Input models generated by petitCODE

With the retrieval method itself having been verified, I proceeded to test its performance when retrieving input spectra which are more complex, and self-consistent with regard to the physical processes that appear necessary for describing the atmospheric state. Such tests are necessary to assess how well the idealized retrieval model assumptions (constant abundances, limited set of absorbers, no scattering in emission) allow to infer atmospheric properties, or whether more complex assumptions for the retrieval model are necessary. To carry out this test I considered the spectra of the the clear, globally averaged insolation case of TrES-4b, calculated self-consistently as described in Chapter 5, also see Mollière et al.

(2017). This spectral calculation included scattering, and used the full set of molecular opac- ities available within the petitCODE framework (except for TiO, VO and O3, see Table3.1). Moreover, because the molecular abundances are determined from assuming chemical equi- librium, the input spectrum will arise from vertically varying molecular abundances.

I show the result for the retrieval of TrES-4b’s synthetic emission spectrum in Figure6.6. In addition to the plots shown for the previous test cases (see figures6.3and6.5), there now is a Panel (e) in the figure, which shows the vertically varying molecular abundances of the input model, as well as the fully marginalized posterior distributions of the retrieved, vertically constant abundances.

The results indicate that the retrieval code can correctly estimate the atmospheric struc- ture and abundances, at least for the input test case shown here. The retrieved temperature structure (see Panel (b)) contains the input temperature profile within its 5 to 95 % enve- lope, and within its 15 to 85 % envelope in the atmospheric region where most of the flux is stemming from, see Panel (c). The residual distribution between the model and input spectrum does not show any systematic features, and scatters symmetrically about zero. The fact that the best-fit and noiseless input spectrum exhibit an offset in the region around

(a)

(b)

(c)

(d)

(e)

FIGURE6.6: Retrieval results of the full self-consistent petitCODE emission spectrum generated by the retrieval model. Panel (a), top: emission spectrum of the input model (cyan line), and its synthetic observation (black crosses with error bars). The best-fit retrieved spectrum is shown as a red line, with its residuals to the synthetic observation shown in the bottom sub panel. Panel (b): atmospheric temperature profiles: the input profile is shown as a cyan line, the best-fit retrieved profile is shown as a red line, and the 5-95 % and 15-85 % envelopes of the sampled structures are shown in gray and black, respec- tively. The red dashed line denotes the maximum pressure that can be probed in the wavelength range of the spectra shown in Panel (a). Panel (c): emission contribution function of the best-fit retrieved atmosphere. The red dashed line means the same as in Panel (b). Panel (d): posterior distribution of the log10(molecular mass fraction)s. The three dashed lines denote the values corresponding to 15, 50, and 85 % of the cumulative distribution of the values. Panel (e): colored solid lines: ver- tically varying abundances of the self-consistent petitCODE input model. Different colors denote different species, as indi- cated in the legend. The red dashed line means the same as in Panel (b). The colored histograms show the fully marginalized abundance posteriors of the respective species, assuming vertically constant abundance profiles. The distributions have been offset and rescaled for clarity. The vertical dashed lines indicate the abundances at 15, 50, and 85 % of the cumulative abundance distributions.

178 Chapter 6. Atmospheric parameter retrieval from spectral observations

1 µm must therefore stem from ‘bad luck’ when sampling the noiseless spectrum for the synthetic observation. The retrieved vertically constant abundances of the molecules corre- lates well with the input profiles, see Panel (e). All molecular abundances, except for CH4, are retrieved within the 15 to 85 % abundance envelopes.

From the test shown here I conclude that my retrieval implementation, in its current form, may be used to retrieve the emission spectra of hot jupiters, at least in the JWST wave- length regime. If the data were to also contain the shorter, optical wavelength regime, then the contribution of reflected stellar light may be problematic for the retrieval algorithm, be- cause I do not currently include scattering (and no other retrieval code currently does). The second limitation arises from the fact that I currently neglect clouds, which can modify the emission spectrum both due to absorption and scattering. While taking care of clouds in the retrieval tests presented here was beyond the scope of this work, I plan to include a parametrized cloud model as one of the next steps (also see Section6.6.1). Additionally, as discussed in the ‘Clouds’ part of Section 4.2.1, and Section5.4.2, hot jupiter emission may still be described well by cloud-free spectra, even when their transmission spectra show evidence for clouds.

The retrieval result based on the transmission spectrum is shown in Figure6.7. Panel (a) shows the synthetic observation, the noiseless input spectrum, the best-fit retrieved trans- mission spectrum, as well as ten spectra generated from sampling the posterior parameter distribution. As can be seen from the lower sub-panel of Panel (a), the posterior distribution exhibits no systematic features and scatters symmetrically about zero, indicating a good fit to the synthetic observation. While Panel (a) looks promising, it can be seen from panels (b) that the idealized retrieval model has problems to correctly constrain the atmospheric state for a fully self-consistent input spectrum: only between 10 3_{and 4⇥10} 2_{bar does the input} temperature profile fall within the retrieved 5 to 95 % envelope, and only between 4 ⇥ 10 3 and 10 2 _{bar within the 15 to 85 % envelope. In addition, the whole posterior temperature} envelope seems to be offset to larger pressures, when compared to the input profile. This can also be seen from Panel (d): the default value for P0 inMollière et al.(2017) was 10 bar, but the retrieval constrains it to be 66 bar. As discussed in the transmission spectrum results in Section6.5.1, too large P0values will result in too small abundances, such that H2O, CO, and K are less abundant than in the input model. The CO2and Na abundances are retrieved within the 15 to 85 % envelope. It is interesting to see that the deep atmosphere is too cool when compared to the input profile, which counteracts the effect of the too large P0 some- what (because the density enters in the hydrostatic equilibrium equation, see Equation2.3). Apparently, however, this effect is not strong enough to lead to the correct abundances in the part of the atmosphere which is probed by observations. Note that the best-fit temperature profile falls somewhat outside of 5 to 95 % temperature envelope, at least between 4 ⇥ 10 4 and 4 ⇥ 10 2 _{bar. To ensure that other likely solutions, with P –T structures within the re-} trieved uncertainty envelope, also provide a good fit to the input spectrum, I sampled ten additional atmospheric structures from the full parameter posterior distribution, which give rise to the aforementioned ten sampled spectra in Panel (a), which are shown as yellow solid lines. As can be seen these spectra do indeed provide a good fit to the spectrum. Hence, for

1 2 3 4 5 6 7 8 9 10 Wavelength (micron) 1.95 2.00 2.05 2.10 2.15 2.20 2.25 Transmission radius (RX ) Input, noiseless Best-fit case Input 1 2 3 4 5 6 7 8 9 10 -2 0 2 Residuals (a) (b) (c) (d) (e)

FIGURE6.7: Same as Figure6.6but for the transmission case. The yellow lines in Panel (a) are ten spectra derived from randomly sampling the posterior parameter distribution.

180 Chapter 6. Atmospheric parameter retrieval from spectral observations

FIGURE6.8: Left panel: retrieved temperature envelope for the same case as discussed in Figure6.7, but with the set of opaci- ties restricted to only contain the retrieval model opacity species. Right panel: retrieved abundances for the case shown in the left panel. Lines mean the same as in Panel (e) of Figure6.6.

transmission spectra, the idealized approach of vertically constant molecular abundances, and potentially not including all necessary absorber species, starts to break down in the case presented here: the retrieval code is still finding a good fit, but it does so by finding atmospheric parameters which are offset from the true values, thus trying to make up for its lack of flexibility in the retrieval abundance model. Moreover, it is interesting to see that the retrieved temperature envelope is much narrower than the one retrieved in the constant abundance case (compare figures6.5and6.7), indicating that the P0–abundance degeneracy is broken, and that the P0 and abundance values are ‘better’ constrained by the increased input model complexity.

Note that vertically constant abundances are a standard assumption in most retrieval codes in the literature, except from the self-consistent retrieval code by Benneke (2015). Apart fromRocchetto et al.(2016), no such tests as presented here seem to have been carried out before. WhileRocchetto et al.(2016), who only studied transmission spectra, show that the use of non-isothermal temperature profiles helps to better constrain abundances, their results, using vertically constant abundances, indicate similar difficulties when retrieving the atmospheric state from transmission spectra. Additionally, they show that whether or not the retrieval yields good results depends on factors like the C/O ratio of the atmosphere being studied. For C/O⇠1 the atmospheric opacity is minimal (also see Section4.4) and one probes deep into the atmosphere, in regions were the abundances vary strongly. For these cases, the quality of the retrieval results degrades strongly, because vertically constant abun- dances represent an even worse approximation. Both their and my results thus indicate that transmission spectral retrieval models need to be improved in the future.

Finally, I carried out the same test once more, using the same restricted set of line opac- ity species for the input spectrum as included in the retrieval model (i.e. CH4, H2O, CO, CO2, Na, K, H2 and He). For this case the results on the retrieved temperature structures improved, see the left panel of Figure6.8, and so did the retrieved abundance values, see the

right panel of Figure6.8, although CO is still not retrieved within the 15 to 85 % envelope. This result is somewhat puzzling, because no major absorbers have been left out when using only the restricted set of atmospheric absorbers. For obtaining a clear picture on the influ- ence of specific retrieval model assumptions, it is thus necessary to carry out more detailed tests in the future. Unfortunately, this was beyond the scope of this work, and requires more investigation in the future.

6.5.3 Summary

To summarize, my tests show that the retrieval implementation is working well, and that synthetic observations generated with the input model can be retrieved correctly. If the input model spectrum arises from a more complicated atmospheric treatment (I used input spectra generated with petitCODE), then, within the JWST wavelength range, the abundances and temperature structures of cloud-free hot jupiters seem to be well retrievable using emission spectroscopy. Even atmospheric structures of hot jupiters appearing cloudy in transmis- sion may be well retrievable with the current emission spectrum implementation, see the ‘Clouds’ part of Section4.2.1, and Section5.4.2.

Note that petitCODE additionally includes scattering, variable abundances, and addi- tional atmospheric absorbers, when compared to the retrieval atmospheric model. How- ever, a more systematic suite of tests will need to be carried out in the future, to identify pathological cases in which the retrieval code assumptions may break down. The reason for being able to retrieve the atmospheric emission spectra well, while neglecting scattering, is that the largest scattering contribution of cloud-free hot jupiters stems from Rayleigh scat- tering. Rayleigh scattering only becomes dominant in the optical wavelengths, and is thus largely outside the JWST range.

If such a complicated model is used to generate transmission spectra, then the results of the retrieved atmospheric abundances and temperature structure are to be treated with caution: the influence of the free parameters of the (possibly too simple) retrieval model may be misused to account for the larger complexity of the input model. Here tests will have to show what the best retrieval approach is (e.g. allowing for vertically varying abundance profiles, as well as including more atmospheric absorbers).