April/May 2000 Newsletter

Stockton Astronomical Society

Valley Skies

Valley Skies - April/May 2000 Issue

The Telescope Nut
by Jeff Baldwin

Resolution

When we look through telescopes we not only see things closer and brighter, but we also see details that we couldn't see without the telescope. The sharpness of the image is important because there is information in the detail of objects that would be lost if the image were blurry.

Imagine looking at a six-sided die from a long distance through a telescope. If the telescope could not resolve the dots on the face, we wouldn't know which side we were looking at. A 'two' would be indistinguishable from a 'three'. We don't look at dice, though. We look at double stars, globular clusters, planets and the Moon, all of which require sharpness to enjoy. There might be a crater on the moon with the Sun shining nearly horizontally across its floor, and without high resolution we might miss subtle details, like ridges or smaller craters within. Sunspots would be dark blotches with darker centers with low resolution, while with higher resolution we might see radial striation or nearby granulation, a totally different view.

Factors that might make resolution high or low are aperture, seeing, optical quality, f ratio, observer's health and vision, and imagination.

Let's start with the theoretical resolution limit of the optic. Telescopes focus stars, and stars are so far away that they are nearly points of light, even with the largest telescopes. However, there is a rule in optics that causes points of light to focus only as small as a disk of radius r where r = 1.22lf/d, l being the wavelength of the light, f/d being the focal ratio of the optic. This disk is called the Airy disk, named after a guy named Airy. This is a result of the wave nature of light, and it is unavoidable. As l gets smaller, the radius of the Airy disk gets smaller, and as the focal ratio gets smaller, the radius of the Airy disk gets smaller. Let's look at the Airy disk for a couple of telescopes, and we will use 550nm as the wavelength of our light (1/50,000 inch, or .00055mm). A Celestron C-8 is an f/10 optic, even though resolving a point for a C-8 might be far-fetched, but its Airy disk should have a radius of 1.22 x .00055 x 10mm, or .00671mm. Now let's look at another telescope with the same focal length only with larger optics, say a 16" f/5. Its Airy disk has a radius of 1.22 x .00055 x 5 = .003355mm, half the size. This means that the 16" should be able to have an Airy disk that is half the size of the 8-inch given the same focal length. Since their focal lengths are the same, this means that the 16" can make smaller disks of resolution than the 8-inch.

Here is a picture of two stars that are very close: the left two are through the 8-inch and the right two are through the 16-inch. The magnifications are the same, and the stars (centers of the disks) are equally spaced.

If we were to examine each of these images with the same eyepiece so that we would be seeing at the same magnification, notice the 16" telescope would out-resolve the 8" telescope as seen by having black sky between the stars. Also, if these optics were perfect, there would be rings around each of these disks, but they are omitted for this illustration.

Now let's compare telescopes of equal aperture but with different focal ratios. We'll use the C-8, so its Airy disk has a radius of .00671mm. Let's also try an 8" f/5. The radius of its Airy disk would be 1.22 x .00055 x 5 = .003355, the same as the 16" scope! However, it's at half the focal length, so to see the objects at the same power we'd have to use an eyepiece of half the focal length. When we look at the two stars this time with the higher power eyepiece, it will look twice the size, making the Airy disks look the same size as the C-8. This means that when you make the two 8" telescopes the same power, they will have the same resolution! Bottom line: telescopes of the same aperture will have the same resolution if everything else is equal. A 6" f/4.5 theoretically should be able to resolve as well as a 6" f/15. This would require both telescopes to be figured perfectly, and this only applies to the center of the field of view.

In Newtonian optics, the field of view has a curve, and the radius of this curve is equal to the focal length of the telescope. This means that a 6" f/5 has field curvature with a radius of 30", and a 6" f/10 has a field curvature of 60". Since the f/5 has a shorter radius on its field curvature, the outermost portion of the field of view is more comatic, and the resolution out there will fall. However, given equal magnifications, they should appear nearly the same.

In the illustration above [left] it can be seen that with a longer radius of curvature of the field of view the focus deviation in the outer regions is less than with a shorter radius of curvature of the field.

So, how tightly can your telescope resolve? Dawe's limit of resolution states a = 120/D where D is the aperture in mm and a is in seconds of arc. If the aperture is known in inches, then a = 4.72/D. Let's try one. A C-8 is 200mm in aperture, so a = 120/200 = 0.6 arcsec. With aperture expressed in inches, a = 4.72/8 = 0.59 arcsec., which, rounded, is 0.6 arcsec. They both agree.

Wow! That means that Dave's 22" f/4.8 should be able to resolve 4.72/22 = 0.2 arcsec! If the optic was perfect and the atmosphere didn't exist, yes, it could. The atmosphere rarely allows better than 0.5 arcsec. viewing, and most likely not better than 1arcsec. If the atmosphere is turbulent, a small optic will be viewing through less turbulence than a larger optic, so lots of planetary viewers use smaller telescopes, say 5" Takahashi refractors. You don't need 40" telescopes to see Saturn, so the smaller scope may have an advantage over the larger scope when the seeing is less than 1arcsec. for the big scope.

Sometimes you see observers place aperture masks over the front of their telescopes to make planetary viewing sharper. There are at least two reasons they do this. First, the telescope is gulping in less turbulent air by having a smaller aperture. Secondly, if the mirror has a peak-to-valley error over its diameter, then using a mask will eliminate lots of mirror, and therefore lots of error, and the result will be a smaller optic with less peak-to-valley error. In the following illustration you can see that when the whole mirror is exposed to the star's light (top) there is a peak-to-valley error that is much larger than when only a portion of the mirror is exposed to the star's light, such as when an aperture mask is used.

Refractors, Newtonian Reflectors, Schmidt Cassegrains, Classical Cassegrains, Ritchey Chretiens and Maksutovs all have differences in their fields of view. For a color-corrected point-perfect focus at the center of the field of view, a Newtonian is the ticket, but suffers in the outermost areas of the field of view. Ritchey Chretiens have a less than perfect focus, but it is fairly equivalent throughout the field of view, and is therefore good for photographic purposes. (The Hubble is a Ritchey Chretien.) Refractors have excellent focusing throughout the field of view, but are prey for color aberrations. Three-element refractors (Apochromats) are the best for both flat field and color corrections (see Glen Youman's web site). Schmidt Cassegrains are on the low end but are inexpensive and have spherical optics. Maksutovs exceed almost all others in apertures less than 3.6", but fall behind refractors and Newtonians with larger apertures. There are other designs.

The f ratio can also alter clarity if the optic is imperfect. If the optic is perfect, then there shouldn't be any difference other than field curvature. If two telescopes are compared and are the same in every way except for the f ratio, then imagine an error in each of them that causes a ray to the focus to miss by the same amount. The longer focal ratio telescope will have a smaller lateral error than the faster telescope as seen in the following illustration.

Both of the telescopes in the illustration have the same axial error, but notice that the faster telescope has a larger lateral error.

For this reason, many first-time telescopes are optically slow in order to perform well even though they may have errors that would really goof up a faster telescope. One of the reasons why large, fast mirrors are difficult to produce is due to this reasoning. An 8" f/6 1/20 wave mirror may perform better than a 1/20 wave 40" f/3.6 mirror since equal errors are unseen on the 8" and are quite visible in the 40". Larger, faster mirrors require a more perfected figure.

Human health is also a factor in seeing. If corneas were spherical they'd be great. However, they are usually lumpy, occasionally contain cataracts, and are usually astigmatic. Retinas occasionally have blood flowing through them that contains chemicals that reduce visual acuity, such as alcohol, sugar, caffeine, nicotine, and other junk. Diseases such as diabetes and glaucoma can also mess up the visual acuity.

Lastly, certain individuals boast about their own optics above other equivalent optics. That makes them feel better, and they may actually think they have superior optics and visual acuity, when in fact a double-blind experiment might find them equivalent. Claims of resolvability are historically found in such famous individuals as Lowell. As far as I know, none of the missions to Mars have found even a single canal!

Correction:
In March's Valley Skies, page 5, bottom left paragraph:
"M^2.51188" should read "2.51188^M"; it occurs twice.

Clear Skies...Jeff Baldwin
For more information on Telescope Making jump to the ATM page.