Careful examination of Polaris is a great way to figure out the shape of the rock we live on. All one needs to do is compare empirical measurement, the flat earth model, and the globe earth model. Polaris has long been used as a way to determine one's latitude. All you need to do is measure the angle between your horizon and a line from you to Polaris (hereby refereed to as Polaris's

*altitude*). The correlation is simple. Your latitude is equal to the measured altitude of Polaris. Another way to look at this is that Polaris's altitude is a

*linear* function of the distance to the north pole. Below, I provide an excel sheet I made that compares the predicted altitude as a function of distance to the north pole for both models. Here's some pictures to wet your appetite.

And here's the link:

https://docs.google.com/spreadsheets/d/1pf5tYh6II-T_kYaXvY5m7jrZX4di38JtrRn7KlkupGs/edit?usp=sharingThe globe model agrees exactly with reality. The altitude is equal to the latitude and it is a linear function of distance to the north pole. The flat earth model does not. We can see that when we are less than about 5,000,000 m from the north pole, the flat model

*underestimates* altitude. After 5,000,000 m, the flat model greatly

*overestimates* altitude. This calculation was done on the assumption that light travels in straight lines (I will call it the "straight line model"). So, we are left with only one way to justify the flat earth model in this case.

The incident light from Polaris must be somehow bent on its journey to the observer.

If it was not bent, then the light would be coming in at the angle predicted by my straight line math, and it would not match with reality. Now I know refraction is a thing. In fact, the globe earth relies on it. So in order to be fair to the flat earth, we will take this to its logical conclusion. In order to simplify the complex refraction that happens in our atmosphere, I'm only going to use Snell's Law here. In reality, refraction is a gradual process through the atmosphere. All we need though, is a qualitative answer, so it does not matter to us if the light is bent gradually or all at once.

I'm going to examine the first part of the graph, less than 5,000,000 m. Before 5,000,000 m, the light must be getting bent

*downward*. If refraction is truly responsible for correcting the straight line, flat earth model, one of two scenarios is happening. Either the air gets less dense as the light gets closer to the observer, or the air gets more dense as the light descends through the atmosphere. The former is unlikely to happen reliably and for every observer on earth, and should be discarded. The later does happen, and so seems like a reasonable explanation for now.

Now let's look at what happens after 5,000,000 m. The light from Polaris must be getting bent

*upward*. Again, two things could cause this. Either the air gets more dense as the light gets closer to the observer, or the air gets less dense as the light descends through the atmosphere. The former will, again, be discarded because it could not happen for every observer on every measurement. The later is not only contradictory with our conclusion for <5,000,000 m, it is contradictory with empirical data. The atmosphere gets thicker as you go down, not thinner.

The problem is apparent. The atmosphere must have a different profile, depending on the observers position. Sometimes it must have a profile that is opposite of reality. It must necessarily flip flop at 5,000,000 m. The effect must get

**quite** powerful at large distances. It makes no logical sense. Meanwhile, the globe models it perfectly.

**Thanks for reading. Sorry about all the hand drawn stuff**