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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

4 Answers 4

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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-1 This seems a bit defeatist and fails to mention all the other ways you can estimate a stellar age. –  Rob Jeffries Dec 3 at 18:43
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@RobJeffries which other ways that are not indirectly based on isochrones? –  Envite Dec 3 at 18:44
    
The 6 I list in my answer. –  Rob Jeffries Dec 3 at 19:45
    
You say: "You can plot the star in the log g vs Teff plane and compare it with theoretical isochrones appropriate for the star's metallicity." so you are using isochrones. The question is "Without using isochrones". –  Envite Dec 4 at 6:58
    
Naturally I included the method which is actually used for Sunlike stars in this scenario, since it appeared the OP (and perhaps you) were unaware that an absolute mag. Is not required to match with isochrones. I then listed six other techniques that don't require isochrones. While I'm here - how do you find the abundance of He from a red dwarf spectrum and how does the "spectrum matching" technique lead to 0.5Gyr precision? Can you point me to an example? –  Rob Jeffries Dec 4 at 7:48

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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The spectra of giants and dwarfs are completely different at high resolution. How does this address the question? –  Rob Jeffries Dec 3 at 18:51
    
"how might you guess a star's age and evolutionary status?" –  adrianmcmenamin Dec 3 at 22:33

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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The spectra of a red giant and a red dwarf are completely different, so there isn't really too much to say about this. For example, alkali lines are almost non-existent in red giants, but strong in red dwarfs. The theory as to why this happens is the stuff of a standard graduate/undergraduate course on stellar atmospheres, not an SE answer.

The fact is that a R=50,000 spectrum with decent signal to noise ratio will quite easily give you the temperature (to 100K), surface gravity (to 0.1 dex) and metallicity (to 0.05 dex), plus a host of other elemental abundances (including Li) to precisions of about 0.1 dex.

What can you do with this:

You can plot the star in the log g vs Teff plane and compare it with theoretical isochrones appropriate for the star's metallicity. This is the best way to estimate the age of a solar-type (or more massive) star, even if you don't have a distance and is the most-used method. How well this works and how unambiguously depends on the star's evolutionary stage. For stars like the Sun, you get an age precision of maybe 2 Gyr. For lower mass stars, well they hardly move whilst on the main sequence in 10Gyr, so you can't estimate the age like this unless you know the object is a pre-main sequence star (see below).

You can look at the Li abundance. Li abundance falls with age for solar-mass stars and below. This would work quite well for sun-like stars from ages of 0.3-2Gyr and for K-type stars from 0.1-0.5 Gyr and for M-dwarfs between 0.02-0.1 Gyr - i.e. in the range where Li starts to be depleted in the photosphere and where is is all gone. Typical precision might be a factor of two. A high Li abundance in K and M dwarfs usually indicates a pre main sequence status.

Gyrochronology is not much help - that requires a rotation period. However you can use the relationship between rotation rate (measured in your spectrum as projected rotation velocity) and age. Again, the applicability varies with mass, but in the opposite way to Li. M-dwarfs maintain fast rotation for longer than G-dwarfs. Of course you have the problem of uncertain inclination angle.

That brings us to activity-age relations. You can measure the levels of chromospheric magnetic activity in the spectrum. Then combine this with empirical relationships between activity and age (e.g. Mamajek & Hillenbrand 2008). This can give you the age to a factor of two for stars older than a few hundred Myr. Its poorly calibrated for stars less massive than the Sun though. But in general a more active M-dwarf is likely to be younger than a less active M dwarf. It should certainly distinguish between a 2Gyr and 8Gyr M dwarf.

If you measure the line of sight velocity from your spectrum, this can give you at least a probabilistic idea of what stellar population the star belongs to. Higher velocities would tend to indicate an older star. This would work better if you had the proper motion (and preferably the distance too, roll on the Gaia results).

Similarly, in a probabilistic sense, low metallicity stars are older than high metallicity stars. If you were talking about stars as old as 8Gyr, these would be quite likely to have low metallicity.

In summary. If you are talking about G-dwarfs you can ages to precisions of about 20% using log g and Teff from the spectrum. For M dwarfs, unless you are fortunate enough to be looking at a young PMS object with Li, then your precision is going to be a few Gyr at best for an individual object, though combining probabilistic estimates from activity, metallicity and kinematics simultaneously might narrow this a bit.

As an add-on I'll also mention radio-isotope dating. If you can measure the abundances of isotopes of U and Th with long half lives and then make some guess at their initial abundances using other r-process elements as a guide then you get an age estimate - "nucleocosmochronology". Currently, these are very inaccurate - factors of 2 differences for the same star depending on what methods you adopt.

Read Soderblom (2013); Jeffries (2014).

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