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Appendix A: Cooling rate and correction factor

The purpose of this appendix is to show that for one-dimensional temperature perturbations, the correct cooling rates are obtained with just two rays if the D/ 3 correction factor in Eq. (15) is applied. Similar considerations apply also to the case of two-dimensional problems. Cooling rates are important for understanding temporal aspects such as the approach to the final state (Sect. 3.1) or the thermal adjustment time (Sect. 3.7). Thus, the equilibrium solutions discussed in the other sections are not affected by the following considerations.

The source of the problem lies in the fact that the 4π angular integration in Eq. (4) becomes inaccurate in one dimension and dependent on the optical thickness. In the optically thick regime, the diffusion approximation holds, so the cooling rate is proportional to K, which has a 1/3 factor in Eq. (7). In one dimension, one uses only the two rays in the vertical direction, so one misses the 1/3 factor and has to apply it afterwards to account for the “redundant” rays in the other two coordinate directions that show no variation. This is what is done in Eq. (15). However, in the optically thin limit, the mean free path becomes infinite and cooling is now possible in all three directions. In that case, the one-dimensional approximation is not useful. To explain this in more detail, we begin by considering first the general case of three-dimensional perturbations with wavevector k (Spiegel 1957). In that case, one can use the Eddington approximation to solve the transfer equation for the mean intensity, J = ∫I dΩ/4π, (A.1)so the cooling rate (for three-dimensional perturbations) is (Unno & Spiegel 1966; Edwards 1990) (A.2)It is convenient to introduce here a photon diffusion speed as (A.3)and to write Eq. (A.2) in the form (A.4)where c_γℓ/ 3 = χ is the radiative diffusivity, as defined in Eq. (25), and ℓ = 1 /κρ is the local mean-free path of photons.

Solving Eq. (5) for two rays corresponds to solving Eq. (A.1) without the 1/3 factor. We would then obtain Eq. (A.4) without the two 1/3 factors. This would evidently violate the well-known cooling rate χk² in the optically thick limit, but in the optically thin limit it would be in agreement with Eq. (A.4), because the two 1/3 factors would cancel for large values of ℓ. However, we have to remember that temperature perturbations are here assumed one-dimensional, so the intensity can only vary in the z direction, while the rays still go in all three directions. This means that under the sum in Eq. (15) only one third of the I − S terms give a contribution, and that the cooling rate is therefore (A.5)which has now only a single 1/3 factor. Likewise, if we had two-dimensional perturbations such as in two-dimensional convection considered in Sect. 3.13, only 2/3 of the terms under the sum in Eq. (15) would contribute. However, in a two-dimensional radiative transfer calculation, the additional 1/3 would be absent, which explains the D/ 3 correction factor with D = 2 in this case.

Fig. A.1 Dependence of the cooling rates computed from models with different values of (from 102 to 105 Mm-1 cm3 g-1) and kz (=1, 2, and 4 indicated by diamonds, triangles, and squares, respectively). 2D models with four and eight rays are indicated by crosses and circles, respectively, while 3D models with six rays are shown as plus signs. The red solid line corresponds to Eq. (A.5), the dashed blue line to Eq. (A.4), and the dotted line with open circles to the case without correction factor.

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We have verified that with the correction factor in place, the code now yields the same cooling rates in both the optically thick and thin regimes, regardless of the numbers of rays used. This is shown in Fig. A.1, where we plot cooling rates for different values of in a domain of size 2π (in Mm), so the smallest wavenumber is 1 Mm^-1. With ρ = 4 × 10^-4 g cm^-3 the photon mean-free path varies from 0.025 to 25 Mm as is decreased from 10⁵ to 10² Mm^-1 cm³ g^-1. For the Kramers opacity, we use the exponents a = 1 and b = 0. (No gravity is included here, so there would be no convection.) The temperature is 38,968 K, as before, which yields c_γ = 3.87 km s^-1 for the photon diffusion speed. There is excellent agreement between 1D cases with correction factor and the 3D calculation (with one-dimensional perturbation). However, the 2/3 correction factor in the 2D calculation (both with four and with eight rays) seems to be systematically off and should instead by around 0.8 for better agreement. However, as discussed before, the correction factor does not affect the steady state and therefore also not the results presented in Sect. 3.13. The diffusion approximation would imply λ = (c_γk/ 3)ℓk = χk², which corresponds to the diagonal in Fig. A.1 and agrees with the red solid line for ℓk ≲ 0.5.

For three-dimensional perturbations, the correct cooling rate in the optically thin regime is three times faster than for one-dimensional perturbations. This is because now the radiation goes in all three directions. Solutions to three-dimensional perturbations clearly cannot be reproduced in less than three dimensions. However, for one-dimensional perturbations, the correct cooling rate is now obtained with a one-dimensional calculation both in the optically thin and thick regimes.

Appendix B: Expressions for Pr Ra and χ_mid

In Table 6 we listed the values of Pr Ra and χ_mid in the middle of the layer. The purpose of this appendix is to give the explicit expressions and to demonstrate the calculation with the help of an example. Since n = 1 was assumed, we have ∇ = (1 + n)^-1 = 1/2. Considering the case , Eqs. (31)−(33) yield F_rad = 0.00131 g cm^-3 km³ s^-3, T_top = 12 320 K, and d = 2.70 Mm. Next, given that the temperature varies linearly, we compute the mid-layer temperature as . This allows us to compute ρ_mid = ρ_bot (T_mid/T_bot)ⁿ = 2.2 × 10^-4 g cm^-3, where ρ_bot = 3.3 × 10^-4 g cm^-3 is smaller than ρ₀ by a factor ρ_bot/ρ₀ = 0.83; see Appendix C. Thus, χ_mid = F_rad/ (ρ_midg∇/∇_ad) = 0.0175 Mm km s^-1, as well as . This yields , where ∇ − ∇_ad = 0.1.

Appendix C: Final to initial bottom density ratio

Initially, the stratification is isothermal, so the density is given by and the initial surface density is (C.1)In the final state, the stratification is polytropic, so the density is given by ρ(z) = ρ_bot [T(z) /T_bot] ⁿ and the surface density is (C.2)

Here, ρ_bot is the bottom density of the final state, which is different from the initial value ρ₀, as explained in Sect. 2.6. Integrating Eq. (C.2) and using dz/ dT = K₀/F_rad from Eq. (28) yields (C.3)Using Eq. (29) together with ∇ = 1/(1 + n) and , we have (C.4)Using mass conservation, we have Σ_fin = Σ_ini, so we obtain from Eqs. (C.1) and (C.4) (C.5)for the final to initial bottom density ratio.

© ESO, 2014