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by Exoplanetology Tech: Research on the Study of PlanetsFebruary 28th, 2024

This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Eleonora Alei, ETH Zurich, Institute for Particle Physics & Astrophysics & National Center of Competence in Research PlanetS;

(2) Björn S. Konrad, ETH Zurich, Institute for Particle Physics & Astrophysics & National Center of Competence in Research PlanetS;

(3) Daniel Angerhausen, ETH Zurich, Institute for Particle Physics & Astrophysics, National Center of Competence in Research PlanetS & Blue Marble Space Institute of Science;

(4) John Lee Grenfell, Department of Extrasolar Planets and Atmospheres (EPA), Institute for Planetary Research (PF), German Aerospace Centre (DLR)

(5) Paul Mollière, Max-Planck-Institut für Astronomie;

(6) Sascha P. Quanz, ETH Zurich, Institute for Particle Physics & Astrophysics & National Center of Competence in Research PlanetS;

(7) Sarah Rugheimer, Department of Physics, University of Oxford;

(8) Fabian Wunderlich, Department of Extrasolar Planets and Atmospheres (EPA), Institute for Planetary Research (PF), German Aerospace Centre (DLR);

(9) LIFE collaboration, www.life-space-mission.com.

Appendix A: Scattering of terrestrial exoplanets

Appendix C: Bayes’ factor analysis: other epochs

Appendix D: Cloudy scenarios: additional figures

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.*

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