Rotational Broadening Kernel

This repository documents the code and progress of a 10-week project investigating whether rotational broadening in exoplanet atmospheres can be exploited using simple time-dependent models to improve spatial resolution.

The code implements a time-dependent rotational broadening kernel that models how the visible hemisphere of an exoplanet evolves with time as the planet rotates. This accounts for changes in the observed radial velocity of the planet due to its spin, in addition to its bulk orbital motion. Note:

To run the spectrum generation directory one must have the petitRADTRANS package installed. https://petitradtrans.readthedocs.io/en/latest/

This isn't neccessary to run the simulations directory, as the output from spectrum_generation is in the data directory. This is so one can demonstrate the rotational broadening kernel without generating their own spectra.

How it works

For a given spectra over time (in this case over an exoplanets "orbital phase"), it follows a Keplerian (sinusoidal) motion as the bulk velocity of the planet red and blue shifts the spectrum. Here is a simulated spectrum from the data directory of H2O in the planet WASP-121b.

This spectrum is then convolved with a kernel. The kernel accounts for the illumination phase of the planet (not to be confused with orbital phase), i.e. how much of the planet’s surface is visible and illuminated by the star at a given time. In the emission spectrum of the planet, the flux measured is highly dependent on the temperture by Stefan-Boltzmann Law, the flux has a $T^4$ dependence. Since the bright sun facing side of the exoplanet would be much hotter, most of the flux comes from the day-side of the exoplanet. In thermal emission spectroscopy, the measured flux is highly temperature-dependent. Since the star-facing side of the planet is much hotter, the majority of the observed flux originates from the day side.

By shaping the kernel in velocity space, rotational broadening can also be incorporated. As the planet rotates, different regions of the day side contribute Doppler-shifted emission. The kernels therefore evolve with time, as shown below:

This considers all the flux to be coming from the day-side of the planet.

After convolution with the spectrum we get;

At zero orbital phase we recieve no signal as the day side is totally facing the star.

Double Atmosphere Model (Day-Night Model)

A more complex "day-night" model is also implemented by generating two separate kernels and two spectra, each corresponding to different atmospheric species (e.g. H₂O on the day side and CO on the night side). This allows the two hemispheres to be analysed independently.

The results of these models seem to be a breakdown in the Keplerian model of the exoplanet as the change in strength and direction of the rotaional broading breaks the sinusoidal pattern of the planet motion. This can be seen by adding the bulk velocity to the kernels and plotting them directly over time;

While the overall motion still follows a sinusoidal trend, small deviations caused by rotational broadening lead to inconsistencies that make traditional analysis methods, such as $K_p$ - $v_{\text{sys}}$ plots, less effective. Below is an example $K_p$ - $v_{\text{sys}}$ plot produced from the model:

By summing the cross-correlation of the spectrum over time, the maximum radial velocity $K_p$ can be estimated. Performing this separately on both sides of the eclipse and overlaying the results produces the plot above. However, deviations from a pure sinusoid can contaminate this velocity estimate, increasing uncertainty.

With more careful modelling, particularly by dividing the planet into more than just two regions, it may be possible to mitigate these effects and recover more accurate velocity measurements.

Conclusion

This model demonstrates that, with a better understanding of rotational broadening effects, it may be possible to extract additional information about planetary rotation and improve spatial resolution. Ultimately, this could reduce uncertainties in atmospheric species measurements and enhance constraints on exoplanet dynamics.