Origin of the 30 THz Emission Detected During the Solar Flare on 2012 March 13 at 17:20 UT

Abstract

Solar observations in the infrared domain can bring important clues on the response of the low solar atmosphere to primary energy released during flares. At present, the infrared continuum has been detected at 30 THz (10 μm) in only a few flares. SOL2012-03-13, which is one of these flares, has been presented and discussed in Kaufmann et al. (Astrophys. J. 768, 134, 2013). No firm conclusions were drawn on the origin of the mid-infrared radiation. In this work we present a detailed multi-frequency analysis of the SOL2012-03-13 event, including observations at radio-millimeter and submillimeter wavelengths, in hard X-rays (HXR), gamma-rays (GR), $\mathrm{H}\alpha$, and white light. The HXR/GR spectral analysis shows that SOL2012-03-13 is a GR line flare and allows estimating the numbers of and energy contents in electrons, protons, and $\alpha$ particles produced during the flare. The energy spectrum of the electrons producing the HXR/GR continuum is consistent with a broken power-law with an energy break at ${\sim}\,800~\mbox{keV}$. We show that the high-energy part (above ${\sim}\, 800~\mbox{keV}$) of this distribution is responsible for the high-frequency radio emission (${>}\, 20~\mbox{GHz}$) detected during the flare. By comparing the 30 THz emission expected from semi-empirical and time-independent models of the quiet and flare atmospheres, we find that most (${\sim}\,80~\%$) of the observed 30 THz radiation can be attributed to thermal free–free emission of an optically thin source. Using the F2 flare atmospheric model (Machado et al. in Astrophys. J. 242, 336, 1980), this thin source is found to be at temperatures T ${\sim}\,8000~\mbox{K}$ and is located well above the minimum temperature region. We argue that the chromospheric heating, which results in 80 % of the 30 THz excess radiation, can be due to energy deposition by nonthermal flare-accelerated electrons, protons, and $\alpha$ particles. The remaining 20 % of the 30 THz excess emission is found to be radiated from an optically thick atmospheric layer at T ${\sim}\, 5000~\mbox{K}$, below the temperature minimum region, where direct heating by nonthermal particles is insufficient to account for the observed infrared radiation.

Appendix

To compute the energy deposited by accelerated electrons, protons, and $\alpha$ particles in the chromosphere as a function of depth, we injected particles with an energy spectrum $F_{0}(E)$ ($\mbox{MeV}^{-1}$) at the top of the atmosphere. We parametrized the position in the atmosphere using the hydrogen column depth $N$. The energy deposited ($\mbox{erg}\,\mbox{cm}^{-1}$) at depth $N$ is (following Emslie, 1978)

$$ \Theta(N) = \int_{E_{\min}(N)}^{\infty}\frac{F_{0}(E_{0})}{v(E)} \biggl\vert \frac{\mathrm{d}E}{\mathrm{d}t}(E,N)\biggr\vert \mathrm{d}E_{0}. $$

(4)

Here $E_{\min}(N)$ is the minimum energy of a particle that can reach depth $N$. In Equation (4) $E$ should be understood as depending on $E_{0}$ and $N$: it is the energy that a particle of initial energy $E_{0}$ has when it has reached depth $N$.

An ion of energy $E$ and charge $z$ loses energy with depth at a rate given by (Gould, 1972b; Emslie, 1978; Olive and Particle Data Group, 2014)

$$ \frac{\mathrm{d}E}{\mathrm{d}N} = -\frac{K}{\beta^{2}} \Lambda_{\mathrm{eff}}(E,N). $$

(5)

Here $\beta$ is the particle speed in units of the speed of light, $K$ is given by

$$\begin{aligned} \begin{aligned} K &= \frac{4 \pi e^{4}}{m_{e}c^{2}} z^{2},\\ \Lambda_{\mathrm{eff}}(E,N) &= x \Lambda + (1-x) \Lambda', \end{aligned} \end{aligned}$$

(6)

where $x$ is the degree of ionization and $\Lambda$ and $\Lambda'$ are the Coulomb logarithms appropriate for slowing on electrons and on neutral atoms, respectively. For ions, expressions for $\Lambda$ are given e.g. by Butler and Buckingham (1962), Gould (1972a). For $\Lambda'$ we used the Bethe-Bloch form (Olive and Particle Data Group, 2014), neglecting the correction for the polarization of the medium, which is negligible in the low-density conditions of the solar atmosphere. We estimated the effects of the atmospheric chemical composition on energy loss by multiplying $\Lambda'$ by

$$\sum_{n} \frac{a_{n}}{a_{H}} \frac{Z_{n}}{A_{n}} = 1.16, $$

where $n$ sums over atomic species and $A_{n}$, $Z_{n}$ and $a_{n}$ are the atomic number, atomic weight, and abundance relative to hydrogen of species $n$. We also generalized $x = n_{e}/n_{H}$ so that it is no longer quite the degree of ionization and may take values ${>}\,1$, dependent on the local electron density.

$\Lambda$ and $\Lambda'$ for electrons are given by Gould (1972a), Emslie (1978). By using only Equation (5), we neglected pitch-angle scattering. Scattering in angle is unimportant for fast ions and relativistic electrons – the only particles that can reach the $\mathrm{LC7}_{\mathrm{l}}$ layer. For nonrelativistic electrons it reduces the mean range by two thirds (Brown, 1972), although dispersion means that electrons will stop and deposit energy over a range of depths up to their full vertical stopping depth (Bai, 1982). For the present purposes of estimation, we completely neglected pitch-angle scattering.

Figure 7 shows the value of $\Lambda_{\mathrm{eff}}$ throughout the chromospheric portion of the C7 atmosphere of Avrett and Loeser (2003) for protons of energies 1, 10, and 100 MeV. The importance of retaining the energy dependence of $\Lambda$ has been emphasized elsewhere (MacKinnon and Toner, 2003). The variable degree of ionization means that $\Lambda_{\mathrm{eff}}$ also strongly depends on position. Thus the energy loss rate of Equation (5) is not separable in energy and depth, and analytical heating rates like those of Emslie (1978) may not be used (although an energy loss rate appropriate to a completely neutral atmosphere would give a good approximation below about 1000 km).

Figure 7

The effective Coulomb logarithm, $\Lambda_{\mathrm{eff}}$ through the atmosphere for protons of energy 1, 10 and 100 MeV.

To evaluate the energy deposition $\Theta$, we need to assume a form for $F_{0}(E_{0})$, e.g. a single power-law:

$$ F_{0}(E_{0}) = \frac{\mathcal{F}_{0}}{E_{*}} (\delta-1) \biggl(\frac{E_{0}}{E_{*}} \biggr)^{-\delta}, $$

(7)

where $\mathcal{F}_{0}$ is the number of particles injected above energy $E_{*}$. Then applying Equation (5) in Equation (4) gives

$$ \Theta(N) = 8.16 \times10^{-31} z^{2} n_{H}(N) \mathcal{F}_{0} E_{*}^{\delta-1}(\delta-1) \int _{E_{\min}(N)}^{\infty}E^{-\delta} \frac{\Lambda_{\mathrm{eff}}(E,N)}{\beta^{2}(E)} \mathrm{d}E_{0} . $$

(8)

Again, $E$ in the integrand of Equation (8) is understood to be a function of $E_{0}$ and $N$: the energy at depth $N$ of a particle that had energy $E_{0}$ at injection. Since we cannot obtain analytical expressions for $E(E_{0},N)$, we calculated it by numerical integration of Equation (5), with numerical interpolation as necessary between the tabulated points of a given atmosphere model, as needed for each of the abscissae of a numerical evaluation of the energy deposition (Equation (8)). More elaborate forms of $F_{0}(E_{0})$, e.g. the broken power-law deduced for electrons, are straightforwardly substituted in Equation (4) as necessary.