2.6 Atmospheric parametrisation
2.6.1 Temperature-pressure profiles
The determination of the vertical atmospheric temperature profile is one of the key chal-lenges in the retrieval of atmospheric emission spectra. Typically two approaches exist in the retrieval of the TP-profile: 1) Layer-by-layer retrieval; 2) Analytic parameterisation. In TauREx we include both approaches, together with a “hybrid” method. We also include an isothermal profile, and a “3-point” and “4-point” profile.
2.6.1.1 Layer-by-layer
In the layer-by-layer method the temperature of each atmospheric layer is fitted indepen-dently. However, this method can converge to a solution only in the presence of extremely high signal-to-noise and broad wavelength coverage observations, such as those obtained for the Earth’s atmosphere and solar system planets (Rodgers, 1976; Hanel et al., 2003).
In a low signal-to-noise and resolution scenario, such as that expected from the observa-tions of exoplanets, this method has extremely poor convergence properties. A common solution to this problem is to impose a “regularisation” of the temperature (e.g. Irwin et al., 2008), based on the fact the adjacent atmospheric layers should exhibit some correlation in temperature.
2.6.1.2 Parametric model
Another approach consists of parametrising the temperature profile based on some analytic models. This method attempts to model the underlying physics of the thermal structure, while minimising the number of parameters. However, despite the clear advantage of re-ducing the parameter space compared to a layer-by-layer approach, the solution will always be constrained within the bounds of the model assumed.
Several analytical models exist in the literature, ranging from radiative-convective ap-proximations to global circulation models (Liou, K N, 2002; Hubeny et al., 2003; Burrows et al., 2008; Hansen, 2008; Madhusudhan & Seager, 2009; Showman et al., 2009; Guil-lot, 2010; Pierrehumbert, 2010; Robinson & Catling, 2012; Heng & Workman, 2014; Heng et al., 2014). Following Guillot (2010), the mean global temperature profile, as a function of the thermal optical depth (see Equation 2.6.8), for a simple radiative downstream-upstream approximation can be expressed as
T4(τth) =3Tint4 where Tintis the planet internal heat flux, Tirrthe stellar flux at the top of the atmosphere and E2is the second-order exponential integral.
We note that similar parameterisations exist in the literature (e.g. Robinson & Catling, 2012). We also include the variation by Line et al. (2013) and Parmentier et al. (2015) including two optical opacity sources κV1 and κV2and a weighting factor between optical opacities (left as free parameter), αV,
T4(τth) =3Tint4
This parametrisation with two visible streams allows more freedom for a temperature inver-sion (Line et al., 2013).
The temperature as function of opacity τthcan be mapped to a pressure grid by assum-ing the followassum-ing relation
τth=κIRP
g . (2.6.8)
2.6.1.3 Other TP-profiles
In addition to the above TP-profiles, we include an isothermal profile as well as a 3-point and “N-point” profile. The 3-point profile is purely geometric and keeps the top of atmo-sphere temperature the tropopause temperature and pressure, T1, P1, and the surface (or 10 bar pressure) temperature T10bar as free variables. The temperature profile is then linearly interpolated in ln(P). The N-point profile add extra “nodes” to the parametrised profile.
2.6.1.4 The hybrid approach of TauREx
TauREx implements a “hybrid” approach, combining the parametrised method to a layer-by-layer approach. This method consist of two retrievals, or “stages”. In the first stage a parametric model retrieval is computed, then, in the second stage, the retrieval solution is used to guide a layer-by-layer model, thus relaxing the parametric model constraint of the first stage retrieval.
After the first stage retrieval, the error on the sampled parametric model parameters is converted to a one sigma lower and upper temperature bounds for each atmospheric layer.
These bounds are obtained by computing the temperature profiles corresponding to all sam-pled parametric models in the Bayesian retrieval, and then calculating the standard deviation of the temperature value of each atmospheric layer. The following matrix is then calculated:
D2i, j=| ˆTi− ˆTj|2+ (σi+ σj)2, (2.6.9) where i and j are subsequent atmospheric layers, ˆT is the maximum likelihood temperature estimator of the parametric fit, and σi is the error in temperature for each layer calculated as described above. This matrix is then normalised in terms of the minimal and maximal temperature variations found in the TP-profile:
Ci, j= 1−D2i, j− argmin(D)
argmax(D) (2.6.10)
which can be thought as a temperature correlation matrix with layers most similar in
tem-perature featuring the highest correlation.
A second correlation matrix is then defined, imposing an exponential correlation length across pressure levels (Rodgers, 1976):
Si, j= (Si,iSj, j)1/2exp
−|ln(Pi/Pj)| C
, (2.6.11)
whereC is a correlation length in terms of atmospheric scale heights. Larger values of C correspond to a stronger smoothing in the TP-profile, and might be preferable in low signal-to-noise, low resolution observations. This correlation matrix is the same used by Irwin et al. (2008). A hybrid correlation matrix is then built combining the previous two matrixes:
Qi, j(α) = Ci, j+ (1− α)Si, j, (2.6.12)
where α is a scaling factor ranging from 0 to 1. In the second stage retrieval, the parameter α is set as a free parameter. This allows us to dynamically relax the parametric model solution from a model-constrained solution (i.e. stage 1) to an unconstrained solution.
Lastly, in order to optimise the number of free parameters, the sparsity of the tempera-ture profile solution of the first stage is used to compute a nonlinear sampling of the profile used in the second stage. This compression algorithm uses the correlation matrix C to only retain layers corresponding to a change greater than 2% in the temperature gradient with re-spect to the previously retained layer. Whenever no change in thermal gradient is detected for> 10 layers, a new layer is included nevertheless. With this approach a 100-layers at-mospheric model can typically be reduced to 15–25 free parameters. Such number of free parameters is still very high, therefore requiring high signal-to-noise and high resolution spectra.