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The concept of penumbras and umbras also applies when we have an eclipse and two shadows are cast ~ the umbra shadow, the dark center of the eclipse shadow, grows smaller as it goes away from the sun, and the penumbra shadow grows larger as it goes away from the sun. (There's also antumbra, which is the lighter part of the shadow that begins where the umbra tapers to a point.) During some total solar eclipses, the umbra's path width reaches over 1000 km/ 600 mi at the poles!


This situation with an occluder shows an example where using Lambert's formula becomes difficult, which gives a motivation for the Monte Carlo technique in this lecture.