NavList:

David Pike, you wrote:
"This is getting more complicated than I imagined."

Well, we're playing around with it in different directions. In principle, I think we all agree it's really very simple! 

You also wrote:
"The problem for an Atlantic crossing was that it might not be possible to correct the altimeter sub scale for changes in surface pressure when out of contact with watch ships, so there was a chance of the airship flying into the sea at night or in cloud or fog. Therefore, opportunities had to be taken to update the sub-scale if a shadow was visible."

Sure. Makes sense. Of course if the barometer fallls significantly during the night... well, tough luck! Do you know if this technique was actually used to any significant extent? It clearly would work in principle (but I really think a sextant for the observations is overkill). Was it a suggestion in a flight manual, more theory than practice? Did they try it experimentally a few times? Or is there documented evidence in a logbook that this method for measuring height was employed with some regularity on airships?

You added:
"As the rays from the sun are essentially parallel by the time they reach the Earth the length of the shadow will always be the same to an observer on the ground, irrespective of the height of the airship."

Yes, that much is clear, and it's also the basis of the penumbra method that I have described. The length of the shadow measured on the ground is the length of the aircraft, but the penumbra grows with distance to the aircraft. That is, the linear size of the penumbra measured on the ground grows. If we turn things around and view it from the aircraft, the penumbra is always the same size -half a degree- while the angular diameter of the whole shadow decreases in direct proportion to distance (when seen on a perpendicular surface). No matter how you look at it, at some point the penumbra eats up the umbra from both sides, and the shadow becomes diffuse. For the 75-meter long "Zeppelin NT" in the photo over Greenwich (and in another photo I'll post later), this happens when the airship is a bit more than a mile away. At that distance, the diameter of the zeppelin is not quite enough to eclipse the Sun. It's all penumbra for greater distances, and the shadow on the ground becomes an indistinct and progressively fainter "blob".

You added:
"It’s just the angle the shadow makes at the sextant in the airship that changes with the height of the airship and the position of the sun."

And it's easy enough geometry to account for that, right? Or you could handle it with a sort of "model" of the airship on a stick --like a cross-staff where the crossing piece could be rotated to remain parallel with the long axis of the airship when looking at the shadow at some angle away from beam.

The trick with shadow penumbras is more effective with photographs of shadows since there's no instrinsic angular scale in a photo. The penumbra from Sun or Moon illumination is always half a degree at ground level, and also half a degree from the photographer's perspective if the photographer is in nearly the same location as the objects casting the shadow, which is the normal case with an aircraft shadow.

No matter what, the limiting factor in either of these approaches for determining altitude from a shadow will be the indistinct edge of the shadow. That is, the uncertainty in the penumbra itself would limit your ability to measure the aricraft's angular size to about +/- 0.25°. If the shadow is, for example, 5° across, then the "slant height" (distance to the shadow along the line of sight) could not be trusted to better than +/- 5%. Clearly, if the angular size of the shadow is larger, the calculated altitude is more accurate in direct proportion.

Frank Reed
ReedNavigation.com
Conanicut Island USA