#### Silent Service

• 28
##### Polaris, Latitude, and Perspective: The Math Doesn't Add Up
« on: April 25, 2017, 07:39:46 PM »
Hey what is up guys so recently I found out about this whole “flat earthers” thing.  It’s definitely intrigued me since it’s a wildly different view than the one I’ve held and I like talking to people who have different opinions.  I know I don’t have all the answers and the only way I can become more knowledgeable is to share information and debate others.  In this post I will try to be as respectful as possible and I hope you will do the same because we can’t have any meaningful conversation if we are throwing stones at each other.  Excuse my poor drawing skills and let me know if anything doesn't make sense.

So, on to the meat of the topic.  My biggest issue with pretty much all the arguments or videos I’ve seen for flat earth is that none of them use math or repeatable experiments.  There’s a lot of hand waving to explain things that don’t fit into their world view but when you crunch the numbers that hand waving winds up creating more problems than it solves.

There are many, many examples of this that I can go into but I think my favorite has to be an analysis of Polaris, Latitude, and Perspective and how all of them relate together.  The math that’s needed for this is pretty basic, mainly geometry and trigonometry that we all got in high school (or earlier.)

We all know that the angle of Polaris above the horizon is equal to the latitude of the observer.  Sailors and soldiers have used this principle for hundreds of years in order to navigate on the high seas or in unknown terrain.  This works great in a round earth model but there are some serious math issues when you trying to explain this with a flat earth model.  When faced with these issues the most common response I hear is “perspective” but that actually just creates more problems than it solves.

Let’s start with a flat earth where there is no “perspective” and calculate where we would expect Polaris to be based on our Latitude.  The assumptions for these calculations are as follows:
• The equator is at ½ the radius of the entire earth.  This means that every 10 degrees of latitude correspond to 1/18 of the radius from the north pole to the ice wall (or every degree of latitude corresponds to 1/180 of the radius of the earth.)
• The star dome has a radius at least equal to the earth’s radius.  This is required because if the radius of the star dome was less than the radius of the earth then there would be a point in the earth where you could pass through the star dome.  I know some flat earthers think that the stars are a plane instead of a dome and I’ll address that possibility later.

Here are my variables and the abbreviation I will be using for each:
R = Radius of the Earth (miles)
D = Distance from Observer to North Pole (miles)
L = Latitude (degrees)
S = Radius of Sky Dome (miles)
Theta = Expected Angle (degrees)

Basically what we’re going to do is set up a right triangle and solve for Theta.  One side of the triangle is S and the other side of the triangle is D.  D is calculated as R * (90 – L) / 180.  Now, the angle we would expect to observe Polaris (Theta) in a flat earth model would be ArcTangent(S/D).  S has a minimum distance of R and if you set S = R you get the minimum amount of error possible between expected angle and actual angle.

Picture link if it doesn't show: http://imgur.com/RgM1hEA

Latitude…..Expected Angle….Error Between Expected and Actual
90…………….90……………………….0
80…………….87……………………….7
70…………….84……………………….14
60…………….81……………………….21
50…………….77……………………….27
40…………….74……………………….34
30…………….72……………………….42
20…………….69……………………….49
10…………….66……………………….55
0………………63……………………….63

You can run the math for any value of S but this is literally the best you can do with a dome sky.  If you want to say that the stars are in a plane then you can calculate the height that the sky has to be in order for expected angle to equal actual angle.  This is calculated as D * Tangent(L).  If you do this you’ll see that the sky needs to be at a different height for every latitude.

Latitude…Height stars must be at for actual to match expected
90…………..Any height
80…………..0.315R
70…………..0.305R
60…………..0.289R
50…………..0.265R
40…………..0.233R
30…………..0.192R
20…………..0.142R
10…………..0.078R
0…………….0

So it seems like a planar sky is impossible since Polaris (and the rest of the stars) obviously can’t be in multiple places at once for different observers.  Let’s go back to the dome sky and try to see if we can resolve the discrepancies with Perspective.

As I understand it, perspective in this context causes you to see less of the night sky due to the angle your height of eye makes with the Land Horizon.  Here are the variables for these parts of the equations:

HE = Height of Eye (feet)
LH = Distance to Land Horizon (feet)
Lambda = Angle made a straight line from your HOE to the LH and the earth (the “perspective angle)
SH = Distance Between Land Horizon and Sky Horizon (feet)
TH = Total Horizon (LH + SH)
SC = Amount of sky that is “cut off” due to perspective
Phi = Angle of sky that is “cut off” due to perspective

Picture link if it doesn't show: http://imgur.com/vmtWGsN

So in order for perspective to explain the discrepancy between expected angle and actual angle to Polaris in the flat earth model, Phi needs to be equal in magnitude to the error between expected and actual angle.

Lambda = ArcTangent(HE/LH)

The triangles formed by the “perspective” angle are similar so we know that we can use Lambda to also find the relationship between SH and SC.  The equation for SC is SC = HE * SH / LH

So from here we can find Phi since that angle is made up of a triangle where one side is TH and the second side is SC – HE.

Phi = ArcTangent[ (SC – HE) / TH ]

What we can find is that regardless of whatever height of eye, land horizon, or sky horizon we put we will never get a Phi greater than 1 degree.  This means that at most perspective can account for a 1 degree shift in the night sky.  This is with me trying to create the most optimal conditions possible for the flat earth model.  I cannot see any way to make the data fit.

If you believe in a flat earth please help me out and provide me with some numbers that actually make the math work.  If I forgot to account for a variable let me know what the mathematical equations are to explain that variable.  If you are unable to solve these equations for your flat earth model then that is very strong evidence that the earth is not flat.  I've included my math so you can manipulate the variables and try to find something that works.  Thanks for your help.