Path: newsfd02.forthnet.gr!HSNX.atgi.net!headwall.stanford.edu!newshub.sdsu.edu!postnews2.google.com!not-for-mail From: canopus56@yahoo.com (Canopus) Newsgroups: sci.astro.amateur Subject: Re: Image Luminosity vs magnification (long explanation) Date: 29 Aug 2004 23:30:39 -0700 Organization: http://groups.google.com Lines: 377 Message-ID: References: <1093598175.922815@athnrd02> <1093633886.990864@athnrd02> <1093641120.500668@athnrd02> <1093642681.105451@athnrd02> NNTP-Posting-Host: 216.190.205.196 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit X-Trace: posting.google.com 1093847439 8637 127.0.0.1 (30 Aug 2004 06:30:39 GMT) X-Complaints-To: groups-abuse@google.com NNTP-Posting-Date: Mon, 30 Aug 2004 06:30:39 +0000 (UTC) Xref: newsfd02.forthnet.gr sci.astro.amateur:518684 Ioannis wrote in message news:<1093642681.105451@athnrd02>... > The calculations still show that I should see M33, for example, 0.42 > times less bright in the 20x100 binos than in the 11x80, whereas in > reality I see it about 2-3 times as bright in the larger pair. > Why the apparent discrepancy? Ioannis, Table 6 in this long explanation will help you order the perceived _unit brightness_ seen in telescopes and binoculars applying differing aperature _and_ magnification. This is a long follow-up explanation to my prior post for those wanting more background regarding the general equation for the relative brightness of the telescopic image - RB = ( D_obj^2 * theta_transmission factor) / (D_exit_pupil)^2 * M^2 Eq. 1.0 The amount of light collected by the telescopic, its _light grasp_, is the ratio of the area of the objective divided by the area of the human eye pupil, or about 7.62 mm at its maximal dilated pupil. Using the basic forumla for the area of a circle - A= (D/2)^2 * pi gives - G=((D_obj^2/4)*pi* theta_transmission factor )/( D_exit_pupil^2/4 * pi) Eq. 2.0 Ignore the transmission factor for now, by making it equal to 1 or 100%), giving: G = ( (D_obj^2/4 * pi) ) / ( D_exit_pupil^2/4 * pi ) Eq. 2.1 or by cancelling, simply - G = D_obj^2 / D_exit_pupil^2 Eq. 2.2 This light grasp brightness can be perceived by removing the eyepiece from a Newtonian reflector and observing the brightness of the image of the full Moon as seen in the primary mirror. Here you are comparing the relative brightness of the object as seen by the naked eye (the full Moon in this case), with its unmagnified light-grasp brightness as seen in the primary mirror. But this is not the brightness of the telescopic, or magnified, image seen in the eyepiece. When viewing an extended object, the radius of the area of light collected is increased from the smaller true field to the larger apparent field. For example, the greater amount of light gathered by the objective (its light grasp), looking at let's say 100 arcsecond true field, and is magnified and spread out over the 30 degree field seen in the eyepiece. This dilutes the brightness of the light collected by the objective. This effect can be seen by pointing a Newtonian reflector at the Moon. Remove the eyepiece and look at the brightness of the image in primary mirror. Insert the eyepiece, focus and compare the brightness of the telescopic, or magnified, image, to the light grasp brightness. So the _brightness of the telescopic image_ seen in the eyepiece is given by: B_tel = ( (D_obj/M)^2/4 * pi ) / ( D_exit_pupil^2/4 * pi ) Eq. 3.0 which simplifies to: B_tel = D_obj^2 / ( D_exit_pupil^2 * M^2) Eq. 3.1 The _relative brightness of the telescopic image_ is the ratio of the telescopic image brightness, given by Eq. 3.1, above, to the brightness of the naked eye image. RB = ( D_obj^2 ) / (D_exit_pupil)^2 * M^2 Eq. 4.0 Putting the transmission factor back in we get: RB = ( D_obj^2 * theta_transmission factor) / (D_exit_pupil)^2 * M^2 Eq. 4.1 But again, we'll ignore the transmission factor for now, set it to 1 or 100%, giving simply: RB = ( D_obj^2 ) / (D_exit_pupil)^2 * M^2 Eq. 4.2 To see this effect, compare the brightness of the full Moon as seen with the naked eye, to its brightness as seen in the eyepiece. Repeating the light grasp and relative brightness comparisions on the full Moon, note that: Light_grasp_brightness > Brightness_naked_eye > Brightness_magnified_telescopic or G > B_naked_eye > B_tel Eq. 5.0 In Equation 4.1 above, we have the variable of magnification, or M. But magnification can be expressed in terms of a uniform scale, the magnification per aperature inch, times the aperature of the telescope in question. David Knisely, often posts a Usenet summary of useful magnifications based on based magnification per aperature inch: ================= Table 2 - Knisely's Useful Magnifications LOW POWER (3.7 to 9.9x per inch of aperture)(6.9mm to 2.6mm exit pupil) MEDIUM POWER (10x to 17.9x per inch of aperture)(2.5mm to 1.4mm exit pupil) HIGH POWER (18x to 29.9x per inch of aperture)(1.4mm to 0.8mm exit pupil) VERY HIGH POWER (30x to 41.9x per inch of aperture)(0.8mm to 0.6mm exit pupil) EXTREME POWER (42x to 75x per inch)(0.6mm to 0.3mm exit pupil) EMPTY MAGNIFICATION (100x per inch and above) From D. Knisely, 5/14/2004 sci.astro.amateur Usenet post, Thread "Eyepiece advice, again". ================= Sidgwick's Amateur Astronomer's Handbook (1971 3ed.) at pages 29-30 has a similar table, which is expanded here at Table 3: ================= Table 3 - Magnitudes per aperature inch M_ap_inch times D_obj = M Mai D M 3.3 D 3.3D Maximum eye pupil size; minimum useful magnification 3.4 D 3.4D Maximum eye pupil size approx. per Sidgwick 3.5 D 3.5D 3.7 D 3.7D 3.9 D 3.9D 4.1 D 4.1D 4.3 D 4.3D 4.5 D 4.5D 4.8 D 4.8D 5.1 D 5.1D 5.5 D 5.5D 5.9 D 5.9D 6.3 D 6.3D 6.9 D 6.9D 7.5 D 7.5D 8.3 D 8.3D 9.3 D 9.3D 10.5 D 10.5D 12.1 D 12.1D 14.3 D 14.3D 17.4 D 17.4D 22.2 D 22.2D 30.8 D 30.8D 33.3 D 33.3D - Minimum exit pencil from eyepiece perceivable by 50.0 D 50.0D human eye; maximum useful magnification 66.7 D 66.7D ================= Substituting in the magnitudes per aperature inch into Eq. 4.2, we get: gamma = Magnitudes per aperature inch Eq. 6.0 RB = ( D_obj^2 ) / (D_exit_pupil)^2 * ( gamma * D )^2 Eq. 6.1 This simplifies to: RB = ( D_obj^2 ) / ( D_exit_pupil^2 * gamma^2 * D_obj^2 ) Eq. 6.2 RB = 1 / ( D_exit_pupil^2 * gamma^2 ) Eq. 6.3 The maximum exit pupil size (7.620mm;0.30in) is a constant. So Equation 6.2 further simplifies to: RB = ( (25.4^2) / 7.620^2 ) * ( 1 / gamma^2 ) Eq. 6.4 11.1 / gamma^2 Eq. 6.5 or, in inches: RB = ( 1/ 0.3^2 ) * ( 1 / gamma^2 ) Eq. 6.6 11.1 / gamma^2 Eq. 6.7 Again, where gamma is the magnitudes per aperature inch. We now have a method of expressing the relative brightness of the telescopic image as a function of magnitudes of aperature per inch. We usually do not think of magnification in terms of magnification per aperature inch. Rather at a particular scope, we think in terms of the magnification being used. Therefore, the following table shows the magnification applied (lines of constant magnification) and not magnification per aperature inch. ================= Table 5 Relative telescopic brightness (eyepiece brightness / naked eye brightness) and magnification Object size D_obj_in 1 3 3.5 4 4.75 6 8 10 12 D_obj_mm 25.4 76.2 88.9 101.6 120.7 152.4 203.2 254.0 304.8 Magnification applied 6 0.31 8.32 13.22 19.73 33.03 66.58 157.82 308.24532.64 8 0.17 4.68 7.43 11.10 18.58 37.45 88.77 173.39299.61 10 0.11 3.00 4.76 7.10 11.89 23.97 56.82 110.97191.75 20 0.03 0.75 1.19 1.78 2.97 5.99 14.20 27.74 47.94 30 0.01 0.33 0.53 0.79 1.32 2.66 6.31 12.33 21.31 40 0.01 0.19 0.30 0.44 0.74 1.50 3.55 6.94 11.98 50 0.00 0.12 0.19 0.28 0.48 0.96 2.27 4.44 7.67 60 0.00 0.08 0.13 0.20 0.33 0.67 1.58 3.08 5.33 70 0.00 0.06 0.10 0.14 0.24 0.49 1.16 2.26 3.91 80 0.00 0.05 0.07 0.11 0.19 0.37 0.89 1.73 3.00 90 0.00 0.04 0.06 0.09 0.15 0.30 0.70 1.37 2.37 100 0.00 0.03 0.05 0.07 0.12 0.24 0.57 1.11 1.92 110 0.00 0.02 0.04 0.06 0.10 0.20 0.47 0.92 1.58 120 0.00 0.02 0.03 0.05 0.08 0.17 0.39 0.77 1.33 130 0.00 0.02 0.03 0.04 0.07 0.14 0.34 0.66 1.13 140 0.00 0.02 0.02 0.04 0.06 0.12 0.29 0.57 0.98 150 0.00 0.01 0.02 0.03 0.05 0.11 0.25 0.49 0.85 ================= Table 5 shows the brightness of the image of an object seen in the eyepiece as compared to the brightness of the object as seen with the naked eye. For simplicity, Table 5 (and Table 6, below) omit the effect of transmission factors of various scopes, i.e. - about 95% for refractors and 70% for reflectors. Table 5 would be hard to plot as its relative brightnesses range between a high of 532 to a low of 0.01. It is easier to plot in terms of the percent of the brightest image, the 12" telescope used at the lowest magnification being the reference value of 100%: ================= Table 6 Relative telescopic brightness and magnification expressed as percent of brightest image (the 12" telescope) Object size D_obj_in 1 3 3.5 4 4.75 6 8 10 12 D_obj_mm 25.4 76.2 88.9 101.6 120.7 152.4 203.2 254.0 304.8 Magnification applied 6 0.06% 1.56% 2.48% 3.70% 6.20% 12.50% 29.63% 57.87%100% 8 0.03% 0.88% 1.40% 2.08% 3.49% 7.03% 16.67% 32.55%56.25% 10 0.02% 0.56% 0.89% 1.33% 2.23% 4.50% 10.67% 20.83%36.00% 20 0.01% 0.14% 0.22% 0.33% 0.56% 1.13% 2.67% 5.21% 9.00% 30 0.00% 0.06% 0.10% 0.15% 0.25% 0.50% 1.19% 2.31% 4.00% 40 0.00% 0.04% 0.06% 0.08% 0.14% 0.28% 0.67% 1.30% 2.25% 50 0.00% 0.02% 0.04% 0.05% 0.09% 0.18% 0.43% 0.83% 1.44% 60 0.00% 0.02% 0.02% 0.04% 0.06% 0.13% 0.3% 0.58% 1.00% 70 0.00% 0.01% 0.02% 0.03% 0.05% 0.09% 0.22% 0.43% 0.73% 80 0.00% 0.01% 0.01% 0.02% 0.03% 0.07% 0.17% 0.33% 0.56% 90 0.00% 0.01% 0.01% 0.02% 0.03% 0.06% 0.13% 0.26% 0.44% 100 0.00% 0.01% 0.01% 0.01% 0.02% 0.05% 0.11% 0.21% 0.36% 110 0.00% 0.00% 0.01% 0.01% 0.02% 0.04% 0.09% 0.17% 0.30% 120 0.00% 0.00% 0.01% 0.01% 0.02% 0.03% 0.07% 0.14% 0.25% 130 0.00% 0.00% 0.01% 0.01% 0.01% 0.03% 0.06% 0.12% 0.21% 140 0.00% 0.00% 0.00% 0.01% 0.01% 0.02% 0.05% 0.11% 0.18% 150 0.00% 0.00% 0.00% 0.01% 0.01% 0.02% 0.05% 0.09% 0.16% ================= A graphic of this table can be found at: http://members.csolutions.net/fisherka/astronote/astromath/RelativeBrightness/RelativeBrightnessMagnificationPercentChart.gif (The project directory is at: http://members.csolutions.net/fisherka/astronote/astromath/RelativeBrightness/ The above graphic can be "surfed to" via this link.) Two implications of Tables 5 and 6 and the foregoing chart, confirming general experience, is that small telescopes cannot be used with much magnification to view faint objects before the image dims and the object can no longer be seen. Larger telescopes can be used at relatively higher magnifications on faint objects before the image dims below visibility. For example, in Table 6, an 8-power 25mm binocular, a NexStar 4 at 40 power, might be expected to produce images of comparable brightness as an 8" telescope used at 120 power. My personal test was to compare the _unit brightness_ of image produced by a 10x50 binocular, a 20x70 binocular, a 6" newtonian at 40x and 100x and the 6" newt's primary mirror. The ordering of brightness generally followed that suggested by Table 6. Another method to plot Table 5 is by the log base 10 of the relative brightness. A graphic plotting by the log of the relative brightness can be found at: http://members.csolutions.net/fisherka/astronote/astromath/RelativeBrightness/RelativeBrightnessMagnificationLogChart.gif If Table 5 is expressed as the percentage of relative brightness, with the reference 100% being the highest relative brightness in each row (the 12" scope), then a constant ratios of brightness are found, as summarized in Table 1, in the prior post, reproduced here: ================= Table 1 Relative telescopic brightness as a percentage of the brightess image within a magnification class Objective size D_obj_in 1 3 3.5 4 4.75 6 8 10 12 D_obj_mm 25.4 76.2 88.9 101.6 120.7 152.4 203.2 254.0 304.8 Relative Brgt 0.06% 1.56% 2.48% 3.70% 6.20% 12.50% 29.63% 57.87%100% ================= This applies when comparing different aperatures operated at the same magnification. An Excel worksheet generating the charts and tables in this post can be found at: http://members.csolutions.net/fisherka/astronote/astromath/RelativeBrightness/20040828_RelativeBrightness.xls This is amateur work. Any corrections or pointing out things that are flaty wrong are welcomed and appreciated. Enjoy - Canopus (Kurt) References: Kitchin, Chris. 2003. Telescopes and Techniques. Springer. At 35-36. Knisely D. 5/14/2004 sci.astro.amateur Usenet post, Thread "Eyepiece advice, again" Sidgwick, J.B. 1971 (3rd ed). Amateur Astronomer's Handbook. Dover. New York. At 29-30. Resources: Project directory: http://members.csolutions.net/fisherka/astronote/astromath/RelativeBrightness/