Ask Dr. SETI ®

Dear Dr. SETI:
I understand that light is a form of electromagnetic radiation. So are radio waves. Now if a telescope can magnify light 400x, couldn't it also do the same with radio waves? It seems to me you should be able to focus your telescope on a star, put your reciver at the eye piece, and get a signal 400x more powerful! If this would work, it would cost less than a dish, and take up less space. So, why don't radio astronomers do this?

CC, Scotland

The Doctor Responds:
You have just described exactly how both (Newtonian) optical telescopes and (parabolic dish) radio telescopes work. The reason the optical telescope mangifies light hundreds or thousands of times is that its mirror is large relative to the wavelength of light being gathered. A radio telescope similarly "magnifies" its "light" hundreds or thousands of times, because its mirror (the parabolic dish -- which focuses light to its eyepiece, the feedhorn) is large relative to the wavelength it is focusing. The only problem is, the radio telescope is dealing with electromagnetic radiation about half a million times longer that visible light wavelengths, so for equivalent performance, its "mirror" needs to be about half a million times larger than the equivalent optical telescope's.

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)
where we use a base 10 logarithm.

So let's put all this together and run some examples.

First: my optical telescope (a Celestron model C-8) 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!
Theoretical power gain is (1,000,000)^2 = 1,000,000,000,000!
Converting to dBi, that's 10 log (10^12) = +120 dBi

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.
Theoretical power gain is (5,000)^2 = 25,000,000.
Converting to dBi, that's 10 log (2.5 * 10^7) ~ +74 dBi

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)
Voltage gain ~ (2 pi * 2 mm) / (500 nm) = 25,000
Power gain = (voltage gain)^2 ~ 600,000,000 = +88 dBi

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)
Telescope Gain (dBe) = (+120 dBi) - (+88 dBi) = +32 dBe

[notice that the i's in the dBi units above cancel]

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 Celestron-class telescopes, we have every reason to expect significant microwave astronomical discoveries to be within the grasp of our Project Argus-class amateur radio telescopes.