On a clear summer night, look up at the Milky Way. Gigantic clouds of innumerable stars form a stairway to heaven. Grabbing binoculars, the clouds of stairs reveal light and dark streaks, and fuzzy condensations . But only with a telescope do these clouds reveal their true nature: countless stars forming the spiral arms of our galaxy. What telescope design best stirs our passion?
Or to rephrase the question, what aperture and magnification will show the greatest number of stars in an eyepiecce? Larger apertures reach fainter magnitudes but have narrower fields of view. The answer hinges on counts of star at particular magnitudes.
Here are star counts based on the same ratio of aperture to magnification as the unaided-eye:
aperture magnification
true-field star-count
eye (7mm)
1x
70 deg 1500
50
mm 7x
9 deg 2900
4
inch 15x
4.4 deg 3200
6 inch
21x
3 deg 3300
8
inch 30x
2.2 deg 3400
10 inch
35x
1.8 deg 3300
16 inch
60x
1.1 deg 3200
32 inch
120x
0.6 deg 2600
As you can see, best results come with a 4 inch to a 16 inch
aperture used with a magnification of 3.5x per inch of aperture.
As many stars visible to the unaided-eyes across the entire sky are
visible in a single eyepiece field of view. The 3.5x per inch
of aperture is the ratio of the eye's pupil opening in inches to a
magnification of 1x. We keep the same ratio for all apertures
since this maximizes the light received by the eye. These 4
inch to 16 inch aperture scopes used at the appropriate magnifiations
are what we call Richest Field Telescopes - the richest field of
stars possible in any single view.
Hmm, Ok, then double the 4-16 inch recommendation to 8-32 inch or
even greater. The star count is about the same, but the stars
will be brighter in the larger aperture.
Then the best views will be with the greatest aperture. The
counts of the dimmest Milky Way stars do bottom out, so an aperture
of 24-60 inches works best.
A complicating factor is the issue of magnification. With the unaided-eye at a dark sky site, the faintest stars visible are about 6.5 magnitude. But if we could use a telescope with the same aperture of the eye at greater than unit power, we would see stars down to 8 magnitude. How is this so? The increased magnification dims the sky background but does not dim the stars. The background glow is spread over a larger area thanks to the increased magnification. Spreading the same amount of light over a larger area dims it. Stars are not dimmed because they are point sources - so far away that no practical magnification can reveal their disks. The diffractive nature of light and turbulence in the atmosphere combined with instrument defects do enlarge a star's pinprick of light, but only extreme magnifications make this detectable. So, the blackened background enable us to see fainter magnitudes.
To achieve adequate sky background darkening, we need to magnify
the view at least three times more. This is 1/9 as much field
of view, consequently we must reach 9 times greater star count to
stay even. Assuming star counts that stay roughly constant
across varying aperture, we need to triple our effective aperture.
On the other side, 1.5 magnitude deeper penetration (8 mag - 6.5 mag
= 1.5 mag) computes to doubling the aperture or 4x greater star
count. So, we fall short. Conclusion: use the lowest
power possible.
Human eyes when dark adapted have a pupil size from 5mm to 9mm; 7mm being taken as an average. Here's a chart depicting exit pupil versus age: http://www.egroups.com/files/amastro/Pupil.jpg. All the light from the optics must fit into this 7mm opening, otherwise the light is wasted. The eye's ratio of aperture in inches to magnification is 3.5x. We must keep this ratio regardless of aperture or waste light.
But what if we do waste light in this manner? We do gain
extra field by going to lower power. How does the extra field
compare to the aperture the eye can use? The effective aperture
is found by using our old friend, the 3.5x per inch of aperture
ratio. If we lower our 3.5x per inch of aperture magnification
by 1/3, the telescope's exit pupil increases from 7mm to 9mm.
The aperture actually used by the eye decreases by 1/3. Our
field of view increases in width by 1/3. It's as if we started
with a telescope of 1/3 less aperture and used the 3.5x per inch of
aperture magnification for that lowered aperture. So, we get
the richest field possible for that lowered aperture. If we
have a 9 inch telescope but instead of using it at 32x, we use it at
21x, thus giving us an effective aperture of 6 inches, and the
corresponding wider field of view of a 6 inch rich field telescope.
It's like having two richest field scopes in one! We
cannot stretch these lowered magnifications and increasing exit
pupils too far because our newtonian diagonal shadow eventually
becomes too large and because the physical size of the eyepieces, in
order to maintain the wide apparent field of view, becomes too large
and heavy and expensive.
To maintain our 3.5x per inch of aperture, and maintain a 65 degree apparent field of view, we need to use the following eyepieces based on focal ratio:
f/4 28mm
1.2" diameter
f/4.5 32mm
1.3" diameter
f/5
35mm 1.5" diameter
f/6
42mm 1.8" diameter
f/7
49mm 2.1" diameter
f/8
56mm 2.4" diameter
Given that 2 inch focusers are the largest commercially available,
and that the lowest power widest angle eyepieces are about 35mm to
40mm in diameter, f/6 is the slowest focal ratio we can use and still
achieve richest field views.
They can be! My 6" f/4 will resolve the Galilean moons at 300x. I did spend close to two months working on the mirror, summoning all my skill to figure the mirror as close to perfect as possible. It is possible but difficult to make a f/4 mirror perform as well as a f/6. All factors such as collimation and focusing accuracy are all that might tighter with a f/4 compared to a f/6.
What about those large diagonals that fast wide angle telescopes
call for? First of all, slightly larger diagonal sizes do not
significantly impact image quality, rumors to the contrary.
Also, at f/4, the illumination fall off, that is, how much does the
light dim as you look towards the edge of the field, is very
gradual. So, a diagonal size that fully illuminates only the
center of the field will do nicely. For instance, an aperture
of 18 inches calls for a diagonal to eyepiece distance of 12",
and if the mirror happens to be a f/4, then a 3" diagonal will
work fine. This is a diagonal to primary ratio of 6:1. At
this ratio, image degradation due to the diagonal obstruction is
practically insignificant.
Ok, so we have a fabulous figure on our f/4 mirror. What other magnifications should we use?
Dimming the sky background a bit so that we can see more of the extended nebulosity but without sacrificing too much of our field of view calls for an eyepiece size that gives us an exit pupil of 4mm to 5mm, in many observer's opinions.
Dimming the sky background completely calls for an eyepiece size yielding an exit pupil of 1mm to 2mm.
For highest powers, grab an eyepiece that gives an exit pupil of 0.5mm to 1mm.
Here are the eyepiece sizes for different focal ratios and exit pupils:
|
exit pupil |
f/4 |
f/5 |
f/6 |
|
7mm |
28mm |
35mm |
42mm |
|
5mm |
20mm |
25mm |
30mm |
|
2mm |
8mm |
10mm |
12mm |
|
1mm |
4mm |
5mm |
6mm |
Try Sky and Telescope magazine, March 1980, pages 192-4
Also
ATM III, pages 389-417, by Ingalls, publisher Willman-Bell
