Dense Loop Flares

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Consider F(E, s) as the electron flux spectrum at longitudinal position s (s=0 for the center of the acceleration region near the loop top). Various physical processes especially Coulomb collisions with ambient electrons can in principle contribute to the behavior of F. Assume that electron acceleration is most efficient near the loop top (s=0) and energetic electrons with a power-law distribution (spectral index δ) are injected into a target with uniform density n. The form of F(E,s) can be deduced from continuity and energy loss equations as:
Consider F(E, s) as the electron flux spectrum at longitudinal position s (s=0 for the center of the acceleration region near the loop top). Various physical processes especially Coulomb collisions with ambient electrons can in principle contribute to the behavior of F. Assume that electron acceleration is most efficient near the loop top (s=0) and energetic electrons with a power-law distribution (spectral index δ) are injected into a target with uniform density n. The form of F(E,s) can be deduced from continuity and energy loss equations as:
-
[[File:eq1.png]]
+
[[File:eq1.png|300px|]]
The longitudinal extent of the source external to the acceleration region can be found by considering the standard deviation:  
The longitudinal extent of the source external to the acceleration region can be found by considering the standard deviation:  
-
[[File:eq2.png]]
+
[[File:eq2.png|400px|]]
For a model in which electrons are accelerated within a region extending from [−L0/2, L0 /2] and injected into an external region where they are collisionally stopped via bremsstrahlung, the observed longitudinal source extent L and the electron energy E have the following relationship:
For a model in which electrons are accelerated within a region extending from [−L0/2, L0 /2] and injected into an external region where they are collisionally stopped via bremsstrahlung, the observed longitudinal source extent L and the electron energy E have the following relationship:
-
 
+
[[File:eq3.png|300px|]]
-
[[File:eq3.png]]
+
The above equation can be easily understood: higher energy electrons are able to propagate further from the acceleration region and hence produce hard X-ray emission over a greater spatial extent than in lower-energy bands. For each flare, the values of the acceleration region length and the loop density are obtained by best-fitting the above equation to the loop length L(E) (obtained from UVS maps) with electron energy E in the nonthermal domain. Fitting results of one of the selected flares are shown in Fig.2 (from Guo et al. 2012a). The source lengths are well-fit by a quadratic form.   
The above equation can be easily understood: higher energy electrons are able to propagate further from the acceleration region and hence produce hard X-ray emission over a greater spatial extent than in lower-energy bands. For each flare, the values of the acceleration region length and the loop density are obtained by best-fitting the above equation to the loop length L(E) (obtained from UVS maps) with electron energy E in the nonthermal domain. Fitting results of one of the selected flares are shown in Fig.2 (from Guo et al. 2012a). The source lengths are well-fit by a quadratic form.   

Revision as of 15:17, 3 August 2012


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Number: 181
1st Author: Jingnan Guo
2nd Author: eminent person
Published: 6 August 2012
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Contents

Introduction

For decades, there has been much discussion of flare models in which copious non-thermal electrons are accelerated in a reconnecting current sheet in the solar corona. In such models the electron transport and bremsstrahlung processes are decoupled from the acceleration site. However, recently we have found a group of flares showing extended coronal sources which seem to combine the processes of electron acceleration, propagation and energy loss due to Coulomb collisions (Xu et al. 2008, Kontar et al. 2011, Guo et al. 2012b, Torre et al. 2012). Based on a collisional model with an extended acceleration region, we fit the observed variation of loop length with photon/electron energy, resulting in estimates of the plasma density in, and longitudinal extent of, the acceleration region. These quantities also allow inference of the number of particles within the acceleration region and hence the filling factor ‘f’ : the ratio of the emitting volume to the volume that encompasses the emitting region(s). Further, coupling information on the number of particles in the acceleration region with information on the total rate of acceleration of particles above a certain reference energy (obtained from spatially-integrated hard X-ray data) also allows inference of the specific acceleration rate (ratio of electrons accelerated per second). The result has implications both for the global electrodynamics associated with the replenishment of the acceleration region and for the nature of the particle acceleration process.


Mathematical Tools

We construct photon and mean-electron-flux maps by processing observationally-deduced photon and electron visibilities respectively(Fig. 1, Piana et al. 2007). Several image-processing methods are used: a visibility-based ForWarD-fit (FWD) algorithm, a maximum entropy (MEM) procedure, and the UV-Smooth (UVS) approach. For the selected flares, all three methods reveal the flare structures as elongated loops, but we focus on the results obtained with the UVS algorithm which is very accurate when reconstructing relatively localized extended sources as considered here (Guo et al. 2012a).

Figure 1: Imaging spectroscopy using RHESSI visibilities.


Nonthermal Collisional Model

Consider F(E, s) as the electron flux spectrum at longitudinal position s (s=0 for the center of the acceleration region near the loop top). Various physical processes especially Coulomb collisions with ambient electrons can in principle contribute to the behavior of F. Assume that electron acceleration is most efficient near the loop top (s=0) and energetic electrons with a power-law distribution (spectral index δ) are injected into a target with uniform density n. The form of F(E,s) can be deduced from continuity and energy loss equations as: Eq1.png

The longitudinal extent of the source external to the acceleration region can be found by considering the standard deviation: Eq2.png

For a model in which electrons are accelerated within a region extending from [−L0/2, L0 /2] and injected into an external region where they are collisionally stopped via bremsstrahlung, the observed longitudinal source extent L and the electron energy E have the following relationship: Eq3.png

The above equation can be easily understood: higher energy electrons are able to propagate further from the acceleration region and hence produce hard X-ray emission over a greater spatial extent than in lower-energy bands. For each flare, the values of the acceleration region length and the loop density are obtained by best-fitting the above equation to the loop length L(E) (obtained from UVS maps) with electron energy E in the nonthermal domain. Fitting results of one of the selected flares are shown in Fig.2 (from Guo et al. 2012a). The source lengths are well-fit by a quadratic form.

Figure 2: Flare (15-Apr-2002) loop lengths obtained from UVS method for each 2-keV electron energy bin. Fitting results of five different time intervals throughout the flare impulsive phase are shown in solid curves.

Results and Interpretations

The above analysis technique has been applied to 22 coronal loop events observed by RHESSI (Guo et al. 2012b). The values of acceleration size (length L0 , width W and volume V0), loop density n, the number of particles inside the acceleration region N , the specific acceleration rate η(>20 keV) and the filling factor f are deduced from the model for each event. We have calculated the value of the (geometric) mean value of each quantity as shown in Table 1.

Table 1: Acceleration Region Characteristics

The mean value of the filling factor obtained is consistent, within a logarithmic standard deviation or so, with unity. It validates the assumption used by many authors (e.g., Emslie et al. 2004) that most of the observed flare volume contains bremsstrahlung-emitting electrons. The observationally deduced value of the specific acceleration rate (~0.01 per second) implies that all available electrons would be energized and ejected toward the footpoints within a few hundred seconds. This result has significant implications for the supply of electrons to the acceleration region, current closure, and the global electrodynamic environment in which electron acceleration and propagation occur (see, e.g., Emslie & Henoux 1995). The value of the specific acceleration rate for Event 4 is consistent with that determined independently by Torre et al. (2012), who used a continuity equation analysis of the variation of the electron flux spectrum throughout the source.

The values of the filling factor f and the specific acceleration rate deduced herein are broadly consistent with stochastic acceleration models which generally involve a near-homogeneous distribution of scattering centers (e.g. Bian et al. 2012).

References

Bian, et al. 2012, ApJ, 754, 103

Emslie, A. G., & Henoux, J.-C. 1995, ApJ, 446, 371

Emslie, A. G. et al. 2004, J. Geophys. Res. (Space Phys.), 109, 10104

Guo, J., Emslie, A. G., et al. 2012a, A&A, 543, 53

Guo, J., Emslie, A. G., et al. 2012b, ApJ, 755, 32

Kontar, E. P., Hannah, I. G., & Bian, N. H. 2011, ApJ, 730, L22

Piana, M., Massone, A. M., Hurford, G. J., et al. 2007, ApJ, 665, 846

Torre, G., Pinamonti, N., Emslie, A. G., et al. 2012, ApJ, 751, 129

Xu, Y., Emslie, A. G., & Hurford, G. J. 2008, ApJ, 673, 576

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