Has stellar evolution ever been modeled analytically?

I'll still remember the handout my astronomy professor gave us more than several decades ago; Our Friends the Polytropes. We spent a lot of time learning how polytropes, simple analytical power-law models could reproduce some basic thermodynamic properties of stars, and how those properties would tend to vary with mass (and perhaps other parameters, I don't remember now).

Have polytrope models of stars (or other simple analytical models of stars) ever been "evolved" analytically in time? In other words, in addition to variations on radius, have a set of equations ever been written which also included a $$\frac{\partial}{\partial t}$$ in such a way that they were amenable to an analytic solution?

• Isn't the title too broad? I was answering at first glance "of course" (though most of deep modelling has numerical solutions) but then I've realised you have something specific in mind. Feb 25 '19 at 9:45
• @Alchimista No, the way I have written the question it isn't broad at all! "Has X ever happened?" requires only a single example to be answered in the affirmative. If you can cite and show a reasonable example, consider posting it as an answer. Thanks!
– uhoh
Feb 25 '19 at 9:50
• Modelling of stars interior and evolution is astrophysics. Of course is done. I am not aware of model that do not require a numerical procedure, though. By the way the title does not ask about the role of polytropes, and my answer/comment refer to the title. I am not a star physicist and it is quite possible that among the material I am slowing going trough there is buried the idea of polytropes. It should be so I guess, but sure it is not emphasised in my book. Feb 25 '19 at 9:58
• @Alchimista it would not have to be polytropes, I'll edit a bit and relax that constraint. It's just the way I thought a purely analytical approach to time evolution would have started. I'll add the history tag to emphasize I'm talking about "the old days" an not modern techniques. Thanks!
– uhoh
Feb 25 '19 at 10:05

Analytic models have been applied to various phases of the evolution, though it would be impossible to apply a single model to all phases because the physics changes so much. Also, a star will often have very different physics going on in various different parts of the star, so analytic models sometimes have to treat different parts separately and then unify the results across boundaries (just as numerical simulations do). For example, I've seen different polytrope indices used in the convection, radiative, and core regions of the Sun.

I realize that your interest is when there is not a steady-state assumption, but rather time is a dynamical variable. But it is not always necessary to use time as a dynamical variable to do evolution. You can simply do a steady-state analytical model (like a polytrope, perhaps supported by a numerical piece that solves for some parameter like core pressure), and allow the parameters in your solution to be time varying in some simple analytical way. For example, you could study how mass loss affects a star by doing steady-state analytical models and just let the mass change with time according to some analytical mass-loss prescription (such as "Reimers' Law", something simple). So steady-state analytical models can be elevated into evolutionary models if you simply have an analytical way to change the parameters with time.

Another simple example is the evolution of a fully convective protostar. You simply fix the mass and initial radius, and you solve the interior by assuming it's all at the same entropy (a reasonable approximation for fully convective stars). Then you fix the surface temperature to lie on the Hayashi track, which for a very simple model could mean just pegging the surface T to 3000 K. That and the initial radius determines the luminosity, and the constant entropy assumption fixes the internal structure, so then you simply let it lose energy at the rate of the luminosity, and use the internal energy as the variable that changes with time, always updating the radius to be consistent with the new internal energy (a la the virial theorem), and that gives the new luminosity and so forth. A simple fully analytic model until the star is no longer fully convective (and we then might call it a pre-main-sequence star).

I've also seen white dwarf cooling done analytically. It's a similar idea-- take the surface temperature as an initial condition and just let it lose heat via its luminosity. The radius doesn't really change, so all you need is to keep updating the surface temperature as a function of the internal energy, which can be handled via some analytic internal heat transport treatment. If you are willing to make approximations that address the key physics, you can do almost anything analytically.

• This is a great answer, thank you! If you happen to think of a link with an example to add, publication, class notes online, etc. or even add a few equations here, that would be great!
– uhoh
Feb 25 '19 at 15:18
• Feb 26 '19 at 6:47
• @RobJeffries omg look at that! ;-) Since comments are considered temporary, it would be great for that to be captured in an answer (this one or separate), this is exactly the kind of thing I was thinking about!
– uhoh
Feb 26 '19 at 11:17

Well yes, it is still a useful tool because it can give much more insight than the output from a black box computer code. However, you have to pick your problems or the complexity of the analysis can lose this advantage.

A particular case is the evolution of low mass pre main sequence stars along the Hayashi track. Since these stars are fully convective they can be treated as single polytrope a (to first order). Myself and a colleague have used such analytical calculations to study the effects of spots and magnetic fields on the evolution of PMS stars and on the rate at which they "burn" lithium in their cores (e.g. Jackson & Jeffries 2014a; Jackson & Jeffries 2014b).

In turn, this work was inspired by a much earlier analytical treatment of the Li depletion problem by Bildsten et al. (1997).