Is there a way to factor age into the mass-luminosity relationship for stars?

Question

I'm wondering if there's a way to formulate the luminosity of a star not only as a function of mass, but of age, too, and if so, what the formula for luminosity would look like. In the case of the Sun I'm aware it gets 10% brighter every billion or so years, but I doubt this relationship holds true for all mass ranges of stars, just like you have to adjust the formula for the luminosity mass relationship for stars of different masses.

If anyone could "shed some light" on this matter, it would be very much appreciated.

Answer 1

What you are looking seems to be the vertical path of a star in the Hertzsprung-Russel diagram (HRD).

The only problem is that stellar evolution is pretty complicated. See here a few simulated trajectories for different masses and chemical compositions from Bertelli et al. 2008

Especially before and after the main sequence (i.e., regular boring hydrogen burning) the luminosity evolution becomes extremely variable.

From the simulations above one can also derive respective age-luminosity relations (taken from Danchi 2013):

Again, you can see that the relation is very complicate for young or old stars and is fairly constant during the main sequence. The problem is, that it changes too little during most of the main sequence. So for most stars even with good mass and metallic measurement you won't be able to accurately estimate age just from luminosity. In addition to that, there are still uncertainties in our models.

Another thing one could do is trying to improve the mass-luminosity-relation by including age. I think this is also what the title of your question implies. The problem here it simply that we generally don't know the age of a star. But if we do, as Rob Jeffries explained in his answer, it is in principle possible.

One think that is done to get age from luminosity is to go all in and calculate so called isochrones in the HRD. These are lines with stars of same age but varying masses and metallicities and can be derived from simulations. If one then measures the luminosity, temperature and metallic one can look on which isochrone the star falls in the HRD (and therefore what age it has). This however is still pretty inaccurate, especially on the main sequence, and is mostly done with whole star cluster, where statistics makes things easier.

This is however not my field of expertise so I would be happy it an actual expert could chime in. :)

Answer 2

I basically agree with spacebread that it is complicated, but then so is the basic luminosity-mass relation if you start including stars that aren't on the main sequence.

If we do restrict ourselves to the main sequence, then you can see that mass is the dominant variable, with the luminosity perhaps increasing by about a factor of two over the course of the main sequence lifetime.

So how to put this into an equation? Well say we have $$ L/L_{\odot} \simeq 0.7 (M/M_{\odot})^{a},$$ as the basic relationship between luminosity and mass for a "zero age main sequence" (ZAMS) star. The index $a$ is something like 3.5, but actually is different in differing mass ranges.

We now need to multiply this luminosity by $f(t)$, where $f$ is an approximately linear function of time $t$. $$ f(t) \simeq 1 + (t/t_{\rm ms}),$$ where $t_{\rm ms}$ is the total main sequence lifetime.

Now we can use an approximation for the main sequence lifetime, which is $$t_{\rm ms} \simeq 10^{10} (M/M_{\odot})^{-2.5}\ {\rm yr} $$

Thus our linear correction factor to the ZAMS luminosity is $$ f(t) \simeq 1 + 10^{-10}(M/M_{\odot})^{2.5} t\ ,$$ where $t$ is in years and the relationship is valid until the end of the main sequence.

To emphasize, this is super-approximate and a more accurate approach would involve numerical interpolation of actual stellar models.

Answer 3

I've had some success by running a given star's mass and assumed metallicity through the stellar evolution model on this web site:

http://user.astro.wisc.edu/~townsend/static.php?ref=ez-web

The only problem being that the choices of metallicity are fixed and rather coarse. Note, also, that you have to give the maximum age in years, not billions of years--I didn't know whether it would accept exponential notation of any kind, so I wrote out 8000000000 (eight billion).

I then fit polynomial approximations to the results (L = f(t)). Usually I have to throw out the first point, which is an outlier compared to the others, but that only affects the first 50,000 years or so.

So far, for six primaries of terrestrial planets in the habitable zone, I've achieved very close fits (R² > 0.999) for quadratics with three stars. For the other three I had to use sixth-degree polynomials. Whether these fits have any physical significance is another question.