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I am sure this varies given the distance from moon to earth varies, but a range would be sufficient. I am trying to explain to a flat earther how there is not a lunar eclipse every full moon.

My approach is to show that the 5 percent inclination of the moon's orbit to the ecliptic will result in a large enough gap for the umbra to miss the moon when projected at a great distance.

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The angle of the cone is due to the angular extent of the light source (the sun). This is approximately 0.53 degrees when near the orbit of the earth.

So the length of this cone is: $$ \tan\left(\frac{\theta}{2}\right) = \frac{R_{earth}}{L}$$ $$L = \frac{R_{earth}}{\tan\left(\frac{\theta}{2}\right)} $$ $$L = 1.4\times 10^9 \text{m}$$

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The diameter of the umbra varies linearly as your displacement from the vertex. The moon at perigee is about $0.36\times 10^9\text{m}$ from earth or about $1.0\times 10^9\text{m}$ from the vertex.

Therefore the diameter of the umbra at that distance is approximately $\frac{1.0}{1.4} D_{earth}$ or about $9100\text{km}$. Smaller than that when the moon is further away.

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  • $\begingroup$ Thank you. Help me understand if I am thinking of this correctly in applying this knowledge. The moon is declined 5 degrees from the ecliptic. The approximate circumference of the orbital track of the moon is 2.4 million kms. 5 degrees represents 1.38% of of the orbital distance, which is about 33,313 kms. 33,313 miles would be the maximum "gap" that the earth shadow might pass through on a given full moon. Thus, it is very easy to see why 9,100 diameter doesn't always intersect with the moon. Is that accurate? $\endgroup$ Oct 22, 2018 at 22:28

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