# [Physics] Why don’t flying birds have shadows

I saw a flock of birds flying around today, and noticed that they didn't cast any shadows on the ground. I thought this to be rather strange, so I tried to resolve this mystery.

My first idea was that the sun might actually be wide enough such that both 'ends' of the sun might cut underneath the birds, as the ground would still be illuminated by one or both sides of the sun as the bird flew in front. I calculated as follows:

I assumed the distance from the earth to the sun to be $d = 150\times 10^6 \text{km}$, and the radius of the sun to be $R = 6.957 \times 10^5 \text{km}$. The angle $\alpha$, as shown in my poorly drawn figure, is the half angle between the two sides of the sun, and can be easily calculated:
$$\tan\alpha=\frac{R}{d}$$
$$\alpha = \tan^{-1}\left(\frac{R}{d}\right) \approx 0.0046 \text{rad}$$
This means that the radius of an object flying at $20\rm m$ height must have
$$r = 20\tan\alpha = 0.093\rm m$$
I assume the birds to have a wingspan greater than $20\rm cm$, so in this case they must cast a shadow.