*A model of a planetary environment doesn't spring forth in all of its detail. Typically we start with the simplest model that captures the essential physics, but which also leaves out important details. Sometimes the description of such a model even fits on the back of an envelope! We then build in the complexity piece by piece. This is a process that PhD student Giang has been pursuing over the past couple of years as his models of K2-141b becomes ever more sophisticated. At each stage, we learn something new as we proceed from a solution accurate to a particular order of magnitude, to a 10% level solution to a 1% level solution. There is benefit in the complexity - but it's important not to outrun the data by too much. If we make a prediction or add a minor process that cannot be verified through the data, we run the risk of inventing stories about these worlds that are mere delusions.*

*By Giang Nguyen*

In my previous post, I showed what happened when I introduce UV radiation absorption to K2-141b’s atmosphere. The results from the model went bizarre as the atmosphere kept heating up to essentially become plasma. Although numerically sound within our mathematical construct, this ultra-hot atmosphere simply isn’t realistic as that would make the atmosphere on the planet even hotter than its star.

As I suspected, there was an issue with how I dealt with radiative cooling. The original way for the atmosphere to cool would be exclusively through infrared emissions. Although most of the energy does radiate in the infrared wavelengths, the emissivity of silicon monoxide in that spectral range is very small compared to UV light. Therefore, there is some UV emission that is unaccounted for that would significantly cools the atmosphere.

The solution to this problem is to separately calculate the blackbody radiation of the atmosphere in both infrared and UV. This is done by integrating the Planck function over the desired wavelength range and multiply it with the corresponding emissivity. Here’s the thing with blackbody radiation, especially for hot temperatures of thousands of kelvins. Most of the radiance comes from a very small sliver of wavelengths, and it is pretty much negligible in comparison at every other wavelength. Therefore, when you have low spectral resolution, the estimate of the radiance becomes very inaccurate once you do your integration.

My next step was to do the Planck integration separately solely as a function of temperature with adequate spectral resolution and then to fit that integration to a polynomial. As the integration process now becomes a single line of calculations instead of a bunch of for loops, we’re back to our old speedy model. However, we are at the mercy of our fit coefficients and it seems that our temperature range is too large for a polynomial fit to be accurate; note that our temperature can range from 0 – 3000 K.

All hope seemed to be lost. I was going to have to run the slow model which I estimate will take weeks to pump out a solution, which might not even be correct solution. Thankfully, some scientists in the 1970s ran into the same problem and were able to solve it themselves. When you integrate the Planck function by parts, you end up with an infinite sum (a little bit of math identities is needed here as well). Computing this infinite sum is much faster than the classic way as this sum converges much faster. Finally, with the Planck finite integral taken care of, we can deal with radiative cooling.

As expected, UV emissions capped the temperature of the atmosphere – but it was still hot. The temperature hovers around 2900K across the dayside almost uniformly. Because UV emission only becomes significant when the atmosphere is hot, it never forces the temperature to drop further at low temperatures. When UV absorption and emission cancel each other out at a specific temperature, a very stable sort of radiative balance occurs. This turns out to be important as the atmosphere becomes too thin for IR radiation to take effect.

A warm SiO atmosphere is expected, but for it to be so horizontally consistent and warmer than the surface is a surprise. A welcoming surprise. For emission spectra, a warmer atmosphere means a brighter signal. Using SiO spectral features, we could ultimately see K2-141b’s atmosphere instead of the ground beneath it. Also, the scale height is thicker, even near the terminator (where on the planet you would see the star on the horizon). This means that during a transit, the planet’s atmosphere is optically thick enough to absorb the star’s light that travels through that atmosphere on the way to Earth. With supersonic winds, this might induce an observable Doppler shift when measuring K2-141b’s transmission spectra.

Ultimately, when considering UV absorption and emission, the atmosphere on K2-141b is easier to detect, for either low-resolution and high-resolution spectral instruments. This is very good news as K2-141b is slotted for observation time with the James Webb Space Telescope (JWST). Along with possible future observations from ground-based telescopes, we may definitively detect and characterize K2-141b’s atmosphere - a first for terrestrial exoplanets.

This concludes my update for my current research project. Using a convenient numerical method to evaluate definite Planck integrals, we solved the problem of dealing with K2-141b’s atmospheric radiative cooling. The resultant atmosphere with the full radiative transfer is almost uniformly hot across the planet’s dayside. This suggests that K2-141b’s atmosphere is a lot easier to detect than anticipated. This is exciting as K2-141b is a high valued target for observation, and it might be the first terrestrial exoplanet where we have observed an atmosphere. Although a small step, it is still a step towards finding habitable worlds and life beyond the solar system.