What
is a Richest Field Telescope?
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, delighting us
at the eyepiece?
Or to rephrase the question, what aperture will
show the greatest number of stars in an eyepiece? Larger
apertures reach fainter magnitudes but have narrower fields of
view. The answer hinges on counts of star at particular magnitudes, or
what is called star density.
Here are star counts based on the same ratio of aperture
to
magnification as the unaided-eye: (figures adapted from Glenn, Sky and Telescope,
1980, adjusted for increased field of view of the latest eyepieces.)
| Aperture |
True field |
Star count |
| eye (7mm) |
100 deg |
3600 |
| 50mm |
14 deg |
6900 |
| 4 inch |
6 deg |
7600 |
| 8 inch |
3.4 deg |
7800 |
| 16 inch |
1.8 deg |
7600 |
| 32 inch |
0.9 deg |
6200 |
As you can see, the best result is broadly centered on 8 inches
aperture.
As many stars visible to the unaided-eyes across the entire sky are
visible in a single eyepiece field of view. Telescopes used with widest
angle eyepieces that yield an exit pupil of 6-7mm are what we call
Richest Field Telescopes - the richest field of
stars possible in any single view.
A Little
History
The idea was originally proposed by Mr. Walken in Knowledge (now Discovery) in 1916.
He writes in part, "My chief part was in perceiving and publicly
pointing out how every
aperture could be made 'an' RFT, of that aperture, for a given observer
and that there was one
of all these which, in connection with the curve of star density
against magnitude of stars, was uniquely 'the' RFT for the observer, in
respect of maximum countable number of star per apparent square
degree." He concludes, using star counts published in Knowledge,
1914, that a 2.5 inch aperture has the greatest star count, though he
goes on to say that the view through a 6 inch is "little inferior" and
that the views through larger apertures "is decidedly more attractive
and "richer."" He calculated a star count of 423 given a 50 degree
eyepiece. Incidentally, he concludes that a magnification that yields
the maximum possible exit pupil is best: too low of magnification
wastes light and too high of magnification narrows the field of view
dropping the star count.
It's
plausible that the table of star counts or density in 1914 set the
stage to conceive of richest field telescopes. If so, then this is an
example of information driven revolution. So perhaps we can set
the origination date to 1915.
In 1936 an updated star count is used to revise the
RFT aperture to 4 inches with the count of stars as high as 711.
Further discussion about the values of star density resulted in the
conclusion that the.RFT
aperture is closer to 7 inches. Finally there's a note that a
brightened background due to the Moon (today, we sadly have to add
light pollution) dramatically increases the optimum aperture perhaps to
10 to 12 inches.
Clyde Tombaugh built a 5 inch RFT in 1935.
Clyde describes the views as "truly marvelous", mentioning how "dark
nebulae in Sagittarius stand[s] out beautifully, as it does on a
moderate exposure photograph." While Clyde extols the virtues of
observing in Sagittarius, he says that the most beautiful and richest
star fields are in the Cygnus region.
H.R. Suiter notes in
1996 that the best RFT aperture has varied from 1.5 inches to 12 inches
depending on revised star density values. Because of this to and fro
with aperture, he concludes that a rich-field telescope is best defined
as one that can be used with an eyepiece yielding the maximum exit
pupil that your eye can open to. He recommends personal experimentation
since the aberrations of the eye are so substantial that star counts
may not increase beyond 5 to 5.5 mm exit pupil. He makes a most
interesting comment that earlier observers would be awe struck by
today's short focal ratio refractors and high quality high angle
eyepieces, but that modern observers would envy earlier observers' dark
skies even more!
In a 1980 Sky and
Telescope
article, Glenn Shaw takes up the question of the best RFT aperture,
using the latest star density values. He concludes that an aperture of
9 inches is best though he employs a scope of 5 inches aperture. He
adds an interesting graph where only stars brighter than 5th magnitude
in the eyepiece are counted. This favors apertures closer to 24 inches.
Since Glenn's numbers are the latest that I'm aware of, I use
these, adjusted for wider angle eyepieces available today.
Faint
Stars Don't Impress Me
Hmm, OK, then increase the aperture. The star count is
about
the same, but the stars
will be brighter with larger aperture, the best aperture being broadly
centered on 24 inches.
How
About Counting the Total Starlight, Not Just the
Star
Count?
Then the best views will be with the greatest
aperture. The
counts of the dimmest Milky Way stars do eventually bottom out: an
aperture
of 24-60 inches works best.
What
About Increasing the Magnification?
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: doubling the aperture doesn't make
up
for the required tripling, or, 4x the number of stars doesn't make up
for 1/9th the number of stars. Conclusion: use the
lowest
power possible.
OK,
So Increasing the Magnification Doesn't Work - How
About
Decreasing the Magnification?
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 3.6x. 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 the formula: lowest power magnification = 25.4/7 =
3.6x per inch of aperture
ratio.
This formula is derived from:
magnification = telescope focal length * 25.4 / eyepiece focal length
magnification = telescope aperture * focal ratio * 25.4 / eyepiece
focal length
magnification = telescope aperture * focal ratio * 25.4 / (focal ratio
* 7mm exit pupil)
magnification = telescope aperture * 25.4 / 7mm exit pupil
magnification per inch of aperture = 25.4 / 7mm exit pupil = 3.6
If we lower our 3.6x 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 3.6x 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
22x, then we get 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.
Widest
Fields: Not All Focal Ratios are Created Equally
and
Why F3 is the Ultimate Focal Ratio for Richest Field Observing
Currently
the widest apparent angle eyepieces on the market are TeleVue and
Explore Scientific eyepieces with 100
degrees apparent field of view. The lowest power eyepiece
in the TeleVue Ethos set is the 21mm. In the Explore
Scientific 100 deg line it is the 20mm. The focal ratio that
gives the
widest field of view using this eyepiece is about f/3 (assuming an eye
that opens to 6mm exit pupil and a coma corrector that increases
the focal ratio by 15%). This creates a huge "WOW" factor when
viewing. It's
unlikely that monster 100 degree eyepieces will ever be commonly
available in significantly longer focal lengths. Imagine such an
eyepiece: it would be
a foot long and weigh 20 pounds not to mention
costing thousands of dollars. As the following table and
images
illustrate, there is a major difference in viewable field area between
widest fields of view
at various focal ratios.
Richest Field
Telescope
Field of View versus Focal Ratio
The old rule of thumb that telescopes with a range of focal ratios can
achieve Richest Field performance as long as a suitably matched
eyepiece is used is no longer valid. Eyepieces of extreme apparent
field of view are now available, but only in shorter focal lengths.
Here's a table showing how eyepiece apparent field of view and focal length impact the RFT experience for varying focal ratios.
Table generated for aperture =
13 inches, exit pupil = 6mm.
Telescope focal ratios optimized for several popular eyepieces.
| Telescope
Focal Ratio |
Eyepiece |
Coma corrector X |
Eyepiece Focal Length mm |
Apparent FOV deg |
Telescope Focal Length inches |
Eyepiece Field Stop mm |
Actual FOV from Field Stop deg |
Actual FOV from Field Stop with Coma Corrector X
deg |
FOV area deg^2 |
Magnification |
2.5
|
Ethos
|
1.15
|
17
|
100
|
32
|
29.6
|
2.1
|
1.8
|
2.5
|
55
|
| 3.0 |
Ethos |
1.15 |
21 |
100 |
40 |
36.2 |
2.1 |
1.8 |
2.5 |
55 |
| 3.8 |
Nagler |
1.15 |
26 |
82.0 |
49 |
35.0 |
1.6 |
1.4 |
1.5 |
55 |
| 5.2 |
Nagler |
1 |
31 |
82.0 |
67 |
42.0 |
1.4 |
1.4 |
1.6 |
55 |
| 6.3 |
Orion Q70 |
1 |
38 |
70.0 |
82 |
44.0 |
1.2 |
1.2 |
1.1 |
55 |
Notes on derivation:
Most columns are published values from the manufacturer.
The "Coma corrector X" is the magnification factor built into the coma
corrector.
The
exit pupil is the eyepiece's focal length divided by the focal ratio,
further divided by the coma corrector magnification factor.
The "Actual FOV from Field Stop deg" is given by the formula: field
stop in inches / focal length in inches * 57.3
There are three keys that work in concert:
1.
Shorter eyepieces allow faster scopes to maintain 6mm exit pupil.
2. Wider apparent fields of eyepieces allow shorter eyepieces to
achieve the same field stop as longer focal length eyepieces.
3. Since the field stops are essentially the same, the faster focal
ratio
results in a shorter telescope focal length which results in a larger
field.
Here
are the widest fields possible (each at 6mm exit pupil) for
the above
focal ratios through 13 inches aperture observing M31 (image from
Stellarium):
F/2.5
or F/3 - 2.5 square deg field
100 deg Ethos
 |
F/3.8
or F/5.2 - 1.5 square deg field
82 deg Nagler
 |
F/6.3
- 1.1 square deg field
70 deg wide field
 |
Another
interesting way to look at it is to calculate the maximum aperture
possible for
different focal ratios given a field of view. The focal ratios are
optimized for widest angle eyepieces.
field of view = 1.8 deg, exit pupil = 6mm
| Telescope
Focal Ratio |
Eyepiece |
Eyepiece
Focal Length mm |
Apparent
FOV deg |
Eyepiece
Field Stop mm |
Coma
corrector X |
Max
Mirror Diameter |
| 2.5 |
Ethos |
17 |
100.0 |
29.6 |
1.15 |
13.1 |
| 3.0 |
Ethos |
21 |
100.0 |
36.2 |
1.15 |
13.0 |
| 3.8 |
Nagler |
26 |
82.0 |
35.0 |
1.15 |
10.1 |
| 5.2 |
Nagler |
31 |
82.0 |
42.0 |
1 |
10.2 |
| 6.3 |
Orion
Q70 |
38 |
70.0 |
44.0 |
1 |
8.7 |
Going
down to f/3.0 or f/2.5 means jumping up in aperture from 10
inches to 13 inches. In other words, what we could see previously with
8 inch scopes and wide angle Erfle eyepieces in the 1960's to 1990's
and with 10 inch scopes equipped with Naglers in the 1990's and 2000's
is now seen with 13 inches aperture. This increase in aperture
increases the limiting magnitude by a whole number.
Formula
is: mirror diameter = eyepiece field stop * exit pupil * 57.3 / (field
of view * eyepiece focal length * 25.4)
from: field of view = eyepiece
field stop / telescope focal length; focal length = focal ratio *
mirror diameter; eyepiece focal length / exit pupil = focal ratio
A
New Relationship:
Maximum Aperture or Field of View Based on Varying Focal Ratio While
Holding Exit Pupil Constant
Impact of Increased Aperture on Visibility
Recalculating
Blackwell's visual detection data as presented in Clark's Visual
Astronomy shows that the increased aperture for the same field of view
results in significantly brighter views. The first chart is for the
difficult small Horsehead Nebula and the second chart is for the
somewhat difficult very large California Nebula. They are arranged so
that matching exit pupils yield the same field of
view (which means that magnification is greater for the larger
aperture). The larger aperture for the same field of view that F/3 with
the 21mm
Ethos yields results in a consistent 0.15 log contrast gain. That’s an
increase in apparent brightness roughly equivalent to the ratio of
the apertures.
Are
Richest Field Telescopes Good for Anything Else?
Planetary
Viewing
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 much 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 and particularly at f/3, 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. My 13 inch f/3.0 RFT uses the same
diagonal size as Coulter's original 13 inch f/4.5. An 18 inch scope
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.
You've Convinced Me
About
Stars. How About Nebulae?
I
understand that the greatest number of stars are seen with the largest
exit pupil that my eye opens to. And that the star counts don't vary
that much with common apertures. But I'm uncertain how best to view
nebulae: lowest power, medium power, high power; more aperture, less
aperture? We need satisfy two conditions: 1) that the nebula be
recognizable and 2) that it be detectable.
For the nebula to be
recognizable it has to fit substantially within the field of view.
Consider perhaps the grandest nebula of all, our Milky Way Galaxy.
Imagine looking skyward on a summer night from atop a ladder in field
with distant horizons. Looking up at our galaxy overhead we imagine
ourselves floating in space 30,000 light years from the monster black
hole that dwells in its center. We imagine the unimaginably immense
galaxy stretching out in front of us. Though we cannot sense it, we are
spinning around the black hole at 150 miles per second. We can see the
central hub peaking up and the giant spiral arms twisting in front of
us. At a particular time in the spring, we can see the entire galaxy
rim the horizon. It's impossible to make these wonderful observations
using a high powered telescope pointed at a tiny fragment of our
galaxy. So the nebula has to fit in the field of view to be
distinguished.
Studying the eye's response at night reveals that
the eye sees the faintest objects best at the maximum exit pupil that
the eye can open to. The eye's ability to detect drops off as exit
pupil is shrunk. As the exit pupil shrinks further to 1-2mm, the eye's
detection ability increases as long as the object is at least three
degrees in apparent size. John Dobson along with other observers
through the many years have commented on this ability to more easily
see the object at small exit pupils (high magnification) even with very
large apertures. Best then is to frame the object in the telescope's
field of view while holding maximum exit pupil. For small objects where
the aperture required is obtainable one should select largest possible
aperture and a small exit pupil.
How about some
showcase
objects as examples?
The Sagittarius
Star Cloud is about 2x1 degrees in size. A scope with a 2 degree field
of view can be up to 12 inches aperture with a 100 degree eyepiece at
F/3, 8 inches aperture with a 70 degree eyepiece at F/6. Larger scopes
simply cannot frame the entire star cloud.
The North
American Nebula is about a degree in size. With the popular F/4.5 size
and a 82 degree eyepiece, aperture can be as great as 18 inches yet
still frame the object.
The Horsehead Nebula is about 8x6
arcminutes in size but needs the nebulosity that it's embedded in to be
framed, so let's say something closer to a half degree. An aperture
close to 50 inches looks to be ideal. That we can see it in scopes as
small as 5-6 inches and that we can see it nicely in 20+ inch scopes is
a testament to our eyes. Before we pat ourselves too much on the back
let's analyze the smallest aperture that we successfully detected the
Horsehead with. 5-6 inch aperture at 6mm exit pupil with a 70 degree
eyepiece yields an apparent size close to 3 degrees, which matches our
three degree apparent size rule from above!
The Milky Way is
full of bright and dark nebulae of all sizes. Telescopes of all
apertures will give exciting views if they are used at large exit
pupils and widest angle eyepieces. If aperture is smaller then the
field of view will contain many Messier and NGC objects: the point
being to observe the panorama taking in the
surrounding nebosity
and stars. If the aperture is larger then the field may contain one or
two objects: the point being to observe detail.
Finally, many
experienced observers think of a threshold aperture close to 12 inches
that resolves most globular clusters even at low power into sprinkles
of stars.
Mapping
Exit Pupils to Eyepiece Focal Lengths
Here are the eyepiece sizes for different focal ratios
and
exit
pupils:
|
Exit pupil
|
f/3 |
f/4
|
f/5
|
f/6
|
|
7mm
|
21mm |
28mm
|
35mm
|
42mm
|
| 6mm |
18mm |
24mm |
30mm |
36mm |
|
5mm
|
15mm |
20mm
|
25mm
|
30mm
|
| 4mm |
12mm |
16mm |
20mm |
24mm |
| 3mm |
9mm |
12mm |
15mm |
18mm |
|
2mm
|
6mm |
8mm
|
10mm
|
12mm
|
|
1mm
|
3mm |
4mm
|
5mm
|
6mm
|
At the faster focal ratios, Newtonians need coma correctors. The
TeleVue corrector multiples the focal ratio by 15%. Using this
corrector
alters the values:
brightness
and resolution
|
Exit pupil
|
f/3 |
f/4
|
|
7mm
|
24mm |
32mm
|
| 6mm |
21mm |
28mm |
|
5mm
|
17mm |
23mm
|
| 4mm |
14mm |
18mm |
| 3mm |
10mm |
14mm |
|
2mm
|
7mm |
9mm
|
|
1mm
|
3.5mm |
5mm
|
Where
Can I Find More?
Sky and Telescope magazine, March 1980, by Glenn Shaw,
pages
192-4
Amateur Telescope Making Book Two, by Ingalls, pages 623-630.
ATM III, pages 389-417, by Ingalls, publisher Willman-Bell
last updated Dec 23, 2011
