This article first appeared in the Autumnal Equinox, 2000 issue of the HORIZON newsletter of the WCAC. 

Through an Eyepiece Darkly
by Dave Gill


 


In the astronomy business, we are photon collectors.  We see an object by collecting enough photons to register in our brains.  One of the fundamental arguments that constantly rages within astronomical circles is, “How faint can you see?”  The short answer to this question is that there is no answer. There are many factors that influence the faintest things you can see.  And the answer is ultimately personal and subjective.  I recently got into a discussion of this topic with Matt Oltersdorf when he innocently tried to add a calculation for this “limiting magnitude” to a web program he was developing for the WCAC page. (Stop out and see his eyepiece/telescope performance calculator on the Tutorials page of http://www.twcac.org/Tutorials/tutorials_index.htm ). This led me to tell Matt that he had opened a real can of worms - then I bravely (or foolishly) pushed forward into the fray.  I revisited some of the good references on the subject, and decided that this was an ideal topic for a HORIZON article both in print and on the Web site.  So here goes.

First, the answer is going to differ for point sources (i.e. stars) than for extended objects (galaxies, nebulae, etc.)  In this article I am going to concentrate on stellar objects.  Maybe I’ll try the extended object topic in another article - there is some really good stuff in that topic.  It is extremely useful, and some of it is counterintuitive to the “conventional wisdom”. 

Secondly, I believe that any calculation of “limiting magnitude” should truly represent some sort of limits.  This is philosophical.  But if you present a table of limiting magnitudes for various instruments that are not really limits, you run the risk of  “scaring off” people from trying to push their limits.  As we will see below, the limits are subjective and can be pushed with experience.  But if you see that you have a 13.6 limit with a 6” scope, you may never try to look for Pluto or faint cataclysmic variable stars.  Yet a calculation from another source would show that you can push it a full magnitude, and actual observers suggest that you can push it another whole magnitude - taking that lowly 6” into the mid-15’s.

There ought to be some calculable real limit to how faint you can see.  Our vision functions by well-understood chemical properties, and there are thresholds to the sensitivities of these chemical reactions. What minimum number of photons is required to trigger a visual stimulus?  This is obviously laid over the natural genetic diversity among people - maybe my eyes require 100 photons, while yours only needs 80 and Mr. X only requires 50.  But somewhere in there lies a physical limit for most of the population.  Clark (4) gives “50 to 150 photons of green light arriving over a several second period”.  This translates to a naked eye zenithal limit of about magnitude 8.5.  For years, this has been published as a “theoretical limit”, and thought to be “pie in the sky”.  But recent published articles (1) & (2) quote documented observations in the 7.9 to 8.4 range from dark sites such as the Texas Star Party and Mauna Kea.  So 8.5 is a reasonable naked eye limit after all.

According to Bradley Schaefer (3) there are three primary factors that absolutely affect how faint a person can see:

  • Telescopic aperture

  •  
  • Magnification 

  •  
  • Darkness of sky, as indicated by limiting visual magnitude at the zenith
  • Let’s talk about these one by one.  First, telescope aperture is pretty obvious.  The light gathering power is proportional to the aperture squared.  Telescopes are big visual funnels, pouring more and more photons into the observer’s eye.  Bigger funnels gather more photons.

    Magnification helps because as you magnify a field, the background sky is darkened.  Technically, the surface brightness of the whole field is reduced.  But since stars are point sources, they are reduced much, much less.  So magnification has the effect of increasing contrast.   Because of telescopic performance and atmospheric stability, one cannot keep magnifying indefinitely (Mel Bartels and his Oregon Star Party “Night of stacked barlows” notwithstanding).  I will return to magnification below.

    Sky darkness is also somewhat obvious.  The darker the sky, the more contrast you will have between the background and the star before you start.  But, the effect of magnification can compensate somewhat for a mediocre observing site - at least for stellar objects.  So, if you live in town, don’t despair of Pluto, though you might not want to waste a lot of time on Stephan’s Quintet!

    OK, we can’t put off the math forever.  (If you are a math-phobe, you can skip over the equations without losing much of the overall drift.)

    Clark (4) gives the basic formula as:
     Mt =  Me + 2.5*log10 ( D2*t/ De2)  where
      Mt is limiting telescopic magnitude
      Me is limiting naked eye magnitude
      D is telescope aperture (mm)
      t is transmission factor
                   (assumed at 0.7 - typical)
      De is your pupil diameter, fully dilated (mm)

    Plugging in Me = 8.5 (see above), De = 7.5 and t = 0.7, 

    the formula reduces to:

           Mt = 3.7 + 2.5 * log10 (D2)

    This yields the following table:

    Dia (inches)   Dia (mm)   Limiting magnitude
           2                   51                  12.2
           3                   76                  13.1
           4.5              114                  14.0
           6                 152                  14.6
           8                 203                  15.2
         10                 254                  15.7
         12                 305                  16.1
         14                 356                  16.5
         16                 406                  16.7
         20                 508                  17.2
         31                 787                  18.2
         40               1016                  18.7
    (In various other published references, other limiting magnitude equations and resulting tables can be found.  These basically differ in the initial assumptions made about limiting sky magnitude, and possibly other parameters.  See O’Meara (2) for some of the history behind these older published formulae.)

    In the article referenced above (3), Schaefer had solicited observations (250 reports were received) from amateurs which he tried to include in his algorithm.  But other effects, such as telescope type, glasses, altitude of observing site, humidity, cleanliness of optics, experience, etc. all had secondary or tertiary effects to the “big three” - aperture, magnification and observing site.  He noted a great deal of scatter in the data.  His plots of the data for 6” scopes reported actual limiting magnitudes of from 11.9 to 15.6!  Even correcting for observing sites and magnitude, there was still a scatter of about 1.5 magnitudes.  These secondary and tertiary effects can account for some of the scatter, but not all of it.  Schaefer concluded that the scatter “remains a mystery”.

    In an article (5), Roger Clark revisits the topic based on some of the recent documented observations that push the limits way beyond the above table.  He goes on to explain that the table’s results are based on a 50% probability of detection.  That means that during your observation, the star should be seen about 50% of the time.  Clark goes on to suggest that those who push the limits have trained themselves to operate at a much lower probability threshold.  This means that, for instance, if you have trained yourself to work at the 10% threshold, you can see the object for perhaps 10% of the time, and still be confident of an observation.

                        TELESCOPE LIMITING MAGNITUDE
                                     Probability of Detection

     Diameter
     (in)   (mm)  98%   90%   50%   20%  10%    5%    2%
       2      51    11.2   11.7   12.2   12.7   12.3   13.9   14.7
       3      76    12.1   12.6   13.1   13.6   14.1   14.8   15.6
       4     102   12.7   13.2   13.7   14.2   14.7   15.4   16.2
       6     152   13.6   14.1   14.6   15.1   15.6   16.3   17.1
       8     203   14.2   14.7   15.2   15.7   16.2   16.9   17.7
      10    254   14.7   15.2   15.7   16.2   16.7   17.4   18.2
      12.5 318   15.2   15.7   16.2   16.7   17.2   17.9   18.7
      14    356   15.5   16      16.5   17      17.5   18.2   19
      16    406   15.7   16.2   16.7   17.5   17.7   18.4   19.2
      20    508   16.2   16.7   17.2   17.7   18.2   18.9   19.7
    Clark comments that some of the amazing observations made by S. J. O’Meara, Barbara Wilson and others show that they have developed the ability to observe at the 5% - 10% levels for the apertures they use.  More on this later.

    What is the effect of magnification? In (4) Clark goes through the exercise of calculating limiting magnitudes for an 8” scope under dark skies and typical suburban skies.  The math is not straight forward, but the bottom line here is that under dark skies, the 8” reaches theoretically maximum dark background at 90x.  Under typical suburban skies, the same scope reaches the same dark background at 570x.   Not a particularly convenient power to use, but doable.  So in this case, it will be dependent on seeing - not many nights will allow you to push the magnification to 70x/inch!

    Given all of this, how can you improve your limiting magnification?   Obviously, get a bigger scope under darker skies and push the power.  But there are other things you can do.  One of them is to work at it. I posed the limiting magnitude question to Ernst Mayer.  Some of you may know Ernst.  He lives in Akron, and was an avid and skilled variable star observer for many years, making on the order of 70,000 variable star observations from the Barberton/Akron area.  Ernst told me that when he started observing variable stars with his 6” Newtonian, he had to work about 9 months before making his first “inner sanctum” observation (one below 13.8).  He referred to this as breaking a psychological barrier.  From then on he knew he could do it.  He told me that he had gone down to about 15.5 with a 6” Newtonian.  This puts him in O’Meara’s “10%” category - truly elite company.  He also mentioned that the star field you are studying makes a difference - no stars too bright nearby to distract or disturb good dark adaptation.  The presence of a good guide star sequence helps a lot.  He, as well as Clark, points out that the position of the object in the field makes a difference.  Our averted vision is position sensitive - it is not the same in all directions.  Your comfort and overall condition on a given night make a difference.  So warm clothes, a good observing chair and an adjustable tube are important.

    Clark (5) points out the importance of taking the time to make an observation.  He mentions that O’Meara will often spend 30 minutes trying to see a single faint star - and that he observed the field for 1 to 2 hours when trying to visually recover Comet Halley in 1985 - at mag. 19.6 in  24” telescope.  He also saw a 20.6 magnitude field star - Clark compares this feat to seeing down to 17th magnitude in a 6”! The long observation time sort of allows your eyes to accumulate light like film.  It takes practice to hold your gaze in the right position for even a minute at a time.  Think of the amount of time required to assure yourself of an observation if you are only seeing the object 5% if the time.

    If you cannot get away from the suburban glow, use a black cloth over your head to maintain dark adaptation.  Tapping the telescope can help a very faint star to be detected because we are sensitive to faint motion.  Deep breathing reportedly helps some by increasing oxygen to the brain to aid concentration.  Clean your optics to reduce scattering.  Be sure your telescope is properly baffled to reduce stray light.  Use multicoated eyepieces to eliminate internal reflections.

    Finally, I cannot emphasize enough the importance of practice and gaining confidence in your observations.  Ernst Mayer spoke of the “psychological barrier” which has to be overcome.  O’Meara has often preached the need to trust your observations.  Believe what you see.  To develop this confidence, there are various calibrated starfields published - in Sky and Telescope, in Clark’s book, in O’Meara’s book, and especially on variable star web sites.  Try some and see how deep you can go.  Then keep practicing.

                       REFERENCES

    (1) SKY & TELESCOPE, October, 1991
                  pg. 423+  by S. J. O’Meara

    (2) The Messier Objects,  pp. 29-32,
                  by S. J. O’Meara

    (3) SKY & TELESCOPE, November 1989,
                  pg. 522+  by B. Schaefer

    (4) Visual Astronomy of the Deep Sky, 
                  pp 49-53, by Roger N. Clark

    (5) SKY & TELESCOPE, April 1994
                  pg. 106+  by R. N. Clark

    Phil Hoyle sent me this nifty table for estimating what the naked eye limiting magnitude of you and your sky is.  You don’t need a star chart.  All you need is to look at the bowl of the Big Dipper or the Keystone of Hercules and count stars.  For greatest accuracy, do this when the region is high in the sky.  The number of stars you can count correlates to the magnitude of the faintest stars.

    Limiting Magnitude Estimator
                       Number of stars visible in:
     Limiting       Bowl of        Keystone of
     Magnitude   Big Dipper   Hercules
           8               39                  34
           7.9            32                  27
           7.8            31                  23
           7.7            26                  22
           7.6            23                  20
           7.5            22                  20
           7.4            19                  17
           7.3            16                  15
           7.2            15                  15
           7.1            13                  13
           7.0            13                  11
           6.9            11                  11
           6.8            10                    9
           6.7              9                    9
           6.6              9                    9
           6.5              9                    9
           6.4              7                    8
           6.3              6                    7
           6.2              4                    7
           6.1              4                    6
           6.0              4                    5
           5.9              4                    4
           5.8              3                    4
           5.7              2                    4
           5.6              1                    3
           5.5              1                    3
           5.4              1                    3
           5.3              1                    1
           5.2              0                    0

          3.3    Star connecting handle to bowl disappears
         M13 is 5.7  M92 is 6.4
         M13 is in above counts
    (Above counts do not count the four corner stars of either the bowl or keystone!)


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