Bats, Bugs, and the Moon

When hobby astronomers talk bugs, it is usually to commiserate about being stung and bitten at the telescope or to share tips about repellents. But bugs affect not only skin sensation, they can contribute to the visual experience, albeit indirectly, as a lure for bigger fish. 
When we observe the night sky through a telescope, we peek through swarms of bugs without even noticing because, with the exception of fireflies, bugs are too dark, too small or too much out of focus to register. Their mammalian predators, being many times larger, are a different matter. Bats, with some luck, are eminently visible. You can see them at the right time of year by observing the Moon, as I happened to be doing on 8 September, 2022, from my backyard in Lexington, Kentucky. 
At around 10 PM, every thirty seconds or so, a bat silhouette would flutter across the Moon’s face. Some silhouettes were large and out of focus, most were small and sharper. The smallest had a wingspan of probably no more than 1/25 of the diameter of the Moon. These bats had to be far away and high up in the sky. Most were flying reasonably straight paths, which brings to mind migration. If this were indeed the case, I would expect them to fly in more or less the same direction. But they didn't: they kept crisscrossing the Moon’s face, and over quite a range of altitudes. Since bats aren’t known for soaring high in search of a mate, I reckon there must have been bugs up there to make the climb worth their while. I did see one or two bats fly erratic paths much like the ones they fly closer to the ground in pursuit of prey.
How high is high? It’s easy to calculate the Altitude above Ground Level using this formula (derived in the appendix):
AGL = wingspan × sin (Moon's altitude angle) ÷ (2⋅tan (silhouette-to-Moon ratio × 0.25°))
The Moon's altitude angle can be obtained from a stargazing app; on this occasion, it was about 24°. The ratio of bat silhouette to the Moon's diameter I estimated to be 1/25; the 0.25° in the formula is the the Moon's radius expressed as the angle it subtends when viewed from Earth. The wildcard is the bat’s wingspan, which varies from species to species. The calculated AGL is directly proportional to it. Let’s assume that the species I saw was the big brown bat, which is ostensibly the most common one in my neighborhood and has an average wingspan of 13". Cranking these numbers through the formula gives an AGL of 1,262 ft (385 m).
I had never heard of bats flying that high and began to doubt my estimates. More solid photographic evidence was called for. I tried to cobble together a scope attachment for my phone until I remembered that I had some adapters in the closet that could be used to attach a real camera to the telescope. The next night’s skies were clear, and the bats were out again, in lesser force than the night before, but enough for a decent sample. Below is the video to prove it. The Moon jumps around whenever I push my non-motorized scope to follow it. I wish the bats were in better focus, but the Moon was my only focusing target, and without a good idea of the bats' distance I didn't know how far out I should rack the focuser. They look sharper when seen directly through the telescope than they appear in the video because the eye can accommodate a little while the camera cannot.
I located the frames in which the bats were most stretched out and took wingspan measurements at 4x magnification. This puts most of the silhouettes in the clip at between 1/30 and 1/60 of the Moon’s diameter; so my earlier estimate of 1/25 wasn’t far off. The video timestamps yield accurate numbers for the Moon’s changing altitude during the hour I was filming (actual recording time was maybe 20 minutes; there was a lot of fiddling involved as I did this for the first time). For an assumed wingspan of 13", these numbers imply an AGL range from about 1,500 ft to 3,500 ft (457 m to 1067 m). You’ll notice some shadows in the video that are much closer. Some that I didn’t include were closer still. The high end of the range has to do with the fact that, by the end of the hour, the moon had risen from 24° to 28.5°.
And then there is the highflier. Fifty-one seconds into the clip, a tiny silhouette enters at one o’clock, heads diagonally across the Moon and, eighteen seconds later, exits at seven o’clock. You may have to replay the clip in full-screen mode to see it. Below is a composite image made from 537 individual video frames that shows the entire transit. The silhouette measures at best 1/120 of the Moon’s diameter (I try to err on the side of size overestimation so as to not overstate the AGL). The Moon’s altitude at the time was 24.5°. Again assuming a 13" wingspan, the corresponding AGL is an astounding 6,178 ft (1,883 m). 
Now what if this wasn’t a 13" big brown bat? For any other wingspan v (in inches), the AGL is obtained by multiplying 1,883 m by the factor v/13". How big or small are alternative candidates? I contacted Gary McCracken, who is one of the foremost experts on high-flying bats. The species he proposed are the hoary bat with a wingspan of 12" to 15", the 13" red bat, the 11.5" silver-haired bat, and the 12" Brazilian free-tailed bat. If it was any one of these, it couldn't have flown much below 6,000 ft because it wouldn't have been much smaller than a big brown bat. It might in fact have been a bigger hoary bat and thus proportionally higher. Decades ago, Brazilian free-tailed bats in Texas were reported to fly at altitudes around 10,000 feet, but then everything is bigger in Texas. (https://www.batcon.org/article/bats-aloft-a-study-of-high-altitude-feeding/)
The video was a good first stab, I thought. But if would be nice to get a clearer picture of the bats, perhaps good enough to narrow down the range of possible species. The video is in 4K, which is the highest definition of which the MFT camera I am using is capable. Still images from that camera have more than twice the resolution of 4K, and I can shoot at 60 frames per second. So I decided to try my luck with bursts of still images for which I would guess the focus. This had to wait until the next lunar cycle. Conditions conspired on 5 October, 2022. Amid a lot of misses, I got some decently sharp images. Having a less-then-perfect telescope turned out to be an advantage for dialing in the focus. The scope produces thin green or purple fringes around objects that are out of focus, and the color indicates in which direction the focusing dial has to be turned. The price for a bat in focus is of course a blurry Moon because the telescope's depth of field is too shallow to encompass both. The bat in this photograph was at about 2,000 feet above ground and 4,300 feet away from me.
The following animations are made from bursts of cropped still images and play at about fifteen percent slow motion. If they are a bit choppy, it is because there aren't enough frames per second for smooth transitions, and each frame is a sharp image without motion blur blending into the next frame because they were shot at 1/1000th of a second in order to help with identification. One of the creatures in the clip is a bird. See if you can spot it. I apologize on behalf of my website provider about the occasional video flicker. Replaying the clip should fix it. 
There were more than just this one bird visible that night, but the bats outnumbered them. The following night, it was the opposite: hardly any bats but a good number of birds of different shapes and sizes. Birdwatching friends suggest owls as candidates for some of the larger ones. If this is right, then the bats were well advised to lie low. See if you recognize a player in this composite image (assembled from 60 fps bursts). If you do, I would be delighted to hear from you. 
A bat at the same rate of 60 frames per second looks very different. Birds travel two or more body lengths in 1/60th of a second, while the bat travels about one body length. 
It is now December, and I haven't seen any bats since early October. I hope they will soar again next spring. 
Appendix
To fix ideas, let
w = the bat’s actual wingspan
d = the bat’s distance along the line of sight
𝛂 = the visual angle subtended by the bat’s outstretched wings (assumed to be orthogonal to the line of sight)
𝛃 = the Moon’s altitude angle at the time of observation
a = the bat’s altitude above ground (presupposing that the ground below the bat is at the same level as the ground under the observer; if it isn’t, add in a correction term)
We also need
0.5° = the visual angle subtended by the Moon
r = the ratio of the bat’s wingspan to the Moon’s diameter.
Then we have
(1)     𝛂 = r⋅0.5°                                by elementary geometry
(2)     tan (𝛂/2) = w/2d                     from the first diagram
(3)     sin 𝛃 = a/d                              from the second diagram
(4)     d = w/(2⋅tan (r⋅0.25°))           by using (1) to substitute for 𝛂 in (2) and rearranging terms
and finally
(5)     a = w × sin 𝛃/(2⋅tan (r⋅0.25°))        by using (4) to substitute for d in (3) and rearranging
When you use this equation, keep in mind that the altitude comes out in the units of the bat’s wingspan and wants to be converted into something more useful.
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