Explore the degeneracy of Amplitude ratio, f_spot, T_spot, and T_ambient

[gully]

All we measure for a given star is an amplitude ratio between TESS and K2. There is a subtle "self dilution effect" that we allude to in the proposal. Most practitioners have not contemplated this effect, so there is not a great reference for it. This paper can help you think about the degeneracies though. Here is my attempt. Imagine three stars:

The simple spot blocks some amount of bright yellow light compared to the spot-free scenario, let's say 2% of the area with an 0.5 contrast, so 1% of the flux is lost in the simple spot case compared to the spot-free case.

But now consider the star with a major polar cap of spots that is seen at all rotational phases. The simple spot tends to block the brighter-on-average flux. So that same 2% for the disk area is blocking say 2.8% of the yellow stuff.

So mapping between amplitude of modulation ratio, f_spot, T_spot, and T_ambient is subtle and partially degenerate.

There is a way to quantify degeneracy: Simply brute-force compute a grid cross all permutations of f_spot and T_spot that are consistent with:

1. The TESS-to-K2 amplitude ratio
2. The T_eff reported in the catalog we are using (set T_eff = T_ambient for simplicity)

For each star we will now have a heatmap of $f_{spot}$ and $T_{spot}$. It is unweildy to work with these heat maps for many stars! So we have to combine them in some way. That will be another task. For now let's get familiar with this idea of degeneracy, and think about how we may scale out this analysis.