DERIVATION OF STOCHASTIC ACCELERATION MODEL CHARACTERISTICS FOR SOLAR FLARES FROM RHESSI HARD X-RAY OBSERVATIONS

It is well established that the impulsive phase hard X-ray (HXR) emission of solar flares is produced by bremsstrahlung of nonthermal electrons spiraling down the flare loop while losing energy primarily via Coulomb collisions (Brown 1971; Hudson 1972; Petrosian 1973). Thus, HXR observations provide the most direct information on the spectrum of the radiating electrons and perhaps on the mechanism responsible for their acceleration. The common practice to extract this information has been to use the parametric forward fitting of HXR spectra to emission by an assumed radiating or injected electron spectrum (e.g., Holman et al. 2003), usually a power law with breaks and cutoffs (or plus a thermal component). A more direct connection was established between the observations and the acceleration process first by Hamilton & Petrosian (1992), fitting to high spectral resolution but narrowband observations (Lin & Schwartz 1987), and later by Park et al. (1997), fitting to broadband observations (e.g., Marschhäuser et al. 1994; Dingus et al. 1994). This was done in the framework of stochastic acceleration (SA) by plasma waves or turbulence.

However, it is preferable to obtain this information nonparametrically by some inversion techniques first attempted by Johns & Lin (1992). Recently, Piana et al. (2003) and Kontar et al. (2005) applied regularized inversion techniques to obtain the radiating electron flux spectra from the spatially integrated photon spectra observed by RHESSI (Lin et al. 2002). This is an important advance but it gives the spectrum of the effective radiating electrons summed over the whole flare loop, but not the spectrum of the accelerated electrons. This difference arises because high spatial resolution observations, first from Yohkoh (Masuda et al. 1994; Petrosian et al. 2002) and now from RHESSI (e.g., Liu et al. 2003), have shown that, essentially for all flares, in addition to the emission from the loop footpoints (FPs) (e.g., Hoyng et al. 1981), there is substantial HXR emission from a region near the loop top (LT). Thus, the total radiating electron spectrum is a complex combination of the accelerated electrons at the LT and those present in the FPs after having been modified by transport effects.

It is therefore clear that separate inversion of the LT and FP photon spectra to electron spectra would provide more direct information on the acceleration mechanism. More recently, Piana et al. (2007) have applied the regularized inversion technique to the RHESSI data in the Fourier domain (Hurford et al. 2002) to obtain electron flux spectral images. The goal of this Letter is to demonstrate that with the resulting spatially resolved electron flux spectra at the LT and FPs one can begin to constrain the acceleration model parameters directly.

In the following section, we present a brief review of the relation between the derived electron flux images and the characteristics of the SA model, and in Section 3 we apply this relation to a flare observed by RHESSI. A brief summary and our conclusions are presented in Section 4.

The observations of distinct LT and FP HXR emissions, with little or no emission from the legs of the loop, point to the LT as the acceleration site and require enhanced scattering of electrons in the LT. Petrosian & Donaghy (1999) showed that the most likely scattering agent is turbulence which can also accelerate particles stochastically. In fact, SA of the background thermal plasma has been the leading mechanism for acceleration of electrons (e.g., Hamilton & Petrosian 1992; Miller et al. 1996; Park et al. 1997; Petrosian & Liu 2004; Grigis & Benz 2006; Bykov & Fleishman 2009) and ions (e.g., Ramaty 1979; Mason et al. 1986; Mazur et al. 1995; Liu et al. 2004, 2006; Petrosian et al. 2009), and is the most developed model in terms of comparing with observations.

2.1. Particle Kinetic Equation

In this model, one assumes that turbulence is produced at or near the LT region (with background electron density n_LT, volume V, and size L). In the presence of a sufficiently high density of turbulence, the scattering can result in a mean scattering length or time (τ_scat) smaller than L or the crossing time (τ_cross = L/v), leading to a nearly isotropic pitch angle distribution (Petrosian & Liu 2004). The general Fokker–Planck equation for the density spectrum N(E) of the accelerated electrons, averaged over the turbulent acceleration region, simplifies to

$$\begin{eqnarray} \frac{\partial N}{\partial t} &=& \frac{\partial ^2}{\partial E^2} [D_{\rm EE}N] -\frac{\partial }{\partial E}[(A(E)-\dot{E}_{\rm L}(E))N]\nonumber\\ &&- \frac{N}{T_{\rm esc}(E)}+\dot{Q}(E), \end{eqnarray} \tag{ 1 }$$

where D_EE(E) and A(E) are the diffusion rate and the direct acceleration rate by turbulence,² respectively, $\dot{E}_{\rm L}$ is the electron energy loss rate, and $\dot{Q}(E)$ and N(E)/T_esc(E) describe the rate of injection of (thermal) particles and escape of the accelerated particles from the acceleration region. For electrons of energies below ∼1 MeV, which are of interest here, Coulomb collisions³ dominate the energy loss rate:

$$\begin{equation} \dot{E}_{\rm L} = \dot{E}_{\rm Coul}=4\pi r_0^2 m_{\rm e} c^3 n_{\rm LT}\ln \Lambda /\beta, \end{equation} \tag{ 2 }$$

where ln Λ is the Coulomb logarithm taken to be 20 for solar flare conditions. Following Petrosian & Liu (2004), we approximate the escape time as T_esc(E) ≃ τ_cross(1 + τ_cross/τ_scat), which smoothly connects the two limiting cases of τ_cross/τ_scat ≫ 1 and ≪1. The mean scattering time is related to the pitch angle diffusion rates (Dung & Petrosian 1994; Pryadko & Petrosian 1997) due to both Coulomb collisions (D^Coul_μμ and τ^turb_scat) and turbulence (D^turb_μμ and τ^Coul_scat) as

$$\begin{equation} \tau _{\rm scat}(E)=\frac{1}{8}\int _{-1}^1\frac{(1-\mu ^2)^2}{D_{\mu \mu }^{\rm Coul}(\mu,E)+ D_{\mu \mu }^{\rm turb}(\mu,E)} {d}\mu. \end{equation} \tag{ 3 }$$

Similarly, we can define the scattering times τ^Coul_scat and τ^turb_scat for each process alone. For Coulomb collisions, $D_{\mu \mu }^{\rm Coul}=\frac{2(1-\mu ^2)}{\gamma +1}\frac{\dot{E}_{\rm Coul}}{{E}}$. For turbulence, D^turb_μμ, like D_EE, depends on the spectrum of turbulence and on the background plasma density, composition, temperature, and magnetic field (see Schlickeiser 1989; Dung & Petrosian 1994; Pryadko & Petrosian 1997, 1998, 1999; Petrosian & Liu 2004). Since these coefficients determine the spectrum of the accelerated electrons, one can then constrain some aspects of the acceleration mechanism if an accurate spectrum of the electrons can be derived from observations.

2.2. LT and FP Spectra

The accelerated electrons in the (LT) acceleration region with a flux spectrum F_LT(E) = vN(E) produce thin target bremsstrahlung emissivity (photons s⁻¹ keV⁻¹):

$$\begin{equation} J_{\rm LT}(\epsilon)=n_{\rm LT}V \int _\epsilon ^\infty F_{\rm LT}(E)\sigma (\epsilon,E) {d}E, \end{equation} \tag{ 4 }$$

where σ(, E) is the angle-averaged bremsstrahlung cross section (Koch & Motz 1959). The escaping electrons with flux F₀(E) = N(E)L/T_esc produce thick target bremsstrahlung emissivity (coming mostly from the FPs; see Petrosian 1973; Park et al. 1997):

$$\begin{equation} J_{\rm FP}(\epsilon)=n V \int _\epsilon ^\infty F_{\rm FP}(E)\sigma (\epsilon,E) {d}E, \end{equation} \tag{ 5 }$$

where n is the density and F_FP is the effective radiating electron flux spectrum at the FPs,

$$\begin{equation} F_{\rm FP}(E)=vN_{\rm FP}=\frac{v(E)}{\dot{E}_{\rm L}(n)} \int _E^\infty \frac{N(E^{\prime })}{T_{\rm esc}(E^{\prime })}{d}E^{\prime }. \end{equation} \tag{ 6 }$$

Since $\dot{E}_{\rm L}\propto n$, the FP photon spectrum is independent of density. In what follows, we evaluate Equations (5) and (6) using the LT density n_LT.

2.3. Acceleration Model Parameters

Regularized spectral inversion of RHESSI count visibilities gives the electron visibilities (Piana et al. 2007), which can then be used to construct images of electron flux (multiplied by line-of-sight column density). From these images, we extract the spatially resolved spectra, F_LT(E) at the LT and F_FP(E) at the FPs. Thus, we can obtain the accelerated electron spectrum N(E) at the thin target LT. Also from differentiation of Equation (6) we derive the escape time as $T_{\rm esc}=-N(E)/\frac{d}{{d}E}(F_{\rm FP}\dot{E}_{\rm L}/v)$, and by converting the denominator to a logarithm derivative we get

$$\begin{eqnarray} &&T_{\rm esc}(E)=\frac{\tau _{\rm L}(E)(F_{\rm LT}/F_{\rm FP})}{\delta _{\rm FP}(E)+2/(\gamma +\gamma ^2)}\equiv \tau _{\rm L}(E)\xi (E), \end{eqnarray} \tag{ 7 }$$

where the FP index $\delta _{\rm FP}(E)=-\frac{{d}\ln F_{\rm FP}}{{d}\ln E}$, $1/(\gamma +\gamma ^2)=-\frac{{d}\ln v(E)}{{d}\ln E}$, and $\tau _{\rm L}(E)=E/\dot{E}_{\rm L}$ is the Coulomb loss time at the LT. The function ξ(E) is an observable quantity representing the ratio T_esc/τ_L. In the above derivation, we have used the relativistic form of electron velocity v(E).

Given T_esc(E), from its relation to τ_cross and τ_scat, we obtain the mean scattering time as τ_scat ≃ τ²_cross/(T_esc − τ_cross), which is valid for T_esc>τ_cross. Disentanglement of τ^turb_scat from τ_scat is complicated (Equation (3)) at energies when turbulence and Coulomb collisions contribute equally to τ_scat. However, if turbulence dominates the pitch angle diffusion, then to the first order we can write τ^turb_scat ≃ τ_scat(1 + τ_scat/τ^Coul_scat) and obtain some average value of D^turb_μμ. Furthermore, given N(E) we can in principle determine the other Fokker–Planck coefficients, namely A(E) and D_EE (see Equation (1)). Therefore, we can reach a consistent picture of the acceleration process due to turbulence and begin to make inroads into the spectrum and the nature of turbulence itself.

As a first demonstration, we apply our new procedure to the 2003 November 3 solar flare (X3.9 class) during the nonthermal peak, in which we find a hard LT source (extending above 100 keV in HXR) distinct from the thermal loop in addition to two FP sources.⁴ In Figure 1, we show the electron flux images up to 250 keV, which also show a loop at low energies, and one LT and two FPs at higher energies. In Figure 2 (top panel), we show the electron spectra ${\cal N}_{\rm LT}E^2F(E)$, where ${\cal N}_{\rm LT}=n_{\rm LT}L$ is the LT column density. The LT flux spectrum can be fitted by a power law with an index δ_LT = 3.0. The summed FP flux spectrum can be better fitted by a broken power law with the indices δ₁ = 2.1 and δ₂ = 2.8 below and above the break energy E_b = 91 ± 3 keV. It is clear that the total radiating electron spectrum differs significantly from the (LT) accelerated electron spectrum.

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Zoom In Zoom Out Reset image size

Given the above LT and FP electron flux spectra, we derive the energy dependence of the escape time (Equation (7)). The LT density can be estimated as $n_{\rm LT}\simeq \sqrt{{\rm EM}/L^3}\simeq 0.5\times 10^{11}\ {\rm cm}^{-3}$, where the LT size L ≃ 10⁹ cm is obtained from the LT angular size, and the emission measure EM ≃ 0.2 × 10⁴⁹ cm⁻³ is obtained from spectral fitting of the LT thermal emission. As in Figure 2 (bottom panel), the escape time increases slowly with energy and can be fitted by either a power law,

$$\begin{equation} T_{\rm esc}(E) = 0.3\ {\rm s} \left(\frac{E}{100\ {\rm keV}}\right)^{\kappa },\,\,\, \kappa =0.83\pm 0.10, \end{equation} \tag{ 8 }$$

or a broken power law with a break at E_b = 118 ± 37 keV, and the indices κ₁ = 0.62 ± 0.23 and κ₂ = 1.09 ± 0.25. The fact that the escape time should be longer than the crossing time yields an upper limit on ${\cal N}_{\rm LT}$, which is satisfied by the above LT density and size.

We then calculate the mean scattering time in the LT region. Except at the lowest energy, the Coulomb contribution is small so that the scattering time thus calculated can be attributed to turbulence. The scattering time due to turbulence (see Section 2.3) can be fitted by a power law above ∼40 keV:

$$\begin{eqnarray} &&\tau ^{\rm turb}_{\rm scat}=0.016\ {\rm s} \left(\frac{E}{100\ {\rm keV}}\right)^{-\lambda }, \,\,\, \lambda =1.90\pm 0.14.\,\, \end{eqnarray} \tag{ 9 }$$

In this Letter, we describe a new method to directly obtain the model parameters for SA of particles by turbulence in solar flares from regularized spectral inversion of the high-resolution RHESSI HXR data (Piana et al. 2007). We have argued that particle acceleration takes place at or near the LT region. The accelerated electrons produce thin target emission at the LT and then escape downward to the dense FP region undergoing Coulomb collisions and producing thick target emission. In this model, the LT and FP electron spectra are connected by the escape process from the LT region (Equation (6)), thus allowing us to determine the energy dependence of the escape time. Our method has the advantage that one can now constrain the model parameters uniquely rather than just satisfying the consistency between the model and the data as commonly done by forward fitting routines. This method can be applied to flares with simultaneous HXR emission from the LT and FP sources.

We have applied our method to the 2003 November 3 flare, in which we can obtain the electron flux images for both the LT and FPs up to 250 keV. The LT accelerated electron flux spectrum can be fitted by a power law, and the effective radiating flux spectrum at the FPs is better fitted by a broken power law. From these spectra we derive the energy variation of the escape time and the scattering time. As seen in Figure 2, the turbulence scattering time is relatively short and decreases with energy. A short scattering time may arise from a high density of turbulence energy (${\cal E}_{\rm turb}$), with the exact relationship depending also on the magnetic field (B), and the spectral index (q) and minimum wave number (k_min) of turbulence. A high level of turbulence also implies efficient acceleration which generally means a flat spectrum for the accelerated electrons, which is the case for the current flare. The energy dependences of τ^turb_scat and D_EE are also a function of these characteristics of turbulence; at high energies, they are determined primarily by the spectral index of turbulence (see Dung & Petrosian 1994; Pryadko & Petrosian 1997, 1998, 1999; Liu et al. 2006).

For the usually assumed Kolmogorov (q =  5/3) or Iroshnikov–Kraichnan (q = 3/2) turbulence spectra, one expects the scattering time to increase with energy as E^2−q, which translates into an escape time varying roughly as $T_{\rm esc}\propto 1/\sqrt{E}$ at high (but nonrelativistic) energies. The energy dependences of T_esc and τ^turb_scat obtained here require a steeper turbulence spectrum (q>3) at high wavenumbers. Such a steep spectrum can be present beyond the inertial range where damping is important (e.g., Jiang et al. 2009). The electron energies and the wave-particle resonance condition determine the wavevector of the accelerating plasma waves. This relation depends primarily on the plasma parameter $\alpha \propto \sqrt{n}/B$ (e.g., Petrosian & Liu 2004). Thus, given the magnetic field and plasma density we can determine the wavevectors for transition from the inertial to the damping ranges of turbulence.

It should, however, be emphasized that the results obtained here may not be representative of typical flares. More commonly flares have much softer LT emission, which would give an escape time decreasing (and scattering time increasing) with energy, consistent with a low level and a flat spectrum of turbulence.

The exact relation between the derived quantities (N(E), T_esc, and τ^turb_scat) and the turbulence characteristics (${\cal E}_{\rm turb}, B, q$, etc.) is complicated and depends on the angle of propagation of the plasma waves with respect to the magnetic field and other plasma conditions. In future, we will apply these procedures to more flares (Q. Chen & V. Petrosian, 2010b, in preparation) and deal with these relations explicitly.

We thank the referee for helpful comments. We thank Anna Maria Massone and Gordon Hurford for providing the visibility inversion code and valuable discussions about data analysis, and Siming Liu and Wei Liu for various discussions. RHESSI is a NASA small explorer mission. This work is supported by NSF grant ATM0648750 and NASA grant NNX10AC06G.