higher spatial resolution. Based on our peak width diagnostic criterion, this can only be the case if we measure a value of ${w}_{p}\approx 2.5\pm 0.2$ for the peak in the size distribution with the current highest-resolution instruments. The Hi-C measurements with 01 resolution, based on ${n}_{{wid}}=138,965$ individual loop width measurements (Figure 13), indicate a peak width ratio of ${w}_{p}/{\rm{\Delta }}x=7.1$ (Figure 13(h)), which clearly suggests that most detected structures are fully resolved at ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 1$ resolution, while the most common loop width (which is given by the peak $w={w}_{p}$ in the size distribution), amounts to ${\rm{\Delta }}x=0\buildrel{\prime\prime}\over{.} 77$ (or ${\rm{\Delta }}x\approx 550\,\mathrm{km}$; Figure 16(a)). This does not exclude the possible measurement of smaller loops, but smaller loops are expected to be less common than loops with a mean width of ${w}_{p}\approx 550$ km, according to our criterion. Smaller loops have indeed been measured from recent Hi-C studies, e.g., $w=200\mbox{--}1500\,\mathrm{km}$ (Peter et al. 2013), $w=117\mbox{--}667\,\mathrm{km}$ (Brooks et al. (2013), $w=150\mbox{--}310\,\mathrm{km}$ (Morton & McLaughlin 2013), or $w\,=120\mbox{--}150\,\mathrm{km}$ (Brooks et al. (2016). Our prediction is that loops at $w\approx 550\,\mathrm{km}$ are the most frequently occurring loops and form the peak of the size distribution, if sufficient spatial resolution is available, requiring ${\rm{\Delta }}x\leqslant {w}_{p}/2.5\approx 120\,\mathrm{km}$ (or $\approx 0\buildrel{\prime\prime}\over{.} 16$). This implies that a statistical analysis of IRIS data (De Pontieu et al. 2014) should be able to reproduce the peak at ${w}_{p}\approx 550\,\mathrm{km}$ (or 077). Most interestingly, Brooks et al. (2013) show an occurrence distribution of 91 hand-picked loops observed with Hi-C with a minimum Gaussian width of ${\sigma }_{w}=90\,\mathrm{km}$ (FWHM = 212 km) and an average Gaussian width of ${\sigma }_{w}=272\,\mathrm{km}$ (FWHM = 640 km), which is fully consistent with our result of a peak width (FWHM) of ${w}_{p}\approx 550\,\mathrm{km}$ based on three orders of magnitude larger statistics. Winebarger et al. (2014) conducted a statistical analysis of the noise characteristics of a Hi-C image and concluded that 70% of the pixels in each Hi-C image do not show evidence for significant substructures, which confirms the scarcity of thinner loop strands than observed so far (see also Figure 11(b), where no structures except data noise are visible). Moreover, Peter et al. (2013) demonstrate that some of the loops seen in Hi-C are essentially resolved with AIA and appear to have a smooth Gaussian-like cross-section.

There are two schools of coronal heating models: (1) the macroscopic view considers monolithic coronal loops that have heating cross-sections commensurate with the size of magneto-convection cells ($w\approx 300\mbox{--}1000\,\mathrm{km}$), and thus are mostly resolved with current EUV imagers (TRACE, AIA, IRIS, Hi-C) and (2) the microscopic view which postulates unresolved nanoflare strands that cannot be resolved with current instrumentation. Our continuously developing technology has faciliated tremendous improvements in astronomical high-resolution observations, so that the available spatial resolution improved by a factor of about 300 over the last 35 years. In particular, the measurement of coronal loop diameters improved from $w\approx 30\,\mathrm{Mm}$ (Davis & Krieger 1982) down to $w\approx 0.1\,\mathrm{Mm}$ with the current Hi-C observations (Peter et al. 2013; Brooks et al. 2013, 2016). Our new finding of a preferred spatial scale of ${w}_{p}\approx 550\,\mathrm{km}$ for coronal loops calls for a physical explanation of this particular value. This peak width value w_p separates the two regimes of a smooth cutoff range ($w\lt {w}_{p}$) and of a scale-free power law distribution at larger values ($w\gt {w}_{p}$).

One obvious physical scale is the granulation structure of the photosphere, discovered by Sir William Herschel in 1801. The spatial size of the time-varying granulation is measured by a combination of high-pass and low-pass filtering in the spatial frequency domain with subsequent thresholding, which segments granular and intergranular lanes. Alternative procedures are "lane-finding" schemes that are based on local gradient detection (Spruit et al. 1990). With the latter method, the smallest granules, near the observational limit, were found to have the most frequent size of 290 km. The size distribution peaks at w_p = 290 km, then shows a slow decline in number up to a diameter near 1500 km, and then a rapid decline toward the largest granule sizes (Brandt 2001). Modern measurements with the New Solar Telescope at Big Bear show a peak in the occurrence distribution of granule sizes shifted further down to $w\approx 150\,\mathrm{km}$ (Abramenko et al. 2012).

If we adopt the magneto-convection of photospheric granular cells as the primary driver of local magnetic reconnection processes (in the chromosphere, transition region, and corona), for instance see the numerical MHD simulations by Gudiksen & Nordlund (2005) or Bingert & Peter (2013), we would expect a congruence between the magnetic reconnection area and the cross-sectional loop heating area (Aschwanden et al. 2007a, 2007b). The diffusion region of a local magnetic reconnection process essentially defines the spatial scale of a local heating event and therefore the cross-sectional area of a coronal loop. Our new result of a preferred cross-sectional scale of ${w}_{p}\approx 550\,\mathrm{km}$, which appears to be perfectly comparable with the most frequent granular scale, adds an additional constraint to the geometry of the heating process, besides 10 other arguments that have been discussed earlier in the context of the coronal heating paradox (Aschwanden et al. 2007a, 2007b).

The second school of thought deals with unresolved "microscopic" structures. The "nanoflare heating" scenario (Klimchuk 2006) builds on Parker's (1988) conjecture that tangential misalignments of the magnetic field between adjacent flux tubes sporadically reconnect, driven by the braiding random motion of coronal loop footpoints in the photosphere. Since magneto-convection of the photosphere plays an important role as driver for magnetic braiding, which applies to both schools, the major difference between the monolithic and the nanoflare heating scenario is mostly the assumed spatial scale of the heating region, which contrasts the macroscopic view with the microscopic view. There is a general consensus that broad loops are likely to consist of bundles of finer strands, as alluded to in Figure 1, but the crucial question remains what the finest scale of a composite loop is. In the scale-free regime of a width distribution, a power-law like distribution function that extends from w_max down to the peak value w_p, where it breaks down by some unknown cutoff mechanisms, is generally observed. If nanoflares on finer scales exist, we would expect that the power-law function extends to lower values than the observed cutoff, while the apparent peak width occurs at ${w}_{p}/{\rm{\Delta }}x\approx 2.5$ due to the finite resolution. However, our crucial observation of ${w}_{p}/{\rm{\Delta }}x\gg 2.5$ in Hi-C data contradicts this assumption that most frequent nanoflare strands exist at smaller spatial scales than $w\lt 550\,\mathrm{km}$. This implies a physical limit, similar to the observed lower limit of granule sizes. Therefore, we conclude that the Hi-C data are not consistent with a nanoflare scenario with most frequent loop strands at ${w}_{p}\lt 550\,\mathrm{km}$. In addition, current nanoflare scenarios have difficulties explaining the isothermality of macroscopic loops (Aschwanden & Nightingale 2005; Aschwanden 2008), but see recent modifications in terms of "nanoflare storm" scenarios (Klimchuk 2015).

The best efforts to model Parker's nanoflare scenario do not show a clear structuring across the (large-scale) magnetic field. In the turbulent (reduced) MHD model of Rappazano et al. (2008), current sheets (along the large-scale magnetic field) form throughout the computational domain. No obvious structuring of the current sheets into bundles is found, where the heating is concentrated and which (in a more realistic model) could give rise to a coronal loop. So there is the need for an external process that would lead to a cross-field scale producing loops of finite width, and the photospheric magneto-convection is a good candidate for this.

Besides the consequences of loop width statistics on coronal heating models, there are also numerous implications for coronal electron density measurements. The emission measure of optically thin plasma, observed in EUV and SXRs, depends on the column depth, which is set equal to loop diameters for coronal loops with a circular cross-section. Thus, the density in coronal loops can only be determined if their diameter or width is known, such as SXT/Yohkoh measurements of soft X-ray-bright flare loops (Aschwanden & Benz 1997). Therefore, measurements of coronal widths are essential for all theoretical models that require an electron density, such as calculations of the thermal energy, coronal filling factors, gyro-synchrotron emissivity, Alfvén speed, or coronal seismology. Such consequences will be discussed elsewhere.