According to the song, Rudolph’s nose is shiny, which means it reflects rather than emits light. Useless for navigating fog.
To which I responded
Nose also glows & bright. Since it’s red we could determine temperature if a thermal source & estimate Rudolph’s calorie needs
If Rudolph’s nose is a thermal source it will follow the Stefan-Boltzmann power law, which tells us the radiated power depends on the fourth power of temperature. Something red-hot will have a temperature of about 1000 K. Now this is an estimate and since it’s raised to the fourth power, will give us a large error bar on our answer. But let’s go with that because I don’t have a calculator handy. For the emitted power we multiply by the area, a few square centimeters (converted to square meters) and Stefan’s constant. Assuming I did the math correctly, we get about 10 Watts. The temperature should not be as large as 2000 K, which would give us and answer 16 times as large. (I am ignoring the “power absorbed” term in the equation, because at these temperatures it’s going to be small — 300K or less)
There’s also the emissivity. The nose is shiny, meaning the emissivity is not close to 1. So perhaps we double our guesstimate. Tens of Watts, maybe as large as 100 Watts as a probable value.
A thermal source has a maximum luminous efficacy of 95 lumens/Watt, at a temperature of around 6600 K but actual bulb filaments that give us white(ish) light are a lot closer to 10 lumens/Watt. So the nose probably emits around 1000 lumens at best — this is not even as bright as a traditional 100W light bulb, but is around what low-beam halogen headlights emit. However, those have reflectors on them to direct most of the light into a beam. Rudolph’s nose emits into a much larger area.
So perhaps the nose is not a thermal source (unless it’s much larger than I estimated) — the radiation is not because it is hot. We could check this if we knew the spectrum of the light being emitted. Perhaps it is some other type — does Rudolph have an LED nose?