I understand the color temperature of radiation to be the temperature of a blackbody that has a similar spectrum as the radiation. I am puzzled by references in astrophysics to the Lyman-alpha color temperature. Lyα is a line, so how can there be a corresponding blackbody that matches the spectrum?— what spectrum?

Update: Here is an example, taken from Furlanetto, Peng and Briggs (2006):

The Lyα coupling also depends on the effective temperature $T_c$ of the UV radiation field, defined in equation (22). This is determined by the shape of the photon spectrum at the Lyα resonance. That the effective temperature of the radiation field must matter is easy to see: the energy defect between the different hyperfine splittings of the Lyα transition implies that the mixing process is sensitive to the gradient of the background spectrum near the Lyα resonance.

Equation (22) is (from earlier):

We have then defined the effective color temperature of the UV radiation field $T_c$ via $$\frac{P_{01}}{P_{10}}=3 (1-\frac{E_{10}}{k_B T_c})$$

where $P_{01}$ and $P_{10}$ are the UV (Lyα I presume) scattering excitation and de-excitation rates (per atom) respectively.


Note that much of the literature uses a slightly different definition of the color temperature (in terms of $J_\nu$) that does not obey detailed balance (e.g., [133, 135]). Obviously the color temperature is a function of frequency; in detail, it should be harmonically averaged over the line profile [136], but this makes only a tiny difference in practice [135]. Note as well that slightly different definitions of $T_c$ can be useful for certain applications [136].

I'm not sure what they mean by "background spectrum". But I'm getting the sense that there is a trend of defining the color temperature by the equation above, even if it's not strictly the normal definition. It seems to me this was started by Field (1958).

  • $\begingroup$ arxiv.org/pdf/astro-ph/0303395.pdf discusses "color temperature" as related to the slope of the radiation spectrum in the vicinity of the specific wavelength (Ly-alpha). $\endgroup$ – BowlOfRed Apr 11 '17 at 23:02
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    $\begingroup$ Do you have a citation (or any general example) where this term is used? (That does sound non-intuitive!) $\endgroup$ – DilithiumMatrix Apr 11 '17 at 23:32
  • $\begingroup$ @DilithiumMatrix updated $\endgroup$ – binaryfunt Apr 12 '17 at 10:09
  • $\begingroup$ That's not quite the standard formulation, but if you assume that the excitation rates are proportional to the populations then your definition is just the standard Boltzmann statistics. $\endgroup$ – Emilio Pisanty Apr 12 '17 at 11:33
  • $\begingroup$ Ah, I'm thinking they talk about "background spectrum" because of scattering of Lyα photons leading to a broadening, perhaps? $\endgroup$ – binaryfunt Apr 12 '17 at 18:34

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