# Are there formulae for calculating stellar luminosity and effective temperature as a function of age?

Is there a manageable formula or set of formulas or simple algorithms that approximate stellar luminosity and effective temperature (or radius) as a function of stellar age?

I'm aware that accurate modeling of these attributes is complex and is determined by many factors; what I'm looking for is something that serves as a decent approximation of the sort used in numerous illustrations or applets that show example "paths" taken by representative stars as they trace their evolution across the H-R diagram.

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This is not really an answer to your question, but Java applets can be easily disassembled into Java source code using freely available tools. The results are often surprisingly readable. Or you could simply see if the applet is open source, or email the author to ask. – Ben Crowell Jul 22 '11 at 4:07
maybe the links here PSE that relates a star's Period of Rotation-Mass-Age, can help you – Helder Velez Jul 22 '11 at 10:16
Well, everything can be approximated by a multivariate polynomial, one way or another. I believe my old stellar astrophysics textbook, Kippenhahn and Wiegert, went through it in depth. I don't have access to my copy at the moment, though. – Andrew Jul 22 '11 at 20:54
Not quite what I was looking for. I'll happily trade accuracy for an approximation in a form that captures the major thrust of the underlying events. – raxacoricofallapatorius Jul 23 '11 at 15:39

It's a really complicated relationship that depends on the metallicity of the star. There is a paper that does show this though: See New grids of stellar models from 0.8 to 120 solar masses at Z = 0.020 and Z = 0.001

Here are the Geneva Grids:

http://obswww.unige.ch/~mowlavi/evol/stev_database.html

An extensive and homogenous database of stellar evolution models for masses between 0.8 and 120 solar masses and metallicities from Z=0.001 to 0.1 is available. In general the models include evolutionary phases from the main sequence up to either the end of carbon burning for massive stars, the early asymptotic giant branch phase for intermediate-mass stars, or core helium flash for low-mass stars. Pre-main sequence tracks, both canonical (i.e. evolved at constant mass) and accretion scenarios are also provided, as well as horizontal banches for low-mass stars. Predictions regarding the spectral evolution of massive stars can further be obtained from the so-called "combined stellar structure and atmosphere models" (CoStar).

In addition to the evolutionary grids we also provide Fortran codes for the calculation of isochrones and stellar population burst models.

Finally, references are also given to the serie of papers on stellar models with rotation.

There are also older ad hoc models applied to the Sun: see http://adsabs.harvard.edu/full/1981SoPh...74...21G and http://articles.adsabs.harvard.edu/full/1984SSRv...38..243S. They were referenced in 2011 publications though (like Pierrehumbert's 2011 Neoproterozoic Climate paper)

With that being said, it's not perfect. We don't know about variations in luminosity - the Sun's luminosity has varied in cycles (the 11-year sunspot cycle is one of them - but there may be others that last hundreds of years too - that could explain things like the Maunder Minimum). This is something that we might learn more of from Kepler telescope data on stellar oscillation, as described in the Chaplin et al. (2011) paper

It also depends on the angle that we view the star. And we may have to correct our zero-age main sequence luminosities too (since some stars can be unusually bright at zero-age). E.g. Altair is unusually bright for its temperature, but several papers have concluded that it's ZAMS rather than subgiant (it's rotating unusually fast, which is something you mostly see in new stars)

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What I'm looking for is something more compact, say one or two simple functions. It can be a very rough approximation, but I'm wondering if there there is such a thing, and whether it's at all useful in generating even an approximate picture of how stars evolve on the H-R diagram. The answer may be that no such thing exists: that nothing that simple really captures the picture. – raxacoricofallapatorius Sep 22 '11 at 14:37

Well, as you have guessed there is no exact formula to give you the answer here. There are only approximations based on this or other approach. I would recommend you have a look here:

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See comment above. – raxacoricofallapatorius Jul 23 '11 at 15:40

Maybe you're looking for something like this

http://arxiv.org/pdf/astro-ph/0001295v1.pdf

or

https://arxiv.org/pdf/1205.5484v2.pdf

You might also search for "Modelling the evolution of solar-mass stars with a range of metallicities using MESA E.F. Jonesa and P.M. Gore "

They give different results, they cannot all be right, and might all be wrong.

D.O. Gough gave this formula for the sun L = Lsun current = [1/[1- 0.4(1- t/4.6 billion years)] which gives a luminosity of about 0.7 4.6 billion years ago, 1 now, and a luminosity of about 1.9 after 10 billion years.

As was mentioned before, different mass stars process nuclear fuel differently, so there's no one relationship, but for stars between 0.75 Mass sun and 2 Mass sun, luminosity is roughly proportional to the 4.8 power of mass for stars of same metallicity, and ths formula should work for such stars, after adjusting for expected lifetime. For example, a star with an expcted lifttime of 20 bllion years, you'd plug in two times 4.6 billion for the denominator in Gough's formula. For a larger star you might get a lifetime of 5 billion years,in which case you'd substitute 2.3 in plase of the 4.6 in Gough's formula.

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## protected by Qmechanic♦May 26 at 21:09

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