The Role of Energy Diffusion in the Deposition of Energetic Electron Energy in Solar and Stellar Flares

TL;DR

This study uses kinetic modeling to show how energy diffusion and thermalization of electrons in the solar corona influence energy deposition in solar flares, revealing delayed heating effects that can diagnose flare properties.

Contribution

It introduces a full collisional warm-target model (WTM) for electron transport, improving upon the cold thick-target model by including thermalization and plasma properties.

Findings

Energy deposition depends on coronal plasma properties.

Thermalization increases non-thermal energy transfer to the chromosphere.

Delayed heating occurs over seconds, affecting flare diagnostics.

Abstract

During solar flares, a large fraction of the released magnetic energy is carried by energetic electrons that transfer and deposit energy in the Sun's atmosphere. Electron transport is often approximated by a cold thick-target model (CTTM), assuming that electron energy is much larger than the temperature of the ambient plasma, and electron energy evolution is modeled as a systematic loss. Using kinetic modeling of electrons, we re-evaluate the transport and deposition of flare energy. Using a full collisional warm-target model (WTM), we account for electron thermalization and for the properties of the ambient coronal plasma such as its number density, temperature and spatial extent. We show that the deposition of non-thermal electron energy in the lower atmosphere is highly dependent on the properties of the flaring coronal plasma. In general, thermalization and a reduced WTM energy loss rate leads to an increase of non-thermal energy transferred to the chromosphere, and the deposition of non-thermal energy at greater depths. The simulations show that energy is deposited in the lower atmosphere initially by high energy non-thermal electrons, and later by lower energy non-thermal electrons that partially or fully thermalize in the corona, over timescales of seconds, unaccounted for in previous studies. This delayed heating may act as a diagnostic of both the injected non-thermal electron distribution and the coronal plasma, vital for constraining flare energetics.

Tables

Table 1. Table 1: The percentage of available non-thermal electron power deposited in the corona and chromosphere for set (a) shown in Figure 4 . The 100 keV values are shown in italic since 100 keV electrons can travel further in this atmosphere than calculated within ± 30 ​ ″ plus-or-minus 30 ″ \pm 30\arcsec .

Set (a)	CTTM		WTM (T=20 MK)
Energy [keV] (E/T)	Chromosphere ($\%$)	Corona ($\%$)	Chromosphere ($\%$)	Corona ($\%$)
10 (5.8)	0	100	61.1	38.9
20 (11.6)	0	100	21.3	78.7
30 (17.4)	5.3	94.7	31.7	68.3
40 (23.2)	64.2	35.8	63.7	36.3
50 (29.0)	80.7	19.3	79.6	20.4
100 (57.9)	88.7	11.3	88.7	11.3

Table 2. Table 2: The percentage of available non-thermal electron power deposited in the corona and chromosphere for set (b) shown in Figure 5 . The 100 keV values are shown in italic since 100 keV electrons can travel further in this atmosphere than calculated within ± 30 ​ ″ plus-or-minus 30 ″ \pm 30\arcsec .

Set (b)	CTTM		WTM (T=20 MK)
Energy [keV] (E/T)	Chromosphere ($\%$)	Corona ($\%$)	Chromosphere ($\%$)	Corona ($\%$)
10 (5.8)	0	100	46.8	53.2
20 (11.6)	48.3	51.7	55.6	44.4
30 (17.4)	83.8	16.2	83.6	16.4
40 (23.2)	91.5	8.5	91.5	8.5
50 (29.0)	93.5	6.5	93.5	6.5
100 (57.9)	88.7	11.3	88.7	11.3

Table 3. Table 3: The percentage of available non-thermal electron power deposited in the corona and chromosphere for set (c) and set (d) shown in Figure 6 .

Set (c)	CTTM		WTM (T=20 MK)
Temperature [MK] (E=30 keV/T)	Chromosphere ($\%$)	Corona ($\%$)	Chromosphere ($\%$)	Corona ($\%$)
10 (34.8)	47.1	52.9	51.8	48.2
20 (17.4)	47.1	52.9	50.7	49.3
30 (11.6)	47.1	52.9	53.5	46.5
100 (3.5)	47.1	52.9	76.6	23.4

Table 4. Table 4: The percentage of available non-thermal electron power deposited in the corona and chromosphere for set (e) and set (f) shown in Figure 7 .

Set (e; 1″bins)	CTTM		WTM (T=20 MK)
$E_c$ [keV] (Ec/T)	Chromosphere ($\%$)	Corona ($\%$)	Chromosphere ($\%$)	Corona ($\%$)
20 (11.6)	29.5	70.5	47.5	52.6
10 (5.8)	6.5	93.5	36.3	63.7

Equations

$$\begin{split}\mu\frac{\partial F}{\partial z}&=\Gamma m_{e}^{2}\left(\frac{\partial}{\partial E}\left[G(u[E])\frac{\partial F}{\partial E}+\frac{G(u[E])}{E}\left(\frac{E}{k_{B}T}-1\right)F\right]\right)\\ &+\Gamma m_{e}^{2}\left(\frac{1}{8E^{2}}\frac{\partial}{\partial\mu}\left[(1-\mu^{2})\biggl{(}{\rm erf}(u[E])-G(u[E])\biggr{)}\frac{\partial F}{\partial\mu}\right]\right)\\ &+S(E,z,\mu),\,\,\,\,\end{split}$$

$$\mathrm{erf}(u)\equiv (2/\sqrt{\pi })\underset{0}{\overset{u}{\int }}\exp (-t^2)dt$$

$$G(u)=\frac{\mathrm{erf}(u)-u{\mathrm{erf}}^{{}^{\prime }}(u)}{2u^2}.$$

$$z_{j+1}=z_j+{\mu }_j\Delta s;$$

$$\begin{split}E_{j+1}&=E_{j}-\frac{\Gamma m_{e}^{2}}{2E_{j}}\,\bigg{(}{\rm erf}(u_{j})-2u_{j}{\rm erf^{\prime}}(u_{j})\bigg{)}\,\Delta s\\ &+\sqrt{2\,\Gamma m_{e}^{2}\,G(u_{j})\,\Delta s}\,\,W_{E}\,\,\,;\,\,\,\end{split}$$

$$\begin{split}\mu_{j+1}&=\mu_{j}-\frac{\Gamma m_{e}^{2}\biggl{(}{\rm erf}(u_{j})-G(u_{j})\biggr{)}}{4E_{j}^{2}}\,\,\mu_{j}\,\Delta s\\ &+\sqrt{\frac{(1-\mu_{j}^{2})\,\Gamma m_{e}^{2}\,\biggl{(}{\rm erf}(u_{j})-G(u_{j})\biggr{)}}{4E_{j}^{2}}\,\Delta s}\,\,W_{\mu}\,\,\,.\end{split}$$

$$\Delta E_{j+1}(z)=E_j(z)-E_{j+1}(z),$$

$$t=\sum _j\frac{|z_{j+1}-z_j|}{\sqrt{2E_j/m_e}}.$$

$$P(z)={\int }_0^{\infty }EF(E,z)dE.$$

$$P={\int }_{E_c}^{\infty }{E}_{0}F(E_0)d{E}_0=\dot{N_0}{E}_c\frac{(\delta -1)}{(\delta -2)},$$

$$n=10^{12}[\mathrm{c}{\mathrm{m}}^{-3}]+n_{\mathrm{photo}}\exp \left(-\frac{|z|+23\text{arcsec}.5}{h_0}\right),$$

$$\ln \Lambda =\ln \left[\frac{3}{2e^3}{\left(\frac{k_B^3{T}^3}{\pi n}\right)}^{1/2}\right].$$

$${\tau }_D=\sqrt{\frac{8m_e}{\pi }}\frac{Kn{L}^2}{3}(k_{B}T{)}^{-5/2}$$

Full text

Sun: flares – Sun: X-rays – Sun: atmosphere – Sun: chromosphere – Sun: corona – stars: flare

††journal: ApJ

1 Introduction

Solar flares are a product of the Sun’s magnetic energy being released and then ultimately dissipated in different layers of its vast atmosphere. The release of magnetic energy, initiated by magnetic reconnection in the corona (e.g., Parker, 1957; Sweet, 1958; Priest & Forbes, 2000), is partitioned into thermal and non-thermal particle energies (e.g., Emslie et al., 2012; Aschwanden et al., 2015, 2017; Warmuth & Mann, 2016), and kinetic plasma motions (‘turbulence’), a vital energy transfer mechanism in the process (e.g., Larosa & Moore, 1993; Petrosian, 2012; Vlahos et al., 2016; Kontar et al., 2017). However, the bulk of the released energy is eventually transferred to the Sun’s cool and dense low atmosphere (the chromosphere) causing rapid heating, ionization (cf Fletcher et al., 2011; Holman et al., 2011), and an expansion of the lower atmospheric material - “chromospheric evaporation” (e.g., Sturrock, 1973; Hirayama, 1974; Acton et al., 1982; Holman et al., 2011). The heated chromosphere, a thin and complex layer, is a prime source of deposited energy information, mainly radiating in optical and ultraviolet (UV) wavelengths (e.g., Hirayama, 1974; Kretzschmar, 2011; Woods et al., 2006). Energy is likely transferred in a variety of different, but closely connected, ways: by flare-accelerated electrons, as evident from hard X-ray (HXR) observations (Holman et al., 2011), by thermal conduction (e.g., Culhane et al., 1970), and complicated by various plasma waves e.g. acoustic waves (e.g. Vlahos & Papadopoulos, 1979), Alfvén waves (e.g. Emslie & Sturrock, 1982; Fletcher & Hudson, 2008), Langmuir waves (Emslie & Smith, 1984; McClements, 1987; Hannah et al., 2009), whistler waves (e.g. Bespalov et al., 1991; Stepanov et al., 2007), and then dissipated via turbulence, even in the lower atmosphere e.g. Jeffrey et al. (2018).

In the flare impulsive phase, X-ray observations (Benz, 2008; Kontar et al., 2011b) suggest that non-thermal electrons are the main source of low atmosphere heating and radiation. Bremsstrahlung X-rays provide a relatively direct diagnostic of the properties of flare-accelerated electrons (cf Kontar et al., 2011b) in the corona and in the dense lower atmosphere via HXR footpoints (e.g. Hoyng et al., 1981). Higher energy HXRs are observed to be produced in progressively lower regions of the chromosphere (Aschwanden et al., 2002; Kontar et al., 2010) by electron-ion (mainly proton) collisions, and via electron-electron collisions above $\sim$300 keV (Kontar et al., 2007). However, electrons predominantly exchange energy via electron-electron collisions (cf Holman et al., 2011). Flare observations of ‘coronal thick-target’ sources (e.g., Aschwanden et al., 1997; Veronig & Brown, 2004; Xu et al., 2008; Kontar et al., 2011a; Guo et al., 2012; Jeffrey & Kontar, 2013) show that electrons with energies up to $\approx$30 keV can thermalize in the corona in high density conditions. However, more general statistical studies of large flares (e.g. Caspi et al., 2014; Aschwanden et al., 2015) show that the flaring corona, at least within the main phase, is often a highly collisional environment. Further, it is likely that non-collisional transport effects such as turbulent scattering by magnetic fluctuations (Bespalov et al., 1991; Kontar et al., 2014), beam-driven Langmuir wave turbulence (Hannah et al., 2009), electron re-acceleration (Brown et al., 2009) and/or beam-driven return current (Knight & Sturrock, 1977; Emslie, 1980; Zharkova & Gordovskyy, 2006; Alaoui & Holman, 2017) are also operating during flares, complicating the overall transport.

For the last fifty years, the properties of non-thermal electrons (their transport, deposition and the heating of the lower atmosphere), are often determined using the ‘cold-thick-target’ collisional transport model (hereafter CTTM) (e.g., Brown, 1971; Syrovatskii & Shmeleva, 1972; Emslie, 1978). The CTTM assumes that the energy $E$ of non-thermal electrons is much larger than the ambient plasma temperature $T$, and hence ‘cold’ (i.e., $T\ll E$). Although this assumption is valid for high-energy electrons that reach the cool layers of the flaring chromosphere, decades of observational evidence with e.g., Yohkoh (Tsuneta et al., 1991) and the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI; Lin et al. (2002)) show high coronal temperatures of 10-30 MK during flares. However, the lasting appeal of the CTTM is its simple analytic form, that can be readily applied to X-ray data, but its use leads to the well-known ‘low-energy cut-off’ problem, whereby the power associated with non-thermal electrons cannot be constrained from X-ray spectroscopy. Firstly, Jeffrey et al. (2014), building upon Emslie (2003) and Galloway et al. (2005), studied electron transport using a full collisional model including finite temperature effects, diffusion and pitch-angle scattering, and showed the importance of including the properties of the coronal plasma (its finite temperature, density and extent). Critically, the inclusion of both thermalization and spatial diffusion led to the “warm-target model” (hereafter WTM; derived by Kontar et al. (2015)) that can resolve the problems associated with determining the low-energy cut-off in the CTTM, finally allowing the power of flare-accelerated electrons to be constrained (Kontar et al., 2019) from X-ray data. In a WTM, the properties and energy content of non-thermal electrons are constrained by determining the plasma properties of the flaring corona.

Here, using full collisional kinetic modelling, we re-investigate flare-accelerated electron energy deposition. As expected, we show that the coronal plasma properties (e.g. temperature, number density and spatial extent) determine how non-thermal electron power is deposited in the chromosphere. Ultimately, we show for a given non-thermal electron distribution, a greater proportion of the non-thermal electron power can be deposited in the lower atmosphere than predicted in the CTTM.

2 Electron transport and deposition in hot collisional plasma

To determine how the energy of flare-accelerated electrons is transported and deposited in a hot collisional plasma (i.e. in a full collisional WTM), we use the kinetic electron transport simulation first discussed in Jeffrey et al. (2014) and Kontar et al. (2015). We model the evolution of an electron flux $F(z,E,\mu)$ [electron erg*-1* s*-1* cm*-2*] in space $z$ [cm], energy $E$ [erg], and pitch-angle $\mu$ to a guiding magnetic field, using the Fokker-Planck equation of the form (e.g., Lifshitz & Pitaevskii, 1981; Karney, 1986):

[TABLE]

where $\Gamma=4\pi e^{4}\ln\Lambda\,n/m_{e}^{2}=2Kn/m_{e}^{2}$, and $e$ [esu] is the electron charge, $n$ is the plasma number density [cm*-3*] (a hydrogen plasma is assumed), $m_{e}$ is the electron rest mass [g], and $\ln\Lambda$ is the Coulomb logarithm. The variable $u(E)=\sqrt{E/k_{B}T}$, where $k_{B}$ [erg K*-1*] is the Boltzmann constant and $T$ [K] is the background plasma temperature. The functions ${\rm erf}(u)$ (the error function) and $G(u)$ are given by,

[TABLE]

and

[TABLE]

Equation (1) is a time-independent equation useful for studying solar flares where the electron transport time from the corona to the lower atmosphere is usually shorter than the observational time (i.e. most X-ray observations have integration times of tens of seconds to minutes).

Here, Equation (1) models electron-electron collisions only, the dominant electron energy loss mechanism in the flaring plasma111We note that electron-proton interactions are important for collisional pitch-angle scattering, but here we only model electron-electron interactions. Equation (1) can be generalized to model any particle-particle collisions.. $S(E,z,\mu)$ plays the role of the electron flux source function and the properties of the injected electron distribution are discussed in Section 2.3.

Following Jeffrey et al. (2014), and re-writting Equation (1) as a Kolmogorov forward equation (Kolmogorov, 1931), Equation (1) can be converted to a set of time-independent stochastic differential equations (SDEs) (e.g., Gardiner, 1986; Strauss & Effenberger, 2017) that describe the evolution of $z$, $E$, and $\mu$ in It$\hat{\text{o}}$ calculus:

[TABLE]

[TABLE]

[TABLE]

$\Delta s$ [cm] is the step size along the particle path, and $W_{\mu}$, $W_{E}$ are random numbers drawn from Gaussian distributions with zero mean and a unit variance representing the corresponding Wiener processes (e.g. Gardiner, 1986). A simulation step size of $\Delta s=10^{5}$ cm is used in all simulations, and $E$, $\mu$ and $z$ are updated at each step $j$. A step size of $\Delta s=10^{5}$ cm is approximately two orders of magnitude smaller than the thermal collisional length in a dense ($n=10^{11}$ cm*-3*) plasma with $T\geq 10$ MK (or the collisional length of an electron with an energy of $1$ keV or greater, in a cold plasma). The simulation ends when all ‘electrons’ have left the warm-target corona, and reached the cool ‘chromosphere’ (where a CTTM approximation is valid for all studied energies). Using a time-independent equation with a constant source of injection produces output variables with units of [electron s*-1* per output variable] i.e., $E$, $z$ or $\mu$, and the final results are reconstructed by summing over all outputs at each step $j$. The derivation of Equation (1) and the detailed description of the simulations can be found in Jeffrey et al. (2014).

Equation (1) (and Equations (5) and (6)) diverge as $E\rightarrow 0$, and as discussed in Jeffrey et al. (2014), the deterministic equation $E_{j+1}=\left[E_{j}^{3/2}+\frac{3\Gamma m_{e}^{2}}{2\sqrt{\pi k_{B}T}}\Delta s\right]$ must be used for low energies where $E_{j}\leq E_{\rm low}$ using $E_{\rm low}=\left[\frac{3\Gamma m_{e}^{2}}{2\sqrt{\pi k_{B}T}\Delta s}\right]^{2/3}$ – see Jeffrey et al. (2014), following Lemons et al. (2009). For such low energy thermal electrons, $\mu_{j+1}$ can be drawn from an isotropic distribution $\mu\in[-1,+1]$.

2.1 The deposition of non-thermal electron power

Electron energy deposited into the ambient plasma can be determined by considering

[TABLE]

where $E_{j}$ and $E_{j+1}$ are the electron energies before and after each simulation step respectively. Using $\Delta E_{j+1}(z)$, a new ambient background temperature at that location can also be determined. However, we do not examine changes in background plasma temperature, and the background temperature remains constant in all simulations.

Although derived from a time-independent equation, we note that Equations (4-6) are related to a time step $\Delta t$ by $\Delta t=\Delta s/v$, where $v=\sqrt{2E/m_{e}}$ is the velocity of the electron, and the simulations can also be used for a time-dependent analysis. In all simulations, the total time it takes for an electron to deposit all of its energy can be approximated using

[TABLE]

In cases where $\mu_{j}\approx 0$, we use $|z_{j+1}-z_{j}|\approx\Delta s$. Also in the CTTM, this time can be estimated analytically using $t=\frac{E_{0}^{2}}{2Knv}$ where $E_{0}$ is the injected energy of the electron.

During each simulation step $j$, in order to determine the non-thermal electron power $P_{j+1}(z)$ at each spatial location $z$, we weight the output of [electron s*-1* cm*-1*] in each $z$ bin by the total $\Delta E_{j+1}$ deposited in that bin, giving $P_{j+1}(z)$ [erg s*-1* cm*-1*]. Summing over all saved $j$ gives the total non-thermal electron power $P(z)$ deposited at each spatial location $z$, equivalent to

[TABLE]

Further, summing over all $z$ gives the total spatially integrated non-thermal electron power [erg s*-1*], which can be compared with the injected non-thermal electron power $P$ [erg s*-1*] input into the simulation. For an injected non-thermal electron power law distribution (cf. Holman et al., 2011), this can be written as

[TABLE]

for the injected energies $E_{0}$, an acceleration rate $\dot{N_{0}}$ [electron s*-1*], a low energy cutoff of $E_{c}$ (the lowest energy in the non-thermal electron distribution), and the power law spectral index $\delta$.

2.2 Flare plasma parameters

We model the flaring atmosphere using a hot corona–cold chromosphere type model (see Figures 1 and 2). This atmosphere is a simple but reasonable description of most flaring atmospheres. Moreover, a more realistic atmosphere is not required since we only want to compare the results of the CTTM and WTM. This type of atmosphere also ensures that a time-independent stationary solution is reached (e.g., see Kontar et al., 2015, for details). Unlike the CTTM, electrons are no longer lost energetically, but accumulate in the corona as they thermalize. This pile-up of thermalized electrons in the corona is balanced by the spatial diffusion of electrons from the hot corona into the cool chromosphere, which can be still considered a cold-target.

We perform simulation runs for two different “hot corona – cool chromosphere” model atmospheres (see Figure 2), including one model that includes a chromosphere with an exponential density profile (e.g. Vernazza et al., 1981; Battaglia et al., 2012).

The development of the WTM has shown that the plasma parameters (the coronal temperature $T$, the coronal number density $n$, and the coronal plasma extent $L$, where the temperature is high enough to be visible in X-rays) are crucial for determining and constraining the properties of flare-accelerated electrons (Kontar et al., 2019). Here, we show how the plasma properties play a key role in the transfer and the deposition of non-thermal electron power. We test how the energy of non-thermal electrons is transferred and deposited in a range of different coronal plasma conditions. In the corona, we use different number densities ranging from $n=3\times 10^{10}$ cm*-3* to $n=1\times 10^{11}$ cm*-3*, and plasma extents (half-loop lengths $L$) of either 20″or 30″ (see Figure 1) between the hot corona and cooler chromosphere, leading to column depths of $10^{19}-10^{20}$ cm*-2*. In the WTM cases, coronal temperatures range from 10 MK to 30 MK (see Figure 1) for solar/M-dwarf cases and up to 100 MK for comparison with certain extreme stellar cases (e.g. see Figure 3 in Aschwanden et al., 2008). In atmosphere type 1, in the cool ‘chromosphere-type’ region, the number density rises to $n=1\times 10^{12}$ cm*-3* and the temperature falls to $T\sim 0$ MK, i.e., it is approximated as a CTTM. In the more realistic atmosphere type 2, the density at the boundary of the cool “chromosphere-type” region is set at $n=1\times 10^{12}$ cm*-3*, but this rises quickly to photospheric densities of $n=\sim 10^{17}$ cm*-3* over $\sim 3\arcsec$ using

[TABLE]

where $n_{\rm photo}=1.16\times 10^{17}$ cm*-3* is the photospheric density at the optical depth of $\approx 1$, and here $z$ is measured in arcseconds. The scale height $h_{0}$ of the density profile is set at $0\arcsec.18\sim 130$ km (e.g. following the simulations of Battaglia et al., 2012).

In most solar flare coronal conditions, $\ln\Lambda\approx 20$, but we can calculate $\ln\Lambda$ using (e.g. Somov, 2007),

[TABLE]

In the CTTM simulations, we choose $T=T_{\rm corona}$, where $T_{\rm corona}$ is the background corona temperature used in the WTM simulations. In the lower ‘cold-target’ atmosphere, we choose $T=0.01$ MK for the calculation of $\ln\Lambda$.

2.3 The injected electron distribution

The source function $S(E,z,\mu)$ is made up of three separate distributions:

Injected energy spectrum – we input either a mono-energetic distribution or a power law distribution of the form $\sim E_{0}^{-\delta}$. Electron distributions with approximate power law forms are routinely observed via X-ray observations. In the mono-energetic cases, we input electrons with energies of either, the coronal thermal energy, 10 keV, 20 keV, 30 keV, 40 keV, 50 keV or 100 keV. In these cases, we compare the outputs using $\dot{N_{0}}=1$ s*-1*. For the power law case, parameters of $\delta=4$ or $\delta=7$, a low energy cutoff of $E_{c}=10$ keV or $E_{c}=20$ keV and a high energy cutoff of $E_{H}=50$ keV are used222Electrons with energies above $\sim 50$ keV will approximate the CTTM solution using the noted plasma parameters, and low-energy electrons carry the bulk of the power due to steeply decreasing power laws.. In power law cases, the electron injection rate $\dot{N_{0}}$ is set at a value that gives the total injected electron power $P=4.8\times 10^{27}$ erg s*-1*. 2. 2.

Injected pitch-angle distribution – we input a beamed distribution (with half moving in one direction and half moving in the opposite direction, i.e $\mu=+1$ or $\mu=-1$). We also run simulations using a completely isotropic distribution ($\mu\in[-1,+1]$), but for brevity the results are not shown here. In general, the injected pitch-angle distribution is not well-constrained by current solar flare observations (e.g. Kontar et al., 2011b; Casadei et al., 2017). Collisional (electron-electron only) pitch-angle scattering is always modeled in the simulations. Further, it is very likely that other non-collisional (and shorter timescale) turbulent scattering mechanisms are also presence in the flaring atmosphere (Kontar et al., 2014). This will also change how the electron energy is deposited spatially and temporally, and the subject of ongoing work. 3. 3.

Injected spatial distribution – we input the electrons as a Gaussian distribution centred at the loop top apex ($z=0\arcsec$), with a standard deviation of 1″. It is possible that electrons are accelerated to varying levels (dependent on the plasma conditions) at multiple points along a twisted loop (e.g. Gordovskyy & Browning, 2012; Gordovskyy et al., 2014), but again simulating all possible cases is beyond the scope of the paper and it is not required for a CTTM and WTM comparison.

2.4 Timescales for the deposition of non-thermal electron power

Unlike the CTTM case, in a WTM, electrons ‘stopped’ in the coronal plasma thermalize and then diffuse through the coronal region in a random walk continuously exchanging energy with the background population. Ultimately, this means that non-thermal electrons fully thermalized in the hot corona still transfer some fraction of their injected energy to the cool lower atmosphere (see the cartoon in Figure 3 (i)).

Moreover, the time is takes for an injected thermal electron, which is dominated by diffusion, to leave the corona and deposit its energy in a cool low atmosphere (cold-target) can be estimated analytically using

[TABLE]

where $K=2\pi e^{4}\ln{\Lambda}$. This diffusion time ($\tau_{D}$; Equation 13) is comparable to the Spitzer thermal conduction time (Spitzer, 1962). In Figure 3 (ii), we calculate $\tau_{D}$ for a range of different coronal parameters: $T$, $n$ and $L$. $\tau_{D}$ increases with increasing $n$ and $L$, and decreases with increasing $T$ (thermalized electrons in hotter plasma have a higher thermal energy and hence reach the chromosphere quicker). For the range of plasma parameters used in the simulations, $\tau_{D}$ ranges between $\sim 1-100$ s. Although, the thermal electron diffusion time $\tau_{D}$ can be calculated analytically, it is not trivial to determine the deposition timescales for non-thermal electrons injected with $E>k_{B}T$ into different coronal plasma conditions. Therefore, in each simulation run, we determine the time it takes for non-thermal electrons of different injected energies to deposit their energy in the lower atmosphere using Equation (8).

In these simulations, we stress that we do not consider the energy transferred from the hot corona to the cool lower atmosphere from the background thermal plasma by thermal conduction, which will play a varying role in different flares and at different stages as the flare progresses. Here, we only consider the energy transferred by non-thermal electrons, and in particular, the ‘extra’ component of energy that comes from partially or fully thermalized non-thermal electrons now being able to reach the lower atmosphere from diffusion.

In the simulations, we make several simplifying assumptions, such as: (1) use of single temperature and number density in the corona, (2) a previously heated corona and the use of a coronal ‘thermal bath’ approximation, i.e. there are no significant changes in the background energy content due to the transport and deposition of non-thermal electron energy, (3) the use of a ‘step function’ type atmosphere. Here, the assumptions are valid for a CTTM and WTM comparison, as stated. Further, the thermal diffusion of electrons is a fundamental transport mechanism always present in flares (to a varying degree depending on the coronal environment), that is usually overlooked, even in Fokker-Planck type simulations where a full collisional model is used. Hence, more complex simulations are not required here, and they would hinder our comparison of energy transport in the CTTM and WTM, which is the main aim of this study.

3 Simulation results

3.1 Mono-energetic energy inputs: spatial distribution of deposited power

Firstly, we perform simulations where we input different mono-energetic electron distributions into different plasma environments, so that the results of the WTM and CTTM can be easily compared for electrons of different energies in a range of different coronal conditions. Here, in each simulation run, the accelerated electron rate is $\dot{N_{0}}=1$ s*-1*, for each input. We perform four different sets of simulations labelled (a)-(d), using atmosphere 1 (see Figure 2):

Sets (a) and (b): different coronal densities- In sets (a) and (b), we input mono-energetic electrons with energies of either 10 keV, 20 keV, 30 keV, 40 keV, 50 keV or 100 keV. The injected electrons are initially beamed in $\mu$, and spread in $z$ as a Gaussian with a $1^{\prime\prime}$ standard deviation centered at the loop apex. They are injected into atmosphere type 1 (see Figure 2), with either a coronal density of (a) $n=10^{11}$ cm*-3* (Figure 4), or (b) $n=3\times 10^{10}$ cm*-3* (Figure 5), a half loop length of $L=20\arcsec$, and for all the WTM cases, a coronal temperature of $T=20$ MK (giving $E/k_{B}T=5.8,11.6,17.4,23.2,29.0,57.9$). All the WTM and CTTM results are shown in Figure 4 and Figure 5 (showing the spatial distribution of non-thermal power deposition in units of [keV s*-1* arcsecond*-1*] and the ratio of the WTM to CTTM result for each run). Tables 1 and 2 show the percentage of non-thermal power deposited in the corona or low atmosphere in both WTM and CTTM cases. From the results of (a) and (b) we find:

In (a) and (b), the spatial distribution of deposition in the WTM is different from the CTTM for all energies up to and including $\approx 50$ keV, and this difference increases for larger coronal densities and smaller injected electron energies (see Figure 4 and Figure 5). 2. 2.

In (a) and (b), electrons can move further and deposit more energy at greater depths in both the corona and low atmosphere in the WTM. This is most obvious for low energy electrons ($<30$ keV) in a high density corona. Such electrons are collisionally stopped in the corona. The CTTM predicts that they deposit all of their energy in the corona. In the WTM, electrons thermalize (or tend towards a Maxwellian) and eventually deposit some fraction of their original energy content in the low atmosphere (see Table 1 and Table 2 for a comparison of the CTTM and WTM percentages). For example, in (a), an injected 20 keV electron population now deposits $\sim 20\%$ of its available non-thermal power in the lower atmosphere, compared to $0\%$ in a CTTM. 3. 3.

The ratio of WTM to CTTM deposition is an informative parameter that shows, in a single given location, up to 100 times more non-thermal electron power can be deposited (in both the corona and low atmosphere), than predicted by the CTTM, for a given injection of non-thermal electrons (see Figure 4 and Figure 5).

Set (c): different coronal temperatures- In set (c), we compare WTM and CTTM deposition in different coronal plasma temperatures of $T=10,20,30,100$ MK. We inject initially beamed, mono-energetic electrons with an energy of 30 keV only (giving $E/k_{B}T=34.8,17.4,11.6,3.5$), into atmosphere type 1, using $n=7\times 10^{10}$ cm*-3* and $L=20\arcsec$. The results are shown in Figure 6 (left) and Table 3. From the results of set (c), we find that:

For 30 keV electrons, the higher the temperature of the coronal plasma, the greater the fraction of non-thermal electron power transferred to the lower atmosphere, with less deposition in the corona (see Table 3). Electrons tend to a Maxwellian, and in higher coronal temperatures, thermalize at higher energies carrying a higher fraction of their power into the lower atmosphere. This dependence on coronal temperature is completely ignored in the CTTM. 2. 2.

Using atmosphere 1, at higher coronal temperatures, more power is deposited at greater depths in the lower atmosphere, and the ratio of WTM to CTTM deposition shows that at certain $z$ in the lower atmosphere, more than three orders of magnitude more power can be deposited (see Figure 6 (left)). In the WTM, the decreased energy loss rate in the corona means that electrons carry more energy when they reach the chromospheric boundary and hence, they can travel deeper into the lower atmosphere. 3. 3.

It is possible that in high temperature plasma, if a fraction of the injected electrons have energies lower than $k_{B}T$, then the electrons will thermalize in the corona gaining energy and hence, deposit a higher fraction of energy in the lower atmosphere than suggested by their initial injected energy.

Set (d): different coronal loop lengths- In set (d), we compare WTM and CTTM deposition in different loop lengths of $L=20\arcsec$ and $L=30\arcsec$. We inject beamed, mono-energetic electrons of 30 keV only into atmosphere type 1, using $n=7\times 10^{10}$ cm*-3* and $T=20$ MK. The results are shown in Figure 6 (right) and Table 3. From the results of set (d), we find that:

As expected, irrespective of the coronal loop length, more power is transferred to the lower atmosphere in the WTM compared to the CTTM. The larger the loop length (greater column depth), the greater time electrons spend in the corona and hence, more electrons tend towards Maxwellian or fully thermalize in the corona. 2. 2.

Although, more power reaches the lower atmosphere in smaller loop lengths, the difference between the power transferred in the WTM and CTTM is greater for larger loop lengths. For example, for $L=30\arcsec$, only 0.7% of the total non-thermal power is transferred in the CTTM but in the WTM this rises to 28%. Further, the ratio of WTM/CTTM deposition at the chromospheric boundary is $\sim 100$.

3.2 Mono-energetic energy inputs: temporal distribution of deposited power

In all simulation sets, we also determine how long it takes for all of the non-thermal electron power to be deposited in the flaring atmosphere. Firstly, in Figure 8, we plot the results of sets (a) and (b). On each plot, we also add another simulation run where we inject (beamed) thermal electrons into the simulation. Note, that in the simulations, the injected electrons relax to the flux-averaged mean energy $2k_{B}T$ of the background plasma, and a $T=20$ MK corona gives $2k_{B}T=3.4$ keV. Figure 7 shows that in a WTM, lower energy electrons $E/k_{B}T\leq 11.6$ can deposit their power in the lower atmosphere over a large range of timescales, and as expected the WTM result tends to the CTTM result when $E>50$ keV. These timescales are shown in Figure 8 for both $n=1\times 10^{11}$ cm*-3* and $n=3\times 10^{10}$ cm*-3* cases. In the WTM, many electrons still deposit their power at times close to CTTM values, and WTM times converge to CTTM times as the electron energy increases. However, in the WTM, lower energy electrons show a large tail of delayed deposition in the lower atmosphere (of seconds to tens of second), unaccounted for in the CTTM, due to partially and fully thermalized electrons.

As an illustrative example from Figure 8, a 10 keV electron will thermalize quickly in a high density corona ($n=10^{11}$ cm*-3*) over a distance of $<10\arcsec$ (over a time of $<0.1$ s). In the CTTM, 10 keV electrons in this scenario never reach the chromosphere and deposit energy. However, in a WTM, once a 10 keV electron has thermalized, it could travel hundreds of arcseconds in a random walk, in the hot coronal region, before exiting into the cooler and denser chromosphere. Therefore, depending on their path (the amount of scattering) and thermalization time, electrons can now exit the corona over a range of timescales from sub-second to tens of seconds333These times will be even further affected by turbulent scattering in the corona..

3.3 Isotropic injection

For cases where we input an isotropic electron distribution, we find similar results: more energy is deposited in the lower atmosphere in the WTM than in a CTTM. For example, in one simulation where we inject 30 keV electrons into a corona ($T=20$ MK, $n=7\times 10^{10}$ [cm*-3*] and $L=20\arcsec$), independent of whether the injected electron distribution is beamed or isotropic, more energy is deposited at deeper locations in the lower atmosphere in a WTM than in a CTTM, up to 1000 times more in the beamed case, at certain locations, and up to 4 times in the isotropic case. Moreover, initially isotropic non-thermal distributions deposit energy over a greater range of timescales, and more energy is deposited at greater times relative to a beamed injection of electrons, possibly providing a diagnostic of electron anisotropy.

3.4 Solar flare power-law energy inputs

To investigate how the power of flare-accelerated electrons is transferred and deposited in a more realistic solar or stellar flare scenario, we also perform simulation runs using an injected electron power-law energy distribution of the form $F(E_{0})\sim E_{0}^{-\delta}$ (set (e): different $E_{c}$ and set (f): different $\delta$). In these runs, we use the following plasma parameters and injected electron inputs: $T=20$ MK, $n=7\times 10^{10}$ cm*-3* and the more realistic atmosphere type 2. Each electron distribution has a total power of $P=4.8\times 10^{27}$ erg s*-1*, with spectral index $\delta=4$ or $\delta=7$, a low energy cutoff of $E_{c}=10$ keV or $E_{c}=20$ keV and a high energy cutoff of $E_{H}=50$ keV. Again, such a low high-energy cutoff is used since we want to examine low-energy electrons that are incorrectly modelled by the CTTM. The results are shown in Figure 7 and Table 4.

For the power law energy inputs we find the following notable results:

As expected, the differences in CTTM and WTM power deposition follow the results of Section 3.1 and Section 3.2. 2. 2.

For a given injected electron power, we see a larger difference in CTTM and WTM deposition for electron distributions with a smaller low-energy cutoff $E_{c}$. Spatially, more power is deposited at greater depths and up to 12 times more power can be deposited in the lower atmosphere for $E_{c}=10$ keV and 2 times more energy for $E_{c}=20$ keV, at a given location, for the studied conditions. 3. 3.

For a given injected electron power, we see a larger difference in CTTM and WTM deposition for electron distributions with softer spectral indices. Spatially more power is deposited at greater depths and up to 4 times more power can be deposited in the lower atmosphere for $\delta=7$ and up to 2 times for $\delta=4$, at a given location, for the studied conditions. 4. 4.

The simulation runs using $0\arcsec.1$ binning show that thermalized low energy electrons deposit a large fraction of power at the top of the chromospheric boundary. 5. 5.

Electron power law distributions with a higher fraction of low energy electrons (i.e smaller $E_{c}$ or larger $\delta$), that partially or fully thermalize in the corona, deposit more of their energy at greater times (second to tens of second timescales).

4 Discussion

In this work, we show that the cold thick-target model (CTTM) does not adequately approximate the transport and deposition of energetic electrons in flaring, and therefore strongly heated, solar or stellar atmospheres. In the CTTM, neglecting second order effects such as velocity diffusion leads to an underestimate of the energy transferred to the low atmosphere in the majority of cases, especially by lower energy electrons with $E<50$ keV (or equivalently $E/k_{B}T\leq 20$) that can fully or partially thermalize in the flaring corona and transfer energy diffusively. This leads to a difference in the spatial distribution of deposited non-thermal electron power in both the corona and cool layers of the low atmosphere. Understanding energy transfer by low energy electrons is important. Most solar flare non-thermal electron distributions are consistent with steeply decreasing power laws with the bulk of the power held by electrons with $E<50$ keV. Further, the thermalization of non-thermal electrons in the corona leads to the non-thermal electron power being deposited over a large range of times from sub-second to tens of second, producing delayed heating in the lower atmosphere. The temporal distribution of this heating profile could act as a diagnostic of both non-thermal electron properties (even electron anisotropy, since isotropic electrons will spend longer in the coronal plasma, leading to greater thermalization), and the plasma conditions in the corona (temperature, number density and the extent of hot coronal plasma), if it can be extracted from observation. Using a full WTM description of energy transfer and deposition may be especially important for the analysis of stellar flares with higher coronal temperatures and densities. Also, the WTM description of energy deposition may be important for the study of microflares with low-energy $\leq$10 keV accelerated electrons (e.g., Wright et al., 2017) that can easily thermalize in the coronal plasma, but still produce heating in the lower atmosphere.

The development of the warm-target model (WTM) shows the important role coronal plasma properties play in determining the acceleration, transport and now deposition of flare-accelerated non-thermal electron power. Future X-ray observatories must aim to better constrain the plasma properties for this purpose. These simulation results, although applicable to archived RHESSI data, anticipate the launch of direct imaging X-ray missions, that will be able to provide a more detailed picture of the solar flare environment (temperature, density, ‘hot’ plasma extent) in different regions of the flare, using better spatial and temporal resolution. Proposed missions such the Focussing Optics X-ray Solar Imager (FOXSI ; Christe et al. (2017)) will have a high dynamic range and greater imaging spectroscopy capabilities. The delayed energy transfer by thermalized electrons should be observable by imaging low energy soft X-rays ($<5$ keV). Current high spectral, spatial and temporal observations with Interface Region Imaging Spectrograph (*IRIS *; De Pontieu et al. (2014)) in the transition region and chromosphere can be used to study the effects of delayed heating by partially or fully thermalized non-thermal electron distributions.

Jeffrey et al. (2015) show that the flaring corona is made up of multiple loops of varying temperature and density. These varying plasma parameters have a huge effect on both the injected electron parameters and on the resulting energy deposition. Such variation must be taken into account in future modelling. Further, we must study a changing, dynamic atmosphere, since deposition by thermalized non-thermal electrons is modulated by changes in the coronal plasma properties. Importantly, we suggest that hydrodynamic models that use the CTTM approximation as an input should be re-evaluated, and eventually the WTM should replace any CTTM approximations. WTM energy deposition with the inclusion of extended loop turbulence and magnetic trapping will also be the subject of upcoming work. As stated, the appeal of the CTTM is its simple analytic form and the next step is to produce a semi-analytic WTM function that can be used by both the solar and stellar communities to the determine the deposition of energy in flaring atmospheres.

NLSJ, EPK & LF gratefully acknowledge the financial support from the Science and Technology Facilities Council (STFC) Consolidated Grant ST/P000533/1. The work is supported by an international team grant (“Solar flare acceleration signatures and their connection to solar energetic particles” http://www.issibern.ch/teams/solflareconnectsolenerg/) from the International Space Sciences Institute (ISSI) Bern, Switzerland.