How far away is that bird?

bird-wise the Dorset trip finished with an atmospheric Arne reserve in the late evening on Tuesday, with several Nightjars churring away (but unfortunately not seen), and a brief but clear view of the Great-Spotted Cuckoo in flight at Portland Bill. Given the choice of spending a couple of hours hunting for the thing at a cold and windy fog-bound  Portland or heading off to find some sun I headed off. More of where I ended up in subsequent posts.

But back to that Goshawk sat in a tree. It was distant, just visible as a pale blob through the binoculars, then clearly identifiable as an accipiter with the scope turned up to 60x. But just how far away was it?

Technical data on the Kowa 883 is surprisingly difficult to find, but according to a review of the Kowa 883, the field of view at 60x is 1.1 degrees.  The bird filled about 10% of the viewer from top to bottom so that is 0.11 degrees, and the bird itself being a female is around 40 cm high.

So simple geometry is that the distance x times the angle a in radians gives the height h of the bird. Reversing that and dividing the height of the bird by the angle in radians gives the distance. So:

•  0.11 degrees x pi/180 = 0.0024 radians. 
• 40 cm/0.0024 = 166 metres.

So on that calculation the bird was 166 metres away. But there are quite a few sources of error on that. Firstly the FOV at 20x is 2.2 so by my simple calculations the FOV at 60x is 2.2/3 = 0.733 degrees. Perhaps the bird stands 50 cm high and it was only a 15th of the total top-to-bottom distance, so put those numbers in to the calculations and the distance becomes just over 500m. Put 1.1 degrees, 30cm and an eighth of the height and get 125m. So that's somewhere between 125 m and 500m with the middle calculation being 166m

From these calculations a rule of thumb is that at 60x and 10% of the view-finder height, a 10cm object = 50 m distance.

I don't think it was 125m away.  It was nearer the 500m, but maybe only 300?

Anyway, next time I'm in this position I'll think about the size of the image in the viewfinder and the likely size of the bird. Maybe do some tests of cars at a known distance. Something to work on ...

... and if you spot a simple mistake, please let me know in the comments section.

[updated with a correction]