Q:  Do airplanes have the same size shadow, no matter how far they are from the earth, and, if so, why?    (Bob, Surprise, Arizona) 

The left image shows a plane's distinct shadow cast on a runway immediately after take off when the plane was close to the ground.  Frankfurt International Airport.  The right image captures a blurred shadow of a plane cast over German terrain when the plane was high above the ground.  Photos courtesy of googlesightseeing.com, NASA and Google Maps. 

A: No, an airplane doesn't cast the same size shadow, no matter how high it flies.  Before we get into the reason, though, let's review what a shadow is. 

I step outside into the hot, noonday New Mexico sun to check mail.  I stroll to the mailbox, and hold my hand over its top, shiny surface.  My hand blocks (actually, absorbs) the sun's rays, and the absence of light casts a shadow beneath my hand.  Since my hand is close to the mailbox, its shadow is sharp, distinct and about the same size as my hand.   Now I raise my hand, and the shadow blurs somewhat, but still appears roughly the same size. 

A solar eclipse showing the moon's shadow on Earth, including the penumbra (large gray oval) and the umbra (black dot).  Drawing (not-to-scale) is courtesy of MrEclipse.com, and modified by the author.

My hand's shadow, like all Sun shadows, has two parts:  the dark part on the inside (called the umbra) and the lighter part (the penumbra) around the edges of the shadow.  It has two parts because the Sun is not a point source but rather a disk. 

Let's shift our gaze into space.  The drawing illustrates the shadow the Moon casts on earth during a solar eclipse, and shows how the Sun's disk creates the two parts (umbra and penumbra) of the shadow.  Sun rays from both sides of the Sun cast a Moon shadow on Earth.  Where the two shadows overlap is the darkest shadow (the black dot), and where Earthlings see a total eclipse.  The gray shadow around the black dot is the lighter shadow cast by the Moon's blocking rays from each side of the Sun — but we don't get two separate lighter Moon shadows, the way the drawing implies.  Instead, an infinite number of shadows cast by blocking an infinite number of rays from all around the Sun's rim and face creates a single composite lighter-gray circular penumbra shadow shown in the figure. 

Let's consider sizes now.  The darkest shadow (about 170 miles across) on Earth's surface has a diameter about a tenth that of the Moon.  The diameter of the lighter blurred shadow is about twice the Moon's diameter.  So the entire shadow (umbra and penumbra) is twice the Moon's diameter. 

Also, the diagram shows how the size of the umbra and penumbra depend on how far earth is from the Moon, and, therefore, where Earth's surface intersects the shadows.  Moving Earth closer makes the umbra bigger and the penumbra smaller.  Moreover, if we could jam the Moon right against earth, its shadow would consist mostly of the umbra, and would be about the same size as the Moon.  Only a small penumbra around the outside of the umbra would then cast on Earth, enlarging its shadow to slightly more than Moon size. 

The shadow of the Moon jammed against earth is akin to an airplane's shadow soon after takeoff; see photo above, left image.  The shadow is roughly the same size as the airplane (about 200 feet (60 m) long).  The farther above Earth the airplane climbs, the more the penumbra fuzzes and enlarges the shadow.  However, even if an airplane climbed to 40,000 feet (12 000 m), that distance is still miniscule compared to the distance of the Sun from Earth, so the penumbra enlargement is, comparatively, a small effect.  Yes, the shadow gets larger, the higher the plane — but, if you could measure the shadow on the ground as the plane climbs, the shadow would still measure something close to 200 feet, even if the plane climbed to its maximum altitude. 

Thus, the plane's ground shadow gets fuzzier and larger, the higher the plane flies, but not much larger.

Further Reading:

Solar eclipse movie "taken" from the moon's perspective showing the passage of the moon's shadow over Africa, NASA exploratorium

Solar eclipses, Royal Observatory Greenwich

Conceptual physics by Paul G. Hewitt

Science from your airplane window by Elizabeth A. Wood

Comment on an earlier dullest-number question (Martin, San Diego, California)

The arrow points to the smallest dull number (5).  Or is it?  Read on.  Image courtesy of PrimeTime, Argentina, modified by author.

It seems to me that there can be no dull numbers. Here's my logic:

If you actually find a "smallest" dull number, then that property makes the number interesting, thus removing it from the list of dull numbers.

Repeat until the list is empty.

As such, no numbers can be dull.

Further Reading

The Berry Paradox by G.J. Chaitin, IBM Research

(Answered August 15, 2006)