Count the Pleiades
As a small child I remember being driven back to Portland at night after a visit to relatives in the countryside. I laid in the back of the station wagon, peering up at the sky through the rear window. The stars were so brilliant against the darkest black of skies! It hurt my eyes to look at the brightest stars! What a contrast to the washed out city skies of Portland, even in 1960!
Since most astronomical objects are quite dim as seen through the eyepiece, it behooves us to understand the conditions that permit the best visibility. In the retina, light is focused onto cylinder shaped rod cells and cone shaped cells, initiating a photochemical process that results in an electrical signal to the brain. Light sensitive molecules, called chromophores are held by proteins. During darkness, a protein called rhodopsin, in a pigment called visual purple, accumulates in the rods. The amount of visual purple determines the sensitivity of the eye to light, and mostly reaches maximum after thirty minutes, with some improvement of up to two hours. Cone proteins are very similar to rhodopsin, in fact, differing in only one key amino acid. This single difference in the amino acid at position 122 results in the cone proteins resetting 100 times faster than rhodopsin after absorbing light. Visual purple sensitizes our eyes by many thousands of times. The change in light sensitivity when the iris opens and closes is very slight in comparison.
In greater detail: Rod cells are long hot dog affairs about 1/125 inch long with the synaptic and nucleus end towards the incoming light. Each rod cell contains 2000 stacked disks which contain up to 100 million modules of light sensitive pigment rhodopsin. Each rhodopsin molecule has two parts, the opsin protein and the light absorbing substance retinal, derived from vitamin A. Before light hits it, the retinal is in the isomer form 11-cis-retinal. When a packet of light energy is absorbed by rhodopsin, it twists 11-cis-retinal to form another isomer, all-trans-retinal. This changes the configuration of the opsin protein converting the whole molecule from rhodopsin into metarhodopsin II in 1/1000 sec. Each metarhodopsin II molecule activates hundreds of molecules of transducin, a protein. Each of these activates an enzyme, phosphodiesterase, which alters the structure of thousands of molecules of the neurotransmitter cGMP, cyclic guanosine monophosphate. Levels of cGMP which in darkness are high, are thus reduced, closing channels which allow sodium ions to flow through the cell membrane. cGMP's function is to keep these channels open. In darkness, these open channels allow the flow of positively charged sodium ions, called the dark current, to counter the diffusion of positive potassium ions out of the cell, making the inside slightly negative. When light hits the rod cell, sodium entry is reduced, charging the cell's interior more negative, called hyperpolarization. Hyperpolarization reduces the release of neurotransmitters from synaptic vesicles, resulting in signals being sent to the brain. All chemicals are cycled within 1/5 second. Rods and cells form the retina's outer layer. Within this layer are incredibly complex layers of interconnected neurons. Hundreds of rods and cones feed into a dozen or so bipolar cells, which then signal a single ganglion cell, carrying out what might be called data compression. The ganglion cell's firing rate is determined by the sum of the signals from all of its photoreceptor cells, the rod and cone cells. 130 million rod and cone cells are compressed into 1 million ganglion cells in the retina. Nerve signals from these 1 million ganglion cells travel along the optic nerve to the cross over junction called the optic chiasma. The signals continue onto part of the thalamus known as the lateral geniculate nuclei (LGN). They then continue onto the visual cortex of the occipital lobes. The main reception area here is called V1, essentially a replica of the retina. Areas V2 and V3 separately process various aspects of vision like shape, color and movement. These areas communicate with other cortical areas including the temporal lobe and the language center. Here we become consciously aware of what we see. Pathways from the LGN also bypass V1, connecting to the brain's motor system, making it possible to sense and react to visual signals without the extra time needed to become consciously aware, this is called blindsight. Other subconscious pathways include the superior colliculi which cause the visual startle reflex to anything unusual.
Astronomers measure brightness in units of magnitude. The eye roughly follows a logarithmic response, with a magnitude equal to a change in brightness of about 2.5 times. A difference of five magnitudes equals a change of brightness of 100 times.
Rod cells are about four magnitudes more sensitive to light than
cone cells. Cone cells are concentrated in the very center of the
retina, with rod cells reaching a peak concentration at 20 degrees
off center and tapering to half their numbers at 60 degrees off
center. Interestingly, rod cells are a bit more concentrated away
from the nose. This leads to the following graphic, which shows
where to best place a dim object for observation in the eyepiece.
You will want to look at a point about 15 degrees away from the
object, slightly above it and with the object aimed at your nose.
Increasing aperture concentrates more light into the star image, hence making it brighter. Here is a table of aperture and magnitude values, from a Sky and Telescope magazine article table published elsewhere on the net:
Telescope Limiting Magnitude
Aperture Probability of Detection
Inches 98% 90% 50% 20% 10% 5% 2%
9.7 10.2 10.7 11.2
11.7 12.4 13.2
2 11.2 11.7 12.2 12.7 13.2 13.9 14.7
3 12.1 12.6 13.1 13.6 14.1 14.8 15.6
4 12.7 13.2 13.7 14.2 14.7 15.4 16.2
5 13.2 13.7 14.2 14.7 15.2 15.9 16.7
6 13.6 14.1 14.6 15.1 15.6 16.3 17.1
7 13.9 14.4 14.9 15.4 15.9 16.6 17.4
8 14.2 14.7 15.2 15.7 16.2 16.9 17.7
10 14.7 15.2 15.7 16.2 16.7 17.4 18.2
12.5 15.2 15.7 16.2 16.7 17.2 19.9 18.7
14 15.5 16.0 16.5 17.0 17.5 18.2 19.0
16 15.7 16.2 16.7 17.2 17.7 18.4 19.2
18 16.0 16.5 17.0 17.5 18.0 18.7 19.5
20 16.2 16.7 17.2 17.7 18.2 18.9 19.7
22 16.4 16.9 17.4 17.9 18.4 19.1 19.9
24 16.6 17.1 17.6 18.1 18.6 19.3 20.1
30 17.1 17.6 18.1 18.6 19.1 19.8 20.6
36 17.5 18.0 18.5 19.0 19.5 20.2 21.0
A website that calculates limiting magnitude http://www.astro.columbia.edu/~ben/star.html
For extended objects, things are not so simple. For starters, it is not possible to increase the surface brightness of an extended object by increasing the aperture. An example: take an object of 10 magnitude/ square arcsecond as seen by the unaided eye at night, exit pupil open to 7mm. Now, look at the object through a 10" scope. If there is no magnification to the image, the surface brightness will increase by the ratio of the scope's aperture to the eye's aperture squared, or, (10"/0.3")^2 =~ 1000x. However, in order to fit all of the light from the 10" aperture into the eye's exit pupil, we must use at least 33x. 33x will dilute the image brightness by 33^2 =~ 1000x, so we are back where we started. In fact, because of mirror coatings not reflecting 100%, and the small obstruction caused by a diagonal, the image brightness per area will actually be a little less than with the unaided-eye!
This leads to the interesting conclusion that sky background brightness as seen in the eyepiece is entirely dependent on exit pupil. At a given location on a given night, no matter the size of scopes, if they are giving the same exit pupil, then the sky background brightness will be very similar.
So how can we see the object in the scope? The eye is a marvelous detector of low contrast faint objects, but the light must fall on large numbers of rod cells so that the eye-brain can detect the slight contrast difference between object and background. The slighter the contrast, the more rod cells that the object's light must fall on in order to generate a signal difference between object and background. By increasing the telescope magnification, the object is magnified so that its light falls on many rod cells. The ratio of brightness between object and background, or contrast between object and background, stays constant because the increase in magnification that dims the object also dims the sky background in equal amounts. The best magnification to detect an object is the magnification that gives the best eye detection contrast value for a given apparent object size and sky background.
The best reference on visual astronomy is Clark's book, titled "Visual Astronomy of the Deep Sky". It is a marvelous book, and is worth many readings. Clark has added additional comments since the book's publication, at http://clarkvision.com/visastro/omva1/index.html
For Nils Olof Carlin's analysis of Blackwell's original data, please see http://www.melbartels.com/visual/nils/blackwel.html. Here, Nils shows that the best contrast comes when the background is dimmed below visual detection, and the object is about one degree in apparent size.
In extensive discussions between Roger Clark, Harold Lang, Nils Olof Carlin, and myself, it became clear that there is more than one way to define 'optimum detection magnification'. I like to think of it in terms of a particular telescope and object and then ask what is the magnification that gives the largest contrast difference between object and sky background.
For my odm.zip that calculates this magnification, go to http://www.melbartels.com/dnld/odm.zip
From Clark and Nils, we can make several conclusions. There are several ways to make an object detectable. Most important is to decrease the sky background. Do this by traveling to darker skies at higher altitudes. Properly baffle your telescope. When you look through the focuser with the eyepiece removed, you should only see optics and blackness. Make sure you eat healthy with sufficient vitamin A, do not smoke, do not drink, give yourself two hours to dark adapt, and avoid bleaching your eyes by avoiding bright direct sunlight for a day or two beforehand. You must block all stray light from your observing site, indeed, even viewing the sky unaided-eye can cause a loss of sensitivity. Avoid momentary lights except for the dimmest possible. Put a black cloth over your head when at the eyepiece. Memorize as much as possible so that you don't have to read charts or books during the evening. Use sufficient magnification to make the background invisible and the object about one degree in apparent size. Most amateurs today use too low of power because their scopes don't track, and because 'that's what everyone else does'.
John Dobson was the first large aperture observer to point out the advantages of high magnifications. Al Nagler, and Brian Skiff, among others, have recommended high magnifications. By increasing magnification, you are decreasing sky background brightness, and making the object larger in apparent size, both crucial to detectability. For small extended objects, you may exceed the old double star observers' rule of 50x per inch of aperture.
How dark is your sky? See Brian Skiff/s excellent article at http://www.astropix.com/HTML/L_STORY/SKYBRITE.HTM
So far we have been talking about decreasing the sky background brightness by finding darker observing sites, by baffling the scope, by preparing yourself physically, and by using higher magnification. Only now can we bring increasing aperture into the picture. A minimum aperture is needed to detect an object. Larger aperture means that the object can be magnified to a larger apparent size, making it more visible, and making ever smaller and fainter detail in the object pop into visibility. So, a larger aperture means that a wider range of magnifications can be employed on an object. This is one of the great attractions of large aperture besides the ability to detect fainter stars.
Larger aperture in city skies will always outdo smaller aperture, though many large amateur scopes are not properly baffled, causing their performance to lag particularly in city skies. However, a small scope in very dark skies will outdo even a giant scope in city skies. Besides, dark adaptation is practically impossible in city skies.
Bill Ferris has generated a series of ODM matrices that compare the variables with each other: http://members.aol.com/billferris/odm.htm
Molecular Basis of Dark Adaptation in Rod Photoreceptors http://hodgkin.physiol.cam.ac.uk/staff/lamb/Papers/Eye_Bleach/LRL98.html
The USAF Flight Surgeon manual on night vision: