ACTIVE REGION EMISSION MEASURE DISTRIBUTIONS AND IMPLICATIONS FOR NANOFLARE HEATING

The ability to carry out multiwavelength observations of active regions (ARs) has led to major advances in understanding how the coronal loops there may be heated. In particular, studies of so-called AR core loops (Warren et al. 2011, 2012; Winebarger et al. 2011; Tripathi et al. 2011; Schmelz & Pathak 2012; Bradshaw et al. 2012; Reep et al. 2013) have, for the first time, imposed serious and credible constraints on their heating. These AR cores were identified as being relatively simple locations, where the radiation from the corona was unlikely to be contaminated by that from the transition region (TR) at the loop footpoints (the "moss"), thus permitting a direct comparison with predictions from coronal heating models.

Using data from the EUV imaging spectrometer instrument on the Hinode spacecraft, with support from the Hinode X-Ray Telescope and the Atmospheric Imaging Assembly instrument on the Solar Dynamics Observatory spacecraft, Warren et al. (2011) were able to determine the dependence of the emission measure (EM) on temperature in an AR core, where EM = ∫n²dh, dh is directed along the line-of-sight and EM(T) is built up by considering many emission lines (in excess of 20 in Warren et al. 2011). EM(T) quantifies the distribution of coronal plasma as a function of temperature and can also be calculated readily from theoretical models.

In the AR cores, EM(T) peaks at around 10^6.5–6.6 K (e.g., Warren et al. 2011, 2012). Between this peak and a few 10⁵ K, it has been suggested on observational and theoretical grounds that EM(T) ∼ T^a (e.g., Jordan 1976; Withbroe 1978; Cargill 1994; Cargill & Klimchuk 2004) and indeed Warren et al. (2011) obtained a = 3.1. Subsequently, Tripathi et al. (2011) looked at another AR and found a value of a closer to 2, and surveys by Warren et al. (2012) and Schmelz & Pathak (2012) found 2 < a < 5, so that the amount of plasma around 1 MK differs considerably between different ARs. Table 3 of Bradshaw et al. (2012) provides a summary. (The earlier result of Antiochos et al. 2003 should also be noted. On the basis of the presence of 1 MK TR "moss" at the base of a loop, they inferred material at a few MK in the high corona, where no emission at 1 MK was seen.)

These authors have all sought to interpret their results in terms of nanoflare models of coronal heating. Nanoflares are believed to arise when a localized bundle of magnetic flux is sheared (or braided) with respect to its neighbors by motions in the photosphere (e.g., Parker 1988). Provided the braiding proceeds for long enough before dissipation occurs, nanoflares with energies in the range 10²³–10²⁵ erg can occur often enough to account for AR radiation losses, when summed over an entire loop structure and many nanoflares. Cargill (1994) and Cargill & Klimchuk (1997, 2004) proposed a nanoflare heating model in which a loop, or sub-element thereof, is heated rapidly and then cools before being reheated by the next nanoflare (hereafter, low frequency or LF nanoflares). In this regime, a ∼ 2 is expected (Cargill & Klimchuk 2004; Section 3 of this paper). The larger value of a obtained by Warren et al. (2011) led them to conclude that nanoflare heating occurred on a faster timescale, with reheating occurring before the loop cooled below 1 MK (hereafter, high frequency or HF nanoflares), so reducing the amount of plasma in the lower part of their observed temperature range. However, the other surveys suggest that values of a consistent with both LF and HF nanoflares arise, and atomic physics uncertainties are probably not significant enough to account for this spread of values of a with either just a LF or HF model (Bradshaw et al. 2012; Reep et al. 2013; Guennou et al. 2013).

An important tool in interpreting these results is hydrodynamic loop models that study the field-aligned plasma response in a sub-element of a loop to heating in the form of a "train" of nanoflares, where the "train" is defined as a semi-infinite sequence of nanoflares applied to the same field line. Through two such simulations, Warren et al. (2011) demonstrated the different EM(T) profiles to be expected from LF and HF nanoflares, with the former showing a ∼ 2 and the latter being very sharply peaked around the temperature of maximum EM. Bradshaw et al. (2012) carried out a more extensive set of LF nanoflare simulations and showed that a was indeed of order two while Reep et al. (2013) presented a "tapered" HF nanoflare train in which the plasma was allowed to cool to below 1 MK after the train terminated. They found 1.5 < a < 4. Thus, the LF trains cannot account for the range of values of a seen, while the tapered HF trains may account for a broader range of a.

This paper explores another option that can account for the range of a. An important previous limitation is the assumption of equally spaced nanoflares having the same energy. We remove this assumption and present a series of zero-dimensional (0D) hydrodynamic models to address how the observed range of EM(T) scalings arise. In particular, we will show that it is difficult for equal-energy, constant-separation nanoflare trains to produce the entire range of observed values of a in a convincing manner. However, distributions in which the nanoflare energy is not constant, especially when the time between each nanoflare is taken to be proportional to the energy released in the latter event, has more success. Section 2 outlines our methodology, and Section 3 outlines the results. Section 4 outlines the implications for magnetohydrodynamic (MHD) coronal processes involved in impulsive heating by both our results and those of Reep et al. (2013).

The major assumption of this paper is that the corona in AR loops is heated by small energy releases (nanoflares) that form a semi-infinite sequence, referred to as a "nanoflare train." While discussions of nanoflares are sometimes restricted to events of the order of 10²⁴ erg, we have no particular precondition as to their size. The nanoflare is assumed to heat a sub-element of the loop with cross-section A_h. This is sometimes referred to as a flux bundle or strand; here we use the term "sub-loop." It is assumed that A_h is much smaller than the cross-section of the observed loop and that the observed coronal signature is the convolution of many sub-loops.

2.1. EBTEL Methods

2.2. Radiative Losses

2.3. Input Parameters

Before presenting new results, we discuss briefly the EM distribution arising from a single nanoflare. Figure 1 shows EM(T) for τ_H = 100 s and H₀ = 0.38 erg cm⁻³ s⁻¹ in a sub-loop of half-length 40 Mm and an assumed line-of-sight of 10% of the total length, as used in all our results. Warren et al. (2011) used 17%. Three loss functions are shown: RTV (dots), EBTEL08 (dashed)b, and RL12 (solid). The EM peaks at approximately T = T(EM_max) = 10^6.6 K. For T > T(EM_max), the coronal part of the sub-loop cools predominately by thermal conduction to the TR and chromosphere (e.g., Antiochos & Sturrock 1976, 1978). Below T(EM_max), cooling is by a combination of optically thin radiation from the corona to space and an enthalpy flux to power the TR radiative losses (Bradshaw & Cargill 2010a, 2010b). The RTV, EBTEL08, and RL12 losses have a = 1.7, 2.3, and 2.7, respectively, between 1 MK and T(EM_max).

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Cargill (1994) noted that below T(EM_max), EM(T) ∼ n²τ_rad, where τ_rad ∼ T^1−α/n is the instantaneous radiative cooling time. Assuming that T ∼ n² in the radiative phase (Serio et al. 1991; Jakimiec et al. 1992; Reale et al. 1993; Cargill et al. 1995), the result EM ∼ T^3/2−α ∼ T² arises from a simple fit to the RTV losses. More generally, the radiative phase has T ∼ n^l, with l ∼ 1 for very long loops (Bradshaw & Cargill 2010b) and l ∼ 2.5 from short ones (Reale et al. 1993; Bradshaw & Cargill 2010b) and one finds EM ∼ T^{1/l + 1 − α} (Cargill & Klimchuk 2004; Bradshaw et al. 2012). So steeper loss functions (more negative α) give larger values of a, as seen in Figure 1. The largest value of a to be expected from a single nanoflare is of the order of three for the case of a very long loop (L > 10¹⁰ cm) and steep loss function. Figures 2 and 3 show both the EBTEL08 and RL12 losses. Figure 4 shows EBTEL08 losses.

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3.1. Equally Separated Nanoflares

3.2. Power Law Distributions of Nanoflares

We next consider the nanoflare distribution to be a power law, N(Q) ∼ Q^−m. It is well known that when the power law extends over several decades, the value m = 2 corresponds to the transition between the input power dominated by large (small) events for m < 2 (m > 2): Hudson (1991). Data from solar flares suggests that m < 2 (Crosby et al. 1993; Crosby 2011). On the other hand, firm evidence for the value of m at nanoflare energies is difficult to obtain. Different analysis methods give values of m > and <2 for observed non-flaring brightenings (Parnell 2004). There is also ambiguity about how N(Q) is modified by plasma cooling processes prior to any observation of intensity fluctuations at EUV wavelengths (Parenti et al. 2006). Further, given that nanoflares may be <10²⁴ erg (see below and Testa et al. 2013), any plasma brightening due to a single nanoflare may be close to unobservable.

Such power law distributions also arise from theoretical and computational modeling of a driven corona (see Charbonneau et al. 2001 for a review). For example, Lu & Hamilton (1991) and Lu et al. (1993) obtained m ∼ 1.4 for a simple model of the coronal field in a self-organized state. Vlahos et al. (1995) used different "rules" governing field dissipation to show that m > 2, at least for smaller events. Numerical simulations of a nanoflare-heated corona (e.g., Rappazzo et al. 2008; Bingert & Peter 2013) also lead to power law distributions generally with m < 2. An important caveat to all these results is that one is unsure whether the correct dissipative processes are modeled and/or resolved. We subsequently consider values of m > and <2 but continue to restrict Q to be distributed over a decade, with the individual nanoflare energies being calculated to ensure the correct total power (8.3 ×10⁻³ erg cm⁻³ s⁻¹) is going into the sub-loop. The lower panels of Figures 3 show results for m = 2.5 and are quite similar to the random distribution discussed above.

Next, we propose that the time between consecutive nanoflares depends on the energy of the second nanoflare, since larger nanoflares will take longer to build up their energy. Two scalings are considered: Q ∝ T_N and Q ∝ T_N², with the motivation discussed fully in Section 4 where we will argue that the former is the more relevant scaling for AR core loops. The results for m = −2.5 (−1.5) are shown in the upper (lower) parts of Figure 4, where the horizontal axes in the right hand column now show an average T_N, so that, for example, when 〈T_N〉 = 1000 s, the delay between individual nanoflares can range from a few hundred to 3000 s. The left and upper right panels show examples with Q ∝ T_N only. The lower right panel, which shows the dependence of a on T_N, has stars denoting Q ∝ T_N and diamonds Q ∝ T_N².

Instead of the abrupt transition in T_N over a few hundred seconds between a being of the order of 2–3 and having no physical value, a now slowly increases from ∼2 for large T_N to ∼5 for T_N over a few thousand seconds. The transition to sharply peaked EM(T) occurs at larger T_N for Q ∝ T_N². The lower panels show a similar result holds for the flatter power law (m = 1.5). It seems possible that the range of a observed could correspond to differences between the various ARs (e.g., magnetic field strengths and small-scale field topologies) leading to different detailed dissipation processes (a range of T_N and hence Q) within a generic nanoflare scenario. The important point is that one is no longer tied to a very narrow range of T_N to give the observed range of a.

The reason behind this result is seen in Figure 5 which shows the temperature and density evolution for two cases with a power law nanoflare distribution, Q ∝ T_N, and m = 2.5, where the upper (lower) rows having equally spaced nanoflares (a "waiting time" between nanoflares). The important point is that when a waiting time is included, the plasma heated by a nanoflare prior to a much larger event cools through the entire temperature range to and below 1 MK, so contributing to EM over a broad temperature range. This does not happen when the nanoflares occur at equal intervals. The slow increase in a as 〈T_N〉 decreases is because there are an increasing number of nanoflares and so fewer cases where a long delay occurs prior to a large nanoflare. Eventually, for some small T_N, the delay between nanoflares is always short enough that in no cases is there cooling to 1 MK. (Note that the cooling time is relatively independent of the energy in the nanoflare; See the Appendix and Cargill et al. 1995). For Q ∝ T_N², there is a smaller range of times between nanoflares for a given 〈T_N〉, so that as 〈T_N〉 decreases, the probability decreases of a sub-loop cooling below 1 MK prior to reheating. This is evident in Figure 4, with larger values of a between 1000 and 2000 s.

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3.3. Parameter Variations

We now study the effect of changing L and H_n. The important quantity in understanding these results is the ratio of T_N to the time taken for the sub-loop to cool to below 1 MK from its peak temperature (the cooling time, τ_cool: typically in the range 1000–3000 s). In general terms, if τ_cool < (>) T_N, one is in the LF (HF) nanoflare regime. An expression for τ_cool is derived in the Appendix for the case of a radiative loss function that has a single power law dependence on temperature. It is linearly proportional to L and depends only weakly on the nanoflare energy (see also Cargill 1993; Cargill et al. 1995).

We define the quantity $T_{N^{\ast}}$ to be the value of T_N below which no valid value of a is obtained (i.e., when τ_cool > T_N). Table 2 summarizes the results for Q ∝ T_N. The upper three rows show $T_{N^{\ast}}$ for a constant nanoflare train (as in Figure 2) and when a waiting time between nanoflares is introduced (Figure 4) for three loop lengths. The scaling between $T_{N^{\ast}}$ and L is not precisely linear, as predicted in the Appendix, but is adequate given the approximations made in obtaining Equation (A2); see Cargill et al. (1995) for more details. (For very short loops, $T_{N^{\ast}}$ is <250 s when a waiting time is included.) The last three rows show a as a function of H_n and L for T_N $\gg T_{N^{\ast}}$; this is independent of the form of the nanoflare train distribution. Here, a lies in a fairly narrow range but does increase as L increases, as we discussed at the start of this section, due to the influence of gravitational stratification in long loops. Further, the net effect of increasing L and/or H_n is to raise the entire range of a over all T_N.

Table 2. Summary of Parameter Variations

2L (Mm)	40	80	120
$T_{N^{\ast}}$(s): constant train	1000	2250	4500
〈$T_{N^{\ast}}$〉(s): waiting time	...	250	1000
Hn (10−3 erg cm−3 s−1)	...	...	...
4	a = 1.7	1.9	2.5
8	a = 2	2	2.6
16	a = 2.2	2.2	2.6

Notes. The second and third rows show the critical time between nanoflares ($T_{N^{\ast}}$) as a function of loop lengths for a nanoflare train with equally spaced nanoflares and one with waiting time included in a power law distribution. The last three rows and columns show the asymptotic value of a for T_N $\gg T_{N^{\ast}}$ as a function of the loop heating rate and loop length for an equally spaced nanoflare train.

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The results of the previous section can be summarized as follows.

• 1.  
  Uniform nanoflare trains lead to a of the order of 2–3 (depending on the radiative loss function used) above some value of T_N. Below that value, the EM distribution is sharply peaked around the temperature of peaked emission. Random distributions lead to similar results.
• 2.  
  Power law distributions of nanoflares lead to a broad range of a (2 < a < 5) if there is a waiting time before a nanoflare that depends on its energy. Larger values of a arise for smaller T_N.
• 3.  
  The loop length is a key parameter. Long (short) loops have a higher (lower) value of $T_{N^{\ast}}$ and larger (smaller) value of a. The results show little dependence on the magnitude of the energy released in the loop.

Points (1) and (2) suggest that uniform nanoflare trains cannot produce the observed range of EM slopes, and that quite steep power laws are required, though an exception to this occurs when a HF nanoflare train is "tapered" (Reep et al. 2013). It also appears as if the observed values of a require quite small values of T_N. This has important implications, as discussed in a moment. Regarding point (3), we note that Warren et al. (2012) find larger a for larger AR areas. From their images, it is clear that there is no direct one-to-one relation between AR area and loop length, though one would suspect that larger ARs would contain longer loops.

We now discuss the implications for nanoflare heating. Defining B_t and B_a as the reconnecting (transverse) and axial (guide) field components in the corona (following Parker 1988), and v a typical photospheric velocity, then the energy supplied to the corona is roughly B_tB_av/4π erg cm⁻² s⁻¹, with B_t ≈ B_avt/2L. If we assume a nanoflare involves the release of all the stored magnetic energy (assumed to be contained in B_t), then $Q = ({B_{tN}^2 /8\pi })A_h 2L$ where B_tN = B_t(t = T_N), the time taken to build up to a nanoflare energy is T_N = (2L/v)(B_tN/B_a), and Q ∝ ${T}_{N}^{2}$. The nanoflare energy can then be related to H_n (Section 2.3) to give the condition $({{B_{tN} }}/{{B_a }}) = ({{8\pi }}/{{B_a^2 }})({{H_n 2L}}/{v})$that needs to be met to satisfy the coronal energy requirements.¹ If we take typical AR numbers, H_n = 8 × 10⁻³, B_a = 150 G, 2L = 8 × 10⁹, v = 1 km s⁻¹, then B_tN/B_a ∼ 0.7, T_N = 56,000 s and Q is of order 10²⁶ erg for A_h ∼ 10¹⁴ cm² and 10²⁴ erg for A_h ∼ 10¹² cm², with smaller H_n having smaller values of B_tN/B_a and T_N. Two points should be noted for the values of H_n used in Table 2: (1) the build-up time is a factor 5–20 times longer than the T_N required to reproduce the EM(T) results discussed in Section 3 and (2) any model which assumes B_t ≪ B_a (such as those using reduced MHD; Rappazzo et al. 2008; Dahlburg et al. 2012) will have difficulties in accounting for AR heating.

The discrepancy between a simple nanoflare model that implies a long T_N and the observed EM(T) results that require a short T_N can be partially resolved as follows. Assume that during the nanoflare, B_t relaxes to B_t0 = B_t − dB_tN and replenishment of the coronal energy by footpoint motions begins at B_t = B_t0, with dB_t = B_avt/2L and dB_tN = B_avT_N/2L. The rate energy is injected is then vA_h(B_a(B_t0 + dB_t)/4π)which integrates to give, on rewriting in terms of dB_tN, Q = 2LA_hdB_tN(2B_t0 + dB_tN)/8π. For B_t0 = 0, we recover the earlier results. For dB_tN = B_t0, Q ∝ T_N, Q and T_N are both smaller by roughly a factor dB_tN/B_t0, while there are B_t0/dB_tN more nanoflares, ensuring that the total energy requirements are met. If T_N is to be reduced to the range of values required for the observed EM(T), we conclude that the nanoflare must be associated with a small decrease in B_t while B_t0 remains large. Setting T_N = 1000 s, and retaining the same values of B_a, B_t0, L, and v as before, dB_t = 1.9 G, Q = 3 × 10²⁴ erg for A_h = 10¹⁴ cm², and Q = 3 × 10²² for A_h = 10¹⁴ cm². Q ∼ a few 10²³ is inferred from AR moss studies using the Hi-C instrument (Testa et al. 2013).

How this can occur in the frameworks of magnetohydrodynamics is unclear at the moment but one suggestion comes from the idea of helicity conservation. Here, a stressed state does not relax to a potential field but that corresponding to a "constant-α force-free field" (Taylor 1974; Heyvaerts & Priest 1984). Were that state to be close to the stressed one, a small release of energy could be the outcome. Further studies are needed.

The relation between Q and T_N is important for our results, and an early discussion of its consequences for a range of astrophysical X-ray sources can be found in Rosner & Vaiana (1978). We note that models of a driven corona such as initiated by Lu and Hamilton (e.g., Lu et al. 1993) and flare observations (e.g., Crosby et al. 1998) suggest that there is little if any connection between Q and T_N. However, the waiting times in these cases are evaluated over a large volume (e.g., an entire AR or computational box; see also Charbonneau et al. 2001) and so yield no useful information about T_N in a sub-loop such as we are concerned with. Other theoretical models suggest various scalings for "local" waiting times (e.g., Berger 1994; Craig 2001; Wheatland & Craig 2003). Berger (1994) and Sturrock & Uchida (1981) argue that rotational twisting gives Q ∼ T_D while Wheatland & Craig (2003) discuss magnetic flux pile-up at a separator leading to Q ∼ T_D². However, none of these examples consider the consequences of injecting energy into a loop that already has a quite stressed field.

We have studied the expected EM distribution as a function of temperature for models of nanoflare heating of AR loop cores, with emphasis on the temporal distribution of nanoflares. The two main results are (1) the best option for obtaining agreement with observed values of the slope of the nanoflare distribution EM(T) ∼ T^a lies in the incorporation of a waiting time prior to a nanoflare that is proportional to its energy and (2) the time separating nanoflares has to be of the order of a few hundred to somewhat over 2000 s, thus ruling out the possibility that a nanoflare involves the relaxation of the coronal field to a near-potential state. We thus argue that nanoflares occur in a corona that is continually stressed, with a small energy release permitting a subsequent rapid replenishment of energy.

Our work has focused solely on slow photospheric driving. However, other authors (van Ballegooijen et al. 2011; Asgari-Targhi & van Ballegooijen 2012; Asgari-Targhi et al. 2013) have developed "turbulent" models that also involve very bursty energy release that one could term nanoflares. In this work, Alfvén waves are injected into the chromosphere and corona by photospheric footpoint motions of the order of 1 km s⁻¹, with a correlation time of the order of a minute. Both footpoints are set in motion and the interaction of counter-propagating waves leads to a rapid turbulent cascade. Although the reduced MHD assumption is made, these models do not suffer from the "energy deficiency" problem discussed in Section 4 (see Table 2 of Asgari-Targhi & van Ballegooijen 2012).

Space does not permit a full discussion of their results but Figure 7 of Asgari-Targhi & van Ballegooijen (2012) is a reasonable sample and shows that for loop lengths of the order 100 Mm, a maximum temperature of the order of 2–3 MK is obtained, consistent with AR observations. In all cases, there is evidence for intermittent energy release, with bursts being separated by a hundred seconds or so, leading to small temperature fluctuations around a constant value (see Figure 9 of Asgari-Targhi & van Ballegooijen 2012). This suggests that such a turbulent heating model cannot account for the observed EM slopes, instead giving a very sharply peaked EM(T). However, with improved numerical resolution, finer spatial structure may appear, perhaps leading to the dissipation occurring in smaller regions.

The Hinode data described earlier in this paper thus appear to impose important constraints on how nanoflare heating may be operating, and it is in a different way from commonly supposed. Indeed, one can finally say that quantitative analysis of coronal heating is at last possible. The other result from this work, and that of Testa & Reale (2012) and Testa et al. (2011), concerns the presence of hot coronal components with temperatures in excess of T(EM_max). This has long been a prediction of nanoflare models (e.g., Cargill 1994, 1995) but is very difficult to interpret due to ionization nonequilibrium (Bradshaw & Cargill 2006), the predicted small EMs, and plasma physics issues associated with heat conduction in very hot plasmas (West et al. 2008). This would appear to be a productive area to focus on following the insights obtained at lower temperatures, as described here.

This work was performed as a contribution to a team supported by the International Space Science Institute (ISSI) and led by Helen Mason and Steve Bradshaw. I am grateful to Harry Warren, Jim Klimchuk, and Steve Bradshaw for helpful discussions and to Aad van Ballegooijen for providing additional information about his work. The referee raised an important issue that led to an improved paper.

Cargill et al. (1995) evaluated the time taken for a loop to cool from an initial temperature T₀ and n₀. The analysis assumed that the loop cooled first by conduction and then by radiation, the change between cooling processes occurring at a temperature T_*. Analytic solutions due to Antiochos & Sturrock (1978) were used for the conductive phase and the scaling T ∼ n² (Serio et al. 1991; Reale et al. 1993; Cargill et al. 1995) was used in the radiative phase. As presented in Cargill et al. (1995), the solution was "hard wired" to a radiative loss function R_L = χT^α = 6 × 10⁻²⁰ T^−1/2. We have discussed earlier that recent work has led to greatly increased losses below a few MK, so that in any single function fit, the value of α will be more negative, with χ being adjusted accordingly to give a larger loss at 1 MK. By following the same analysis as Cargill et al. (1995), it can be shown that the total cooling time is

$$\begin{equation} \tau _{{\rm cool}} = \left({\frac{{2 - \alpha }}{{1 - \alpha }}} \right)\left({\tau _{c0}^{4 - 2\alpha } \tau _{r0}^7 } \right)^{1/(11 - 2\alpha)}, \end{equation} \tag{ A1 }$$

where τ_c0 and τ_r0 are the cooling times defined at t = 0: $\tau _{c0} = 3kL^2 n_0 /\kappa _0 T_0^{5/2}$, τ_r0 = 3kT^{1 − α}/χn, and τ_r0 ≫ τ_c0. One can tidy up Equation (A1) in terms of the loop length and initial pressure (p₀):

$$\begin{equation} \tau _{\rm cool} = \left({\frac{{2 - \alpha }}{{1 - \alpha }}} \right)3k\left({\frac{1}{{\kappa _0^{4 - 2\alpha } \chi ^7 }}\frac{{L^{8 - 4\alpha } }}{{(n_0 T_0)^{3 + 2\alpha } }}} \right)^{1/(11 - 2\alpha)}. \end{equation} \tag{ A2 }$$

Setting α = −1/2 recovers the scalings in Equation (14E) of Cargill et al. (1995). If we now relate the initial loop pressure to the energy in a nanoflare using Q = 3n₀kT₀H₀τ_HLA_h/2, Equation (A2) can be used to show that τ_cool ∼ LQ^{−(3 + 2α)/(11 − 2α)}, so τ_cool scales exactly linearly with L, independent of the loss function slope, and depending weakly on Q.