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 | + | [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: | ||
<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> | |
| + | This was both reasonably accurate for the relevant energy range, and simple enough algebraically to allow a largely analytical discussion. | ||
| + | |||
| + | Write ''J(ε)'' for the observed energy flux at photon energy ''ε'' (erg keV<sup>-1</sup> cm<sup>-2</sup> s<sup>-1</sup>). John showed that | ||
| + | <math> | ||
| + | J(\epsilon) \sim \int_\epsilon^\infty F(E) \frac{N(E)}{\sqrt{E-\epsilon}} \mathrm{d}E | ||
</math> | </math> | ||
| + | Here ''N(E)'' is the electron energy distribution in the source. This is Abel's integral equation, which has a solution for ''N(E)'' in terms of ''J(ε)'' and its derivatives: | ||
| + | <math></math> | ||
| + | |||
| + | What a wonderful discovery! The electron distribution in the source can be deduced directly, in a non-parametric way, rather than e.g. assuming some parametric form for ''N(E)'' and finding the values of the parameters that give the best fit. | ||
Revision as of 15:49, 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 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: 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.
Write J(ε) for the observed energy flux at photon energy ε (erg keV-1 cm-2 s-1). John showed that 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
Here N(E) is the electron energy distribution in the source. This is Abel's integral equation, which has a solution for N(E) in terms of J(ε) and its derivatives:
What a wonderful discovery! The electron distribution in the source can be deduced directly, in a non-parametric way, rather than e.g. assuming some parametric form for N(E) and finding the values of the parameters that give the best fit.
