Fast electrons relaxing
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| - | As the parameter <math>\kappa \rightarrow \infty</math> this distribution becomes identical with a Maxwell-Boltzmann, with thermal speed <math>v_{0}</math>. | + | As the parameter <math>\kappa \rightarrow \infty</math> this distribution becomes identical with a Maxwell-Boltzmann, with thermal speed <math>v_{0}</math>. For smaller values of <math>\kappa</math> the distribution still looks maxwellian at small speeds but displays a 'tail' at high energies (<math>v > v_0</math>). |
| - | We can wave our hands and concoct explanations for the ubiquity of these distributions: there's turbulence, all sorts of microscopic, fluctuating electric fields, a huge range of conditions at its base in the region where it becomes collisionless, it's too tenuous for collisions to re-assert the maxwellian distribution.... This is all very fine and good, but is there a better explanation, something more fundamental | + | We can wave our hands and concoct explanations for the ubiquity of these distributions: there's turbulence, all sorts of microscopic, fluctuating electric fields, a huge range of conditions at its base in the region where it becomes collisionless, it's too tenuous for collisions to re-assert the maxwellian distribution.... This is all very fine and good, but is there a better explanation, something more fundamental? Possibly. |
| - | + | To obtain the Maxwell-Boltzmann distribution we can imagine dividing our velocity space into lots of tiny boxes, dropping electrons one by one randomly into these little boxes, and asking how likely, statistically, a particular distribution across little boxes is. The Maxwell-Boltzmann distribution is the most likely of these possibilities, overwhelmingly so as the number of particles becomes realistically large. But this argument supposes that we really can drop our electrons randomly into the velocity space boxes, independently of what has happened previously. Suppose instead that the probability of an electron being dropped randomly into one velocity space box depends on how many electrons have already been dropped into the neighbouring boxes. Then the Maxwell-Boltzmann distribution would no longer be the most likely. Instead we would find a kappa distribution, where now κ parametrises the degree of inter-dependence of velocity space elements and also determines the steepness of the power-law tail. As <math>\kappa \rightarrow \infty</math> we recover the maxwellian form. This essential idea has of course been put much more elegantly and powerfully, in terms of <a href="http://adsabs.harvard.edu/abs/2004PhPl...11.1308L">Tsallis entropy</a> and <a href="http://adsabs.harvard.edu/abs/2008PhRvL.100o5005T">Gibbsian statistics</a>. Kappa distributions are indicated now by arguments of such generality that we can expect them to result from many electron acceleration scenarios, even if their physical details vary. Jana Kasparova and Marian Karlicky have <a href="http://adsabs.harvard.edu/abs/2009A%26A...497L..13K">compared it with RHESSI observations</a> and found that it might describe X-ray spectra seen in some coronal sources (which makes sense; we wouldn't expect a single kappa distribution to represent the variety of conditions and transport processes involved in the entirety of a flare). | |
| - | + | ||
| + | == Our dream hard X-ray source == | ||
| + | Ever since the 1970's people have looked longingly at 'thermal' hard X-ray sources, in which all the electrons have similar energies and collisions between them, that make thick target models so inefficient, serve only to redistribute energy and drive the distribution to maxwellian form. In principle the evolution of such a source is mathematically tricky, a diffusion (Fokker-Planck) type problem in which the diffusion coefficients change all the time, with the function we want to know. We found a way of appealing to energy and number conservation that lets us describe most of what is going on without solving this hard nnumerical problem. We assumed that all our electrons are trapped in a box, given an initial, kappa-like state (not quite kappa but a similar, "core+tail" distribution that lets us do everything analytically) and allowed to evolve by colliding with one another. A sequence of resulting hard X-ray spectra are shown here; these are strongly reminiscent of the evolution of the 'superhot source' observed in the flare of blah 1981 by Bob Lin and colleagues. We estimate that such a source might be 7 - 10 times more efficient at producing hard X-rays than a thick target. | ||
Revision as of 10:28, 18 August 2010
| Nugget | |
|---|---|
| Number: | 134 |
| 1st Author: | Alec MacKinnon |
| 2nd Author: | |
| Published: | 2010 August 23 |
| Next Nugget: | The X-ray limb |
| Previous Nugget: | Chromospheric plasma parameters |
| List all | |
James Clerk Maxwell
James Clerk Maxwell first wrote down the distribution of velocities in a gas of fixed temperature, a contribution to theoretical physics that most of us now use routinely, without reflection, on a daily basis. The Maxwell-Boltzmann distribution represents a state of maximum entropy, the overwhelmingly most likely state given that the temperature of the gas has been fixed. It is such a standard ingredient of our thinking about gases and plasmas that "non-maxwellian" is one of the trendiest words you can use. A non-maxwellian distribution is of course what we would expect to find in a coronal hard X-ray source, like the one Säm Krucker has described. How would it relax to maxwellian form? Might we recognise this in observations? How efficient is it as a source of hard X-rays? Ross Galloway, Per Helander, John Brown and I have been thinking about this here. I'll describe some of what we've been doing and why it might be interesting for solar hard X-rays. First some more non-maxwellian words.
Kappa distribution
Nonmaxwellian distributions are found ubiquitously in the plasma of the solar wind. They are measured in situ in space and are found to be well described by so-called kappa distributions:
As the parameter 
this distribution becomes identical with a Maxwell-Boltzmann, with thermal speed v0. For smaller values of κ the distribution still looks maxwellian at small speeds but displays a 'tail' at high energies (v > v0).
We can wave our hands and concoct explanations for the ubiquity of these distributions: there's turbulence, all sorts of microscopic, fluctuating electric fields, a huge range of conditions at its base in the region where it becomes collisionless, it's too tenuous for collisions to re-assert the maxwellian distribution.... This is all very fine and good, but is there a better explanation, something more fundamental? Possibly.
To obtain the Maxwell-Boltzmann distribution we can imagine dividing our velocity space into lots of tiny boxes, dropping electrons one by one randomly into these little boxes, and asking how likely, statistically, a particular distribution across little boxes is. The Maxwell-Boltzmann distribution is the most likely of these possibilities, overwhelmingly so as the number of particles becomes realistically large. But this argument supposes that we really can drop our electrons randomly into the velocity space boxes, independently of what has happened previously. Suppose instead that the probability of an electron being dropped randomly into one velocity space box depends on how many electrons have already been dropped into the neighbouring boxes. Then the Maxwell-Boltzmann distribution would no longer be the most likely. Instead we would find a kappa distribution, where now κ parametrises the degree of inter-dependence of velocity space elements and also determines the steepness of the power-law tail. As we recover the maxwellian form. This essential idea has of course been put much more elegantly and powerfully, in terms of <a href="http://adsabs.harvard.edu/abs/2004PhPl...11.1308L">Tsallis entropy</a> and <a href="http://adsabs.harvard.edu/abs/2008PhRvL.100o5005T">Gibbsian statistics</a>. Kappa distributions are indicated now by arguments of such generality that we can expect them to result from many electron acceleration scenarios, even if their physical details vary. Jana Kasparova and Marian Karlicky have <a href="http://adsabs.harvard.edu/abs/2009A%26A...497L..13K">compared it with RHESSI observations</a> and found that it might describe X-ray spectra seen in some coronal sources (which makes sense; we wouldn't expect a single kappa distribution to represent the variety of conditions and transport processes involved in the entirety of a flare).
Our dream hard X-ray source
Ever since the 1970's people have looked longingly at 'thermal' hard X-ray sources, in which all the electrons have similar energies and collisions between them, that make thick target models so inefficient, serve only to redistribute energy and drive the distribution to maxwellian form. In principle the evolution of such a source is mathematically tricky, a diffusion (Fokker-Planck) type problem in which the diffusion coefficients change all the time, with the function we want to know. We found a way of appealing to energy and number conservation that lets us describe most of what is going on without solving this hard nnumerical problem. We assumed that all our electrons are trapped in a box, given an initial, kappa-like state (not quite kappa but a similar, "core+tail" distribution that lets us do everything analytically) and allowed to evolve by colliding with one another. A sequence of resulting hard X-ray spectra are shown here; these are strongly reminiscent of the evolution of the 'superhot source' observed in the flare of blah 1981 by Bob Lin and colleagues. We estimate that such a source might be 7 - 10 times more efficient at producing hard X-rays than a thick target.
| RHESSI Nugget Date | 23 August 2010 + |
| RHESSI Nugget First Author | Alec MacKinnon + |
| RHESSI Nugget Index | 134 + |
