how to estimate the phase parameter of a complex function?

What is a first-order approximation for the equation of state of a main sequence star?

• Say I want a simple, dumb model of a star on the main sequence. It has a single parameter, which I'll take to be its mass M. If I assume it's spherically symmetric (negligible angular momentum) and made entirely of ionized hydrogen, then it is described by a pressure, temperature, and density, all as functions of radius from the center of the star. In order to find these functions, I need some equation of state for a hydrogen plasma. What should I use? Then I need to balance the heat loss (found via Stefan-Boltzmann) to the nuclear fusion rate, so how do I estimate the fusion rate as a function of the state? I'm basically looking for the crudest, order-of-magnitude way to do this that roughly reproduces basic features such as radius being approximately proportional to mass, luminosity for a solar mass coming out about right to within a factor of two or so, lifetime of a solar-mass star being on the order of 10^10 years etc.

• Answer:

  Essentially the plasma can be modeled to first order as a non-relativistic ideal gas for a light star such as the sun. However it will be important to take account of the initial 23% helium fraction, and 2% metals. A polytropic equation of state is often used as a first approximation to a stellar eos. It takes the form: P=Kρ1+(1/n)P = K \rho^{1+(1/n)}. For an ideal gas n \rightarrow \infty and K=\frac{kT}{\mu m_p}. From this it is not hard to derive an estimate of the central temperature, which follows from an estimate of the central pressure, which is also not hard to derive, based on hydrostatic equilibrium and dimensional considerations. \mu is a constant describing the mean molecular weight. For a pure hydrogen plasma the value would be \mu = 1/2, neglecting the electron mass. But in the actual sun the average value is more like \mu=0.59, because there is also the helium and some fraction of still heavier elements. Also in the core, helium and certain other elements are gradually building up in mass fraction due to fusion, so \mu is changing over time. The helium fraction currently is approaching 63% in the core, hydrogen is 35%, and metals are about 2%. The simple polytropic form has the advantage that analytic solutions can be found to the equations of hydrostatic equilibrium for certain values of n. For a star near the mass of the sun, a polytrope, with n=3, gives a reasonable mass distribution in the core, which extends out to some 20-30% of the solar radius, beyond which point there is practically no fusion. Eventually, as the radius from the center increases, the heat transfer becomes dominated by convection rather than radiation, and the plasma becomes less fully ionized starting at about 50% of the radius. The very outermost layers of the sun are not fully ionized at all. However at about 5 times the solar radius you are already in the corona, and there is the solar wind - which is a very dilute, highly ionized plasma, again at very high temperature. So the sun doesn't really have a well defined boundary - in a sense it actually extends all the way out to the heliopause. But these details are not important for what you want to do. The core densities and temperatures are all that matter for the overall fusion rates. The starting point, then, is a numerical solution to the equations of hydrostatic equilibrium. The fusion rates are very strongly temperature dependent, since temperatures in the sun's core imply that fusion is happening only far out on the tails of the Maxwell-Boltzmann distribution. Being two-body reactions, the rates are also strongly density dependent. The rate limiting step is the p(p,\gamma e^+ \nu_e)D reaction, since it requires a weak interaction to form deuterons. The slowness of this step is what accounts for the very long lifetime of the sun. Then Helium-3 is formed from deuterons and protons, and He-3 reacts with He-3 to produce He-4 and two protons, completing a cycle. The volumetric production of energy in the core is actually surprisingly low, given the high temperatures that obtain there. The rate of energy production is actually less than in a human body. But heat transport out of the core is very slow. So there is a huge temperature gradient. To work out the details of the time evolution, you will need to put in a simple reaction network and solve the Saha equations, and account for the opacity of the plasma to photons to get the rate of radiative heat transfer, and the temperature distributions, self-consistently You also need to take account of the energy loss to neutrinos, but that's not going to make an order of magnitude difference.