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Lockheed Martin Advanced Technology Center, Solar and Astrophysics ... High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, ...
The Astrophysical Journal, 566:L59–L62, 2002 February 10 䉷 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

EFFECTS OF TEMPERATURE BIAS ON NANOFLARE STATISTICS Markus J. Aschwanden Lockheed Martin Advanced Technology Center, Solar and Astrophysics Laboratory, Department L9-41, Building 252, 3251 Hanover Street, Palo Alto, CA 94304; [email protected]

and Paul Charbonneau High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307; [email protected] Received 2001 November 15; accepted 2002 January 4; published 2002 January 16

ABSTRACT Statistics of solar flares, microflares, and nanoflares have been gathered over an energy range of some 8 orders of magnitude, over E ≈ 10 24–1032 ergs. Frequency distributions of flare energies are always determined in a ˚ filters are used from an extreme ultraviolet limited temperature range, e.g., at T ≈ 1 –2 MK if the 171 and 195 A telescope (the Solar and Heliospheric Observatory/EUV Imaging Telescope or the Transitional Region and Coronal Explorer). Because the electron temperature Te and the thermal energy E p 3ne k B TeV are statistically correlated in flare processes, statistics in a limited temperature range introduce a bias in the frequency distribution of flare energies, N(E) ∝ E ⫺aE. We demonstrate in this Letter that the power-law slope of nanoflare energies, ˚ ), corresponds to a corrected value e.g., aE ≈ 1.9, as determined in a temperature range of T ≈ 1.1 –1.6 MK (195 A of aE ≈ 1.4 in an unbiased, complete sample. This corrected value is in much better agreement with predictions from avalanche models of solar flares. However, it also implies that all previously published power-law slopes of EUV nanoflares, covering a range of aE ≈ 1.8 –2.3, correspond to unbiased values of aE ! 2 , which then poses a serious challenge to Parker’s hypothesis of coronal heating by nanoflares. Subject headings: Sun: corona — Sun: flares — Sun: UV radiation — Sun: X-rays, gamma rays distributions (e.g., Krucker & Benz 1998; Parnell & Jupp 2000; Aschwanden et al. 2000) and is investigated in this Letter. Corrections for the temperature bias and other truncation biases are important for a proper observational inference of the powerlaw logarithmic slope (aE) of frequency distributions of nanoflare energies, since a critical value aE 1 2 is required for nanoflares to make a dominant contribution to coronal heating (Parker 1988; Hudson 1991).

1. INTRODUCTION

What is known as the Malmquist bias in the determination of the Hubble diagram at high redshifts (e.g., Teerikorpi 1998), which is a distance- and redshift-dependent threshold effect that biases the galaxy number counts as a function of distance, has its analogy in the wavelength or temperature dependence of solar nanoflare statistics. If the thermal energy of a flare would be completely independent of the flare temperature, we could sample flare energies in an arbitrary temperature or instrumental wavelength range and would still retrieve the same representative statistics of flare energies. However, the thermal energy content of a flaring volume V (with an average electron density ne and temperature Te ), has a number of explicit and implicit potential dependencies on temperature: E(Te ) p 3ne (Te )k B TeV(Te ).

2. DATA ANALYSIS

We approach the problem of the temperature bias in nanoflare statistics by performing Monte Carlo simulations. From a complete sample of synthetic flares covering flare temperatures spanning T ≈ 1–100 MK, we sample EUV fluxes in the much narrower temperature range T ≈ 1 –2 MK, in order to mimic the sensitivity of the most commonly used Solar and Heliospheric Observatory/EUV Imaging Telescope (SOHO/EIT) and Transitional Region and Coronal Explorer (TRACE) filters. We then compare the frequency distribution of the incomplete (temperature-biased) sample with the complete (unbiased) sample. We assume that the flaring volumes are looplike structures of length l and width w. The synthetic distributions used in the Monte Carlo simulations are constrained by the observed (temperature-biased) distributions of flare loop lengths N(l), loop widths N(w), flare areas N(A), and emission measures N(EM) and the observed correlations between these parameters, based on a selection of 281 nanoflares studied in Aschwanden et al. (2000). The construction of the unbiased distribution begins with a power-law distribution of loop lengths:

(1)

A compilation of many flare statistics, from nanoflares with E ⲏ 10 24 ergs to the largest flares with E ⱗ 10 32 ergs, reveals statistical relations of the form ne (Te ) ∝ Te2 and V(Te ) ≈ L(Te ) 3 ∝ Te3, which yields a dependence of E(Te ) ∝ Te6 (Aschwanden 1999).1 Because of this strong dependence of the flare energy E on the flare temperature Te, statistics of nanoflare energies in a restricted temperature range [T1 , T2 ] constitute incomplete samples that are not representative of the entire flare distributions. This leads to a sampling bias and thus to a frequency distribution N(E)dE different from that characterizing the underlying complete distributions. This temperature bias has not been considered in previously published studies on nanoflare frequency

N(l)dl ∝ l ⫺a,

l1 ! l ! l 2 ,

(2)

1

One does need to keep in mind that this general trend was derived from samples that are themselves incomplete, and thus may over- or underestimate the temperature dependence.

in the range of l1 p 2.5 Mm to l 2 p 100 Mm, which we choose as the complete sample range of flaring loop sizes. The distriL59

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volume V are A(l) p

p lw(l), 4

(4)

V(l) ≈ lw 2 (l).

(5)

To every flare volume we assign a flare temperature with a random scatter of jTe/ ATe S p 1 and a statistical correlation of Te (l) ∝ l c cf. Aschwanden 1999), with Te (l) ≥ 1 MK, Te (l) p T0

[

c l (1 ⫹ 0.5r) . 5 Mm

]

(6)

Similarly, we assign flare densities with a random scatter jne/ Ane S p 1 and impose a statistical correlation ne (Te ) ∝ Ted (again as found in Aschwanden 1999): ne (Te ) p n 0

[

d Te (1 ⫹ 0.5r) . 1 MK

]

(7)

The resulting emission measure is defined by EM(l, Te ) p



ne2 (Te )dz ≈ ne2 (Te )w(l),

(8)

and the thermal energy is defined according to equation (1). The peak flux of the events is then obtained from the calibrated response function of the instrument, which can be approximated by a Gaussian in temperature:

[

F195 p f195 Dtexp EM(T ) exp ⫺ Fig. 1.—Monte Carlo simulations of frequency distributions of flaring loop parameters: length N(l), width N(w), volume N(V), temperature N(Te), emission measure N(EM), and thermal energy N(E). The simulated distributions of complete samples are shown with histograms and power-law fits as thin lines. Distributions of the biased subsets (thick black lines), have been truncated to the same temperature range (T p 1.1–1.6 MK), flux threshold, and minimum area requirement as the observations (thick gray lines). Note that all the biased distributions (with incomplete temperature range) have a steeper slope than the complete (unbiased) samples. In particular, the logarithmic slope of the inferred frequency distribution of flare energies, N(E), changes from aE p 1.87 to aE p 1.35 when the temperature bias is taken into consideration.

bution of N(l) is generated with random values drawn from a uniform distribution [N(x) p constant] in the range of x p ⫺ l 1⫺a )x]1/(1⫺a). [0, …, 1], transformed by l(x) p [l11⫺a ⫹ (l 1⫺a 2 Next we generate the numerical values of loop widths w. These are statistically correlated with the loop lengths l, with w ≤ l. We simulate a corresponding distribution with a correlation w ∝ l b, including multiplicative random scatter: w(l) p w0

[

b l (1 ⫹ 0.25r) , 1 Mm

]

(3)

where r is a random number with a mean of ArS p 0 and a standard deviation of jr p 1. The factor 0.25 is chosen to reproduce the observed scatter. Following Aschwanden et al. (2000), the (elliptical) flare area A and approximate loop

(T ⫺ T195 ) 2 , 2jT,2 195

]

(9)

with f195 p 0.81 # 10 26 DN pixel⫺1 s⫺1 cm5, T195 p 1.37 MK, ˚ filter. An exposure jT, 195 ≈ 0.25 MK, for the TRACE 195 A time of Dtexp p 64.3 s was used, as in the analyzed observations. We run a Monte Carlo simulation for N p 128,000 flare events and sample subsets of events with the same criteria as in the previous observational analyses, i.e., a flux threshold with 3 j above the noise (F195 1 8 DN) and a minimum flare area of 2 macropixels (l 1 4 # 0⬙. 5 # 0.725 Mm). These two criteria introduce subtle truncation biases in every parameter distribution. The flux threshold (eq. [9]) essentially truncates values outside of the temperature range of T p 1.1–1.6 MK. The simulated distributions are shown in Figure 1. The unbiased distributions (containing all events) are shown with thin lines, the biased distributions with thick black lines, and the observed distributions in gray (taken from Aschwanden et al. 2000). We determine the unknown coefficients a, b, c, d, w0, n 0, and T0 of the unbiased distributions (in eqs. [2], [3], [6], and [7]) by fitting the power-law coefficients as well as the truncated parameter ranges of the biased distributions N(l), N(w), N(V ), N(EM), and N(E) to the observed power laws in the same parameter ranges (Fig. 1; see also Fig. 4 in Aschwanden et al. 2000). The best-fit solution is found to be a p 2.8, b p 0.78, c p 1.0, and d p 1.0, w0 p 0.5 Mm, n 0 p 2 # 10 8 cm⫺3, and T0 p 1.0 MK. This leads, via equation (1), to an unbiased frequency distribution of flare energies

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(Fig. 1, bottom right), N(E)unbiased ∝ E ⫺1.35,

(10)

which is significantly flatter than the simulated biased distribution, N(E)biased ∝ E ⫺1.87,

(11)

the latter having been optimized to match the observed distribution, N(E) obs ∝ E ⫺1.88Ⳳ0.08. As a cross-check we plot also the correlated parameters versus each other in Figure 2 and compare the associated power-law fits to the observationally inferred correlations (see also Fig. 5 of Aschwanden et al. 2000). The results are found to be fairly consistent with the observations: V(A) ∝ A1.44.V obs (A) ∝ A1.44,

(12)

E(A) ∝ A1.52.E obs (A) ∝ A1.47,

(13)

E(l) ∝ l 2.85.E obs (l) ∝ l 2.30,

(14)

E(V ) ∝ V 0.99.E obs (V ) ∝ V 1.01.

(15)

The largest disagreement is in the relation of equation (14), which is not surprising because the parameter l covers only a small range of less than a decade. We also find that the observed distributions are consistent with a length scale distribution N(l)dl ∝ l ⫺2.8 (cf. eq. [2]), and the correlations w(l) ∝ l 0.78 (cf. eq. [3]), Te (l) ∝ l 1.0 (cf. eq. [6]), ne (Te ) ∝ Te1.0 (cf. eq. [7]), which altogether yields the following dependence of the thermal energy on the temperature (see also Fig. 2, bottom right), E(Te ) ∝ Te4.5. Note that the correlation Te (l) ∝ l 1.0 is identical to that found in Aschwanden (1999) by combining a variety of data sets. The other correlation found here, ne (Te ) ∝ Te1.0, is somewhat different from that found in Aschwanden (1999), i.e., ne (Te ) ∝ Te2, probably because of a big-flare sampling bias of the underlying flare studies made in soft X-rays. Regarding the flaring volume, we find a scaling of V ∝ lw(l) 2 p l 2.56 p A1.44, which can be interpreted as a fractal scaling (McIntosh & Charbonneau 2001; McIntosh et al. 2002). 3. DISCUSSION AND CONCLUSIONS

The Monte Carlo simulation presented herein demonstrates that statistical distributions of flare energies obtained in a restricted temperature range have quite different power-law slopes than those of a complete sample spanning the entire temperature range. The correction for a particular case of TRACE-observed nanoflares in the T ≈ 1.1 –1.6 MK range reduces the power-law slope dramatically, from aE ≈ 1.9 down to aE ≈ 1.4 in the unbiased sample. Other studies of nanoflare statistics in the same temperature range, obtained with the ˚ filters of SOHO/EIT or TRACE reported power171 and 195 A law slopes between aE p 1.8 –2.6 for the thermal energy. After consolidating assumptions on the line-of-sight depth in geometric models (either h p const or h ∝ 冑A), the range of reported power-law slopes could be narrowed down to aE p

Fig. 2.—Correlations between the following flare parameters: thermal energy E, area A, length l, volume V, and temperature Te. The data points and orthogonal distance regression fits for the complete sample are shown with dots and thin lines, respectively; those of the incomplete sample from the temperature range of Te p 1.1–1.6 MK are shown with black crosses and thick lines; ˚ data (Aschwanden et al. 2000) are shown with those of the TRACE 195 A gray crosses and thick lines. Note again the differences in the power-law indices of the fits for complete versus incomplete samples. For clarity, only a random subset of 281 data points is shown from the simulated 128,000 data points.

1.8–2.1 (McIntosh & Charbonneau 2001; Benz & Krucker 2002). Additionally, we find that the temperature bias implies even a larger correction of DaE ≈ ⫺0.5, which would bring the observed range of values down to aE ≈ 1.3–1.6. The Monte Carlo simulations rest on three modeling assumptions. First, the observationally inferred power laws can be extended to lower and higher flare temperatures; this is supported by the power-law form of other flare parameters, which is indicative of self-similarity in the flaring process. Second, the loop model used to go from equation (2) to equation (9) must be a suitable representation of the flaring volume; this at first seems to be a restrictive assumption. However, the fact that other models relying on distinctly different geometrical assumptions can yield very similar V-A relationships (see, e.g., McIntosh & Charbonneau 2001) suggests that the loop model is more robust than one might anticipate a priori. Third, the seven free parameters defining our unbiased distributions are adjusted by fitting the resulting observables to observational inferences that themselves might have suffered from sampling biases similar to those we are attempting to correct for in the first place. This tricky issue is currently being investigated. Further support for our implicit assumption that the correlations Te (l) ∝ l 1.0 and ne (Te ) ∝ Te1.0 hold over the entire temperature range of flares (Te p 1–30 MK) can be sought in flare statistics reconstructed from observations made with broadband filters. Such observations are expected to be less subject to temperature biases because broadband instruments such as Yohkoh are sensitive to all temperatures above T ⲏ 2 MK. Statistics of transient soft X-ray brightenings yield indeed lower powerlaw slopes than in EUV, i.e., aE ≈ 1.6–1.7 (Shimizu 1997,

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1995), which is systematically lower by DaE ≈ ⫺0.4 than those observed in EUV. A complementary parameter is the nonthermal energy in hard X-ray–producing flare electrons, which yielded power-law slopes of a X p 1.53 Ⳳ 0.02 from the Solar Maximum Mission/Hard X-Ray Burst Spectrometer (Crosby, Aschwanden, & Dennis 1993), a X p 1.67 Ⳳ 0.02 from the International Cometary Explorer (Bromund, McTiernan, & Kane 1995), or a X p 1.39 Ⳳ 0.02 from Granat/WATCH (Crosby et al. 1998). The effects of temperature bias can thus account for what had remained the most glaring discrepancy between extant flare observational inferences and predictions of the avalanche model of Lu et al. (1993), which yield aE ⯝ 1.5 (Charbonneau et al. 2001). To sum up, both the nonthermal flare energies and the temperature-bias–corrected thermal nanoflare energies yield powerlaw slopes in the range of a ≈ 1.3 –1.6. This is in good agreement with the prediction of avalanche models of solar flares, but far

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below the critical limit of a 1 2 required for nanoflares to make a dominant contribution to coronal heating. This result poses a serious challenge to Parker’s conjecture of coronal heating by (detectable) nanoflares because of their insufficient energy budget to heat the bulk of the corona. At the same time, however, it offers further support to Parker’s picture of photospherically driven, complexly tangled coronal magnetic field as a nonlinear energy dissipation process that is capable of producing flares of all sizes. The observed power laws in flare parameters then become a natural consequence of the self-similarity characterizing the avalanching process. Part of this work was supported by NASA contracts NAS 5-38099 (TRACE) and NAS 8-00119 (Yohkoh/Soft X-ray Telescope). The National Center for Atmospheric Research is supported by the National Science Foundation.

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