Electron re-acceleration and HXR emission

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{{Infobox Nugget
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|name = Nugget
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|title = Passages of Electron  Beams
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|number = 194
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|first_author = Heather Ratcliffe
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|second_author = Marian Karlický and Eduard Kontar
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|publish_date = 2013 February 18
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|next_nugget = [[Burst-on-Tail (BOT)]]
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|previous_nugget =[[Passages of Electron Beams]]
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}}
== Introduction ==
== Introduction ==
-
During solar flares, vast amounts of energy are released from the Sun's magnetic field, and a fraction of this goes into particle acceleration. A fast electron beam thus produced can propagate along a coronal loop, and produce Hard X-ray (HXR) emission at the loop footpoints by collisional bremsstrahlung in the dense chromosphere. This HXR emission is one of the primary diagnostics of such electron beams, and is usually interpreted using the Cold Thick Target Model (CTTM, Brown 1971). This assumes, among other things, that the emitting electrons are modified only by collisions.
+
During solar flares, vast amounts of energy are released from  
-
 
+
[http://solarscience.msfc.nasa.gov/the_key.shtml the Sun's magnetic field], part of which leads to
-
However, when such fast electron beams propagate in plasma, Langmuir (plasma) waves are generated. These waves are themselves strongly affected by density inhomogeneity, or by wave-wave interactions, and it is exactly these processes which can lead to modification of the electron distribution. These processes are complicated and non-linear, and are thus best addressed via simulations. Several simulation methods are applicable in this situation. In this nugget we show some quasilinear and PIC (particle-in-cell) simulations of the electron evolution and briefly discuss the possible effects on HXR emission.
+
[http://en.wikipedia.org/wiki/Fermi_acceleration particle acceleration].  
 +
Fast electrons thus produced can propagate along a coronal magnetic loop, and produce Hard X-ray (HXR) emission at its footpoints by collisional  
 +
[http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xrayc.html bremsstrahlung] in the dense  
 +
[http://solarscience.msfc.nasa.gov/chromos.shtml chromosphere].  
 +
This HXR emission is one of the primary diagnostics of energetic electrons, and is usually interpreted using the  
 +
[http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xrayc.html  "thick target"] model.
 +
This assumes, among other things, that the [http://en.wikipedia.org/wiki/Distribution_function distribution function] of the emitting electrons is modified only by collisions.  
 +
However, when such fast electron beams propagate in a
 +
[http://en.wikipedia.org/wiki/Plasma_(physics) plasma],
 +
[http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1932/langmuir-bio.html Langmuir] (plasma) waves are generated.
 +
These waves are strongly affected by density inhomogeneities, or by wave-wave interactions, and this evolution can have significant effects on the electron distribution. Several simulation methods are applicable in this situation.
 +
In this Nugget we show some quasilinear and PIC (particle-in-cell) simulations of the electron evolution and briefly discuss the effects on HXR emission.
== Quasilinear Simulations ==
== Quasilinear Simulations ==
-
One model situation describing these electron beams is to consider the collisional relaxation of an electron beam. We may reasonably assume the electron and Langmuir waves dynamics are along a single direction, the direction of beam propagation, and so need only 1-dimensional simulations. We begin from the well established quasilinear equations (Melrose, 1980), and add collisional terms, and terms describing plasma density inhomogeneity and wave-wave interactions (Kontar et al, 2012). The former are treated using a diffusive treatment, derived in (Ratcliffe et al, 2012).
+
Our model (based on ref. [1]) describes collisional relaxation of the energetic power-law distribution. We consider firstly a simplified model including only collisions, and then the relaxation including wave generation, and evolution.
 +
 +
[[File:RatcliffeKontarKarlickyFig1.png|thumb|center|800px|
 +
Figure 1:
 +
Time-averaged electron fluxes. In each panel the black line shows the case without Langmuir wave generation, while the blue lines show, respectively: left panel-with Langmuir wave generation, middle panel-with plasma density fluctuations, and right panel-fluctuating density plasma plus wave-wave interactions. (From ref. [2])
 +
]]
-
The primary quantity of interest is the time-averaged electron flux as a function of electron energy, as this directly related to the observed HXR emission. In Figure 1 we show this distribution for the simulation models with and without Langmuir wave generation (left panel), with Langmuir wave generation and evolution due to density inhomogeneities (middle panel), and finally with wave-wave processes also included (right panel).
+
The primary quantity of interest is the time-averaged electron flux as a function of electron energy, which is closely related to the observed HXR emission.  
-
 
+
We show this for the simulation models with and without Langmuir-wave generation (left panel), with Langmuir-wave generation and evolution due to density inhomogeneities (middle panel) (see ref. [3] for details), and finally with wave-wave processes also included (right panel).
-
We see from Figure 1 that the Langmuir wave generation alone has no effect on this distribution, which confirms a well known previous result (Smith and Emslie, 1976??). However, evolution of the Langmuir waves can produce significant changes in the HXR emission. For the model parameters chosen, this occurs primarily between 20 and 200 keV (above 200keV the simulations cannot be trusted as relativistic effects become very important). Moreover, we can expect an enhancement of the HXR emission of several times, or perhaps an orer of magnitude within this range.  
+
 +
We see from Figure 1 that the Langmuir-wave generation alone has a very weak effect, which confirms a well-known previous result.
 +
However, the time evolution of the Langmuir waves can produce significant changes in the HXR emission.
 +
For the model parameters chosen, this occurs primarily between 20 and 200 keV, and within this range we can expect an increase in HXR emission of an order of magnitude (Figure 1).
== PIC simulations ==
== PIC simulations ==
-
 
+
<!--Comment
In Karlický and Kontar (2012) a 3-D particle-in-cell code is used to consider
In Karlický and Kontar (2012) a 3-D particle-in-cell code is used to consider
a similar problem, that of the monoenergetic beam injected into plasma, with
a similar problem, that of the monoenergetic beam injected into plasma, with
Line 33: Line 61:
$v_b/c = 0.666$. The ratio of the beam to the plasma densities was
$v_b/c = 0.666$. The ratio of the beam to the plasma densities was
$n_{b}/n_{e}= 1/8$. The periodic boundary conditions were used.
$n_{b}/n_{e}= 1/8$. The periodic boundary conditions were used.
 +
-->
 +
It is also possible to use a 3D
 +
[http://en.wikipedia.org/wiki/Particle-in-cell "particle-in-cell"] (PIC) simulation to consider
 +
the problem of a monoenergetic beam injected into a plasma, with the effects of wave-wave interactions included.
 +
In these PIC simulations (ref [3]) the proton-electron mass ratio was taken to be 16 for computational reasons, but that is sufficient for these simulations.
 +
The initial electron beam was homogeneous throughout the numerical box of the simulation, and an appropriate return current introduced to
 +
keep the total current in the system zero.
 +
Periodic boundary conditions were used.
 +
 +
[[File:RatcliffeKontarKarlickyFig2_2.png|thumb|center|800px|
 +
Figure 2: The electron energy distributions (solid lines) at (normalized) time t = 200.
 +
The magnetic field is zero in model A and increases through models B to F. For comparison in each panel we plot the initial electron plasma
 +
distribution together with the initial monoenergetic beam (dashed lines).
 +
]]
 +
 +
<!--Comment
Using this model we made several computational runs, in which we also changed
Using this model we made several computational runs, in which we also changed
-
the magnetic field, oriented in the beam propagation direction, see Karlick\'y
+
the magnetic field, oriented in the beam propagation direction, see Karlický
and Kontar (2012). As an illustration, in Fig. 2 we present the electron energy
and Kontar (2012). As an illustration, in Fig. 2 we present the electron energy
distributions at the time $\omega_{pe} t$ = 200, for the magnetic field
distributions at the time $\omega_{pe} t$ = 200, for the magnetic field
Line 46: Line 90:
beam-plasma system with low magnetic fields the Weibel instability reduces this
beam-plasma system with low magnetic fields the Weibel instability reduces this
acceleration process.
acceleration process.
 +
-->
 +
Using this model we made several computational runs, in which we also changed
 +
the magnetic field, oriented in the beam propagation direction.
 +
In Fig. 2 we present the the resulting electron energy
 +
distributions at the (normalized) time t = 200, for the magnetic field
 +
expressed through the ratio  of the
 +
[http://sprg.ssl.berkeley.edu/~hhudson/plasma/webpage/plasma.html electron-cyclotron frequency] to the
 +
[http://sprg.ssl.berkeley.edu/~hhudson/plasma/webpage/plasma.html plasma frequency], equal to 0.0, 0.1, 0.5, 0.7,
 +
1.0, and 1.3, respectively (models A-F).
 +
As can be seen here, there are electrons
 +
accelerated above their initial energy, and the number of these increases with the magnetic field strength.
 +
This is due to the
 +
[http://en.wikipedia.org/wiki/Weibel_instability Weibel instability], which in the 3-D beam-plasma system with low magnetic fields reduces this
 +
acceleration process.
== Complementary approaches ==
== Complementary approaches ==
-
The two simulation methods presented here are very different, and each have their advantages and disadvantages. In brief:
+
The two simulation methods presented here are very different, and each has its advantages and disadvantages.  
-
Quasilinear simulations as used here consider only a 1-dimensional model, using weak turbulence theory. The magnetic field is ignored, except as a guiding force for the electron beam. Computationally they are simple and fast, and the beam-plasma interaction is well treated by such a model. Moreover, we can argue in favour of almost 1-d electron dynamics.
+
Quasilinear simulations use weak turbulence theory. Computationally such simulations are fast, and the beam-plasma interaction is well treated by such a model. PIC simulations are computationally demanding and limited to short time and spatial scales, and thus require such approximations as a small electron-proton mass ratio, and a small number of particles.
 +
However, the effects of magnetic field can be included, and the treatment is fully 3-D and self-consistent. Thus the two methods offer very good independent confirmation, and together give a strong argument for such an acceleration effect occurring.
-
PIC simulations are computationally demanding, and thus require such approximations as a small electron-proton mass ratio, and a small number of particles. However, the effects of magnetic field can be included, and the treatment is fully 3-dimensional and self-consistent.
+
== Conclusions ==
-
The fact that these two very different approaches return similar effects in terms of electron acceleration is perhaps surprising, and is a strong argument in favour of such an acceleration effect occuring.  
+
The effects of Langmuir waves on HXR emission from an electron beam were considered a long time ago, but only as an energy loss process for the beam, where they were found to have no effect on the time-averaged electron spectrum.
 +
However our simulations found significant electron acceleration, due to the redistribution of energy from below to above 20 keV. This shows that one needs far less electrons to produce the HXR spectrum as observed by, for example, RHESSI. When this HXR spectrum is analysed to deduce the electron spectrum generating the HXR emission via use of standard inversion techniques, this could be substantially overestimated (refs [2] and [5]).
-
== Conclusions ==
+
== References ==
 +
 
 +
[1] [http://adsabs.harvard.edu/abs/1967PlPh....9..719V Oscillations and instability of a weakly turbulent plasma]
 +
 
 +
[2] [http://adsabs.harvard.edu/abs/2012A%26A...539A..43K Wave-particle interactions in non-uniform plasma and the interpretation of hard X-ray spectra in solar flares]
-
The effects of Langmuir waves on HXR emission from an electron beam were considered a long time ago, but until now a full self consistent treatement of the Langmuir wave evolution was not performed. In the former case, the Langmuir waves were found to have no effect on the time-averaged electron spectrum. However in our simulations, Langmuir wave evolution is seen to lead to significant electron acceleration, and thus a modified HXR spectrum.
+
[3] [http://adsabs.harvard.edu/abs/2012ApJ...761..176R Density Fluctuations and the Acceleration of Electrons by Beam-generated Langmuir Waves in the Solar Corona]
-
We may still recover the electron spectrum generating the HXR emission using standard inversion techniques, but this will only return the electron distribution as injected into the dense region from which bremsstrahlung originates, and this will NOT be the same as the originally accelerated electron spectrum. Moreover, the complex nature of the beam-wave interactions means we cannot easily, if at all, recover this original spectrum.
+
[4] [http://adsabs.harvard.edu/abs/2012A%26A...544A.148K Electron acceleration during three-dimensional relaxation of an electron beam-return current plasma system in a magnetic field]
-
To conclude, by answering the question posed in our abstract- yes, Langmuir wave generation and evolution can affect the electron spectrum in such a way as to be visible in the HXR emission from loop footpoints.
+
[5] [http://adsabs.harvard.edu/abs/2013A%26A...550A..51H Effect of turbulent density-fluctuations on wave-particle interactions and solar flare X-ray spectra]

Latest revision as of 10:03, 23 February 2013


Nugget
Number: 194
1st Author: Heather Ratcliffe
2nd Author: Marian Karlický and Eduard Kontar
Published: 2013 February 18
Next Nugget: Burst-on-Tail (BOT)
Previous Nugget: Passages of Electron Beams
List all



Contents

Introduction

During solar flares, vast amounts of energy are released from the Sun's magnetic field, part of which leads to particle acceleration. Fast electrons thus produced can propagate along a coronal magnetic loop, and produce Hard X-ray (HXR) emission at its footpoints by collisional bremsstrahlung in the dense chromosphere. This HXR emission is one of the primary diagnostics of energetic electrons, and is usually interpreted using the "thick target" model. This assumes, among other things, that the distribution function of the emitting electrons is modified only by collisions.

However, when such fast electron beams propagate in a plasma, Langmuir (plasma) waves are generated. These waves are strongly affected by density inhomogeneities, or by wave-wave interactions, and this evolution can have significant effects on the electron distribution. Several simulation methods are applicable in this situation. In this Nugget we show some quasilinear and PIC (particle-in-cell) simulations of the electron evolution and briefly discuss the effects on HXR emission.

Quasilinear Simulations

Our model (based on ref. [1]) describes collisional relaxation of the energetic power-law distribution. We consider firstly a simplified model including only collisions, and then the relaxation including wave generation, and evolution.

Figure 1: Time-averaged electron fluxes. In each panel the black line shows the case without Langmuir wave generation, while the blue lines show, respectively: left panel-with Langmuir wave generation, middle panel-with plasma density fluctuations, and right panel-fluctuating density plasma plus wave-wave interactions. (From ref. [2])

The primary quantity of interest is the time-averaged electron flux as a function of electron energy, which is closely related to the observed HXR emission. We show this for the simulation models with and without Langmuir-wave generation (left panel), with Langmuir-wave generation and evolution due to density inhomogeneities (middle panel) (see ref. [3] for details), and finally with wave-wave processes also included (right panel).

We see from Figure 1 that the Langmuir-wave generation alone has a very weak effect, which confirms a well-known previous result. However, the time evolution of the Langmuir waves can produce significant changes in the HXR emission. For the model parameters chosen, this occurs primarily between 20 and 200 keV, and within this range we can expect an increase in HXR emission of an order of magnitude (Figure 1).

PIC simulations

It is also possible to use a 3D "particle-in-cell" (PIC) simulation to consider the problem of a monoenergetic beam injected into a plasma, with the effects of wave-wave interactions included. In these PIC simulations (ref [3]) the proton-electron mass ratio was taken to be 16 for computational reasons, but that is sufficient for these simulations. The initial electron beam was homogeneous throughout the numerical box of the simulation, and an appropriate return current introduced to keep the total current in the system zero. Periodic boundary conditions were used.

Figure 2: The electron energy distributions (solid lines) at (normalized) time t = 200. The magnetic field is zero in model A and increases through models B to F. For comparison in each panel we plot the initial electron plasma distribution together with the initial monoenergetic beam (dashed lines).


Using this model we made several computational runs, in which we also changed the magnetic field, oriented in the beam propagation direction. In Fig. 2 we present the the resulting electron energy distributions at the (normalized) time t = 200, for the magnetic field expressed through the ratio of the electron-cyclotron frequency to the plasma frequency, equal to 0.0, 0.1, 0.5, 0.7, 1.0, and 1.3, respectively (models A-F). As can be seen here, there are electrons accelerated above their initial energy, and the number of these increases with the magnetic field strength. This is due to the Weibel instability, which in the 3-D beam-plasma system with low magnetic fields reduces this acceleration process.

Complementary approaches

The two simulation methods presented here are very different, and each has its advantages and disadvantages.

Quasilinear simulations use weak turbulence theory. Computationally such simulations are fast, and the beam-plasma interaction is well treated by such a model. PIC simulations are computationally demanding and limited to short time and spatial scales, and thus require such approximations as a small electron-proton mass ratio, and a small number of particles. However, the effects of magnetic field can be included, and the treatment is fully 3-D and self-consistent. Thus the two methods offer very good independent confirmation, and together give a strong argument for such an acceleration effect occurring.

Conclusions

The effects of Langmuir waves on HXR emission from an electron beam were considered a long time ago, but only as an energy loss process for the beam, where they were found to have no effect on the time-averaged electron spectrum. However our simulations found significant electron acceleration, due to the redistribution of energy from below to above 20 keV. This shows that one needs far less electrons to produce the HXR spectrum as observed by, for example, RHESSI. When this HXR spectrum is analysed to deduce the electron spectrum generating the HXR emission via use of standard inversion techniques, this could be substantially overestimated (refs [2] and [5]).

References

[1] Oscillations and instability of a weakly turbulent plasma

[2] Wave-particle interactions in non-uniform plasma and the interpretation of hard X-ray spectra in solar flares

[3] Density Fluctuations and the Acceleration of Electrons by Beam-generated Langmuir Waves in the Solar Corona

[4] Electron acceleration during three-dimensional relaxation of an electron beam-return current plasma system in a magnetic field

[5] Effect of turbulent density-fluctuations on wave-particle interactions and solar flare X-ray spectra

Facts about Electron re-acceleration and HXR emissionRDF feed
RHESSI Nugget Date18 February 2013  +
RHESSI Nugget First AuthorHeather Ratcliffe  +
RHESSI Nugget Index194  +
RHESSI Nugget Second AuthorMarian Karlický and Eduard Kontar  +
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