<p>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.</p>

<p>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.</p>

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

<p>We now need to <em>multiply</em> this luminosity by <span class="math-container">$f(t)$</span>, where <span class="math-container">$f$</span> is an approximately linear function of time <span class="math-container">$t$</span>.
<span class="math-container">$$ f(t) \simeq 1 + (t/t_{\rm ms}),$$</span>
where <span class="math-container">$t_{\rm ms}$</span> is the total main sequence lifetime.</p>

<p>Now we can use an approximation for the main sequence lifetime, which is
<span class="math-container">$$t_{\rm ms} \simeq 10^{10} (M/M_{\odot})^{-2.5}\ {\rm yr} $$</span></p>

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

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

CC BY-SA 4.0
2019-07-25T07:00:17.103Z
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
CC BY-SA 4.0
2019-07-25T07:04:21.363Z
@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.
ProfRob
CC BY-SA 4.0
2019-07-25T07:14:37.470Z
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
CC BY-SA 4.0
2019-07-25T07:33:25.463Z
@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.
ProfRob
CC BY-SA 4.0
2019-07-26T20:32:54.157Z
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: https://thehappykoala.github.io/Harmony-of-the-Spheres/#/category/TESS/scenario/L%2098-59
Happy Koala
CC BY-SA 4.0
2019-07-24T19:04:15.850Z
2019-07-24T19:04:15.850Z
ProfRob
<p>I've had some success by running a given star's mass and assumed metallicity through the stellar evolution model on this web site:</p>
<p><a href="http://user.astro.wisc.edu/%7Etownsend/static.php?ref=ez-web" rel="nofollow noreferrer">http://user.astro.wisc.edu/~townsend/static.php?ref=ez-web</a></p>
<p>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).</p>
<p>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.</p>
<p>So far, for six primaries of terrestrial planets in the habitable zone, I've achieved very close fits (R<sup>2</sup> &gt; 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.</p>

CC BY-SA 4.0
2022-12-13T15:16:52.280Z
2022-12-14T19:47:38.487Z
2022-12-14T19:47:38.487Z
bpl1960
<p>What you are looking seems to be the vertical path of a star in the Hertzsprung-Russel diagram (HRD). </p>

<p>The only problem is that stellar evolution is pretty complicated. See here a few simulated trajectories for different masses and chemical compositions from <a href="https://www.aanda.org/articles/aa/pdf/2008/24/aa9165-07.pdf" rel="noreferrer"> Bertelli et al. 2008 </a></p>

<p><a href="https://i.sstatic.net/vp4NL.png" rel="noreferrer"><img src="https://i.sstatic.net/vp4NL.png" alt="enter image description here"></a></p>

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

<p>From the simulations above one can also derive respective age-luminosity relations (taken from  <a href="http://%20file:///home/thalhammer/Downloads/Effect_of_Metallicity_on_the_Evolution_of_the_Habi.pdf" rel="noreferrer">Danchi 2013</a>):</p>

<p><a href="https://i.sstatic.net/kjtsF.png" rel="noreferrer"><img src="https://i.sstatic.net/kjtsF.png" alt="age-luminosity"></a></p>

<p>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. </p>

<p>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.</p>

<p>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. </p>

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

CC BY-SA 4.0
2019-07-24T15:51:08.207Z
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
CC BY-SA 4.0
2019-07-24T17:49:51.710Z
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!
ProfRob
CC BY-SA 4.0
2019-07-24T17:52:30.463Z
What is true is that for main sequence stars, the luminosity is much more dependent on mass than age.
ProfRob
CC BY-SA 4.0
2019-07-24T19:30:49.397Z
@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.
SpaceCore
CC BY-SA 4.0
2019-07-25T04:37:09.643Z
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.
ProfRob
CC BY-SA 4.0
2019-07-25T04:40:17.653Z
In fact the main problem with using the luminosity to estimate an age *is* that the mass is unknown, not that the luminosity changes relatively little on the main sequence.
ProfRob
CC BY-SA 4.0
2019-07-24T15:32:29.533Z
2019-07-24T19:18:38.427Z
2019-07-24T19:18:38.427Z
SpaceCore
<p>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. </p>

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

<star>
Is there a way to factor age into the mass-luminosity relationship for stars?
Happy Koala
# Is there a way to factor age into the mass-luminosity relationship for stars?

(Asked by: username_0 on 2019-07-24)

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.

**Tags:** star

---

## 3 answers

### Answer by username_3 on 2019-07-24. Score: 7 | Accepted answer

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](https://www.aanda.org/articles/aa/pdf/2008/24/aa9165-07.pdf)

[![enter image description here](https://i.sstatic.net/vp4NL.png)](https://i.sstatic.net/vp4NL.png)

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](http://%20file:///home/thalhammer/Downloads/Effect_of_Metallicity_on_the_Evolution_of_the_Habi.pdf)):

[![age-luminosity](https://i.sstatic.net/kjtsF.png)](https://i.sstatic.net/kjtsF.png)

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. :)

> **Comments:**
> **username_0**: 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.  
> **username_1**: 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!  
> **username_1**: What is true is that for main sequence stars, the luminosity is much more dependent on mass than age.  
> **username_3**: @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.  
> **username_1**: 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.  
> **username_1**: In fact the main problem with using the luminosity to estimate an age *is* that the mass is unknown, not that the luminosity changes relatively little on the main sequence.  

---

### Answer by username_1 on 2019-07-24. Score: 6

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.

> **Comments:**
> **username_0**: 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.  
> **username_1**: @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.  
> **username_0**: 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* ).  
> **username_1**: @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.  
> **username_0**: 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: https://thehappykoala.github.io/Harmony-of-the-Spheres/#/category/TESS/scenario/L%2098-59  

---

### Answer by username_2 on 2022-12-13. Score: 0

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](http://user.astro.wisc.edu/%7Etownsend/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.


---

