Astronomy 112: The Physics of Stars Class 9 Notes: Polytropes With our discussion of nuclear reaction rates last time, we have mostly completed our survey of the microphysical properties of stellar matter – its pressure, how energy flows through it, and how it generates energy from nuclear reactions. For the next few weeks we will be using those microphysical models to begin to make our first models of stars. I. The Stellar Structure Equations To begin we will collect the various equations we have developed thus far to describe the behavior of material in stars. As always, we consider a shell of material of mass dm and thickness dr, which is at a distance r from the center of the star and has a mass m interior to it. The shell has density ρ, pressure P , temperature T , and opacity κ. The radiation flux passing through it is F , and the shell generates energy via nuclear reactions at a rate per unit mass q. We will write down the equations describing this shell, under the assumption that the star is in both hydrostatic and thermal equilibrium, so we can drop all time derivatives describing change in position, energy, etc. Since we are assuming equilibrium, for now we will also assume that the composition is fixed, so that we know X, Y , Z, and any other quantities we need that describe the chemical makeup of the gas. The first equation is just the definition of the density for the shell, which says that ρ = dm/dV . Writing this in Eulerian or Lagrangian form (i.e. with either radius or mass as the independent variable), we have dm dr 1 = 4πr2ρ = . dr dm 4πr2ρ The second equation is the equation of hydrostatic balance, which we can also write in either Eulerian or Lagrangian form: dP Gm dP Gm = − ρ = − . dr r2 dm 4πr4 This just equates the change in gradient in pressure with the force of gravity. The third equation is the equation describing how the temperature changes with position within a star as a result of radiative diffusion: dT 3 κρ F dT 3 κ F = − = − . dr 4ac T 3 4πr2 dm 4ac T 3 (4πr2)2 Finally, we have the equation of energy conservation, which for material in equilibrium just equates the change in energy flux across a shell to the rate at which nuclear reactions generate power within it: dF dF = 4πr2ρq = q. dr dm 1 These are four coupled, non-linear ordinary differential equations. As we discussed a few weeks ago, by themselves they are not a complete system, because by themselves they contain more unknowns than equations. The unknowns appearing are ρ, T , P , F , κ, and q. If we adopt Eulerian coordinates, then m is also an unknown, and r is the independent variable. For Lagrangian coordinates, r is also an unknown, and m is the independent variable. Thus we have seven unknowns, but only four equations. We therefore need three more equations, and that is what we have spent the last few weeks providing. The pressure depends on density and temperature via R 1 4 P = ρT + Pe + aT . µI 3 The first term is the ion pressure, which we have written assuming that ions are non-degenerate, which they are except in neutron stars. The last term is the radiation pressure. The middle term, the electron pressure, takes a form that depends on whether the electrons are degenerate or not, but which is a known function of ρ and T . The opacity and nuclear energy generate rate, we have seen, are in general quite compli- cated functions. However, we have also seen that they can be approximated reasonably well as powerlaws: a b m n κ = κ0ρ T q = q0ρ T . Regardless of whether we make the powerlaw approximation or not, we now know how to compute κ and q from ρ and T . Thus we have written down three more equations involving the unknowns. We are therefore up to seven equations for our seven unknowns, which is sufficient to fully specify the system. The only thing missing is boundary conditions, since differential equations produce constants of integration that must be determined by boundary con- ditions. Since there are four differential equations, we need four boundary conditions. Three of them are obvious. In Lagrangian coordinates, we have r = 0 and F = 0 at m = 0, and P = 0 at m = M. In words, the first condition says that the innermost mass element must reside at radius r = 0, and it must have zero flux (F = 0) entering it from below. The third condition says that the pressure falls to zero at the boundary of the star, m = M. The fourth condition is slightly more complicated. The simplest approach is to set T = 0 at the star’s surface. This is actually not a terrible approximation, since the temperature at the surface is very low compared to that in the interior. A better approach is to specify the relation between flux and temperature at the stellar surface as F = 4πr2σT 4. An even better approach is to make a detailed model of a stellar atmosphere and figure out how the flux through it depends on its temperature and pressure, and use that as a boundary condition for the stellar model. The set of seven equations and four boundary conditions we have written down now fully specifies the structure of a star. Solving those equations, however, is another matter entirely. There is no general method for solving sets of coupled non-linear 2 differential-algrebraic equations subject to boundary conditions specified at two points. There is every reason to believe that such equations cannot, in general, be solved in closed form. Today the standard approach is to hand the problem to a computer. A computer can integrate the equations and find solutions to any desired level of accuracy, and this problem is sufficiently simple that the calculations will run on an ordinary desktop machine in a matter of seconds. However, in the days when people first approached these problems, there were no such things as computers. Instead, people were forced to come up with analytic approximations, and it turns out that one can understand a great deal about the behavior of stars using such approximations. (It turns out that a significant fraction of being a good physicist consists in the ability to come up with good approximations for intractable differential equations.) II. Polytropes A. Definition and Motivation The first two stellar structure equations, describing the definition of density and hydrostatic equilibrium, are linked to the second two only via the relationship between pressure and temperature. If we can write the pressure in terms of the density alone, without reference to the temperature, then we can separate these two equations from the others and solve them by themselves. Solving two differential equations (plus one algebraic equation relating P and ρ is much easier than solving seven equations. As a first step in this strategy, we can combine the first two first-order ODEs into a single second-order ODE. To do so, we start with the equation of hydrostatic equilibrium and multiply by r2/ρ to obtain r2 dP = −Gm. ρ dr Next we differentiate both sides: d r2 dP ! dm = −G . dr ρ dr dr Finally, we substitute for dm/dr using the definition of density, dm/dr = 4πr2ρ. Doing so we obtain 1 d r2 dP ! = −4πGρ. r2 dr ρ dr This is just another form of the equation of hydrostatic balance, this time with the definition of density explicitly substituted in. Thus far everything we have done is exact. Now we make our approximation. We approximate that the pressure and density are related by a powerlaw γP P = KP ρ . 3 Equations of state of this sort are called polytropes. For historical reasons, it is common to define 1 1 γP = 1 + or n = n γP − 1 where we call n the polytropic index. Before going any further, it is important to consider whether an equation of state like this is at all sensible. Why should a star ever obey such an equation of state? The answer to this question becomes clearer if we recall that, for an adiabatic gas, the equation of state reads γa P = Kaρ , where Ka is the adiabatic constant and adiabatic index. It is important to un- derstand the difference between this relation and the polytropic relation. The polytropic relation describes how the pressure changes with density inside as one moves through a star, while the adiabatic equation of state describes how a given gas shell would respond to being compressed. The constant Ka depends on the entropy of the gas in a given shell, so different shells in a star can have different values of Ka. If different shells have different Ka values, then as I move through the star the pressure will not vary as P ∝ ργ, because different shells will have dif- ferent constants of proportionality. Thus a star can be described by a polytropic relation only if Ka is the same for every shell. While this condition might seem far-fetched, it is actually satisfied under a wide range of circumstances. One circumstance when it is satisfied is if a star is domi- nated by the pressure of degenerate electrons, since in that case we proved a few 0 5/3 0 4/3 classes back that Pe = K1(ρ/µe) for a non-relativistic gas, or Pe = K2(ρ/µe) . 0 0 The proportionality constants K1 and K2 depend only on constants of nature like h, c, the electron mass, and the proton mass, and thus do not vary within a star.