User:Michael Retriever/Earth radius mean

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A radius mean for the Earth which isn't used but is interesting to know is the one that you would get distributing all the area of the Earth's ellipsoid over an adapted sphere.


Given the equatorial radius


a = 6,378,137 \ m \,


and the polar radius


b = 6,356,752 \ m \,


the Earth's great ellipse eccentricity would be


e = \sqrt{1 - \frac{b^2}{a^2}} = 0.08181979099211440805067099056516 \,


Being the formula for the longitude of an ellipse


l = 2\pi a z \,


where

z = \left[{1 - \left({1\over 2}\right)^2e^2 - \left({1\cdot 3\over 2\cdot 4}\right)^2{e^4\over 3} - \left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{e^6\over5} - \dots}\right]\!\,


we can say that


2 a\cdot 2 a z \pi = 2 E_r\cdot 2E_r\pi \,


thus


a \sqrt {z} \ = E_r \,


After calculating the decreasing function z with a precision of 8 diminishing terms, we find that


E_r = 6,372.79075377 \ km \,


Note that the result is extremely similar to the quadratic mean.