Remembering John Brown

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[https://ui.adsabs.harvard.edu/abs/1971SoPh...18..489B/abstract John's first paper] was published in 1971. As of 22 November 2019, [https://ui.adsabs.harvard.edu/ ADS] says it has been cited 889 times. It introduced several ideas that became key to the study of flares and their X-radiation. A previous nugget dealt with the [http://sprg.ssl.berkeley.edu/~tohban/wiki/index.php/John_Brown_and_the_thick-target_model thick target model]. Another new idea was the possibility of "inverting" observed X-ray spectra to deduce the energy distribution of the emitting electrons. Electrons are accelerated to 10s of keV energy in the flare. The hard X-rays they emit via bremsstrahlung yield information on their number and energies.
[https://ui.adsabs.harvard.edu/abs/1971SoPh...18..489B/abstract John's first paper] was published in 1971. As of 22 November 2019, [https://ui.adsabs.harvard.edu/ ADS] says it has been cited 889 times. It introduced several ideas that became key to the study of flares and their X-radiation. A previous nugget dealt with the [http://sprg.ssl.berkeley.edu/~tohban/wiki/index.php/John_Brown_and_the_thick-target_model thick target model]. Another new idea was the possibility of "inverting" observed X-ray spectra to deduce the energy distribution of the emitting electrons. Electrons are accelerated to 10s of keV energy in the flare. The hard X-rays they emit via bremsstrahlung yield information on their number and energies.
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The bremsstrahlung cross-section ''dσ/dε'' gives the probability that an electron of energy ''E'' will emit a photon of energy ''ε''. At any instant, the flux at Earth is  ''I(ε)'':
+
<p>The bremsstrahlung cross-section ''d&sigma;/d&epsilon;'' gives the probability that an electron of energy ''E'' will emit a photon of energy ''&epsilon;''. At any instant, the flux at Earth is  ''I(&epsilon;)'':
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[[Image:eqn1.jpg|500px]]
+
<br>[[Image:eqn1.jpg|500px]]
-
Here ''d'' = 1 AU, ''v'' is the speed of an electron of energy ''E'' and
+
<br>Here ''d'' = 1 AU, ''v'' is the speed of an electron of energy ''E'' and
 +
 
 +
<br>[[Image:eqn2.jpg|500px]]
 +
 
 +
<br>i.e., the angle-averaged electron distribution ''f'' weighted by ambient density ''n'' and integrated through the whole of the source region. In the limiting, thick or thin target cases, ''F(E)'' can be simply related to the flux of electrons being injected to the X-ray emission region - information that should be intimately related to the behaviour and physical character of the energy release process.
-
[[Image:eqn2.jpg|500px]]
 
-
i.e., the angle-averaged electron distribution ''f'' weighted by ambient density ''n'' and integrated through the whole of the source region. In the limiting, thick or thin target cases, ''F(E)'' can be simply related to the flux of electrons being injected to the X-ray emission region - information that should be intimately related to the behaviour and physical character of the energy release process.
 
-
(E)
 
How do we use Equation (1) and the observed spectrum to learn about the fast electron population ''F''? One approach is to assume a parametric form for ''F(E)'', e.g. the power-law ''AE<sup>-&delta;</sup>'' and adjust the parameters (in this example, &delta; and ''A'') to obtain the best fit to ''I(&epsilon;)''. An alternative approach regards (1) as an integral equation and attempts to solve for the unknown ''F(E)''. John employed the Born approximation bremsstrahlung cross-section due to Bethe and Heitler, in its non-relativistic limit, and showed that (1) reduced to an instance of [https://www.encyclopediaofmath.org/index.php/Abel_integral_equation Abel's integral equation]. Since this equation has a known, analytical solution, this opens up the possibility of deriving the form of ''F(E)'' directly, non-parametrically from the form of ''I(&epsilon;)''. What a discovery!  
How do we use Equation (1) and the observed spectrum to learn about the fast electron population ''F''? One approach is to assume a parametric form for ''F(E)'', e.g. the power-law ''AE<sup>-&delta;</sup>'' and adjust the parameters (in this example, &delta; and ''A'') to obtain the best fit to ''I(&epsilon;)''. An alternative approach regards (1) as an integral equation and attempts to solve for the unknown ''F(E)''. John employed the Born approximation bremsstrahlung cross-section due to Bethe and Heitler, in its non-relativistic limit, and showed that (1) reduced to an instance of [https://www.encyclopediaofmath.org/index.php/Abel_integral_equation Abel's integral equation]. Since this equation has a known, analytical solution, this opens up the possibility of deriving the form of ''F(E)'' directly, non-parametrically from the form of ''I(&epsilon;)''. What a discovery!  
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If we use more precise expressions for the cross-section the analytical solution of Abel's integral equation is no longer applicable but a numerical (matrix) process can still be constructed, deducing a vector of values (''F<sub>1</sub>,F<sub>2</sub>,F<sub>3</sub>...F<sub>M</sub>'') from a histogram of values (''I<sub>1</sub>, I<sub>2</sub>,...,I<sub>L</sub>).
 
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if the observations giving the spectrum ''J(&epsilon;)'' are sufficiently free from noise, and sufficiently well resolved. The scintillators in use in the 1970s had a typical energy resolution &delta;&epsilon;/&epsilon; of about 10%, inadequate for implementation of this approach but the possibility might be realisable with other sorts of X-ray spectrometer.
+
If we use more precise expressions for the cross-section the analytical solution of Abel's integral equation is no longer applicable but a numerical (matrix) process can still be constructed, deducing a vector of values '''F''' = (''F<sub>1</sub>,F<sub>2</sub>,F<sub>3</sub>...F<sub>M</sub>'') from a histogram of values '''I''' = (''I<sub>1</sub>, I<sub>2</sub>,...,I<sub>L</sub>).
 +
 
 +
 
 +
In practice this analytical solution is not so easy to use. It involves the first and second derivatives of ''I(&epsilon;)'', difficult to obtain from noisy data. Also the scintillators used for hard X-ray spectroscopy in the 1970s, NaI(Tl) etc., had a typical energy resolution &delta;&epsilon;/&epsilon; of about 10% or worse, inadequate for implementation of this approach. The possibility might be realisable with other sorts of X-ray spectrometer, however.
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== Ill-conditioned problems ==
== Ill-conditioned problems ==
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A worse difficulty lay just below the surface.
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The relationship of (1) may be approximated as

Revision as of 22:37, 1 January 2020

John C Brown (1947-2019)

John Brown passed away unexpectedly on 16 November 2019, a great sadness to all who knew him and a huge loss particularly to the Glasgow Astronomy and Astrophysics group. His many contributions to solar physics were highly influential and we were prompted to think back over some of the big themes of his research career. In this nugget we look at one of those big themes, inverse problems in astrophysics. This particular topic spans the whole of John's research career and combines a technical side with a more philosophical approach to the character of data and the limits on what one can deduce from them.


Brown(1971)

John's first paper was published in 1971. As of 22 November 2019, ADS says it has been cited 889 times. It introduced several ideas that became key to the study of flares and their X-radiation. A previous nugget dealt with the thick target model. Another new idea was the possibility of "inverting" observed X-ray spectra to deduce the energy distribution of the emitting electrons. Electrons are accelerated to 10s of keV energy in the flare. The hard X-rays they emit via bremsstrahlung yield information on their number and energies.

The bremsstrahlung cross-section dσ/dε gives the probability that an electron of energy E will emit a photon of energy ε. At any instant, the flux at Earth is I(ε):
Eqn1.jpg
Here d = 1 AU, v is the speed of an electron of energy E and
Eqn2.jpg
i.e., the angle-averaged electron distribution f weighted by ambient density n and integrated through the whole of the source region. In the limiting, thick or thin target cases, F(E) can be simply related to the flux of electrons being injected to the X-ray emission region - information that should be intimately related to the behaviour and physical character of the energy release process. How do we use Equation (1) and the observed spectrum to learn about the fast electron population F? One approach is to assume a parametric form for F(E), e.g. the power-law AE and adjust the parameters (in this example, δ and A) to obtain the best fit to I(ε). An alternative approach regards (1) as an integral equation and attempts to solve for the unknown F(E). John employed the Born approximation bremsstrahlung cross-section due to Bethe and Heitler, in its non-relativistic limit, and showed that (1) reduced to an instance of Abel's integral equation. Since this equation has a known, analytical solution, this opens up the possibility of deriving the form of F(E) directly, non-parametrically from the form of I(ε). What a discovery! If we use more precise expressions for the cross-section the analytical solution of Abel's integral equation is no longer applicable but a numerical (matrix) process can still be constructed, deducing a vector of values F = (F1,F2,F3...FM) from a histogram of values I = (I1, I2,...,IL). In practice this analytical solution is not so easy to use. It involves the first and second derivatives of I(ε), difficult to obtain from noisy data. Also the scintillators used for hard X-ray spectroscopy in the 1970s, NaI(Tl) etc., had a typical energy resolution δε/ε of about 10% or worse, inadequate for implementation of this approach. The possibility might be realisable with other sorts of X-ray spectrometer, however.

Ill-conditioned problems

The relationship of (1) may be approximated as

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