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article published in Webb Society Quaterly Journal no 108 (Avril 97)
Comparing Deep-Sky observations is not a new problem. However, it is only one more problem among others in different Amateur Astronomy fields: planetary surfaces, comets, variable stars, etc. What follows is a new approach suggesting the use of an instrumental criterion in order to improve comparison, especially that of either written or drawn observations of the Deep-Sky and generally between observations from all amateurs' fields of interest (photography, ...).
The observer is a system which is hard to calibrate: there is no easy and popular test among amateurs that is able to describe precisely the sensibility of the eye to low-light levels, and its variations between individuals. Such a variable factor is thus hardly comparable. Transparency and seeing can be estimated with several well-known techniques: estimating the eye's visual limiting magnitude, the visual aspect of the Milky Way or various Deep-Sky objects obtained with the eye alone (M31, M33, M13, etc.), as well as estimating the seeing derived from single and double stars' twinkling with or without telescope, etc. Yet these methods depend on the characteristics of one's eye, and we have already stated that the eye is really hard to evaluate in that regard. So, we can only assert that the quality of an observer's eye and the quality of an observing site are so dependent on each other that they are impossible to separate !
Concerning the telescope, there seems to be no problem: it is the diameter of the main collecting lense or mirror that tells best the collecting power of the instrument. The latter is easy to define and compare: a refractor of 100mm in diameter will perform 33% less than a telescope of 150mm (always from the light-collecting point of view). This is not always the case. In fact, knowing the diameter of an instrument is not knowing its collecting power. Let's analyze the light-path through the objective of a refractor: light goes through two lenses (in a classical design) and then passes through the multiple lenses of the ocular before arriving at your eye. For a reflector, a certain amount of light is stopped by the secundary mirror (in most classical systems), and the rest reflects itself on the primary and secundary mirrors, and then goes through the ocular before reaching the eye.
One must know that light doesn't reflects on a mirror or doesn't refract through a lens without a part of it being absorbed and/or dispersed. A lense provokes a lightloss every time the light beam passes from air to glass and vice versa (it is the concept of air to glass surface). There is also absorption within the glass itself (2% per centimeter of thickness). A mirror absorbs some light with its so-called "reflecting surface". All these losses from lenses and mirrors depend on the quality of the glass and coatings for refractors, and on the quality of the mirrors' reflecting coatings for reflectors.
An air to glass surface of an uncoated lens reflects (stops) 4,5% of the light. If this surface is "anti-reflection coated" (ARC) with magnesium fluoride, losses are reduced to about 1,5%, and even go down to 0,5% if the lens is "multi-coated" (MC). New coatings called "wide band" (WBMC) are now introduced on certain series of oculars and losses appear to be less than 0,2% per air to glass surface. To test your binocular, refractor or ocular, just check the reflections of white clouds on the glass surface: if the reflections are white and bright, there is no coating at all; if the reflections are blue and light blue, the coating is ARC, if they are greenish, brownish or reddish, the coating is MC, and if they are deep blue or very dark, the coating is WBMC. If you count the air to glass surfaces of your refractor of binocular objective, you will know exactly the amount of light loss in the studied design. A typical refractor has 3 to 4 air to glass surfaces that are ARC with a 1 centimeter thickness , a fluorite refractor has 2 to 4 air to glass surfaces that are MC with 2- or 3- centimeter thickness. A binocular has 10 to 14 air to glass surfaces: models between £40 and £150 ($100-300) are ARC and models beyond £150 ($300) are MC. In any case, do check for yourself !
A=absorption et R=reflection.

A silver mirror surface (AG) absorbs 7,6% of the light but the commonest kind of surface mirror, that is protected aluminium (ALp), retains 11,0% of it. Some "Schmidt-Cassegrain" and other designs are proposed with "enhanced coated" mirrors that absorb only 4,9%. For the reflectors, one must keep in mind that there is another interesting thing for the collecting power, which doesn't apply to refractors or binoculars. The obstruction of the secundary mirror is of major interest because it conditions the amount of light likely to reach the primary mirror.. There are two ways of considering it: you can calculate the obstruction by dividing the secundary mirror's diameter by the primary one's, and also by dividing the secundary mirror's surface by the primary's. The latter solution is the most adequate for the collecting power, which is much better expressed in terms of surface than of diameter.

You can calculate easily the obstruction using the following simplified formula:

where O, d and D stand respectively for obstruction in %, secundary's diameter and primary's diameter expressed in the same unit (centimeters or inches).

For example, a Schmidt-Cassegrain of D=203mm has a secundary of d=65mm; the obstruction is 652/2032=0,102 being 10,2%, -which means 10,2% of the primary mirror remains unused.

The last optical system, the most critical one for light loss, is the ocular. Without going any further into detailed descriptions of the numerous designs, it is sufficient to know that 2 to 8 lenses are to be found in classical and commercialised oculars. That means that 4 to 16 air to glass surfaces should be coated, which often they are not, only the external lenses being coated. The coatings can be external ARC, all surfaces ARC, external MC or total MC, etc.

Here is a list of oculars with their barrels in inches and transmission in percentage:

RAMSDEN 0,965 85%
HUYGENS 0,965 89%
KELLNER 0,965 84%
KELLNER 1,25 91%
ORTHOSCOPIQUE 1,25 91% (98% for the best ones)
PLOSSL 0,965 89%
PLOSSL 1,25 91% (98% for the best ones)
ERFLE 1,25 78 - 85% according to the coating
ULTIMA 1,25 90%
WIDE FIELD 1,25 96%
NAGLER1&2 1,25 96&95%

So far we have been concerned with light losses and absorption. It is convenient to convert the given data in order to express them in term of transmission: a lens or a mirror that loses 5% of the light is actually transmitting 95% of it. We shall now combine all the collected transmission data into a total transmission capacity (CT) of the optical system used (in percentage). This CT can be calculated by multiplying the unobstructed percentage of light (UPL) by all the transmission values of the system, Tobj being the objective's transmission (primary and secundary mirrors, lenses, ...) and Toc the transmission of the ocular used. The whole can be summed up in the following formula:

CT = UPL x Tobj x Toc

For example, we shall take the author's Dobsonian (D=445mm, d=107mm). The surface obstruction here is 5,8% (UPL=94,2%) and the mirrors are aluminized normally (transmission of 89% by air to glass surface), so Tobj is thus equal to 79,2% (=0,89x0,89). The use of a Nagler eyepiece corresponds to a Toc of 96%. The cumulated effects of UPL, Tobj and Toc are giving a CT of 0,942x0,792x0,96 = 71,6%. With the following formula, you can then determine the virtual aperture (AV) of your instrument, -which would apply to an hypothetical instrument without any obstruction, light loss and perfect reflecting or refracting surfaces.

where OV is the virtual aperture and D the diameter of the instrument, both expressed in the same unit. For the Dobson, OV is 377mm.

As a conclusion, I would like to share with you the results that can be obtained from the first example I mentioned in this article, dealing with a comparison between a refractor of diam.100mm and a reflector of diam.150mm: the refractor is a Fluorite with a MC objective and the reflector is a Newtonian obstructed at 6%, and they are both used with the same ocular. For the refractor: CT=96%, OV=98mm. For the reflector: CT= 74.4%, OV=129mm.

Comparing 150 to 100mm and 129 to 98mm is not the same story. The fortunate owner that can compare several instruments, or the club afficionado that has a vast experience of looking through various types of instruments, doesn't need the above formula to "feel" the collecting power of an instrument compared to another one. But the beginner or simply the observer that wants to compare his notes or drawings with others' can find the virtual aperture an easy and handy tool. The arbitrary notion of diameter must not remain the only way to compare the collecting power of astronomical instruments.

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