The Scattering of Terrestrial Exoplanets: What This Means

Table of Links

Abstract & Introduction

Methods

Results

Discussion

Conclusions

Next Steps & References

Appendix A: Scattering of terrestrial exoplanets

Appendix B: Corner Plots

Appendix C: Bayes’ factor analysis: other epochs

Appendix D: Cloudy scenarios: additional figures

Appendix A: Scattering of terrestrial exoplanets

As discussed in Mollière et al. (2020), petitRADTRANS was updated to treat scattering. This was done using the Feautrier method (Feautrier 1964). This is a third-order method that allows the treatment of the radiative transfer equation in the diffusive regime.

The Feautrier method solves the angle- and frequencydependent radiative transfer equation for both the planetary and the stellar radiation field. These can be treated separately, since the radiative transfer equation (Eq. A.1) depends only linearly from the intensity I.

Here, µ = cos θ where θ is the angle between a light ray and the surface normal, τ is the optical depth, I is the intensity, and S is the source function.

Conceptually, for any direction µ of a ray, there also exists a ray in direction −µ, where µ ∈ [−1, 1]. It is possible to instead let µ run from 0 to 1 only, and define rays I+ and I− parallel and antiparallel to this direction. For one of these, the projection onto the atmospheric normal vector (defined by the scalar product) will be positive (going upward), while for the other one it will be negative (that is, going downward). Eq. A.1 can be therefore rewritten as:

To solve these, it is convenient to define other two variables:

So that Eqs. A.2 and A.3 become:

In this paper, we only take into account thermal scattering, i.e. scattering of the planetary radiation. We therefore neglect the scattering of the direct stellar contribution. However, since the radiative transfer equation depends only linearly on Iν, the contribution of the stellar radiation can be treated as an additional term in the calculation (see Mollière et al. 2017). This term is also included in the latest version of petitRADTRANS and we refer to the online documentation for a more detailed description [10] .

Purely considering the planetary radiation, we define the boundary conditions at the top of the atmosphere:

meaning that there is no planetary radiation coming downwards from the top of the atmosphere, and at the surface:

there is no dependence from the (N − 1)th layer in the boundary condition A.13; bN, as a consequence, will be equal to 1.

The iterations stop once the estimate of the flux has reached a convergence value.

[10] https://petitradtrans.readthedocs.io/en/ latest/content/notebooks/emis_scat.html# Scattering-of-stellar-light

This paper is available on arxiv under CC 4.0 license.