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How tall would a tower at the South Pole have to be for the top to get year-around sunshine?

No doubt the height would vary each year since the Tropic of Cancer changes latitude slightly. Perhaps assume 23.5° N for calculation's sake.

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up vote 21 down vote accepted

assuming spherical earth with radius $R$ and axis tilt $\mu$, from geometry reasoning we get: $$ \frac{R}{R+h}=\cos \mu \Rightarrow R\left( \frac{1}{\cos \mu} - 1 \right)=h $$ earth

so your tower height should be $h=((1/\cos(23.5)-1)\cdot 6371 \text{ km} = 576.2 \text{ km}$; greetings to ISS

for ellipsoid-Earth the height is bit different, but the above is a fine estimate

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5  
+1 for drawing the south pole at the top of the image contrary to arbitrary convention ;-) –  joriki Aug 17 '11 at 7:05
    
"greetings to ISS" - I missed this at first reading... :D –  J. M. Aug 17 '11 at 7:15
    
lol, yeah, I probably put myself too much in the Antarctic... –  troyaner Aug 17 '11 at 7:15

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