In the isotropic steady-state model, we treat all quantities in the companion as functions ofr, the distance from its center. No other independent variable enters in this model, astis forbidden by the steady-state assumption andθandφare forbidden by the isotropy assumption. Thus we write temperature asT(r), pressure as P(r), and so on.
To a very good approximation, we may neglect the variation in the composition of the star with position. That is, we treat all compositional variables as global constants, such thatX(r) = X0, the hydrogen mass fraction in the star, and likewise for all other such quantities. In making this approximation we mainly lose accuracy in calculating the properties of convection zones, though there our accuracy is primarily limited by the uncertainty in the choice of mixing length, and so this loss is acceptable.
The remaining spatial variables are then only thermodynamic ones. Of these, one might pick as "fundamental" ones the pressure, temperature, density, and mean
1Richard P. Feynman. Los Alamos From Below. https : / / www . youtube . com / watch ? v = 0ogSC6JKkrY. Feb. 6, 1975. url: http : / / calteches . library . caltech . edu / 34 / 3 / FeynmanLosAlamos.htm.
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2. ONE-DIMENSIONAL MODEL 15 molecular weight2. All other quantities of interest may be derived from these. We may, however, eliminate µ, for it is a direct function of T. This follows from the fact that we have held compositional variables fixed, such that µvaries only through ionization.3. This variation occurs mainly when kBT is comparable to 13.6eV, and is generally taken to happen between 103.8K and 104.1K. The value of 13.6eV, of course, is the ionization energy of hydrogen.
Using the equation of state, we may eliminate yet another function, to reduce the total count of "fundamental" thermodynamic variables at each point to two. The equation of state is most generally written as
P =f(ρ, T), (2.1)
though it is usually well approximated by the form µP =ρkBT +1
3aT4, (2.2)
where the second term is included to accommodate radiation pressure. At low temperatures the second term may be dropped, yielding the familiar ideal gas law.
Regardless of the specific form, we will use the equation of state to eliminate the density from consideration, and hence write
ρ=g(P, T). (2.3)
Our ability to write it in this form comes from P being monotonic in ρ andT. We chooseρ rather than T or P because we generally wish to compute heat transport properties in terms of temperature, and in hydrostatic equilibrium the pressure is computable by a straightforward integral. As a result, we are left with two basic functions, P(r) and T(r), which fully characterize the star to within our various approximations.
It will often be more convenient to replace r withm, the mass above a particular radius, as the independent variable. As m is monotonically decreasing with r this is a perfectly well-defined transformation. We thus writeP =P(m), T =T(m). In this language, the condition of hydrostatic equilibrium may be cast into a convenient form, as
dP
dr =−ρg → dP
dm = g
4πr2. (2.4)
Now over the depth ranges of interest, as will be verified later,r varies only slightly relative to R. As a result, we may neglect its variation in computing quantities
2Other valid choices include specific energy, specific entropy, sound speed, etc.
3At high pressures it may also depend on pressure, and indeed we will account for this
2. ONE-DIMENSIONAL MODEL 16 in which r appears as a multiplicative factor. This is known as the thin-envelope assumption, and has several useful implications. For instance, we may approximate the gravity of the star as being fixed at
g ≡ GM
R2 . (2.5)
As a result, we may write the condition of hydrostatic equilibrium as dP
dm = GM
4πR4. (2.6)
Using the boundary conditionP(r=∞) = 0, m(r =∞) = 0 we find P(m) = GM m
4πR4 . (2.7)
Note that we may also use the variable
Σ≡ m
4πR2 (2.8)
as the independent variable. Given that this is the form in which we know the heating depth, we will often switch to using this rather thanm.
Given T(m), in addition to what we have found so far, we will know the structure of the star to within the bounds of our approximations. As a result, we know that T(m) must depend in some fashion on the luminosity of the star and on the external illumination we hope to investigate, for these quantities appear nowhere else and they seem quite important. To that end, consider the outer boundary condition on the star. There are a variety of models for this4, but most treat the low-m regime by some gas-radiation dilution model and use this to find the optical depth along the radial direction. From there, it is typically asserted that
L= 4πR2σT4 (2.9)
at the place where the optical depth τ = 2/3. This is just an application of the Stephan-Boltzmann radiation law to a gray-body atmosphere, with an effective treatment for the differing rates at which different frequencies of radiation escape at low optical depth. We will not go into the specific details of the model we used, and merely state that they are those described in Ref.5.
4B. Paczyński. “Envelopes of Red Supergiants”. In: Acta Astronomica 19 (1969), p. 1.
5Ibid.
2. ONE-DIMENSIONAL MODEL 17 From this upper boundary condition on T, we may integrate towards higher m using the equation
dT
dm = dlnT dlnP
!T P
dP
dm =∇T P
dP
dm, (2.10)
where the second equality defines the symbol∇and where the derivative with respect to lnP is taken along the radial direction. This last point is not relevant in an isotropic star, where ∇T and ∇p are aligned, but will become important when we move to higher dimensional models.
Of course, there is no physical content in Eq. (2.10): it is simply a true statement regarding differentiable functions. The reason we bother to cast the problem in this form is that∇ may often be expressed simply. In regions of the star where heat is transported radiatively,
∇=∇rad = 3κP L
16πacGM T4, (2.11)
where κ is the Rosseland mean opacity of the stellar material, and is generally a function ofP andT. On the other hand, when the region of interest is unstable against convection, the thermal gradient∇ is somewhat more complicated. If convection is efficient, then the convective gradient matches the adiabatic gradient, such that
∇=∇ad = dlnT dlnP
s
. (2.12)
This gradient is typically 0.4 for monatomic gas and for fully ionized gas, and drops to 0.1−0.2 in the ionization zone. If, on the other hand, convection is inefficient, then matters become somewhat more complex, as then both radiation and convection contribute nontrivially to thermal transport. The full solution for the convective gradient in this case is somewhat complicated, and involves the root of a cubic with a closed form which does not yield much intuition. Various methods of numerical solution have been developed6, and will be employed in the next section. As will be shown later, however, convection is usually highly efficient in the cases of interest, and so setting∇=∇ad in convecting regions is generally accurate.
It is worth noting that the question of convective stability is much simpler in stars than in other contexts. The microscopic viscosity of stellar atmospheres is generally far too low to stop convection7. This is a statement about the typically large value of the Rayleigh number whenever the radiative gradient exceeds the adiabatic one. Thus
6Ibid.
7This will be discussed at length when we examine the properties of fluids in motion for higher dimensional heat transport
2. ONE-DIMENSIONAL MODEL 18 in the absence of shear turbulence the primary criterion determining if convection occurs is
∇rad >∇ad. (2.13)
If this condition is satisfied then convection occurs. Loosely speaking this criterion may be thought of as indicating that the temperature gradient needed to carry the thermal flux through radiation is too high relative to the buoyancy experienced by an adiabatically expanding packet of gas. The result is a convective instability.
The only remaining piece of physics we need to compute stellar structures with the above equations is κ. This we take from tables such that those of OPAL8 and Ferguson9, as discussed in Appendix B.1. A plot of the opacities produced by these tables at X = 0.7, Y = 0.27, Z = 0.03 is shown in figure 2.1.