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I'm trying to construct a solar system, and I'm toying with the idea that the planet capable of sustaining life was initially outside of the habitable zone, but the star's advanced age has caused the luminosity to increase, thereby shifting the HZ further outwards.

How do you calculate the increase of a star's luminosity with age? Is there a fixed correlation across all stars? Does it vary according to type?

Additional details -

  • Star is 0.88 Solar masses
  • Lifespan is 1.38 Solar lifespan
  • Star is still within main sequence, but significantly further along than the Sun, both in total age and in terms of its life cycle.

Any answers would be greatly appreciated.

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  • $\begingroup$ There are a complete of issues with this question; stellar evolution is a largely theoretical field at the best of times. And the luminosity at a given wavelength, say visible blue light can theoretically increase independent of the behaviour of the total stellar luminosity. $\endgroup$ – Ash Sep 10 '18 at 15:52
  • $\begingroup$ HDE 226868's answer probably has everything you need but you might want to know that your star probably has a K0V Spectral Class, at least according to this classification set $\endgroup$ – Ash Sep 10 '18 at 17:59
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I would recommend looking at pre-existing numerical models, rather than computing your own. This has a couple of advantages:

  1. You don't need to use any approximations.
  2. Factors like metallicity, rotation and composition have already been taken into account.
  3. You just need to look up the values in a table - no calculations required.
  4. You can also compare values for a star of the same mass and composition at many points in its life cycle.

I usually point people towards the Geneva grids of stellar models. They're easily accessible and simple to use. Let's say you want to look at stars of approximate solar composition ($X\approx0.76$, $Y\approx0.24$). There's a set of models by Schaller et al. 1992 that should suit your purposes. You probably want Table 43, for $M=0.9M_{\odot}$ - close enough to your star's mass. If you look at the column labels, you can see that Column 2 gives the star's age, in years, and Column 4 gives the logarithm of the star's luminosity, in solar luminosities.

I took the liberty of plotting luminosity against age for this particular model:

Plot of star's luminosity

Notice the steep increase in luminosity at around $\sim10^{10}$ years, when the star leaves the main sequence and enters the red giant phase. Additionally, I calculated the boundaries of the habitable zone. I assumed that the inner edge corresponds to an effective temperature at 273 K, and that the outer edge corresponds to an effective temperature of 373 K - the freezing and boiling points of water.

Plot of star's habitable zone

If you play around a bit and check out different grids of models, you'll indeed see that factors like mass, metallicity and composition strongly affect the evolution of a star, which is why it's important to have fine enough grids of models in the first place.

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  • $\begingroup$ One why table 43 and not table 21? Both are for the same mass star. Is that a total luminosity figure or something slightly more useful? $\endgroup$ – Ash Sep 10 '18 at 17:25
  • $\begingroup$ @Ash No, they're different stars - Table 21 has a higher metallicity ($Z=0.02$), while Table 43 has $Z=0.001$ - effectively $0$. Picking the latter was something of an arbitrary choice on my part, but I think most worldbuilders usually assume zero metallicity. $\endgroup$ – HDE 226868 Sep 10 '18 at 17:27
  • $\begingroup$ I don't go for low metallicity but that's me and I have my reasons, low metallicity makes more sense for an older star though so yeah that's the table you want, I missed the difference in Z ratings. Is that peak wavelength or visible luminosity? $\endgroup$ – Ash Sep 10 '18 at 17:32
  • $\begingroup$ @Ash It should be luminosity across all wavelengths. Stellar structure and evolution codes really have no reason to only list the luminosity in a portion of the spectrum, AFAIK, since the equations of stellar structure require the total luminosity at a given location in the star for the purposes of energy conservation and transport. $\endgroup$ – HDE 226868 Sep 10 '18 at 17:39
  • $\begingroup$ Thanks for the graph. I tried to look at the data, but... wow, that's one big mass of numbers... $\endgroup$ – N Francis Sep 11 '18 at 0:23

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