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. | [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. | ||
| - | The probability of an energetic particle emitting a photon in a close encounter with another charged particle is given by the cross-section. John used the Born approximation cross-section due to Bethe and Heitler, in its non-relativistic limit: | + | The bremsstrahlung spectrum observed at Earth will be given by |
| + | <math>J(\epsilon) = K \int_\epsilon^\infty blah blah \mathrm{d}E | ||
| + | </math> Here J is the energy flux detected at Earth (erg keV<sup>-1</sup> cm<sup>-2</sup> s<sup>-1</sup>) and N is blah blah, after an appropriate average over the source volume. The probability of an energetic particle emitting a photon in a close encounter with another charged particle is given by the cross-section. John used the Born approximation cross-section differential in photon energy due to Bethe and Heitler, in its non-relativistic limit: | ||
<math> | <math> | ||
\frac{\mathrm{d}\sigma}{\mathrm{d}\epsilon} = \frac{\sigma_0}{\epsilon E}\ln\{\frac{1+\sqrt{1-\epsilon/E}}{1-\sqrt{1-\epsilon/E}}\}</math> | \frac{\mathrm{d}\sigma}{\mathrm{d}\epsilon} = \frac{\sigma_0}{\epsilon E}\ln\{\frac{1+\sqrt{1-\epsilon/E}}{1-\sqrt{1-\epsilon/E}}\}</math> | ||
| - | This was both reasonably accurate for the relevant energy range, and simple enough algebraically to allow a largely analytical discussion. | + | This was both reasonably accurate for the relevant energy range, and simple enough algebraically to allow a largely analytical discussion. |
| - | + | With some algebra, this expression can be rewritten | |
<math> | <math> | ||
J(\epsilon) \sim \int_\epsilon^\infty F(E) \frac{N(E)}{\sqrt{E-\epsilon}} \mathrm{d}E | J(\epsilon) \sim \int_\epsilon^\infty F(E) \frac{N(E)}{\sqrt{E-\epsilon}} \mathrm{d}E | ||
</math> | </math> | ||
| - | + | This is Abel's integral equation, which has a solution for ''N(E)'' in terms of ''J(ε)'' and its derivatives: | |
<math></math> | <math></math> | ||
| - | What a | + | What a discovery! Before this people had tried to fit observed hard X-ray spectra by assuming parametric forms for ''N(E)'', e.g. a power-law in electron energy ''E<sup>-δ</sup> and looking for the value of ''δ'' that gave the best fit to the observed spectrum. Now we realise that the electron distribution in the source can be deduced directly, in a non-parametric way - if the observations giving the spectrum ''J(ε)'' are sufficiently free from noise, and sufficiently well resolved. The scintillators in use in the 1970s had a typical energy resolution δε/ε 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 inversion of Abel's integral equation is no longer applicable but a numerical (matrix) process can still be constructed, deducing a vector of values (''N(E<sub>1</sub>),N(E<sub>2</sub>),N(E<sub>3</sub>)...N(E<sub>M</sub>)'') from a histogram of values (''J<sub>1</sub>, J<sub>2</sub>,...,J<sub>L</sub>). | ||
| + | |||
| + | == Ill-conditioned problems == | ||
| + | |||
| + | A worse difficulty lay just below the surface. | ||
Revision as of 22:45, 30 December 2019
John C Brown (1947-2019)
John Brown passed away unexpectedly on 16 November 2019, a great sadness to all who knew him. His many contributions to solar physics were highly influential and we were prompted to think about 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.
The bremsstrahlung spectrum observed at Earth will be given by Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): J(\epsilon) = K \int_\epsilon^\infty blah blah \mathrm{d}E
Here J is the energy flux detected at Earth (erg keV-1 cm-2 s-1) and N is blah blah, after an appropriate average over the source volume. The probability of an energetic particle emitting a photon in a close encounter with another charged particle is given by the cross-section. John used the Born approximation cross-section differential in photon energy due to Bethe and Heitler, in its non-relativistic limit:
Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): \frac{\mathrm{d}\sigma}{\mathrm{d}\epsilon} = \frac{\sigma_0}{\epsilon E}\ln\{\frac{1+\sqrt{1-\epsilon/E}}{1-\sqrt{1-\epsilon/E}}\}
This was both reasonably accurate for the relevant energy range, and simple enough algebraically to allow a largely analytical discussion.
With some algebra, this expression can be rewritten Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): J(\epsilon) \sim \int_\epsilon^\infty F(E) \frac{N(E)}{\sqrt{E-\epsilon}} \mathrm{d}E
This is Abel's integral equation, which has a solution for N(E) in terms of J(ε) and its derivatives:
What a discovery! Before this people had tried to fit observed hard X-ray spectra by assuming parametric forms for N(E), e.g. a power-law in electron energy E-δ and looking for the value of δ that gave the best fit to the observed spectrum. Now we realise that the electron distribution in the source can be deduced directly, in a non-parametric way - if the observations giving the spectrum J(ε) are sufficiently free from noise, and sufficiently well resolved. The scintillators in use in the 1970s had a typical energy resolution δε/ε 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 inversion of Abel's integral equation is no longer applicable but a numerical (matrix) process can still be constructed, deducing a vector of values (N(E1),N(E2),N(E3)...N(EM)) from a histogram of values (J1, J2,...,JL).
Ill-conditioned problems
A worse difficulty lay just below the surface.
