Energy transport

In this section, the energy transport through the atmosphere is tested. The hydrodynamical solver is only considering an energy change caused by emission and absorption of radiation, whereto the result of the radiative transfer equation is needed. All other influences are ne-glected. As a first test, the initial temperature structure changes if the hydrodynamical solver is working on the SN Ia atmosphere. As the initial atmosphere structure is already in ra-diative equilibrium, the hydrodynamical solver should not change the temperature structure significantly, because it also pushes the atmosphere towards a radiative equilibrium state.

In figure 5.3, a comparison of the temperature structure of the hydrodynamical solver to the result of the temperature correction procedure is shown. The differences in the tem-perature structure are for most layers less than 1%. But as can be seen in figure 5.3, the temperature differences of the inner layers are clearly higher. These differences arise in the temperature correction result as it shows a spike in the temperature structure. This may have been emerged due to the boundary condition in the temperature correction. Hence, the resulting temperature structure obtained with the hydrodynamical solver is more accurate.

Here, the temperature structure is smooth. In order to obtain an atmosphere in radiative equilibrium, the energy transport part of the hydrodynamical solver can be used instead of the temperature correction procedure. The main problem is that about a few hundred time steps are needed to obtain the resulting atmosphere structure in radiative equilibrium, while the temperature correction needs fewer iteration steps and is, therefore, significantly faster.

For the next tests, the temperature of the innermost layer is changed in order to get an en-ergy perturbation, which moves through the atmosphere via the enen-ergy transport. Numerous different perturbations can be put into the inner part of the atmosphere to test the energy transport part of the hydrodynamical solver. Whatever the perturbation of the inner bound-ary condition is, the temperature structure is always expected to relax to the new conditions and move back to be in radiative equilibrium. A few test cases with perturbations of the inner boundary condition are presented in the following.

For the first of these test calculations, the temperature of the innermost layer is increased.

The expectation is that this additional radiation energy is moving through the atmosphere.

The temperature should increase everywhere, and the atmosphere adapts to the new inner boundary condition, until it is again in the radiative equilibrium state. In figure 5.4, the re-sults of this test calculation are presented. A plot of the observed luminosity is shown in figure 5.4(a). The luminosity is increasing, as the temperature of the atmosphere increases because of the hotter inner boundary condition. One can also see that it takes some time,

un-Figure 5.3: The temperature structures obtained with the hydrodynamical solver and the PHOENIXtemperature correction are compared in this plot. Both atmospheres are in radiative equilibrium, and therefore the resulting temperature structures should be the same. The differences in the temperature are less than 1%, except for some inner layers.

(a) (b)

Figure 5.4: Result of a test calculation with an atmosphere, where the inner layer is heated.

This additional energy moves through the atmosphere. (a) The increasing ob-served luminosity of the outer layer is shown. (b) The temperature structure at different points in time.

(a) (b)

Figure 5.5: Test calculation with an atmosphere, where the innermost layer is set to a lower temperature. (a) The decreasing observed luminosity is shown. The cooled in-ner part of the atmosphere moves through the atmosphere. (b) The temperature structure on its way to radiative equilibrium is shown here.

til the additional energy gets to the outer part of the atmosphere and is seen by the observer.

The temperature of the atmosphere is increasing everywhere, until the atmosphere is again in radiative equilibrium. The initial and final temperature structure are presented in figure 5.4(b), where the increased temperature structure of the atmosphere that has an increased temperature as inner boundary condition are compared.

For the next test, the temperature of the innermost layer is set to a significantly lower tem-perature. This cooler inner condition leads to a cooling of the whole atmosphere, as it moves back to the radiative equilibrium state. The results of this calculation are presented in figure 5.5. The decreasing observed luminosity is shown in figure 5.5(a). Again, it can be seen that it needs some time before the energy reaches the outer layers of the model atmosphere, and the atmosphere structure has adapted to the new inner boundary condition. The resulting cooler atmosphere structure is shown in figure 5.5(b). The atmosphere relaxes to the changed inner condition and assumes a radiative equilibrium temperature structure.

As the hydrodynamical solver works for changed but then fixed inner condition, now a time dependent temperature of the innermost layer is considered. Thus, for a last test of the en-ergy transport, the temperature of the innermost layer is varying as a sine in time. The time step size is set to a constant value of 2·10⁻²s to have a high time resolution for a whole period of the sine, which takes 400 time steps. Hence, a whole period of the sine needs 8s.

The amplitude of the sine of the inner layer is set to 20% of its initial temperature. This perturbation moves through the whole atmosphere and is expected to make the temperature of the whole atmosphere varying as a sine. A surface plot of the temperature is shown in figure 5.6. As can be seen, the temperature of the whole atmosphere is varying periodically.

It again takes time before the information about the sinusoidally varying inner temperature reaches the outer layers of the model atmosphere. This becomes apparent as a phase shift in the sine. A plot of the temperature structure at different points in time is shown in figure

Figure 5.6: The temperature of the innermost layer is varying with a sine in time. This sur-face plot shows that the whole atmosphere is after some time varying as a sine.

This propagation through the atmosphere needs also some time, which leads to a phase shift of the sine.

(a) (b)

Figure 5.7: Atmosphere with a sinusoidally varying inner temperature. (a) Temperature structure of a few points in time. This perturbation moves the through the whole atmosphere, making it varying as sine everywhere. (b) The temperature of a few layers vary in time and have a sinusoidally shape.

Figure 5.8: Luminosity of the sinusoidally varying atmosphere seen by an observer. After an initial rise, the luminosity is varying with a sine that reflects the sinusoidally varying temperature structure.

5.7(a). One can see the varying temperatures in time. The temperature of a few layers over time is plotted in figure 5.7(b). As it can be seen, the temperature of every layer is varying periodically. The shape is not a sine, but it looks similar. The rising occurs faster than the decline. The deviation may be due to the radiative transfer. It takes time, until the temper-ature change has moved through the atmosphere. The phase shift is about 200 time steps.

The varying luminosity is plotted in figure 5.8. As can be seen, the observed luminosity is varying as a sine, after some initial disturbance.

All the tests indicate that the energy transport part of the hydrodynamical solver works properly. It moves the temperature structure of an atmosphere towards radiative equilibrium.

If disturbed, the temperature structure also adapts to the changed inner boundary conditions of the innermost layer. A radiative equilibrium temperature structure obtained with the hy-drodynamical solver is almost the same as one obtained with the temperature correction procedure.