Distance from Star Winning Smile and Planet Winning Smile

Roblox cosmologists found out today the orbit of Planet Winning Smile. Here is how they calculated it:

They noted that the temperature of Planet Winning Smile was 24.6 kelvins at its farthest descent and 27.1 kelvins at its closest descent.

They then noted Star Winning Smile had a luminosity likened to that of the Roblox Sun, namely 3.848×10²⁶ watts.

From this, they were able to deduce a function relating solar irradiance to distance.

$${\displaystyle I=\frac{\left(3.848\cdot10^{26}\right)}{4\pi r^{2}}}$$Next, they tried find the solar irradiance required for a planet to have that 24.6 kelvins and 27.1 kelvins as their coldest and hottest temperatures. For this, they turned to a formula:

$${\displaystyle T=\sqrt[4]{\frac{K_{s}\left(1-albedo\right)}{4\sigma}}}$$Isolating the solar irradiance gave them this:

$${\displaystyle K_{s}=\frac{400T^{4}\sigma}{71}}$$Plugging it back to (1) and solving for the orbital radius gives:

$${\displaystyle r=\frac{2331370000000}{T^{2}\sqrt{\sigma}}}$$At this point, this problem reduces to a plug and chug. They found out that the planet lies at farthest 1.61789 × 10^13 meters and at nearest 1.33316 × 10^13 meters. This means the semimajor axis is 1.47552 × 10^13 meters.