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Usual methods of estimating stellar ages involve isochrone approximations. It can also help to estimate a star's radius by correlating its absolute magnitude with effective temperature and apparent magnitude. In the absence of these measurements or observations of tell-tale variability, how might you guess a star's age and evolutionary status?

Given a single, high-resolution $(R\gtrsim 50000)$ spectrum as your only data point, how easy is it to accurately infer the age and evolutionary status of a star? For example, how would the spectrum differ between a red dwarf and red giant, both with of $T_{\mathrm{eff}} = 4000~\mathrm{K}$? Or between two red dwarfs of ages $2~\mathrm{Gyr}$ and $8~\mathrm{Gyr}$?

A good answer could describe how surface gravity $(\log g)$ affects spectral lines (and how this relates to stellar mass and radius), what elements we might observe more strongly at different stages of evolution, and some observational results of gyrochronology.

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The point is not to differentiate between red dwarf and red gians (they are very different stars with very different spectrums) but to differentiate between a young red dwarf and an old one. –  Envite Nov 30 '13 at 14:07
Indeed that is true, but perhaps some of those differences are also manifested (albeit more subtly) in a comparison between a 2 Gyr and 8 Gyr old red dwarf - it's these more precise determinations that I am more interested in. Essentially, I am wondering how easy (or possible) it is to make a reasonably precise (say to 1 Gyr) age determination that does not rely on isochrones. –  Moriarty Nov 30 '13 at 14:30

3 Answers 3

In short: you can not.

In length: best you can do is to match up your spectrum with a library of known spectra, and find the best match. But for these spectra to be useful you need to have determined their ages, masses, Y's (contents of Helium) and Z's (contents of metals, that is, evrything beyond Helium). And their ages come from... yes, isochrones, so you would be using isochrones indirectly.

So, in short again, yes, you can determine the mass, age and Y and Z of a star with its spectrum and without its own isochrone, maybe up to 5% of its main-sequence lifetime during main sequence status (e.g. 0.5 Gyr for a 10 Gyr main-secuence lifetime star like our Sun).

And yes again, this match-up of spectra gives additional info like surface gravity, which is not useful on its own but needs previous knowledge of mass and radius.

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I'm no expert on stellar atmospheres, so I have a limited idea of how things like $\log g$ affect the lines. But I work with stellar models, so I can take a stab at that part.

The overall principle is that computing stellar model ages is a kind of optimization problem. We model the structure of stellar interiors by constructing a system of differential equations based on a few simple assumptions. (When I teach stellar structure and evolution, I usually recommend the outstanding and free lecture notes by Onno Pols and Jørgen Christensen-Dalsgaard.) These models depend on many parameters. Some are familiar: the mass, composition, and age. Some less-so: there's usually at least one parameter for how convection is parametrized. e.g. the mixing length. Some are discrete: which opacity data is used, what solar abundances are chosen. And some are relatively inconsequential: there a dozens (or even hundreds!) of numerical parameters used in solving the equations.

So let's just say we have a magical black box that takes five parameters—mass, initial metallicity, initial helium abundance, age and mixing length—and produces $T_\text{eff}$ and $\log g$. What we have to do is select values of the parameters to match the observations, which is a standard problem in optimization, inference, parameter estimation, or whatever you want to call it.

Keep in mind that age is a special parameter. There are ways of measuring things like mass, radius or luminosity relatively directly. But choosing the sequence of models that produces the appropriate star is always depends on which stellar models you use in the first place. Ages are uncertain both because of the uncertainties in the observations, but also because of the intrinsic uncertainty in the models. Although something like interferometry can potentially give an independent radius, we can only get indirect measures of age, and converting these indirect measures to ages also introduces uncertainty.

The trick now is how much data you have...

Given a single, high-resolution (R≳50000) spectrum as your only data point, how easy is it to accurately infer the age and evolutionary status of a star?

I'd say it's very hard to get an accurate (or even precise) age just given a single spectrum. Currently, the spectrum would probably first be used to determine $T_\text{eff}$ and $\log g$, and thus values would then be used as inputs in the stellar model. Remember: I'm talking about interior models, so they don't typically produce a model atmosphere to compare. You've then already got the issue that there are more parameters than observables. This is resolved by supposing that the mixing length parameter is the same as the best-fitting values for the Sun (for which we have much more data) and that the abundances of helium and metals are correlated. (We call this an enrichment law.) This makes the problem tractable, because the high-resolution spectrum should also tell us the metal content.

Knowing the evolutionary state is easier, I think, because the surface gravity should help you to distinguish, especially given a high resolution spectrum. As said, I'm no expert here, and I'm aware that misclassification can happen with multi-colour photometry, but I don't expect it to happen with high-res spectra.

If you'd like to read further, here are some quick resources that might be of interest. First, some lecture notes on determining stellar ages recently appeared on arXiv:

Second, you can play around with synthetic line profiles and other atmospheric data with GrayStar, a web app that computes basic atmosphere data. (I'm not experienced with it, so I'm not exactly sure how it works, but you can play around to get the information you want about e.g. the difference between line profiles in giants and dwargs, I think.)

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For some very large (and hence relatively cool) red giants you might be able to ascertain something from their spectra, as emission lines are sometimes seen - these are typically brighter central patches seen the middle of the more typical absorption (dark) spectral lines - caused by the large size of (realtively!) hot gas clouds that surround the giants. But that would not be a reliable method of red giant detection.

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