star magnitude and binoculars

Harald Lang wrote in message ...
Bino Adler#
7x35 41 12.0 12.5 13.0 13.5
10x50 71 12.8 13.3 13.8 14.3
12x50 85 13.0 13.5 14.0 14.5
16x70 134 13.6 14.1 14.6 15.1
16x80 143 13.8 14.3 14.8 15.3
20x80 179 14.0 14.5 15.0 15.5


These values are all 3 magnitudes too high. Note, A in the
formula (m = 3*Log(A) + 2*Log(X) + 0.6 + v) is aperture in *cm*
not mm.


This agrees with what I'm stating about the results my my testing.
The Binocular Performance article highlights the results of a year of
testing with nearly a dozen pair of binoculars. Although I have not
identified the exact lim mag of every binocular, I have spent enough
time in the field with all the binocs and charts and have taken enough
notes to be able to say these values are too high.


so the difference is rather small -- Adler puts somewhat higher
weight on magnification relative to aperture compared to "my"
formula.

Adler does put more weight on magnification, which my testing has led
me to believe is a step in the right direction. I am inclined to put
even greater weight on magnification. The conclusions I reach in
Binocular Performance testing are aperture may be responsible for a
20% increase in seeing, while increasing magnification at a given
aperture may be responsible for an 80% increase in seeing. These
increases would need to be interpreted in the context of the equipment
I used in my tests and the magnification ranges that equipment
provided.

The most important conclusion I reached is that magnification has a
far greater effect on the ability to see thru a given size binocular
than does the aperture itself, because the higher the magnification in
use, the greater the potential of the aperture is utilized.

There is a big problem in practise in trying to use any formula
predicting the visibility of extended objects.

Contrast is what helps the most in seeing extended objects. The next
step beyond the Adler index, which already accounts for the weight
that should be given to magnification, is accounting for and applying
some factor for the quality of the coatings and baffels. Contrast is
the quality that improves as these components improve. That is why a
16x70 Fujinon binocular will outperform a 20x80 Oberwerk binocular.
Not only does this quality provide the needed performance for seeing
exteded objects but also it improves the performance for viewing point
source limiting magnitude.

Although these formulae do show a relative index of performance, just
the same as the Adler Index, they do not account for that quality that
determines the ultimate performance and will always fall short in that
judgement.

How to incorporate such a quality into a formula that will predict
limiting magnitude I am not sure. I have made my attempt at a much
simpler ranking in the hopes of providing a means to judge and rank
relative performance.

edz

Harald Lang wrote in message ...
Hi Kurt, you wrote [snip table Proposed "rule-of-thumb"]
These values are all 3 magnitudes too high. Note, A in the
formula (m = 3*Log(A) + 2*Log(X) + 0.6 + v) is aperture in *cm*
not mm.

My apologies for the math bungling. The numbers looked high to me, but
I posted anyway. To correct things:

CORRECTED TABLE

Limiting magnitude - binocular
Bortle 6 5 4 3 2 1 H_t
Bino Adl# v_lim 5.1 5.6 6.1 6.6 7.1 7.6 6.6
7x35 41 9.0 9.5 10.0 10.5 11.0 11.5 10.3
10x50 71 9.8 10.3 10.8 11.3 11.8 12.3 11.0
12x50 85 10.0 10.5 11.0 11.5 12.0 12.5 11.0
16x70 134 10.6 11.1 11.6 12.1 12.6 13.1 11.8
16x80 143 10.6 11.1 11.6 12.1 12.6 13.1 11.8
20x80 179 10.8 11.3 11.8 12.3 12.8 13.3 12.1

using the Clark-Blackwell based-equation for binocular limiting
magnitude:

m = 3*Log(A) + 2*Log(X) + 0.6 + v

where
m = magnitude of faintest star (point source) visible in binocular
A = aperture in cm
X = magnification
v = magnitude of faintest star visible to the naked eye

Adler binocular performance descriptive index:

Adl# = Sqrt(A) * m

The column "H_t" refers to Harrington's table
per the Sidgwick-Steavenson equation:

Mag=6.5 - (5 log d_ep) + (5 log D_obj) + 1.8
Mag= 9.1 + 5(log D_obj_in)
Mag_limiting = 1.8 + 5*log(D_mm)

Column "H_t" was included to compare the older Sidgwick-Steavenson
equation results, which were based on empirical studies of
small monocular refractor performance with the more
modern Clark-Blackwell-based equation. Compare the
columns labeled "Bortle-3 - limiting sky magnitude 6.6" and
"H_t - 6.6".

In summary, the older Steavenson equation, based on empirical tests
with telescopic refractors, gives a similar rule-of-thumb estimate to
the Clark-Blackwell-based equation, but Clark-Blackwell-based
equation:

1) is particularized for binoculars, not monocular telescopes;
2) easily incorporates the effect of differences in magnification
between binoculars; and,
3) includes the effect of the background brightness of the night sky.

Data inputted to Clark-Blackwell equation was:

v_lim / Bortle
Bino Adl# A M 6 5 4 3 2 1
7x35 41 35 7 5.1 5.6 6.1 6.6 7.1 7.6
10x50 71 50 10 5.1 5.6 6.1 6.6 7.1 7.6
12x50 85 50 12 5.1 5.6 6.1 6.6 7.1 7.6
15x70 125 70 15 5.1 5.6 6.1 6.6 7.1 7.6
16x70 134 70 16 5.1 5.6 6.1 6.6 7.1 7.6
16x80 143 80 16 5.1 5.6 6.1 6.6 7.1 7.6
20x80 179 80 20 5.1 5.6 6.1 6.6 7.1 7.6

References:

The Clark-Blackwell-based equation is based on the
works of:

Clark, Roger N. 1991. Visual Astronomy of the Deep Sky. Cambridge
Univ. Press

Clark, R. "How faint can you see?", Sky & Telescope, 87(4):106 (April
1994)

Blackwell, R. H., J. Opt. Soc. Am. 36:624 (1946)

and the discussions of:

Mel Bartels, Nils Olof Carlin and Harald Lang circa 1997

See Carlin's web page discussion at:

http://w1.411.telia.com/~u41105032/visual/limiting.htm (1997)
accessed July 2003

The older Sidgwick-Steavenson equation was
published in:

J.B. Sidgwick. Amateur Astronomer's Handbook. (3rd ed. 1980 Dover)

and reproduced by Phillip Harrington in:

Phillips S. Harrington, Star Ware (2d ed. 1994) (John Wiley & Sons,
Inc.) republished at
http://www.stargazing.net/david/cons...manystars.html accessed
July 2003

===============================

Harald, thank you for help in nailing this down.

With Harrington's _Touring the Universe through Binoculars_, Crossen's
& Tirion's _Binocular Astronomy_, and Moore's _Exploring the Night Sky
with Binoculars_, the popularity of binocular amateur astronomy has
experienced some growth, at least as a pleasurable supplement to the
telescope observing.

The question "what is the difference between the limiting magnitude of
binoculars and telescopes" seems to come up more often.

- Kurt

Harald Lang wrote in message ...

There is a big problem in practise in trying to use any formula
predicting the visibility of extended objects. Clark uses angular
size and magnitude as input relevant for the object. But objects
differ in other respects: some (galaxies) have a very bright
core, which is relatively small, so the brightness falls off
gradually. It's not even clear where boundary is (at what
isophote?) hence the "size" is a fuzzy number. Other objects,
have a more even brightness.

I guess this inherent problem (of defining the "size" of the target
object) is why limiting magnitude is usually expressed in terms of
stellar limiting magnitude. The problem and data can be defined
precisely and with more agreement in the observing community, so the
predicted result is more consistent.

Clark's output from the input of "angular size and magnitude" is his
Object Detection Magnitude ("ODM")?

5*Log(A/y) + 0.4*5*Log(k) magnitudes.
Now we just clean up that expression:
[big math snip]
So the faintest star we can see is of magnitude
3*Log(A) + 2*Log(X) - 3*Log y + v

I've already taken up too much of your time (and everybody else's time
in this thread). But if you are up to handling one more question. (If
not, I understand.)

In Clark-Blackwell-based equation and Carlin's discussion at:

http://w1.411.telia.com/~u41105032/visual/limiting.htm

I'm not tracking if there is any adjustment for the fact of binocular
vision through two light gathering devices. For any binocular size of
D_mm, the combined light area of the two tubes is sqrt(2)*D_mm or
1.414*D_mm.

That binocular vision has some increased benefit from the use of two
fovea can be seen using a simple demonstration discussed by Harrington
in his "Why Binocular" introduction to _Touring the Universe with
Binoculars_. Using the binoculars that you where born with - your
eyes - go out and look at an area of the night-sky with few stars.
(The current washed-out sky from the full Moon is ideal for this.)
Cover one eye. You will see a reduction in the background brightness
of the sky as compared with two eyes and some increase in the ability
to see the faintest star. Two eyes see "brighter" and a little
"deeper" than one.

In the Clark-Blackwell-based equation,

1) is an adjustment for binocular seeing implicit in the model because
Blackwell's detection data was based on the ability of naked-eye
observers* (using their natural binoculars) to see faint objects?

2) is there a collecting area adjustment in the equation for the use
of the two fovea and two objectives?

3) is the fact that the collecting area of a binocular is 1.414*D_mm
the area of one monocular side not relevant to determining limiting
magnitude in the Clark-Blackwell model?

* - A group of young women aged 18-21 with 20/20 vision attending an
art school during the 1940s.

BTW, did you download Schaefer's article? I put it temporarily on
my web site, and I'll take it away if you are done with it.

Yes, thank you again. I'm still reading that and studying your
explanation of Clark and Blackwell in your post.

1) Has your and Carlin's equation ever been tested with a group
of
binocular users to assess it's empirical accuracy?
Not that I know of, but see comment later.
[big snip]
In this sense, it has been empirically tested, but in a
different context -- not by applying it directly to observation
of faint stars in binoculars (AFAIK).

I'll add to my list of proposed observing projects, maybe finding a
suitable open cluster for binoculars ( analogous to Clark's clusters
for testing the limiting magnitude of deep sky class telescopes ), and
collect some empirical evidence. If I do carry through, I'll forward
anything I systematically collect to you.

- Thanks again - Kurt

(edz) wrote in message . com...
Proposed "rule-of-thumb"
[big snip of table]

I have mag 5.1 and sometimes mag 5.4 to mag 5.6 skies. Even under the
best conditions in mag 5.4 skies I could not even hope to see 13th
magnitude stars with any binocular, let alone a 50mm binocular. [snip]
I can only say, from all the test results I have recorded, the predicted
results are significantly too high.

Thanks for catching it. I blew the math, for which I apologize. A
corrected post and table has been made.

- Kurt

P.S. - I'll slink off now to the talk. and alt. hierarchies for a
suitable period of penance. -

edz wrote:

Limiting magnitude where faintest
naked-eye visible star is
5.1 5.6 6.1 6.6
Bino Adler#
7x35 41 12.0 12.5 13.0 13.5
10x50 71 12.8 13.3 13.8 14.3
12x50 85 13.0 13.5 14.0 14.5
16x70 134 13.6 14.1 14.6 15.1
16x80 143 13.8 14.3 14.8 15.3
20x80 179 14.0 14.5 15.0 15.5
[snip]
Since I have't yet followed thru the formula to try and determine for
myself which factors are giving the most weight to the result, I am
not sure why the predicted results are so high. I can only say, from
all the test results I have recorded, the predicted results are
significantly too high.


Yes, as I pointed out in another post, they are three magnitudes
too high given the formula I gave.

Cheers -- Harald

Kurt wrote:

Clark's output from the input of "angular size and magnitude" is his
Object Detection Magnitude ("ODM")?


I don't have his book here (I'm currently in my country house),
but as I recall, he is trying to determine two things:

- Is a given object (defined by size and brightness) at all
detectable in a particular telescope (i.e., aperture)
- What magnification "should" be used so as to see the object
"best"?

This latter question isn't as clear as it might seem at first.
Furthermore, Clark slips on the math here, so many of his tables
are out of whack. (For instance, he erroneously draws the
conclusion that a *smaller* aperture in some cases should use a
*higher* magnification.) However, his approach is innovative, and
his book was an eye opener for me. Too bad he is not about to give
out corrected edition, as it seems.

As an aside, Bill Ferris' article in the current S&T is based on
Mel Bartel's programme. He (Mel) is one of the persons who
spotted Clark's error, and his programme was written to correct
for that (his correction is not the same as the one Nils Olof
uses [in another context than this binocular formula], though. We
found that Mel and I had one idea of what was the best procedure,
Nils Olof and Roger -- when Roger eventually recognised his error --
had another.)


In Clark-Blackwell-based equation and Carlin's discussion at:

http://w1.411.telia.com/~u41105032/visual/limiting.htm

I'm not tracking if there is any adjustment for the fact of binocular
vision through two light gathering devices. For any binocular size of
D_mm, the combined light area of the two tubes is sqrt(2)*D_mm or
1.414*D_mm.


First, the data that is used, i.e., Blackwell's data, are all
based on binocular vision (i.e., with two eyes.) Second, for the
binocular formula, I don't think it matters, as long as we use
equally many eyes when we determine the naked eye limiting
magnitude as when looking through the binocular/monocular.

I wouldn't use the binocular formula for telescopes at high
magnification, though, for at even moderately high magnification
(small exit pupil) we come far outside the range of Blackwell's
data (too dark sky background), so we don't know if the relation
holds to a reasonable approximation any longer that it can be
used even as a rule of thumb.


In the Clark-Blackwell-based equation,

1) is an adjustment for binocular seeing implicit in the model because
Blackwell's detection data was based on the ability of naked-eye
observers* (using their natural binoculars) to see faint objects?

2) is there a collecting area adjustment in the equation for the use
of the two fovea and two objectives?

3) is the fact that the collecting area of a binocular is 1.414*D_mm
the area of one monocular side not relevant to determining limiting
magnitude in the Clark-Blackwell model?


I think the answer I gave above applies to all these questions.

IIRC Clark never adresses the question about monocular vs.
binocular vision. Schaefer (in the paper you downloaded) does,
though, and he claims that looking with two eyes increases the
"sensitivity" by 0.38 magnitudes, which corresponds to a factor
of square root of two [he refers to a study by Pirenne, 1943, in
Nature (152).] But as I said, if you use monocular vision when
you determine "v" in the equation

m = 3*Log(A) + 2*Log(X) + 0.6 + v

then m will obviously decrease by the same amount as v, so the
correction will be made automatically, whatever the magnitude of
the correction might be.

Cheers -- Harald
P.S. The discussion between me, Mel, Nils Olof and Roger Clark
took place spring 1999. Nils wrote his web page 1997, so our
discussion had no impact on that. I shouldn't have brought up
that discussion at all -- it had nothing to do with the issue at
hand.

(Paul Schlyter) wrote in message ...
In article ,
Harald Lang wrote:

So is -3*Log(a) + v pretty much a constant? To answer that question
properly would of course require asking a lot of people to measure
both their dark-adapted pupil size . . . .

Harald's simplifying assumption of a 6.3 mm exit pupil is probably
reasonable. In Schaefer's 1990 article, Schaefer notes the following
general formulae for the size of the human exit pupil by age:

D_exit_pupil = 7 mm exp(-0.5[A/100]^2])

which yields:

Age D_ep
10 7.0
20 6.9
30 6.7
40 6.5
50 6.2
60 5.8
70 5.5
80 5.1
90 4.7
100 4.2

Schaefer, B.E., Telescopic Limiting Magnitudes, Pub. Astron. Soc. Pac.
102:212-229 (Feb. 1990)
citing prior work of:
Kadlecova, V., Peleska, M., and Vasko, A. Nature 182:1520 (1958)
Kumnick, L.S., J. Opt. Soc. Am. 34:319 (1954)

The simplier way to handle varying exit pupils is a disclaimer that
younger observers can add 1/2 a magnitude to the result.

To answer that question properly would of course require asking a lot
of people to measure . . . their visual magnitude limit under dark skies.

To me, that's a feature and not a bug. - On the one hand, using mag
6.5 or 6.6 as a "good skies" simplifying assumption gives a useful
starting-point for amateurs to generally discuss "what is the limiting
magnitude of binoculars." At the same time, the ability to break out
predicated limiting magnitudes by the on-site sky brightness with
minimumal computation, or with a simply-generated table or graph,
better reflects an intermediate observer's experience. It can be
useful in explaining occassional disputes over "I can't believe you
claim to have really seen _that_," or when enjoying the night sky,
making the your own honesty check of "file that one in the 'imaginery
averted vision' file." -

Personally, I subscribe to Brian Tung's comment, made earlier in this
thread, to the effect that limiting magnitude guidelines are not going
to stop me from actually trying to eyeball an object through a
telescope or binocular - unless its two or three mag rungs down the
visual acuity ladder. -

- Kurt

Harald, thank you for help in nailing this down.

With Harrington's _Touring the Universe through Binoculars_, Crossen's
& Tirion's _Binocular Astronomy_, and Moore's _Exploring the Night Sky
with Binoculars_, the popularity of binocular amateur astronomy has
experienced some growth, at least as a pleasurable supplement to the
telescope observing.

The question "what is the difference between the limiting magnitude of
binoculars and telescopes" seems to come up more often.

- Kurt

I must admit there is an impressive array of figures here and fomula
to support those figures. However, I respectfully disagree that you
have nailed this down.

Unless you were to gather a collection of binoculars of varying
magnifications and aperture, all by the same manufacturer, then maybe
under those circumstances your tables might be accurate. However,
this is not reality.

Binoculars are of a tremendously wide variety of components and
quality of manufacture. These formula go so far as to take into
consideration the affects of magnification, aperture, sky background
and even eye pupil. And yet they fail to take into account one of the
single most inportant qualities of the binocular that helps provide
the ability to see and that is contrast.

Contrast provided by the optical system of binoculars may be equally
as important as aperture. I have already shown by repeated testing
and have publised results showing that a smaller aperture with better
contrast is capable of seeing more than a larger aperture with lesser
contrast. Your tables would never show this. I don't fully
understand why that is ignored, but it is my opinion that until it is
incorporated into the calculation, you will not have a representative
indication of the predicted outcome.

Of course, you could say that the tabulated data would give a good
indication of the performance of a given set of binoculars of
equivalent quality. However, that limits it's usefullness. It is
more likely to be useful data if the end user were able to apply a
factor allow the placement of their own binoculars into the field of
the data by some means that accounts for the endlessly varying levels
of quality and premium (or lack of) features incorporated into the
manufacture.

edz

(edz) wrote in message . com...

edz, I think we are in basic agreement but just differ on the user
groups for which these mathematical models will have the best utility.
Undoubtedly detailed equations exist that yield more precise results
for the optical-engineering, manufacturing and advanced amateur
observers, by using more user, scope and site specific variables.
Even more precise models could incorporate manufacturer and brand
specific data.

But for the beginning and intermediate amateur, a less precise
mathematical model suffices, as long as the amateur understands its
limitations, assumptions and resulting lack of precision.

For other followers of this thread, sometimes a picture is worth a
thousand words -

Harrington's table is based on a Sidgwick-type model of telescope
optics - the equation Mag_limiting = 1.8 + 5*log(D_mm) discussed in
this thread. It looks like this:

http://members.csolutions.net/fisher...steavenson.gif

Several in this thread have stated, paraphrasing, "That's too low; I
can see deeper than that."

Steavenson's empirical adjustment simply recognizes similar
experiences by other observers using small refractors that an
unadjusted Sidgwick-type equation is too low. Steavenson added
another 1.9 mag to the Harrington-Sidgwick-type equation. It is
depicted as the top line of data, also in:

http://members.csolutions.net/fisher...steavenson.gif

I believe that a couple of people in this thread commented about the
results of the Steavenson equation, "That's too high."

Others, including yourself, Ed, noted that the Steavenson or
Sidgwick-Harrington-type results do not account for the increased
effect of contrast that results from using higher magnifications -
i.e., when you use higher magnifications, the background sky looks
darker and this increased contrast with the object allows you to see
fainter objects.

A Steavenson-type equation also does not include any adjustment for
the loss of contrast from light-polluted skies.

In response to these problems, Schaefer (1989-1990) and Clark (1994)
independently developed improved models. A Clark-Blackwell-type
equation is less precise, but is more easy to compute than the
Schaefer algorithm. For some archtypical binoculars,
Clark-Blackwell-type equations yield results that look like this:

http://members.csolutions.net/fisher...de%20clark.gif

This Clark-Blackwell graph provides much more useful information for
the amateur observer than the Steavenson graph. While the
magnification of the binoculars are fixed, predications are made about
their performance in various skies of improving Bortle qualities - the
magnitudes along the x-axis.

As a matter of practical utility to the amateur binocular observers,
this:

http://members.csolutions.net/fisher...de%20clark.gif

has a lot more practical, daily utility to the amateur observing
community than this:

http://members.csolutions.net/fisher...steavenson.gif

even though this:

http://members.csolutions.net/fisher...de%20clark.gif

may be less precise than a more detailed computation using many of the
Schaefer javascript calculators available on the web or a
brand-specific table obtained from a manufacturer. It is easier for
beginners and other amateur observers to use as a rule-of-thumb, than
this:

http://www.astro.columbia.edu/~ben/star.html

For discussion purposes, the following graph compares the two models
in one view (Steavenson and Clark-Blackwell), although to some extent
this is an invalid comparision of "apples" and "oranges":

http://members.csolutions.net/fisher...k%20limits.gif

This graph again illustrates that the Clark-Blackwell-type equation
gives the amateur more useful information than the Steavenson.

Typical questions amateurs ask themselves and each other a

1) Could my friend really have seen (insert name of object of your
choice) using (insert binocular or telescope and brand name of your
choice) in that a (insert Bortle scale number of your choice) sky?

2) Considering I was half awake at 3:00am, is it reasonable for me to
claim to others that I saw (insert name of object of your choice)
using (insert binocular or telescope and brand name of your choice) in
that a (insert Bortle scale number of your choice) sky?

3) Do binoculars have an intrinsically deeper limiting magnitude than
the same sized monocular, since they have a larger light-gathering
area ( 2*A, instead of 1*A)?

With respect to this third question, Steavenson or
Harrington-Sidgwick-type limiting magnitude equations leads beginners
off on a tangent of trying to construct an "equivalent" sized
monocular of a diameter equal to sqrt(2) times the diameter of the
binocular lens.

A Clark-Blackwell type equation leads beginners to a better
understanding of the counterintutive performance of binoculars and of
differing sizes of monocular telescopes. Binoculars see "brighter" and
a little bit, but not appreciably more, "deeper" than the same sized
monocular even though they have 2x the light-collecting area. The
reason for this counterintutitive performance is the incremental
beneficial effect that contrast gives to limiting magnitude is
principally an effect of increased magnification. Binoculars (except
for some binocular-telescopes with interchangeable eyepieces) have
fixed magnification, and by design, cannot exploit their available,
but unused higher magnification, which would see to a better limiting
magnitude. I believe this is what you, Brain Tung and Harald Lang have
tried to explain to us thick-headed beginners using differing
technical language.

In closing, I hold people with the math and engineering skills,
supplemented by years of experience, like yourself, needed to
understand the nuances of the implementations of these optical
algorithms, in great deal of awe.

A less precise equation might be the "better" engineering equation for
the purposes of utility to the broader amateur observing community,
while a more detailed model better serves the needs of the
manufacturing and engineering community.

Contrast provided by the optical system of binoculars
may be equally as important as aperture. . . . Your tables
would never show this. I don't fully understand why that is
ignored, . . .

If I understand the Clark-Blackwell type formulae correctly ( a
dubious assumption at best - ), the contrast-detection benefit of
increased magnification is incorporated into this rule-of-thumb
equation. That is what the underlying Blackwell study of the ability
of the unaided-eye to detect contrasts between faint objects is all
about.

Regards - Kurt

(edz) wrote in message . com...

Thanks, this is a great plain english summary of governing principles.

Regards - Kurt