Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."
In her Parade Magazine column of April 24, 2005, Marilyn claims that an airplane flying at 40,000 feet casts a shadow on the ground that is approximately the same size as the airplane.
Sorry to tell you this, but this answer is incorrect. You seem to only have considered prenumbral blur in the shadow effect. Airplanes fly by creating air pressure differentials above and below the wings, and they also create pressure gradients around the fuselage. The pressure differential around the fuselage changes the index of refraction of the air flowing close to the plane. Now while this is a small change it has a very significant effect. And since you mentioned a shadow, at 40,000 feet, we only have the sun or the moon as a significant light source to create a potential shadow.
So let's take the easiest case first, the sun. The light from the sun will be refracted by the slight change (it is an increase) in the index of refraction of the denser air flow and, a thin lens effect is created. What this means is that the sunlight that one would normally think would pass straight by the fuselage does not. It is refracted and creates a very interesting optical effect. The image that is formed on any surface below will depend on the distance that the plane is from that surface. With a typical plane length on the order of 200 feet (i.e. commercial airliner) and an image plane 40,000 feet away, the image created will be much larger than the airplane. The image will be rather faint in intensity but will in general be circular. Further since sunlight is composed of the full visual wavelength spectrum ( except for minor dark lines due to elements in the sun) the image will have features similar to a rainbow. This is because the index of refraction is wavelength dependent.
Now this discussion may seem rather strange to you, but as a physicist, I love to sit by the window in airplanes and observe the cloud formations and other physical, meteorological and atmospheric effects. One day on a flight I noticed a very bright spot on a layer of clouds below the plane I was flying in. It moved with the plane. It seemed strange at first and I thought about it. I also noticed how it changed as the height of the cloud layer changed with respect to the plane. Then I figured it out. On many flights after that I have observed the same effect. It can be quite beautiful. And finally when the clouds are about 20,000 feet below the plane, one can see a huge circular refaction pattern.
With respect to moon light the same effect will occur, but be much dimmer.
Now this discussion may seem a little heavy, but if you ask your average reader if they have ever seen a mirage, they will probably answer yes. Mirages are caused by the local heating of the air just above a large flat surface. The heating changes (decreases) the index of refraction of the heated air and light reflection occurs at the boundary of the hot air and the cooler air.
Airplanes cannot fly without air, and everyting above follows naturally.
You may be familiar with a parhelion (also called sun dogs), the refraction of sun light by upper atmospheric clouds that contain ice crystals. They are most often seen in the autumn. Last fall while sailing off the coast of Georgia (about 60 miles offshore and away from the glow of land, I was lucky enough to see a faint moon induced parhelion - it was exactly as one would expect, sunlight reflected from the moon giving rise to the same effect, but only dimmer. A raindow at night! It is rarer than the green flash!
I agree that Marilyn is wrong concerning the size of the shadow of a jetliner. However, if she answered your letter, she probably said that it was a geometry problem and not a physics (i.e., refraction, etc.) problem.
She is still wrong, though. Below is my explanation of why she is wrong, as excerpted from an email to a friend dated 4/25/05.
Since the sun is not a point-source (as viewed from earth), all shadows cast will consist of an umbra and a penumbra. The further the subject is from the object upon which it casts its shadow (i.e., the greater the elevation of the plane above the ground) the smaller the umbra becomes and the larger the penumbra becomes. At a certain distance the umbra disappears, as the plane is too high to completely blot out the sun as viewed from the ground.
As the plane goes higher and the penumbra becomes larger, it becomes less "intense" as well. Surely at 40,000 feet, when the plane appears as not much more than a spec against the sun, the shadow will be too diffuse to be detected by the human eye. The shadow will be nearly circular and will subtend an angle, as viewed from the plane, approximately equal to that of the sun itself.
I'm sure Marilyn will claim her answer is correct if the shadow cast is that from a star other than the sun, or from a nearly point source such as a planet. After all, the questioner did not specify what light source the shadow was relative to.
So, while the physical effects you mention are at odds with Marilyn's answer, I believe my rebuttal to her answer is a much better one in that she can't claim some implicit conditions that would make her answer technically correct, unless she wants to appear to have written a "trick answer".
Perhaps others have already emailed you with my analysis or a similar one since it fairly straight forward and obvious, but I would appreciate your physicist's analysis of my reasoning. It's not like I have never been wrong before, and I usually am grateful when others point out my mistakes (after my initial embarrassment).