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This is a question about the basics of astronomy, which I have never happened to see a good discussion for. It is about how well would we be able to measure effective temperature of a star, if we had any arbitrarily perfect measurement devices.

Here is some context. Canonical definition of $T_{\textrm{eff}}$ of a star is based on its bolometric luminosity $L$ (total electromagnetic energy radiated by the star per unit time) and its photospheric radius R (radius, at which the optical depth at a given wavelength is equal to unity). This way, the definition specifies $T_{\textrm{eff}}$ through $L=4\pi \sigma R^2 T_{\textrm{eff}}^4$, where $\sigma$ is Stefan-Boltzmann constant.

The definition clearly alludes to black-body law. Many stars, including our own Sun, have a spectrum that does not follow it. For this reason, one often talks about another effective temperature, which is the temperature of stellar material at photospheric radius, and which can be determined by examining stellar spectrum. There are a few more complications to that, but let's put them aside.

Determining $T_{\textrm{eff}}$ is extremely important in characterising stars, therefore there exists a variety of methods of measuring it, and naturally researchers strive for obtaining the best possible precision.

Hence, the question: How well can one in principle measure $T_{\textrm{eff}}$, if one could have arbitrarily perfect instruments?

Edit: I would like to see a quantitative estimate in your answer. Is the best possible precision for $T_{\textrm{eff}}$ of order $10\textrm{K}$, or is it $1\textrm{K}$, or some $10^{-4}\textrm{K}$, or can we measure it arbitrarily well?

Here are just a few sources of uncertainty/arbitrariness: convection in stars, dependence of photospheric radius on wavelength, limb darkening, stellar variability, to name a few.

I would encourage the answers to be in the format "Source of uncertainty" - "Simple derivation" - "Estimate of the effect". If there are more than a few estimates, I will add a summary of them in the question or in a separate reply. Please, also feel free to edit the question if you might like to.

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It is quite easy. In fact you do not need a bolometer. You just need to perform Intensity measurements in several parts of the spectrum, and then fit these to a teorethical black body spectrum. Three uses to be enough if it does not happen that you are measuring on a spike or valley in the spectrum caused by an emission or absortion line. The black body spectrum that best fits your measurements will give you Teff.

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First off, part of the uncertainty of the temperature will come from the goodness of the fit (what will it be?). Secondly, do you mean spectrum over the whole disk, or just in the center? If it is the disc, you are measuring T of rather different layers, of it is the center, you are more subject to irregularities (by how much?). Finally, remember about stellar varibility, sunspots, etc. Given all this, no, it is not quite easy. There is a set of phenomena, which limit foundamentally the robustness of the definition of $T$. Main point of my question is - to what degree? –  Alexey Bobrick Nov 24 '13 at 13:50
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It is quite easy, since for most stars out there we can not distinguish disc and atmosphere. All we see is a Airy disc of the whole of the star light. –  Envite Nov 24 '13 at 20:01
    
Perhaps, I should have stressed in the question, but I really mean that we are looking at the star with arbitrarily perfect instruments: with perfect resolution, sensitivity etc. When talking about the disc, because of limb darkening different parts of the disc correspond to different temperatures. –  Alexey Bobrick Nov 24 '13 at 20:14
    
With perfect instruments you would simply measure R with a perfect telescope and L with a perfect bolometer and apply the formula. No trouble in that. –  Envite Nov 25 '13 at 2:16
    
And how do you 'simply' measure R with a perfect telescope? Let me remind you that R is dependent on $\lambda$, and that the stars are not spheres, nor are they isotropic and contant in time. You can argue, that these are all small effects, but it is the value of the errors from them that I am looking for. Whether the definition of $T_{eff}$ is model dependent up to $10 \textrm{K}$, or whether it is $1\textrm{K}$, or some $10^{-4} \textrm{K}$. –  Alexey Bobrick Nov 25 '13 at 10:55
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