6
$\begingroup$

I had a discussion at work regarding a recent fusion experiment in China that resulted in temperatures five times hotter than the Sun. Someone mentioned that no one can know the temperature of the Sun. I replied that we have mathematical models of the Sun, but I didn't have any on hand, and I haven't been able to find much on the internet.

So, where can I find a system of equations for modelling a star?

I found this site, which is about the level of complexity I was looking for, but it looks spotty. For instance, $\partial T / \partial t$ and $\partial P / \partial t$ terms seem to come out of nowhere--most everything else is written in terms of $\partial / \partial M$.

I can code numerical solutions to PDEs, but I haven't had much physics or anything, so I didn't know where or what to look for.

EDIT:

I found this really great post by Dr. Brian Koberlein describing a very (very) simple model of a star. He goes on to build upon that simple model here.

$\endgroup$
2
9
$\begingroup$

The basic problem of modelling a star is covered in a number of textbooks and lecture notes. Try searching for "stellar structure and evolution" or something along those lines. The best readily available lecture notes, IMO, are those of Onno Pols, available here. There was also a similar post on Quora, which you can read too. In the mean time, here's the basic run down.

To construct a reasonable stellar model in a reasonable amount of time, we make several assumptions. We assume that a star is a spherically symmetric, dynamically stable, self-gravitating fluid in local thermodynamic equilibrium. Here's how we unpack all this.

First, spherical symmetry means one spatial co-ordinate. In this case, for a fluid, we can write down the equation of mass conservation: $$\frac{dm}{dr}=4\pi r^2 \rho$$ This just means that an infinitesimal spherical shell of thickness $dr$ at radius $r$ contributes $dm$ to the total mass inside radius $r$. (I'll probably end up calling $m$ the mass co-ordinate.)

Now, if we regard our star as spherically symmetric and dynamically stable, we can kick out the velocity terms and time derivatives in Euler's equation. Supposing gravity is the only external body force, we end up at the equation of hydrostatic equilibrium: $$\frac{dP}{dr}=-\frac{Gm\rho}{r^2}=-\rho g$$

Remember, this follows from the conservation of momentum. To conserve energy, like mass, we say that the contribution to the total luminosity at $r$ is the mass of the shell times the specific energy generation rate $\epsilon$, so we write $$\frac{dL}{dr}=4\pi r^2\rho\epsilon$$

To keep it simple, I've neglected to specify where $\epsilon$ will come from. It generally includes the energy generated by nuclear reactions rates, less the losses due to neutrinos streaming out in some reactions, plus—in some phases—the energy released by contraction. (It can be shown that when a star contracts it heats up but loses energy overall. See the Virial Theorem.) The energy generation depends on the density, temperature and chemical composition of the material. It isn't something we know from first principles. Instead, we use tables of data taken either from detailed calculations or laboratory experiments.

We now have to describe how energy is transported inside the star. The equations are a bit of a mouthful, so I won't write them here, but basically energy can either be transported by radiation or convection, depending on the temperature structure. In either case, you get an equation of the form $dT/dr=$(some right-hand side, see the notes). In the case of radiation, the transport coefficient depends on the opacity of the stellar material, denoted $\kappa$, which itself depends again on the density, temperature and chemical composition. (Strictly speaking, opacity depends on frequency, but we use a specific average opacity: the Rosseland mean opacity.) Like the energy generation rate, this isn't known from first principles: we use tabulated lab data.

Finally, as is usually the case in fluid problems, we have to close the system with an equation of state, which relates the pressure, density, temperature and chemical composition. It's the third equation for which we generally use lab data, although here we do also have some approximate analytic forms.

These four equations (three given + temperature transport) are almost entirely independent of time, so they're sometimes called the structure equations. The three tabulated inputs (energy generation, opacities and equation of state) are sometimes called the matter or microphysics equations.

So, why does a star evolve?

The answer is because the composition changes. Suppose there are $N$ chemical species (${}^1H$, ${}^4He$, etc.), each of whose fractional mass abundance is denoted $X_i$. Then the nuclear reactions convert species $i$ into $j$ at some rate $R_ij$, and we can write a set of equations $$\frac{dX_i}{dt}=\sum_j R_{ij}$$ The rates also depend on the material properties (density, temperature, etc.). Also, in truth, we expect convection to mix material on a dynamical timescale, so we throw in a monstrous diffusion coefficient in those regions.

But that's basically it. Given a composition profile, the structure equations tell you what the star looks like. Then, the reaction rates dictate how the composition changes, and the structure changes accordingly through the matter equations. I haven't gone into details like boundary conditions and whatnot, but if you're still interested, I recommend the notes! They're aimed at a reasonably high level (I'd say late undergrad although there's no reason a second-year couldn't make sense of them) but if you're familiar with other areas of physics it should be a synch.

If you want to build models, you can try using polytropes for very simple (but still useful) models. Or, I'd recommend the Modules for Experiments in Stellar Astrophysics (MESA) package for a fully-fledged, research-grade modelling tool.

$\endgroup$
0
$\begingroup$

The variable $M$ on that page is used instead of the radial $r$ coordinate; $M$ denotes the total mass inside the ball of radius $r$, the cumulative mass. These explanations are clear e.g. from this alternative presentation of the equations of the stellar structure:

https://en.wikipedia.org/wiki/Stellar_structure#Equations_of_stellar_structure

That page also more or less describes where the equations come from and what they mean. On that page you found, $t$ is time and it isn't surprising that the time derivative of the pressure $P$ and absolute temperature $T$ appear in the dynamical equations as well.

There are lots of books on stellar structure

http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=stellar+structure&tag=lubosmotlsref-20

If you had some additional, more particular questions about the equations, their origin, or their independence etc., you may ask again.

$\endgroup$
4
  • $\begingroup$ I read that $M$ is the independent variable, because $r$ changes. I don't know what $a$ and $c$ were in the radiative energy transfer equation. Anyway, I figured out I should be googling "energy transport in stars" instead of "solar model". Thanks. $\endgroup$ Jan 24 '14 at 18:35
  • $\begingroup$ Sorry, the link to your reference page disappeared so I can't meaningfully tell you what $a,c$ meant on the currently unknown page. $\endgroup$ Jan 24 '14 at 18:53
  • $\begingroup$ I figured it out. Wikipedia uses the Stefan-Boltzman constant, $\sigma$, whereas the other page uses this $a \cdot c$ term which is related to the Stefan-Boltzman constant by $a=\frac{4\sigma}{c}$, where $c$ is the speed of light, and $a$ is the radiation constant. $\endgroup$ Jan 24 '14 at 19:51
  • $\begingroup$ Good to hear! Lots of technical subtleties will be hiding in the coefficients etc. $\endgroup$ Jan 24 '14 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.