Quantitative results follow.
I originally said that people at Godman Field *might* have seen
a balloon at Nashville because it was such a large reflector, and
that I did not know how to calculate the apparent brightness. It
turns out that the math involved is pretty simple, at least
calculate an upper limit.
The question of visibility is obviously fundamental to unraveling
who saw what, and Brad was correct to bring it up. 140 miles is
a very long distance to look through the atmosphere, after all.
One question I still have is if the balloon is 73 ft or 73 x 129
inflated. Such a balloon is mentioned at CUFON site. I have used
73 ft in my calculations.
The answer is that the MAXIMUM brightness that a 73 ft diameter
sphere should have is -4.6 magnitude seen from 140 miles away.
That is quite bright, just brighter than Venus, in fact. But, I
think that Venus could be seen in the daylight sky at Godman
and a real balloon would be fainter still than my calculation.
-4.6 is the most optimistic estimate, assuming diffuse (not
reflection from a sphere. The balloon is certainly not a perfect
reflector, nor a exact sphere and there would be a complicated
dependence on phase angle (the angle between Sun, balloon,
and observer), so the correct answer must be less than the maximum
value which I calculated. Also not included is atmospheric
which would be important attempting to see so far from Godman
I think the calculation effectively rules out seeing a 73 ft
balloon 140 miles away.
For a 45 mile range I calculate -7.1 magnitude, and for a 15 mile
range, -9.4 magnitude. Once again these are upper limits, not
actual claims of actual brightness. A balloon would be fainter.
I tested the equation (and myself) by calculating the brightness
the full moon, and got -12.1, which is only half a magnitude
than the correct value.
The equation I used came from:
All that is required do the calculation is the cross-sectional
the object, the reflectivity, the phase function, and distance of
object. I simply used 1.0 for the reflectivity (to get an upper
for the brightness), and reduced the phase function to 2/(3*pi)
which simply forces zero phase angle, once again so as to
the upper limit).
If anyone knows an optical physicist interested in flying saucers,
feel free to run this calculation by him.
I have only replied to Current Encounters since I can't send to
and all active participants are on this list, as far as I can tell.