6.5 Retrieval tests with mock observational data
6.5.1 Input models generated by retrieval code
As said above, the input spectra in this section were calculated with the retrieval code itself, and I assumed vertically constant abundances when calculating the spectrum, i.e. I used the same abundance model as within the retrieval code (see Section 6.4.1). For the abun- dances I assumed the following mass fractions: log10(XH2O) = 2.5, log10(XCH4) = 5, log10(XCO2) = 4, log10(XCO) = 2, log10(XNa) = 5, and log10(XK) = 5. The P – T profile was defined using Equation 6.20, and making the following parameter choices: log10( ) = 5, log10( ) = 0.4, Tint = 600 K, Tequ= 2300 K, Ptrans= 0.01 bar, and ↵ = 0.5. I assumed P0 = 0.01bar for calculating the transmission spectra. The synthetic observation was carried out by assuming one eclipse in each of the three JWST instruments listed in Table5.3.
The retrieval was run by drawing 200000 so-called ‘pre burn-in’ samples to find the global minimum, plus 106samples centered around that minimum, to derive a reliable pos- terior distribution. The latter part is the main MCMC run, and it was set up by sampling the initial walker positions from a Gaussian, centered around the best-fit value of the pre burn-in run. The spread of the Gaussian was set up to be about half to the mean variation of the pre burn-in run. The number of ensemble walkers was 800, such that the total number of samples drawn (106) required 1250 steps. After the 106samples had been drawn I discarded the first 160000 to leave out the ‘burn-in’, i.e. the phase where the walkers relax from the initial Gauss to the actual posterior distribution.
The result of the retrieval run, based on the emission spectrum, is shown in Figure6.3. The noise-less input spectrum, the observed input spectrum, as well as the best-fit result of the main 106 160000MCMC samples is shown in Panel (a). It can be seen that the spectrum fits the input spectrum very well, and the residuals, shown in in the lower sub panel of Panel (a), do not exhibit any discernible features and scatter symmetrically about zero.
The input P –T profile, together with the best-fit profile, and the 5-95 % and 15-85 % envelopes of the cumulative temperature profile distribution is shown in Panel (b) of Fig- ure 6.3. I concentrate on these envelopes because the 15 to 85 % envelope, in particular, approximately corresponds to the 1 envelope if the posterior distribution were to be well represented by a Gauss profile (with envelope boundaries at 15.87 and 84.24 %). One can see that between 10 6and 1 bar the best-fit P –T profile and the input P –T profile agree very 3More species are straightforward to include. However, the computational time is ⇠linear in the number of species, and I expect the molecular species listed here to be the most important ones for warm to hot gaseous planets, such that I want to test how valid this assumption is. For carbon-rich hot jupiters (with a C/O ratio > 1), also HCN (ans possibly C2H2) would have to be included.
(a)
(b)
(c)
(d)
FIGURE6.3: Retrieval results of the constant abundance input 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 cumulative temperature profile distribution are shown in gray and black, respectively. 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 true input mass fraction is denoted by the blue lines, whereas the three dashed lines denote the values corresponding to 15, 50, and 85 % of the cumulative distribution of the values.
172 Chapter 6. Atmospheric parameter retrieval from spectral observations
well, but start to deviate significantly for pressures larger than 1 bar. For pressures lower than ⇠ 2 ⇥ 10 3 bar the input temperature lies outside the 15 to 85 %, but within the 5 to 95 % envelope. For all pressures larger than ⇠ 2 ⇥ 10 3 bar the input temperature profile lies within the 15 to 85 % envelope. Hence I regard the input temperature profile as having been successfully retrieved. The reason for a decreased agreement between the input tem- perature profile and the 15 to 85 % envelope at pressures. 2⇥10 3, as well as the difference between the input and the best-fit P –T profile at pressures > 1 bar, can be understood from looking at Panel (c) of Figure6.3.
Panel (c) of Figure6.3shows the so-called ‘emission contribution function’ of the best-fit atmospheric parameters, which is a measure for how strongly the emission of a layer con- tributes to the atmosphere’s emerging spectrum at the top. It depends on the temperature, and on the optical thickness of a given layer, because the latter determines how strongly it will be able to emit light into a ray passing through it. The contribution of a layer is not only decided by these tho factors, however, but also by the optical depth of the atmosphere above the layer: if the overlying atmosphere is optically thick, then the light emitted by the layer will be absorbed before it leaves the planet, and hence the layer’s contribution to the total emission will become zero. From the sum on the RHS of Equation6.14one can see that the contribution of the region between layer i and i + 1, i.e. of layer i + 1/2, to a ray pass- ing through it is given by (1/2)⇥ ¯B(Ti) + ¯B(Ti+1)⇤ ¯Ti T¯i+1 . This means that the total contribution of a layer to the arising spectrum at the top of the atmosphere is
Cemi+1/2 = PNµ j=1⇥ ¯B(Ti) + ¯B(Ti+1)⇤ ⇣ ¯Tji T¯ji+1 ⌘ µj µj PNµ j=1 n 2 ¯B(Tbot) ¯Tatmo+PNk=1L 1⇥ ¯B(Tk) + ¯B(Tk+1)⇤ ⇣ ¯Tjk T¯jk+1 ⌘o µj µj , (6.22) where µ = cos(#), # being the angle between the ray and the atmospheric normal. Nµis the number of angle grid points. This can be derived from noticing that the emission flux F is 2⇡R01µI(µ)dµ(see Equation3.36).
From Panel (c) in Figure6.3one now sees that most of the emission stems from regions between 2 ⇥ 10 3 and 1 bar. Consequently, this is the region within which the retrieved atmospheric P –T structure agrees best with the input structure. Moreover, as no radiation emitted a pressures larger than 1 bar is able to leave the planet, the 15-85 % temperature envelope at P > 1 bar is fully determined from the parametrized temperature model. For a real planet the atmospheric physics may give rise to deep retrieved temperature profiles which are different from what the parametrized P –T profile predicts. It may then be incor- rect to assume that the retrieved temperature profile below the regions probed by observa- tions, > 1 bar in the case presented here, are representing the actual temperature profile of the atmosphere. Because we cannot know the actual deep atmospheric physics with defi- nite certainty, except for theoretical expectations, this has to be critically kept in mind when drawing inferences derived from the deep temperature profile.
This also explains why the best-fit P –T structure can exhibit temperatures very different from the input profile at P > 1 bar. Additionally, from Panel (c) one sees that the only wavelength region which samples the high atmospheric region at P < 2 ⇥ 10 3 baris the
CO2 feature at 4.2 to 4.6 µm. When I checked the mean flux of the synthetic observation across the 4.5 µm region I found that the observations, by chance, let to a slightly too large flux (also compare red and cyan lines in Panel (a)). Hence the retrieved temperatures there are slightly too high, when compared to the input model.
(a) (b)
FIGURE6.4: Panel (a): retrieved abundance posterior distribution of a case with observational errors half as large as in the case considered in Figure
6.3. Panel (b): retrieved temperature envelope for the case shown in Panel (a).
Finally, the posterior distribu- tion of the retrieved abundances is given in Panel (d) of Figure
6.3. While the full atmospheric model is determined by 12 param- eters (6 abundances and 6 temper- ature structure parameters), and the full posterior distribution is 12-dimensional, I only show the posteriors for the abundances of the line absorbers in Panel (d). This is because one is, in general, not interested in the parameters which describe the parametrized P–T structure; rather the distribu- tion of temperature profiles they give rise to is of interest, which is given in Panel (b) of Figure6.3. The parameter values for the P –T profiles do not carry any physical meaning, they merely help to gen-
erate different temperature profiles. Hence, such parameters are often called ‘nuisance pa- rameters’. The sampled abundance distributions are shown in 2-d and 1-d marginalized spaces, i.e. projected onto the planes defined of the log10(Xi)and log10(Xj)axes for species iand j in the 2-d case, and on the log10(Xi)axis of species i in the 1-d case. One sees that, within the 15-85 % envelopes of the sampled values (which are given as the error values on top of the columns in Panel (d)), all molecular abundances agree with the input value. Only the sodium and potassium abundances cannot be successfully retrieved: the strong sodium and potassium resonance lines lie at ⇠589 nm and ⇠768 nm (see Section3.3.1), and therefore outside the JWST spectral range. However, they can affect the JWST spectra because their broad line wings extend out to 0.9 and 1.15 µm, respectively. This means that their spectral influence reaches out to 0.9 µm, beyond which their opacity contribution is overwhelmed by H2O absorption, which starts at 0.9 µm. Hence it seems that the quite noisy emission spectral data between 0.8 and 0.9 µm is degenerate with respect to the sodium and potas- sium abundances, and favored an increased sodium, and decreased potassium abundance for the observation presented here.
174 Chapter 6. Atmospheric parameter retrieval from spectral observations
For comparison, in Figure6.4, I show the posterior abundances (and temperature en- velopes) retrieved for a case where the observational error was only half as large, corre- sponding to four eclipse observations in each instrument, or a planet which is at half the distance of TrES-4b, i.e. 240 pc. In this case the K input abundance is correctly retrieved, whereas the extent of the uncertainty envelope of the sodium abundance (log10(XNa) = 9.60+5.2737.40) indicates an unconstrained abundance. This test shows that the retrieved results may depend quite strongly on the by-chance sampling of the spectrum during the observa- tion process, at least when parameters are determined by only a narrow spectral range, and affected by non-negligible noise.
In conclusion, emission part of the first retrieval test was successful: Within the 15 to 85 % envelopes, I correctly retrieved the atmospheric abundances, and the temperature profile in the region which generates the spectrum. And, as mentioned above: the distribution of the residuals between the input and best-fit spectrum does not show any systematic features and scatters about zero (see Panel (a)). One property of the retrieved results, which may look worrisome, is that the mean values of the retrieved abundances in Panel (d) in Figure
6.3 all appear to have a positive systematic offset with respect to the input value. In that sense it is reassuring that this is not the case for the retrieved abundances shown in Figure
6.4, such that the particular result for the abundances in Panel (d) of Figure 6.3 may be caused by an ‘unfortunate’ sampling of the noiseless spectra when generating the synthetic observation. While a detailed test of this is clearly needed, it was beyond the current scope of this work, but will have to be investigated in greater detail in the future.
The result from retrieving the atmospheric transmission spectrum for the constant abun- dance case is shown in Figure6.5. Like for the emission retrieval case in Figure6.3, I show the input and best-fit retrieved spectra and temperature structures, along with the retrieved abundances and the transmission contribution function.
The transmission contribution function of layer i at a given wavelength is calculated from Ctri = R 2 0 R2(i = 0) PNL j=1 ⇥ R02 R2(j = 0)⇤ , (6.23)
where R0 is the planet’s transmission radius at a given wavelength, and R(i = 0) is the transmission radius one obtains when setting the opacity in the ith layer to zero. I use squared radii here, because transmission spectra measure the flux decrease of the star as it is transited by its planet, which is proportional to the planet’s area.
From Panel (a) in Figure6.5one can see that the transmission spectrum is correctly re- trieved, with a residual distribution which scatters about zero, with no noticeable features. From Panel (b) one sees that the atmospheric temperature structure is successfully retrieved within the regions which are probed by transmission (see Panel (c)). Likewise, the abun- dances of H2O, CH4, CO2and K are all retrieved, with the true input values falling within the 15 to 85 % envelopes.
In the lowest row of Panel (d) in Figure 6.5 one can see that there exists a correlation between the reference pressure P0 and the molecular abundances of the spectrally active species. This can be understood from the fact that the transmission spectrum is less sensitive
1 2 3 4 5 6 7 8 9 10 Wavelength (micron) 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 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)
FIGURE6.5: Retrieval results of the constant abundance input transmission spectrum generated by the retrieval model.
Panel (a), top: transmission 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, respectively. 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): transmission 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 true input mass fraction is denoted by the blue lines, whereas the three dashed lines denote the values corresponding to 15, 50, and 85 % of the cumulative distribution of the values.
176 Chapter 6. Atmospheric parameter retrieval from spectral observations
to pressure than it is to the temperature of the atmospheric regions being probed. Within the retrieval model, if larger P0 pressure values are assumed, such that r(P = P0) = RPl, then a given pressure value will correspond to a larger radius, which follows directly from inte- grating the hydrostatic equilibrium equation (see Equation2.3). Hence, if too large P0values are put into the model, at correct molecular abundances, then the transmission radii of the planet will become too large. In such cases, reducing the molecular abundances improves the fit, as one can then probe deeper into the atmosphere, which decreases the planetary transmission radii. This effect is well known, and has been described in, e.g.,Lecavelier des Etangs et al.(2008);Benneke & Seager(2012);Griffith(2014);Heng & Kitzmann(2017).
The marginalized abundance distributions in Panel (d) of Figure 6.5 show that there seems not only to be a correlation, but actually a clear degeneracy, because the retrieved abundances for H2O, CH4, CO2 and K, and the reference pressure P0, exhibit a plateau in their posterior distributions. The respective plateaus include the true input values. Again, the quite broad distribution of the sodium abundance uncertainties, log10(XNa) = 3.92+1.232.11, indicate that the sodium abundance is not well constrained in the JWST wavelength range. Interestingly, the CO posterior does not exhibit a clear plateau.