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When speaking of eclipses it is important to first discuss the two possible types. One such type is a lunar eclipse. This occurs when the moon passes through the shadow of the earth created by the sun. The second type of eclipse is called a solar eclipse. This occurs when the earth passes through the moons shadow.
In the case of a lunar eclipse three types are possible: A partial eclipse, a total eclipse, and a penumbral eclipse. The same holds true for a solar eclipse, however the third type of eclipse is not called a penumbral eclipse but instead a annular eclipse.
In order to understand the differences, one first must look at some basic terminology in regards to eclipses. As seen here in this image (Taken from here) one can observe the umbra, the inner area of total darkness where no sun is observed. In this case the umbra is generated by earth's shadow. The penumbra is also important, and is the grey outer area where a portion, but not all, of the sun's light is blocked. In terms of a lunar eclipse, a total lunar eclipse occurs when the entire moon is located inside the umbra of the earth. A partial lunar eclipse occurs when only a small part of the moon is instead located inside the earth's umbra.
A penumbral eclipse occurs when the moon's path takes it through the penumbra of the earth, but it does not cross into the umbra. Solar eclipses are slightly different. Because the earth is much larger than the moon, the moon's umbra can only cover a small portion of the earth. The same holds true for the moon's penumbra which is also much smaller than the diameter of earth. Because of this, the complete darkness of a total solar eclipse falls within a very small area of earth. The same holds true for a partial solar eclipse. In this case the observer is inside the moon's penumbra and can still see a small portion of the sun's light. Interestingly, an annular eclipse occurs when the moon crosses between the earth and sun while at its furthest point from the earth. In this case the moon is not able to create an umbra and instead an observer in the zone of the umbra would witness the sun as a giant ring. Hope this helps!
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