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 
 at least radius
$$
r = 20\tan\alpha = 0.093\rm m
$$
I assume the birds to have a wingspan greater than $20\rm cm$, so in this case they must cast a shadow.
I also considered the possible effects of diffraction, but aren't those effects too small to be observed on such a scale?
Does anybody have an explanation for why birds don't seem to cast shadows, or maybe where my attempt at explaining the phenomenon falls short?
 A: In order to cast a discernible shadow, an object has to block the entire sun (or else some light rays from the unblocked parts will wash out whatever shadow was created from the blocked parts). Since birds tend to be relatively thin, irregular-shaped objects, whereas the sun is a circle with a diameter of about half a degree, it doesn't seem particularly likely that even a bird with sufficient wingspan would be able to block the entire sun at once.
A: Your estimate of the angular size of the Sun is correct in that it is often taken to be $\frac 1 2 ^\circ \approx 0.09$ radians.  
Because the Sun cannot be treated as a point object it will always produce a region of partial darkness (penumbra) and possibly a region of total darkness (umbra).  
To produce an umbra with a $10$ cm diameter circular disc the shadow must be formed less than approximately $10$ m from the disc as shown in the diagram below.  
 
The size of the umbra will be larger the closer the disc is to the ground but never larger than $10$ cm.  
The disc would have to be closer to the ground if it was elliptical in shape and with a major axis of $10$ cm.  
A fully grown Golden Eagle is a large bird of length $66-102$ cm and wing span $180-230$ cm however the width of its body and of its wings are considerably less.
 
To produce an umbra of any reasonable size and a Golden Eagle would have to fly fairly close to the ground.  
All this assumes "ideal" conditions in that there is no other extraneous sunlight being reflected from nearby objects and scattered by the atmosphere which would make any shadow less distinct.  
On a bright sunny day a smaller bird such as a blackbird has to be fairly close to the ground for a shadow to be observed.  

A: No, it is wrong.  You probably thought that(I also thought it was like this) because the light rays converge before the ground, and this results in it inverting.  This is because after converging, both rays go in the opposite direction, forming this X shaped diagram of the sun rays, which inverts the image.  Hence, actually it is a dim, inverted image that is formed, not no image.  
By an inverted image, I mean that the shadow will appear to flip the object.  If the bird’s beak was facing north, the shadow of the bird’s beak will be facing south.
However, I noticed that the above situation is not true, since the sun rays enter earth as parallel beams.  Hence, your answer is actually wrong.  In fact, it is the small size of the bird, and some environmental factors like temperature, etc, that my distort the shadow such that it is not visible.
