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Dear Dr. SETI: CC, Scotland
The Doctor Responds: Now that we agree on the basics, let's run the numbers. A reflecting telescope (optical or radio, it doesn't matter) has a "magnification" which can be described in terms of power gain. At 100% efficiency (which we can never achieve, because the real world isn't perfect), we can calculate that power gain. It's actually easier to calculate voltage gain, and then square it, since power ratio varies with the square of voltage ratio. The relationship is:
Voltage gain ~ (Reflector circumference) / (wavelength) where both are measured in the same units. Of course, circumference equals diameter times pi (for a round mirror), and diameter is twice radius, which is why all the textbook formulae contain a (2 pi * r) factor. Next, power gain = (voltage gain)^2. Think of this as your "magnification" of light. Finally, in radio we usually convert power gain to dBi, a logarithmic shorthand. dBi means deciBels compared to an isotrope. An isotropic radiator is a theoretical (can't actually build one, buy one, or find one in nature) ideal omnidirectional antenna. Omnidirectional means it radiates equally poorly in all directions. Anyway, the conversion is:
dBi = 10 * log (power ratio) So let's put all this together and run some examples. First: my optical telescope (a Celestron model C8) has a 100 mm radius reflector. That mirror has a circumference of (2 * pi * 100 mm) ~ just over a half meter. I use it to magnify visible light which has a 500 nanometer wavelength. The voltage gain is (1/2 m)/(500 nm) = 1,000,000!(only I won't really get anywhere near that performance, because my eyepiece and mirror are quite imperfect.) Next: let us consider the world's largest radio telescope, Arecibo radio observatory, at the hydrogen line. The mirror at Arecibo has a 152 meter radius. Its circumference is (2 * pi * 152 meters) ~ just under 1 kilometer. I use it to receive "light" at a wavelength of 21 cm. So: Voltage gain is (1 km)/(21 cm) ~ 5,000.(only Arecibo won't really get anywhere near that performance, because its eyepiece {the feedhorn} and mirror {the reflector} are quite imperfect.) Well, actually I've misled you a little here, because I'm comparing apples to kumquats. If we level the playing field by restricting each instrument to its intended application, we see that my Celestron telescope and my 12 foot radio telescope are just about equivalent in performance (see the Appendix below). But this doesn't invalidate our... Conclusion: In terms of power gain, my Celestron telescope is tens of thousands of times more sensitive than Arecibo. Which explains why optical SETI is so appealing. Appendix: Optical Magnification and the Eye You may be wondering why the typical amateur optical telescope has a magnification of about 400, while we have just computed its power gain at 1,000,000,000,000. The apparent discrepancy is because antenna gain is calculated relative to that elusive isotrope, while optical magnification is generally specified relative to the naked eye. So if we know the optical telescope's gain, and want to know its magnification, we need also to compute the "gain" (relative to isotropic) of the human eye. Fortunately (although optical physics has its own formulas) straight antenna theory can apply here as well.
Let's consider the human eye to be an antenna, whose aperture to electromagnetic radiation is the pupil. Say the radius of the dilated pupil (it is dark when we use a telescope, after all) is on the order of 2 mm. We can use the same equations we use for a parabolic antenna: Voltage gain ~ (pupil circumference) / (wavelength)which makes the naked human eye an optical Arecibo! Now, as to the question of the magnification of my Celestron, let's assume I have an eyepiece which perfectly couples to my pupil (I don't, which means it's efficiency is less than 100%, but this is the ideal case.) Theoretical magnification would be the ratio of the antenna's power gain to that of the naked eye. That comes to (1,000,000,000,000) / (600,000,000) = 1667. The optics people have simpler formulas for calculating magnification, to be sure (ratio of mirror to eyepiece dimentions being the favored one). We would expect the results obtained that way to correlate well with antenna theory. Next, let's calculate the power gain of that Celestron telescope, not in dBi (deciBels compared to isotropic), but rather in a new unit which I'm going to call dBe (deciBels compared to the human eye). The relationship should be (and in fact is): Telescope Gain (dBe) = Telescope Gain (dBi)  Human Eye Gain (dBi)It has been previously shown in SETI Sensitivity: Calibrating on a Wow! Signal that the "standard" 12foot dish used in Project Argus stations achieves a gain of +31.8 dBi. Comparison to an isotrope is as appropriate for radio telescopes as comparison to an eyeball is for optical telescopes. Thus it appears that, as far as their intended applications are concerned, the average amateur radio telescope (typified by the Argus station) and the typical optical telescope (exemplified by the Celestron C8) are roughly equivalent in performance. Convolusion [Convoluted Conclusion]: Whereas the human eye is the optical equivalent of an Arecibo, the standard Project Argus station is the microwave equivalent of a Celestron! And since significant optical astronomical discoveries have been made with Celestronclass telescopes, we have every reason to expect significant microwave astronomical discoveries to be within the grasp of our Project Argusclass amateur radio telescopes.

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