Implementing absorption regions and parameter space exploration

After fine tuning the PTPL to ensure that an appropriate level of continuum polarisation is being generated, absorption regions can be added to model line polarisation. The ab- sorption region, since it is assumed to have very high optical depth such that all underlying photon packets are removed from the simulation, can be implemented with a simple mask. As previously mentioned, one of our main objectives is to systematically explore param- eter space to find all good solutions and understand potential degeneracies. Our grid search approach for 3 free parameters required thousands of simulated systems (photosphere + absorption region), but due to the nature of Monte Carlo methods, photospheres simulated with the same parameters may yield slightly different results. Additionally, re-generating a million photon packets and their associated polarisation is computationally intensive.

Therefore, we favoured an approach where the photosphere were not re-created for every simulation. Instead, a single instance of the photosphere object was created and its state was saved to a file (“Pickled”). A range of absorption regions with parameters xfree could

then be modelled by calling in the object, masking the appropriate areas, and calculating the integrated polarisation.

5.6

Application to SN 2011hs

We applied the toy model to SN 2011hs for the polarisation features of He i λ5876, Hα and Ca ii at −3, and +2 days, where the assumption of a bi-axial ellipsoid for the geometry of the photosphere seems robust (see Section 4.4). We also applied our models to epoch 3 (+10 days), where this assumption is not expected to be valid, for reasons that are discussed in the next section. For these epochs the asymmetry was ∼10 percent, and the fine tuned PTPL for an axis ratio of 0.87 was found to be 30 percent.

The number of photon packets used in each simulation was determined by the precision required given the uncertainty on the observations that were being modelled. From Section 5.5 we know that for 106 photons the errors on q and u are ∼0.07 percent. Better precision can be achieved with more photons (e.g. Reilly et al. 2016 used 107 photons), but the run time scales linearly with photon number. Additionally, due to Poisson statistics, an increase in run-time by a factor of 10 would only result in an increase in SNR by a factor of ∼√10. Given the errors on the observational data investigated, models with 106 _photons

optimised for shorter run times whilst providing sufficient precision.

Toy model 94

the 3D structure of the line forming regions we selected 2 to 3 wavelength bins per line to model. This is an alternative between modelling only one bin per line feature (resulting in no depth information), and running models for all bins, which is unnecessarily laborious as successive bins will have very similar solutions given the size of the errors on our data. Additionally, if a polarisation feature showed two components, we made sure to select bins sampling them both. Within those guidelines, the choice of bins is arbitrary, and other combinations of bins could have been chosen to obtain similar results. A summary of the target values for the bins chosen is given in Table 5.1.

In Figure 5.7, we show the models that could reproduce the observed polarisation data. In most cases the solutions within 1 σ are shown, although some results are presented to 1.2, 1.5 and even 2 σ (as indicated by the numbers on the visual representation of the models shown in Figure 5.7). These cases arose when parameters could not be found down to the 1-σ level, and were the result of solutions being confined to very small regions of parameter space. Finer grids were explored, but due to time constrains we did not always reach the 1-σ level, and halted simulations when sufficient precision was obtained. As can be seen in Figure 5.7, these cases yielded the most consistent solutions, with best constrained free parameters. This suggests that no solution exist within 1-σ. In these particular cases we find good agreement with the observed parameters, and the fitting issues arise in the line to continuum flux ratio. This is most likely due to the inaccuracy of the continuum flux estimate and the small formal errors on (Fline/Fcont)obs.

This type of visualisation, if accurate, can be complementary with Polar Plots (see Section 4.4), as they provide a view of the distribution of line forming regions as projected on the sky whilst polar plots show the P.A. of polarisation line features with depth.

We want to emphasise here that a model contains only one circular absorption region, and the multiplicity observed in some cases in Figure 5.7 is just a result of the uncertainty on the possible location of the absorption region and shows the degeneracy of the solutions.

T

oy

mo

del

95

Table 5.1: Measured values to be reproduced by the models shown in Figure 5.7.

He I 5876 Hα CaII −3 days 5615˚A 5660˚A - 6048˚A 6152˚A 6257˚A 8046˚A 8136˚A 8195˚A q 0.69 ±0.18 0.51 ±0.17 - 0.62 ±0.18 0.91 ±0.19 0.51 ±0.19 1.22 ±0.22 1.53 ±0.24 0.79 ±0.24 u -0.051 ±0.16 -0.24 ±0.16 - -0.21 ±0.16 0.21 ±0.17 0.09 ±0.17 -0.09 ±0.20 0.27 ±0.22 -0.14 ±0.22 Fline/Fcont 0.91 ±0.02 0.886 ±0.020 - 0.948 ±0.022 0.690 ±0.018 0.592 ±0.016 0.81 ±0.20 0.604 ±0.046 0.604 ±0.007 +2 days 5571˚A 5675˚A 5705˚A 6137˚A 6271˚A 6316˚A 8061˚A 8165˚A 8240˚A q 0.25 ±0.16 0.64 ±0.18 0.55 ±0.18 0.96 ±0.18 0.95 ±0.20 0.66 ±0.19 1.17 ±0.21 1.25 ±0.23 1.25 ±0.27 u -0.66 ±0.15 -0.62 ±0.16 -0.77 ±0.16 -0.42 ±0.16 -0.20 ±0.18 -0.17 ±0.17 -0.63 ±0.19 -0.56 ±0.21 -0.66 ±0.26 Fline/Fcont 0.94 ±0.03 0.72 ±0.02 0.81 ±0.03 0.523 ±0.017 0.556 ±0.018 0.78 ±0.03 0.495 ±0.017 0.443 ±0.015 +10 days - 5720˚A 5779˚A 6167˚A 6301˚A - 8240˚A 8315˚A 8374˚A q - 0.12 ±0.20 0.25 ±0.19 0.55 ±0.19 0.53 ±0.21 - 0.20 ±0.3 0.45 ±0.29 -0.136 ±0.29 u - -0.71 ±0.19 -0.47 ±0.17 -0.41 ±0.17 -0.38 ±0.19 - -0.92 ±0.3 -1.40 ±0.28 -1.046 ±0.27 Fline/Fcont - 0.441 ±0.013 0.630 ±0.018 0.73±0.02 0.450 ±0.014 - 0.36 ±0.01 0.324 ±0.011 0.404 ±0.014

Toy model 96

Figure 5.7: Best solutions of our toy model for He iλ5876, Hα and Ca iiλ8567 at −3, +2 and +10 days. The target values and bins chosen are summarises in Table 5.1. The colour pallet represents the goodness of fit, where the lighter colours indicate the most likely solutions. The grey ellipse shows the photosphere, onto which solutions yielding good fits to the observed polarisation are superposed. The black circles indicate the absorption regions giving the best fit. In most cases the solutions within 1 σ are shown, although some results are presented to 1.2, 1.5 and even 2 σ, as indicated by the numbers on the visual representation of the modelled ejecta. The coloured lines link the modelled bins to their visual representation. The flux spectrum (blue), and the degree of polarisation is shown (purple) are shown for comparison with the models.

Toy model 97