How well can we in principle determine $T_{\textrm{eff}}$ of a star?

Question

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.

Answer 1

The question is compromised by saying that you allow arbitrarily perfect measurements.

If we have a bolometer that can measure the amount of flux from a star, at a distance that is known to arbitrary accuracy, with arbitrarily good spatial resolution, then what we do is measure the bolometric luminosity from a 1 m$^2$ area at the centre of the stellar disk. This flux is $\sigma T_{eff}^4$.

Now of course, stars do not have homogeneous atmospheres (spots, granulation, meridional flows, non-sphericity due to rotation...), so the result you would get would depend on exactly what 1 m$^2$ bit of atmosphere you were looking at. So, with my arbitrarily accurate instruments I would have to measure the luminosity from every 1 m$^2$ patch over the entire surface of the star. Each one would give me another estimate of $T_{eff}^4$; each would be somewhat different. That would be difficult, but the form of your question allows me to ignore those problems.

At this level of precision, the utility of a single $T_{eff}$ for the whole star is questionable, but if you wanted one then it would be the flux-weighted mean of all the above measurements, and as far as I can see one can instantaneously determine it to whatever accuracy you desire. Of course it will then vary if you have a variable star, and it will vary from point to point with time due to granulation; so the accuracy of the $T_{eff}$ could depend on how quickly and by how much it varies compared with how long it takes you to do your arbitrarily accurate measurements.

I think to get a better answer, you do need to specify some realistic observational constraints - such as (a) you cannot resolve the star at all, or (b) that you can resolve it, but observations can only take place from an earth-bound observatory (thus not allowing you to take flux measurements from the whole surface at once).

One thing occurs, is that in unresolved observations, even with an absolutely accurately measured luminosity (assuming isotropic radiation) there is still the issue of what radius to use. The radius at which the radiation escapes the star (at optical depth $\sim 2/3$) is ill-defined and wavelength dependent. An error bar of maybe tens of km is appropriate here, since atmospheres are 100-200 km "thick". For a solar-type star this would limit $T_{eff}$ accuracy to $\sim 0.1 K$ !

Answer 2

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 theoretical 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 absorption line. The black body spectrum that best fits your measurements will give you $T_{eff}$.