Temperature profiles¶

BeAR currently includes the following parametrisations for atmospheric temperature-pressure profiles

Milne’s solution

Guillot temperature profile

Isoprofile

Piecewise polynomials

Cubic b splines

Details on the implementation of the latter three parametrisations are discussed here.

Milne’s solution¶

Milne’s solution is a simple, analytic temperature profile that has been derived for a self-luminous object with a grey atmosphere. In the corresponding forward_model.config of the chosen forward_model.config, this parametrisation is chosen by using the keyword milne:

#Temperature profile
milne

Milne’s profile is given by

\[T^4(\tau) = \frac{3}{4} T_\mathrm{eff}^4 \left(\tau + q(\tau) \right) ,\]

where \(T_\mathrm{eff}\) is the effective temperature of the self-luminous object, \(\tau = \int \kappa(z) \mathrm d z\) the grey optical depth for the wavelength-dependent opacity \(\kappa\) (usually taken to be the Rosseland mean opacity), and \(q\) the Hopf function. In the Eddington approximation, \(q\) would be \(1/3\). Internally, BeAR has a parametrised form of \(q(\tau)\), taken from Mihalas (1979).

The Milne temperature profile parametrisation has two free parameters that have to be listed in the prior config file in the following order:

\(\kappa\), the constant grey opacity in units of \(\mathrm{cm}^2/\mathrm{g}\)

the effective temperature \(T_\mathrm{eff}\)

Guillot profile¶

The analytical profiles by Tristan Guillot (Guillot (2010)) have been derived for irradiated planetary atmospheres. BeAR contains two different implementations of the Guillot profile, one for an incident stellar beam (Eq. 27 from Guillot, 2010) and for isotropic stellar irradiation (Eq. 29 from Guillot, 2010). They are chosen by setting the temperature profile to guillot beam:

#Temperature profile
guillot beam

and guillot iso, respectively:

#Temperature profile
guillot iso

The stellar beam case has five free parameters that have to appear ordered as shown below in the prior config file:

\(\kappa\), the (constant) grey thermal opacity in units of \(\mathrm{cm}^2/\mathrm{g}\)

\(T_\mathrm{irr}\), the temperature of incident stellar radiation (assumed to be black body radiation)

\(T_\mathrm{int}\), the planet’s internal temperature. For self-luminous objects, this would be the effective temperature

\(\gamma\), the ratio of the short-wave to the thermal opacity

\(\mu_*\), the cosine of the zenith angle of the incident stellar beam

When using isotropic stellar insolation, the following five free parameters that have to appear ordered as shown below in the prior config file:

\(\kappa\), the (constant) grey thermal opacity in units of \(\mathrm{cm}^2/\mathrm{g}\)

\(T_\mathrm{irr}\), the temperature of incident stellar radiation (assumed to be black body radiation)

\(T_\mathrm{int}\), the planet’s internal temperature. For self-luminous objects, this would be the effective temperature

\(\gamma\), the ratio of the short-wave to the thermal opacity

\(f\), energy distribution factor. Typical values are \(f=1\) for the sub-stellar point, \(f=1/2\) for the day-side average, and \(f=1/4\) for an average over the whole planetary surface.

Isoprofiles¶

When using an isoprofile, a constant temperature as a function of pressure is set internally. This is the most common temperature profile employed for transmission spectra of exoplanets. In the forward_model.config file it is chosen by using the keyword const:

#Temperature profile
const

The isoprofile parametrisation has a single free parameter: the constant temperature.

Piecewise polynomials¶

If the temperature profile should be described by using the parametrisation with piecewise polynomials, the keyword poly, followed by the number of elements k, and the polynomial degree q need to be added to the forward_model.config file:

#Temperature profile
poly k q

As stated here, the number of free parameters for this parametrisation is \(k q + 1\). The first free parameter in the prior distribution file refers to the bottom temperature. All subsequent \(k q\) parameters are factors \(b_i\), such that the temperature at the control point \(i\) is determined by the one from the previous control point \(i-1\) following \(T_i = T_{i-1} b_i\). The control points are distributed equidistantly in logarithmic pressure space.

The use of the \(b_i\) factors allow some control over the general form of the temperature profile. By not allowing them to exceed unity, for example, temperature inversions can be prevented.

Cubic b splines¶

For a description of the temperature profile by cubic b spline the keyword cubicbspline, followed by the number of control points k need to be added to the forward_model.config file:

#Temperature profile
cubicbspline k

As stated here, the number of free parameters for this parametrisation is equal to the number of control points \(k\) and needs to be at least 5.

The first free parameter in the prior distribution file refers to the bottom temperature. All subsequent \(k-1\) parameters are factors \(b_i\), such that the temperature at the control point \(i\) is determined by the one from the previous control point \(i-1\) following \(T_i = T_{i-1} b_i\). The control points are distributed equidistantly in logarithmic pressure space.