# Background¶

## Perturbed spherical harmonic basis: temperature map¶

The photospheric temperature of an exoplanet, as a function of the planetary latitude \(\theta\) and longitude \(\phi\) can be defined as:

such that \(T_\mathrm{eq} = f T_\mathrm{eff} \sqrt{R_\star/a}\) is the equilibrium temperature for greenhouse factor \(f\), stellar effective temperature \(T_\mathrm{eff}\), and normalized semimajor axis \(a/R_\star\); \(A_B\) is the Bond albedo. The prefactor on the left acts as a constant scaling term for the absolute temperature field, on which the \(h_{ml}\) terms are a perturbation.

The \(h_{m\ell}(\alpha, \omega_\mathrm{drag})\) terms are defined by:

where

is the dimensionless fluid number of Heng & Workman (2014), and is a function of the Reynold’s number \(\mathcal{R}\) and the Prandtl number \(\mathcal{P}\). \(\omega_\mathrm{drag}\) is the dimensionless drag frequency, \(\mu = \cos\theta\), \(\tilde{\mu}=\alpha \mu\), \(H_\ell(\tilde{\mu})\) are the Hermite polynomials:

## Phase curve¶

We can then compute the thermal flux emitted by the planet at any orbital phase \(\xi\), which is normalized from zero at secondary eclipse and \(\pm\pi\) at transit:

given the blackbody function defined as:

where \(T(\theta, \phi)\) is the temperature map described with the perturbed spherical harmonic basis functions in the previous section, and \(\mathcal{F_\lambda}\) is the filter throughput.

The observation that we seek to fit is the infrared phase curve of the exoplanet, typically normalized as a ratio of the thermal flux of the planet normalized by the thermal flux of the star, like so:

where the intensity \(I\) is given by

for a filter bandpass transmittance function \(\mathcal{F}_\lambda\).

## Example temperature fields¶

The first several terms in the spherical harmonic expansion of the temperature map in the \(h_{m\ell}\) basis. Each subplot shows the temperature perturbation (purple to yellow is cold to hot) as a function of latitude and longitude (shown in Mollweide projections such that the substellar longitude is in the center of the plot). The \(m = 0\) terms are always zero. The \(\ell=2\) terms are asymmetric about the equator and therefore do not represent typical GCM results, so we keep all \(\ell=2\) terms fixed to zero in the subsequent fits. These maps were generated with \(\alpha=0.6\) and \(\omega_\mathrm{drag} = 4.5\).

(`Source code`

, `png`

, `hires.png`

, `pdf`

)

Below is the same as above, but this time for \(\alpha=0.9\) and \(\omega_\mathrm{drag} = 1.5\) – note that when the drag is set to a smaller value, the chevron shape becomes more pronounced as a perturbation on the temperature maps with \(\ell \neq 0\).

(`Source code`

, `png`

, `hires.png`

, `pdf`

)