David Grant (smartly not on social media) did an amazing job with this work. I have been dreaming of doing transmission mapping since 2016, but I never got past the hard maths & coding stage. Enter David! He came in with the idea himself & quickly started on how to make it work🙏

This work involved a LOT of maths & new methods David Grant found to make computing Transmission strings fast & precise taking into account other aspects of transits such as limb-darkening. The paper contains 42 equations, resulting in a behemoth table of symbols in the appendix

Harmonica is also incredibly precise! Using a 5-term Transmission string for a typical
@NASAWebb
transit lightcurve (yes we already have typical #JWST lightcurves) & assuming ~1% deviation from a circle Harmonica is orders of magnitude more precise than expected JWST noise floor

Harmonica can calculate transmission strings in fractions of a second and can infer the shape of an occulting object directly from the transit lightcurve in less than one second for any dataset with less than 100,000 data points 🤯 (BTW that is insanely fast 🏎️)

The deviations from a circle can be plotted - here we show the 3000 different transmission strings computed to fit our #JWST-like lightcurve. The degree of change as a function of the angle around the object can tell you about the changes in a feature measured at that wavelength

We tested the idea of Transmission Strings by considering a planet measured with a
@NASAWebb
like transit light curve (465 data points🧐) injecting a shape w/ East/West differences & equator/pole differences. We fit for a 5-parameter string using Harmonica & recover the shape!

For this work David Grant developed a Transmission strings open source package called Harmonica https://harmonica.readthedocs.io/en/latest/
From this you can compute transit lightcurves for irregularly shaped occulter (not just planets!)

Each of the different Fourier series functions result in unique changes to the shape of the planet as a deviation from a circle. In a transit lightcurve these can be measured on ingress as the planet enters the stars area and egress as it leaves

The shape is built up based on a Fourier series, which is the sum of harmonically related sinusoidal functions (or harmonics). As you increase the number of harmonics you can get increasingly more complex shapes