EVOLUCIÓN DEL PROCESO EDUCATIVO RELACIONADO CON EL DESARROLLO VIRTUAL

Since the previous section has confirmed that the radiation component of the new code is able to reproduce the expected initial radius of a model HIIregion, it is necessary to test the

dynamics of the combined code. Similarly to other new radiation hydrodynamical methods (see review of codes by Bisbas et al. 2015), the ability to model the expansion of HIIregions in a

uniform density medium was tested. There have been several analytical solutions to describe this growth, the most famous being that of Spitzer (1978) alongside those by Hosokawa & Inutsuka (2006) these solutions can be used to determine how well simulations reproduce analytic results. While these two solutions are accepted benchmarks for the expansion of HII

regions, they both rely on assumptions that may make them differ from numerical models. For example, the Spitzer (1978) solution assumes that the gas is in an isothermal state as well as neglecting the inertia of the shell, an assumption that may not be valid in HIIregions.

5.3. Tests Similarly, Hosokawa & Inutsuka (2006) derived their solution from the equation of motion of the shell, thereby including the inertia of the shocked neutral gas that is formed as the region expands, making this a more realistic model. This does still however assume an isothermal equation of state. The Spitzer solution is given by:

ri=r0  1+7 4 c_it r₀ ‹4/7 (5.11) where r0 is the radius of the Strömgren sphere, and ci is the sound speed in the ionised gas.

Similarly, the Hosokawa & Inutsuka (2006) solution is given by:

r_i =r0 ‚ 1+7 4 v t4 3 cit r0 Œ4/7 (5.12)

where the extra factor ofq4₃ is a result of the inclusion of the inertia of the shocked gas, lead- ing to a slightly faster expansion than that found by Spitzer (1978). Due to the assumptions implicit in both of these analytic solutions, it is therefore expected that simulated results will not exactly match either solution. This was found by all of the different methods analysed in Bisbas et al. (2015) where all methods were found to more closely match the Hosokawa & In- utsuka (2006) solution, but still with some variation. Bisbas et al. (2015) therefore developed a new equation that accounts for variations from these solutions:

R=R2+  1₋0.733 exp  − t M y r ‹‹ (R1−R2) (5.13)

WhereR2is the Hosokawa & Inutsuka (2006) solution whileR1is the Spitzer (1978) solution.

In order to test the new code, we consider a simple simulation of an HIIregion expanding in to a uniform density medium with densityn=1000 cm−3 including one ionising source with ionising luminosityQ=1048photons s−1, typical of an O5 star.

Figure 5.3 shows results for three dimensional simulations with resolution 323, 643 and 1283 with 106 and 107 Monte Carlo photons. This model is able to reproduce the analytical solution increasingly well with increasing resolution and number of Monte Carlo photons. All of these simulations deviate from the Spitzer and Hosokawa solutions at the start of the simulation, however this deviation decreases as the resolution of the simulation increases. At longer times however, the differences fall to below 10% in line with models of Krumholz, Stone

Figure 5.2: Test of radiation code to reproduce expected Strömgren radius. Black diamonds repre- sent the expected radius, red crosses represent the radius calculated by this code with varying photon number and dimensions.

& Gardiner (2007). Similarly to Krumholz, Stone & Gardiner (2007) we find that the width of the ionised shell decreases with increasing resolution, since the density peak is spread over a few cells due to numerical diffusion, as expected. This can be seen in figure 5.4, which shows an example of the density structure for the expansion of an HIIregion in density n= 1000 cm−3,Q=1048photons s−1at times 5_×104years, 2_×105years, 4_×105years and 6_×105years. As a result, the measured radius varies depending on whether the inside, centre, or outside of the shell is measured. The radii presented here are found by searching for the peak in density, if this is altered the radii increase and decrease when the outer and inner shell positions are measured respectively. It is for this reason the simulations presented here vary slightly from the Bisbas et al. (2015) solution. Figure 5.4 also shows slices through the density (middle panel) and temperature (bottom panel) structure for these simulations. These show that the density and temperature both peak in the shell, and that the shell density increases with time as more material is swept in to the shell.

Altering the resolution of the simulation box increases the accuracy of the simulations since this reduces the errors introduced in the interpolation phase of the Godunov scheme. This occurs because there are likely to be smaller differences between the value of each variable

5.3. Tests

Figure 5.3:Radius of ionisation front as the simulation evolves in three dimensional simulations. The Spitzer (1978) solution is shown in black, the Hosokawa & Inutsuka (2006) is shown in red and the

STAR BENCHsolution in purple. Red crosses indicate a simulation box with resolution 323, blue 643and green 1283.

Figure 5.4: Example maps of density structure at times 5×104years, 2×105years, 4×105years and 6×105years for a simulation withn=1000 cm−3,Q=1048 photons s−1(top panel). Middle panel shows slices through the density structure at times corresponding to those in the top panel wile bottom panel shows temperature slices. Black lines indicate the earliest times and green the latest.

5.4. Diffuse radiation field