Umbra (source code)
= Umbra
An <umbra> is the region from which an opaque intervening object blocks the whole visible disk of an extended source of <light>. In the lunar <umbra>, an observer on <Earth> can see a total <solar eclipse>; the surrounding <penumbra> gives only partial coverage.<Why Do Eclipses Happen? by NASA>[Geometry of total and partial solar eclipses]{ref}
For a spherical source of <radius> $R_s$ larger than an opaque body's <radius> $R_o$, separated by centre distance $d$, the common tangent rays meet on the axis behind the body at
$$
L=\frac{R_od}{R_s-R_o}.
$$
This finite cone explains why a smaller occulting body need not produce totality at every distance. Beyond its tip the source can surround the occulting disk, producing an annular alignment. The <umbra> excludes direct source rays; it need not be perfectly black when scattered or refracted <light> enters the region.
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