The radiative transfer code

The radiative transfer model that is used in this thesis for the interpretation of observations and modelling of exoplanetary signals is an adding-doubling algorithm similar to that used in the study of Karalidi et al. [2013]. The code is based on the description by de Haan et al. [1987] and includes polarisation for all orders of scattering. For a model atmosphere composed of stacked, vertical layers with scattering and/or absorbing gaseous molecules, cloud, and/or haze particles, with a lower bound of either a black or reflecting surface, the code calculates the Stokes vector (see Section 1.2) of the reflected sunlight (starlight in the case of an exoplanet) for pre-defined illumination and viewing angles.

The user defined illumination (incidence) and viewing (emergence) angles are defined as follows: ✓0 represents the angle between the incident flux and the local vertical direction (also known as the solar zenith angle);✓ represents the angle between the reflected flux and local vertical direction; and 0 represents the azimuthal angle between the propagation direction of the incident flux and the reflected flux, which is measured in the local horizontal plane. The atmospheric layers considered in this thesis are horizontally homogeneous, thus only the di↵erential azimuthal angle is relevant. Given a pixel on the planet and a specific phase angle, the local values for✓0,✓, and 0 are computed. The locally reflected flux vector (defined with respect to the local meridian plane, containing the local zenith and the direction towards the observer,F, is calculated by [see Hansen & Travis, 1974]:

F(µ, µ0, 0) =µ0R(µ, µ0, 0)F0, (2.12) whereF0represents the vector of the incident (unpolarised) stellar or solar flux, and

Rrepresents the 4⇥4 local planetary scattering matrix. The parametersµ0 = cos✓0 (0 _ ✓0  90 ), and µ = cos✓ (0  ✓  90 ). All of the models used assume incident unpolarised light from the star or the Sun, meaning that the incident flux vector is alwaysF0 = [F0,0,0,0], whereF0 is the total incident solar or stellar flux measured perpendicular to the direction of incidence divided by ⇡ [see Hansen & Travis, 1974]. Only the first column of the 4⇥4 planetary reflection matrix R is required, since the incident light is unpolarised, meaning Eq. 2.12 becomes

F(µ, µ0, 0) =µ0R1(µ, µ0, 0)F0, (2.13) where R1 represents the first column of the planetary reflection matrix. Given a (local) model atmosphere, R1 is calculated for the given local illumination and

viewing geometries with the adding-doubling algorithm. The circular polarisation is ignored in the computations, which can be done without introducing a significant error [Stam & Hovenier, 2005]. As was mentioned before, the reference plane for the locally computed Stokes parametersQandU is the local meridian plane. The degree of linear polarisation of the reflected flux is independent of the chosen reference plane. Since only the normalised reflected fluxes and degree of linear polarisation are considered, the solar spectrum and the solar flux incident on the objects considered do not have to be taken into account. Also, the distance between the objects and the parent star is irrelevant for the modelling carried out in this work.

2.3.2.1 The model atmospheres

The numerical simulations were carried out for atmospheres composed of stacked plane-parallel layers, all horizontally homogeneous. All layers contain gas molecules and optionally have in addition cloud and/or haze particles. A black or reflecting homogeneous surface bounds the model from below; for atmospheres with high optical thickness, such as that of Jupiter, the extent of the atmosphere precludes an influence of the surface albedo. An atmospheric profile consisting of 20 layers, a tropospheric cloud layer, and up to two haze layers residing above was used in all of the atmospheric models. A basic representation of the model atmospheric layers are shown in Figure 2.4a and Figure 2.4b for atmospheres with both one and two haze types, respectively.

The radiative transfer calculations require knowledge of the optical thickness,b, the single scattering albedo, a, and the single scattering matrix, Ssca, of the mixture

of molecules, cloud, and/or haze particles for each atmospheric layer. The optical thickness of an atmospheric layer, b, is given by the sum of the molecular, cloud, and/or haze extinction optical thicknesses as described in Stam [2008].

The Rayleigh scattering optical thickness is higher at shorter wavelengths, decreas- ing with approximately 4 _{toward longer wavelengths. The molecular scattering}

optical thicknesses of the individual atmospheric layers, bm

sca, are dependent on the

molecular column density (i.e. the number of molecules per m2), the gas refractive index, and the depolarisation factor of the molecular mixture, for which the typ- ical values of the specific atmosphere are used [see Hansen & Travis, 1974]. The molecular column density varies as a function of ambient temperature and pres- sure. The model results presented in this thesis were all calculated using a 20 layer model atmosphere, with pressures and molecular mixing ratios dependent on the body that the light was reflected from. The atmospheric layers are assumed to be in

CLOUD GAS GAS GAS HAZE (a) CLOUD GAS GAS GAS HAZE HAZE GAS (b)

Figure 2.4: Basic schematic of the model atmospheric layers: left is for an atmo- sphere with a cloud and one haze layer, right is for an atmosphere with one cloud layer and two haze layers. Gas layers consist of gaseous molecules, the scattering properties of which are described by Rayleigh scattering theory. The cloud and haze layers contain gaseous molecules and larger particles that fall under the Mie scattering regime. Pressure increases with decreasing height, so the clouds are at a higher pressure (lower altitude) than the haze layers(s).

hydrostatic equilibrium. The wavelength region considered depends on the observa- tions, and is specified in each chapter, but is always between 0.35µm and 1.0µm. The measured CH4 absorption cross sections of Karkoschka [1994] were used for the

modelling of gas and ice giants.

The cloud and haze particle properties (except those of the fractal particles used only in Jupiter models) used are calculated via a Mie algorithm, described by de Rooij & van der Stap [1984], with a standard size distribution defined by Hansen & Travis [1974], as follows:

n(r) =Cr(1 3ve↵)/ve↵_e r/ve↵re↵_{, (2.14)}

where C is a normalisation constant, n(r)dr is the number of particles per unit volume with radii between r and r+ dr, re↵ is the e↵ective radius, and ve↵ is the e↵ective variance [see Hansen & Travis, 1974, for the definitions]. The units of re↵ are microns, whilstve↵ is dimensionless.

The single scattering albedo of the mixture of gas molecules, and either cloud or haze particles, is given by

a( ) = b

m

sca( ) +basca( )

and the scattering matrix of the mixture [see Hovenier et al., 2004] is calculated as

Ssca( ) =

bm_sca( )Sm_sca( ) +ba_sca( )Sa_sca( ) bm

sca( ) +basca( )

, (2.16)

where Sm_sca and Sa_sca are the scattering matrices of the molecules and either the cloud or haze particles, respectively. All of the scattering matrices depend on the wavelength ,but are also functions of the scattering angle, ⇥(with⇥= 180 ↵, where↵ is the planetary phase angle).

Model computations are of course tailored specifically to the planet in question, and each chapter will outline the properties used in the modelling, such as the number of atmospheric layers, and the particle properties and cloud/haze layer optical thickness.