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

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.

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. :)

• Expert or not, that was a top quality answer, so thank you for taking your time to write it down :). Didn't even consider the metallicity of a star as a variable in this equation, but then again I'm not an expert either, evidently, on the subject, so I'll forgive myself for not thinking that far. For now I'll just have to settle for using the mass of a star to estimate its luminosity. – Happy Koala Jul 24 '19 at 15:51
• This is all fair comment and +1. But I don't follow the argument that we can't measure the age so it's not a useful exercise to add age to the L-M relation. Well, we can't in general measure the mass either! – Rob Jeffries Jul 24 '19 at 17:49
• What is true is that for main sequence stars, the luminosity is much more dependent on mass than age. – Rob Jeffries Jul 24 '19 at 17:52
• @Rob Jeffries True. Do you know how complicated it is to get the age of a star? I probably underestimated for how many stars we know their age. – SpaceBread Jul 24 '19 at 19:30
• That is isn't relevant. We don't in general know the mass either. The question isn't asking how do you estimate the age of a star, it is asking how is the L-M relation modified by stellar age. The relationship exists whether you can empirically calibrate it (which you can, by the way - using star clusters) or not. – Rob Jeffries Jul 25 '19 at 4:37

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.

• Thanks! I'm gonna give that a try. My reason for asking to begin with was that I'm trying to estimate the habitable zones around stars around which we have discovered exoplanets. Fortunately the exoplanetarchive, which I use to pull data, has age data on most of the stars in its database. – Happy Koala Jul 25 '19 at 7:00
• @HappyKoala Note that the ages there are possibly estimated from the luminosity for some stars. However, there are independent ways to estimate age (lithium, rotation, activity). The best way to estimate the luminosity is using how bright the star is and its measured distance. – Rob Jeffries Jul 25 '19 at 7:04
• Cheers, I'll keep that in mind. First thing I'll try to do is to try your formula on our solar system to see if setting the age of the solar system to 2 billion years will put Venus in the habitable zone, and then fast forward 1 billion years to see if Earth falls out of the habitable zone (right now it is hugging its inner edge, which had me realise that in cosmic terms Earth is nearing the end of its ability to support life... violin music ). – Happy Koala Jul 25 '19 at 7:14
• @HappyKoala Ah yes, I see. Yes, that will work ok, subject to the caveats above and that habitability doesn't just depend on the luminosity of the star. – Rob Jeffries Jul 25 '19 at 7:33
• Can confirm that it works, so I'll push that with the new build next week. In case you're curious, here's what I'm working on: thehappykoala.github.io/Harmony-of-the-Spheres/#/category/TESS/… – Happy Koala Jul 26 '19 at 20:32