Shadow Cones

By John Reed

Solar eclipses, lunar eclipses and occultations all have one thing in common: shadows shaped like cones.  These cones are sometimes short and fat, sometimes long and thin.  Some stretch almost to infinity, approaching cylinders.

The reason these shadows are this particular shape has its roots in geometry.  Light travels in straight lines (ignoring relativity, which in most eclipses is negligible [see the November, 1983 Sky & Telescope discussing this complex phenomena]) and so it stands to reason that these shadows are such a simple shape.

Lets take a look at one of the most popular shadow-cones: the one cast by the moon.  The moon casts a shadow which starts at its terminator (the line of light and dark) and ends in a point about a quarter of a million miles out directly away from the Sun.  The reason this shadow comes to a point is because the sun is larger than the moon.

Perform the following thought experiment: imagine that the sun were a mathematical point.  Light would radiate outward in lines passing through that point and no other.  Any solid object, such as the moon, would block all the lines of light between itself and that point, casting a shadow that expands forever, getting wider and wider (patience, reader, this really has something to do with shadows that narrow).  Now imagine that the sun were exactly the same size as the moon.  Now the lines of light would radiate from points over the entire surface of the small sun.  The moon would now block some of the lines of light from each point on the hemisphere facing the moon.  If this is hard to imagine try thinking of the point of light sweeping out circles the same size as the moon; the shadow would sweep back and fourth with this motion and there would only be one volume where light would never reach: a cylinder directly behind the moon.

Now that you are totally confused by rays, lines and cylinders, let me continue.  If the sun were much bigger than the moon (as ,of course, it is) the rays would now be sneaking past the moon on all sides.  Again think of the sweeping point of light, this time making huge circles.  Its light would now actually get behind the moon at the extremes of its motion.  The only volume the light would never reach would be a cone pointing away from the center of this circular motion.

So what does all this junk mean?  It means that the moon's shadow can be delineated by drawing lines from the edge of the sun past the edge of the moon.  Remember, light travels in lines.

What is fascinating is the coincidence that the moon's shadow-cone comes to a point at the same distance from the moon as is the earth's surface.  This means that on rare instances when the fantastic motions of the three bodies happen to bring the moon directly in between the earth and the sun, this pointy shadow just touches the curved surface of our planet.  Sometimes.  On occasion the moon is slightly too far (for reasons too complex to go into here) and the shadow doesn’t quite touch.  This is what is happening in May.  On other times the moon is closer and the shadow-cone is still miles wide at the surface.  This is a total eclipse.

What about lunar eclipses?  They are caused by the moon passing through the earth's shadow-cone, which is a huge thing in comparison to the moon's.  The Earth's shadow-cone ends far beyond the orbit of the moon, so it is still quite fat where the moon crosses it.  That’s why lunar eclipses last so long.

Occultations are really strange.  Say an asteroid passes directly between us and a prominent star.  The star casts a shadow-cone off the tiny asteroid and one would think that it would never reach the earth from another part of the solar system.  This turns out not to be true as the star is so much further away than the asteroid is, the shadow-cone is not even perceptibly smaller than the asteroid when it reaches the Earth.  The cone actually ends far beyond the reaches of the solar system!