Of hawks, craters and galaxies

Sometimes I learn things the hard way. No, I always learn them the hard way – sometimes I just carry that approach to the ridiculous. This is one such time. But I think I’m a little wiser, though not a heck of a lot more certain about the answers I seek.

See, for years I’ve been trying to get an intuitive grasp on the size and distance of what I am seeing in binoculars and telescopes. Two specific problems kept nagging me. The first was how far away were the migrating hawks I’ve been watching from Mt. Wachusetts or Watatic for the past 15 years or so? (See ‘Looking for Lift in all the Right Places.” The second is how big an object, such as a crater on the moon or a distant galaxy is, judging from how big it looks in the telescope.

See, I know how big the hawks are. Most of the ones we’re seeing are broadwings and their wingspan averages roughly 34 inches. But as I talk with fellow hawk watchers, I get varying guesses about how far away they really are. We stand on one mountain top, Wachusetts, and look in the direction of another, Watatic and when we’re lucky we catch sight of little dots way off in the distance. If there are several, they look like grains of pepper, swept up in a tornado. These are distant broadwing hawks who have caught a thermal – a column of rising air, and are circling round and round as the air takes them higher and higher.

The distance between the two mountains is 12 miles and these hawks look like they’re over by the distant mountain – maybe 10 miles away. I know they’re not – veteran hawk watchers tell me they’re not, and the veterans are right - but that’s what it looks like and I always have to learn things for myself. So how close are they? I’ve actually gone out on a straight piece of highway, measured a traffic light, then seen how far away I could see it by driving off in my car. Of course a traffic light doesn’t look like a hawk – and it is a light. Hawks don’t shine – unless their wings catch the sun just right – and sometimes that’s the way we see them, winking in and out as the sun catches their wings. So outside of the optical physics here – which I’m largely ignorant about – I have problems with other complications, such as brightness which can make something appear larger. (Certainly does in astronomy. Everyone is fooled by the actual size of the moon and sun in our sky – they both look far larger than they are.)

But as I mulled this stuff over and experimented from time to time and searched the Web for answers, I have never been satisfied until now when I last found there was a simple formula that applies:

“A degree is the apparent size of any object whose distance is 57.3 times it diameter.”

So wrote Robert Burnham, Jr some 40 years ago in his wonderful “Burnham’s Celestial Handbook.” I’ve had this three volume set for about 35 years and my copies are loaded with underlining and highlighting and marginal notes – but I never thought to look at them when I was tackling my angular distance problem, even when they dealt with celestial objects.

From Burnham’s formula you can quickly calculate that if you have a disc 1.5-inches in diameter, it will cover half a degree if you put it about 14-feet 4-inches away. Try it. I used a washer. Why? Because half a degree is the actual angular size of both the sun and the moon and experimenting this way reveals how small they are. Don’t believe your eyes? OK, have a friend hold a penny about 7 –feet away from you and have them cover the full moon with it – that penny at 7 feet represent a half a degree – do the math – and it is tiny. Yet it will cover the moon or sun in our sky. Of course, the real size of the moon and sun is much different – roughly 400 times different – and their real distances from us are much different – roughly 400 times different. Which points out another important relationship here.

Every time you double the distance, you halve the angular diameter.

So yes, my 1.5-inch disc appears to be one degree across if you put it at 7-feet 2-inches away. (57.3 X 1.5) Double that and it appears to cover half a degree. Double again and it would appear to cover 1/4 a degree – or 15 minutes. One you double it enough so it gets down to about one minute you will barely be able to see it with your naked eye.

How does this begin to relate to birds in your binoculars. Well, take my broadwing with his 34-inch wing span. My binoculars have a 5 degree field of view. If my bird seems to be occupying one fifth of that field, then he is covering one degree. 37.3 X 34 - that puts him about 162 feet away.

Of course he’s seldom that close – though occasionally they do come right over us. Don’t think I want to do the math while I’m hawk watching. I don’t. I want to do it now, then make it enough of a part of me so I intuitively have a sense of how far away something is, if I know how big it is. This problem of size/distance is especially vexing when you are looking at things in the sky. All of our normal clues as to distance and size vanish – no trees, no flowers, no people, no buildings – just sky. So the only way to start to get a handle on what you are seeing is to make this business of angular diameter a part of you. And yes, the same principle applies to astronomy.

Burnham provides some wonderful tables that relate to astronomical viewing in this respect. I need to study them , then see if I can relate them to my experience at my particular telescopes using the eyepieces I use. I won’t delve into them here – but if you’re interested and don’t have the book, you can find it in Google Books and search on “Burnham’s Celestial handbook.” Once you have the handbook on your screen, simply go to page 62 , which is where the relevant tables begin, though the discussion of this is back on page 60.

So, how far away are my hawks when I can just barely make them out? I don’t know. Arggggggggggh. . . . OK, I do have a good guess. I’m pretty sure they are about 2.4 miles away at the maximum and more often little more than a single mile! Yeah, you can see a mountain 12 miles away, but it’s a bit bigger than a hawk.

How did I derive the number? Well, hawks are not a disc. So what I did was I took a 3x5 file card and put lines on it of varying lengths from three to one-quarter of an inch. I made these thick enough so they simulated hawk wings and put a blob in the center to represent a hawk body. Then I taped it to a wall and measured off one hundred feet. With my 8X binoculars at one hundred feet I found I could certainly make out the half-inch “hawk” and could detect the quarter-inch one –barely. So if, at 100 feet I can see a 1/4-inch line, all I have to do is keep doubling the size – and the distance – until I get up to 34 inches. Actually, it rounded off nicely at 32-inches. By this method I concluded I could detect a hawk in my 8X binoculars that was roughly 2.4 miles away. But I suspect I am much more likely to detect the “half inch” hawk at 1.2 miles away. Or maybe I can see them farther away than 2.4 miles?

Why do I find these numbers vague? Well, for one thing the hawks are moving and motion makes them easier to detect then a line on a piece of paper. For another, they are wheeling about, presenting constantly changing profiles with the sunlight sometimes glinting off their wings. Both the motion and reflected light would argue for them being detected at a greater distance. And when you see a large group in the distance it goes in and out of view constantly. One moment you see nothing – the next several hawks – and the next they are gone again. I suspect in those case they’re 2-4 miles away.

Bottom line – there’s no simple answer here, but next time I’m hawk watching I know I’m going to have a much better feel for how near or far the swirling “kettles” of broadwings are from me. Is there an analog to this uncertainty in astronomy? Absolutely. As near as I can tell most astronomical distances 0 after the first 150-light years, or so – are educated guesses. And if the distances are guesses, so are the sizes. But an informed guess is better than no idea at all.

Now I need to either do more math on my own, or study Burnham’s table, so that when I look at an extended object in the telescope and I “know” either its size or its distance, I will have a reasonable feel for the missing element – size or distance.

Who cares? I do. Is all this just academic? Abstract meanderings? I don’t think so. The technology we put between ourselves and the rest of the world can sometimes obscure reality. I want the same level of comfort in using my binoculars and telescopes that I have with using my eyes alone. When I use my eyes alone I know – through long experience – how far away something is – such as a car – because I have an idea of how big it is. I need that same intuitive – experiential – feel for the objects I see through the scopes.