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Understanding Surface Brightness: It took a while, but the light bulb finally went on above my head.

IN the June 2017 Astronomer's Workbench (page 38), wrote about building solar filters. When I showed a homemade solar filter to my astronomy club, one of the members asked me a question I couldn't answer: If the filter provides a relatively bright view when you look through it, and the light enters the telescope through the entire aperture of the filter, with it all directed to my naked eye, why doesn't the concentrated light increase the brightness to an unsafe level?

That's a very good question. I didn't have a good answer for it, so I did some research and found out why. The answer has to do with what astronomers call surface brightness.

As counterintuitive as it may seem, an object in a telescope can never be brighter than that same object seen by naked eye--with one exception: distant stars. Distant stars are so far away that they're essentially point sources. If you magnify a point source, it remains a point, so any increase in the amount of light that the telescope gathers will also remain in that tiny point, making the point brighter. But any other object--a planet, a nebula, the Moon, even the Sun--has dimension to it, and that makes a world of difference.

An object with dimension gets bigger when you magnify it. And the area of that object gets bigger as the square of the magnification (area is measured in square units, after all). So if you double the size of an object, its area increases by [2.sup.2] (2 squared), or 4 times. Triple it and its area increases by [3.sup.2], or 9 times. And so on. At a magnification of 100x, an object's surface area is 10,000 times greater than when seen by naked eye. So the telescope gathers more light, but that light is spread out over a larger area. It cancels out.

"What's to prevent you from simply lowering the magnification?" you might ask. Get it low enough and the scope gathers more light than it spreads out. Wouldn't the surface brightness go up?

Nope, because of what's known as the exit pupil. The exit pupil is the diameter of the light beam that leaves the eyepiece aimed at your eye. As magnification goes down, the exit pupil gets bigger. That's what magnification is, actually: the ratio of the incoming light beam compared to the outgoing light beam. If you squeeze a 100-mm telescope aperture (about 4 inches) down to a 5-mm exit pupil, you'll have a 20x (20-power) image (100/5 = 20). You can work the math the other way, too: If you want 25x out of a 100-mm telescope, you'll have a 4-mm exit pupil (100/25 = 4).

If you reduce the power down to 10x, you'd have a 10-mm exit pupil. A magnification of 5x would give you a 20-mm exit pupil. And lx would give you a 100-mm exit pupil. In other words, at lx the telescope wouldn't be doing anything at all. The outgoing beam of light would be the same diameter as the incoming beam.

That leads us to the next key concept: the entrance pupil.

Your eye's iris will only dilate so far. For most of us, it'll open to about 6 mm when fully dark adapted. That means if you look into that 100-mm-wide beam of light coming out of a 1-power telescope, you're only going to see a 6-mm portion of that beam. Sure, the telescope is gathering a lot more light (278 times more light than an eye with a 6-mm pupil diameter), but most of that light is hitting your face. Only 6 mm worth of it is going into your eye--the same amount you'd get if you didn't look through the telescope at all.

Let's crank up the magnification just a little bit to 10x. That gives a 100-mm scope a 10-mm exit pupil (100/10 = 10). The light is definitely more concentrated now--100 times more than when viewed by naked eye. (Compare the squares of the light beam diameters: 1002 mm is 10,000 mm, and 102 mm is 100 mm. Divide 10,000 by 100 and you get 100 times the light intensity.)

That sounds pretty serious until you realize that the eye only sees 6 mm of that 10-mm beam. So it's seeing 6/10 of the beam's intensity, but we're dealing with area here, so we have to square that. Squaring 6/10 gets you 36/100, or 0.36. The eye is getting 0.36 times the 100-fold increase in light intensity, so it's only getting 36 times as much light as when you look at the same object with the naked eye.

Doesn't 36 times still sound pretty significant? That's almost 4 magnitudes. When you're looking at the Sun, couldn't that be dangerous?

Not at all. Remember surface brightness? The surface brightness goes down by the square of the magnification, so at 10x the surface brightness is 100 times dimmer. The telescope delivers 36 times as much light to your eye, but magnification spreads that light out 100 times, for a net loss of light of 36/100, or 0.36. Sound familiar? That's the proportion of the total light entering your eye, because the rest of that 10-mm-wide light beam is hitting the outside of your iris and being wasted.

The worst-case scenario would be when an entire 100-mm telescope aperture is squeezed down into a 6-mm exit pupil so all the light could enter your eye. [100.sup.2] is 10,000, and [6.sup.2] is 36, so you'd be getting 278 times as much light (10,000/36). But 100/6 = 16.7, so you'd need an eyepiece that delivered 16.7 power. That 16.7 power would decrease the surface brightness by 16.[7.sup.2], which is 278 times. It's a wash.

What about a teeny-tiny exit pupil? Say you squeezed all that aperture down to just 2 mm. The increase in total brightness would be [100.sup.2] over [2.sup.2], or 2,500 times more light (10,000/4). That sounds pretty serious! But the magnification at that exit pupil is 100/2, or 50x. The decrease in surface brightness at 50x is [50.sup.2], or ... wait for it ... 2,500.

Again, it's a wash.

You can never get a higher surface brightness of an extended object through the telescope than you get with your naked eye. Stars don't magnify because they're point sources, but everything else --including the Sun--has dimension, so it grows larger with magnification. And thus your eyes are safe when looking through a properly solar-filtered telescope, no matter how big the scope.

I can hear thousands of voices protesting, "But when I look at the Orion Nebula in a 20-inch scope, it's bright enough to read by! It's bright enough to trigger color vision! How can you tell me it's not any brighter than if I were to look at it by naked eye?"

Do the math. I guarantee you it's no brighter per unit of area. But the image in that 20-inch scope fills a much greater area than what you see with your naked eye. And if you're using a magnification high enough to squeeze that scope's entire aperture into your pupil (that would be 85x for a 20-inch scope and a 6-mm dilated eye), the view in your telescope is 7,225 times bigger ([85.sup.2]) than the naked-eye nebula, yet every part of that huge image is just as bright as the original. Go have a look at the Sword of Orion without any optics. That middle "star" in the sword is the nebula. It's pretty bright, isn't it?

But ... but ... but ... distant galaxies? My naked eye is only good to 6th or 7th magnitude at best. If the surface brightness doesn't go up in a telescope, how can I see the 13th-magnitude components of Stephan's Quintet?

That's a very good question. The answer is magnification.

Many people (myself included until recently) believe that magnification increases contrast. As we've just shown above, that can't be true. When you increase magnification, the background gets dimmer at exactly the same rate as the foreground object (by the square of the increase in size). So that's not why magnification helps.

The truth of the matter is, our eyes can see objects much dimmer than magnitude 6 or 7. We just can't see them if they're so small that only a handful of the several million rods and cones in our retinae receive the photons. Our reti-nae will only send a signal to our brains if several adjacent cells are stimulated at once, and the number of cells necessary to trigger a signal increases as the brightness goes down. To see something dim, we need to make it larger so it covers more retinal cells.

You can prove this on any clear night. Go outside and look up at the sky. If you live in town or the close suburbs, the average surface brightness of the sky will be somewhere in the range of 17-18th magnitude per square arcsecond. If you live out of town, it can get down to magnitude 21.5 or so. That's really dim, yet you can easily tell that the sky is brighter than a silhouette of a tree. More to the point: You can tell that a dark patch of sky, devoid of bright stars, is still brighter than a silhouette. The dark patch is probably as low as 22nd or 23rd magnitude per square arcsecond. So you can see very dim objects indeed; they just need to be big enough to trigger your retinal cells.

That's what a telescope does for you. It magnifies that tiny little galaxy until it covers a significant portion of your retina. And if you increase the aperture of the telescope, the galaxy will get brighter, right up to the point where the exit pupil matches the entrance pupil of your dilated eye. At that point you're seeing the galaxy as bright as it can get. More aperture will just spill the extra light out onto your iris--unless you reduce the exit pupil, but that raises the magnification and spreads the light out again.

That leads to one obvious last question: What's the ideal magnification for a given aperture?

That's a very good question. Get out there with your telescope and find out.

* Thanks to aperture and magnification, Contributing Editor JERRY OLTION has viewed Hoag's Object, his favorite 16thmagnitude galaxy. Contact Jerry at j.oltion@gmail.com.

Caption: PARTLY CLOUDY Because of the limitations of resolution and aperture, stars don't appear as point sources in images. The nebulosity of the Pleiades may appear brighter when viewed under magnification in the eyepiece, but it's an illusion. It's no brighter per unit area than it would be in a naked-eye view.

Caption: RISING STARS Point sources like stars do appear brighter in the eyepiece, but it's magnification, not an increase in brightness, that makes the nebulosity more evident in a telescope.

Caption: THE HARDER YOU SQUEEZE ... If the telescope squeezes the light down to 1/10 of its original diameter, the image will be magnified by 10x.

Caption: SIZE MATTERS These three boxes use the same colors, fonts, and gradations. The larger a dim image, the easier it is to see.

Caption: SAME, SAME You can never get a higher surface brightness of an extended object like the Great Orion Nebula with optical aid than you would get with your naked eye. An object's surface brightness decreases by the square of the magnification provided by your telescope and eyepiece.

Caption: DARKER THAN YOU THINK You can see much dimmer objects than the 6th--or 7th-magnitude stars that we often consider our limit. A good dark sky is about 21 magnitudes per square arc-second, but you can easily tell it's brighter than the trees in front of it.
SPREADING OUT Surface area increases with the square of magnification.
The light is spread over a greater area, so the brightness per unit
of area goes down by the same factor.

1x
2x   4 times area
3x   9 times area
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Title Annotation:HOW BRIGHT IS IT?
Author:Oltion, Jerry
Publication:Sky & Telescope
Geographic Code:1USA
Date:Nov 22, 2017
Words:2032
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