Billions and Billions


    Remembrance of things past: Most people of my generation recall Carl Sagan’s ‘billions and billions’, pronounced with an exaggerated hard ‘b’ sound, except that Carl Sagan didn’t actually utter those words.1 The true source of this memory was Johnny Carson spoofing Carl Sagan! Be that as it may, there are billions of stars in our galaxy and billions of galaxies in the universe, just as Carl Sagan said there are. Perhaps a lesser known fact is that our amateur radio transmitters spew forth billions and billions of photons per second. Indeed the word ‘billions’ is much too small to express the numerosity of the energy packets that we routinely cast into our world and beyond.

    It is not necessary to have studied physics to know that x-rays are more energetic than visible light—that’s why the dental technician covers us with a heavy blanket before hastening from the room to shoot the required annual image of our so-called ‘bite’. Similarly we know that visible light is more energetic than microwaves, which are more energetic than VHF/UHF, and so forth through high frequencies, medium and long waves, all the way down to the Schumann resonances.2 Come to think of it, a person does not need to have studied physics in order to learn almost anything about our universe nowadays. The answer can be found in Google 101 or StartPage or DuckDuckGo or wherever you like.

    Physics, or science in general, often deals with numbers and I’ve learned over the years that many people are uncomfortable with numbers. They are okay with reading the word ‘billion’ but the number 1,000,000,000 (which is the same thing) makes them cringe. A sort of workaround for this problem is an extended set of words for bigger and bigger quantities, trillion, quadrillion, quintillion, and so forth. The same works in the other direction, micro, nano, pico, femto, etc.3 But there comes a point where you need numbers, as the words become awkward, like working with Roman numerals—I’m writing this paragraph in MMXXIII.

    Maybe the solution is to promise something. “If you can tolerate a few numbers, I will tell you how many photons a 100 Watt transmitter shoots forth per second at 7.150 MHz.” But wait, there is no need to suffer. With the admittedly unrealistic assumption that the entire 100 Watts power is effectively radiated at the specified frequency, I will state the answer to an absurd degree of precision. It is twenty-one octillion, one hundred seven septillion, five hundred fifty-four sextillion, nine hundred sixty quintillion, thirty quadrillion, ninety-five trillion, six hundred ninety billion, more or less. How satisfying was that? —Not very!

    Biting the numbers bullet: We are ham radio operators, after all. We must deal with some numbers, like how long to cut the dipole wires, or how much reflected power our transmitter is able to tolerate—oops! We must also have passed a license exam, though that may have been in the distant past, and it’s possible we could have forgotten some of those answers. Well, surely we have not forgotten such fundamentals as the fact that wavelength is inversely proportional to frequency. Though it may seem less intuitive, it is just as simple to work out the immense number of photons per second at 7.150 MHz, as it is to convert frequency to wavelength, or vice versa. Instead of the speed of light, a different constant of nature enters the picture. This one describes how the energy of a photon varies with its frequency.

     Planck’s constant is an exceedingly small number. In ordinary decimal notation it is 0.000000000000000000000000000000000662607015 Joule-seconds. Scientists and engineers use a more compact notation, 6.626 × 10-34 J-s, but never mind that just yet. The energy of a photon is Planck’s constant multiplied by its frequency in Hertz. You can see where multiplying this miniscule number by a few billion isn’t going to tip the scale very far! Now, one last piece of arithmetic lore... Dividing by a small number makes a large number. For example, dividing by one-half is the same as multiplying by two. Dividing by one-tenth is the same as multiplying by 10, and so forth. More to the point, dividing power in Watts by the energy of a photon (at a given frequency) gives the number of photons per second being generated. With this one additional fact you can check my arithmetic on the 7.150 MHz problem. —I have been known to be wrong.

    Penetrating the mist: It is one thing to calculate how many photons are generated, but how many of them are getting through to Timbuctoo?—either the one in Mali, or the one in California. Stay tuned.
* * *

S-meter    When the operator says you are 5-9 in Timbuctoo, it is best not to take the report too literally. Surely every station cannot be 5-9 in Timbuctoo! Alas, politeness has overtaken accuracy in the signal reports business. Your true signal is more likely to be the one illustrated pictorially on the left. For the sake of this exercise, let’s assume that the transmitted signal is received as S3 in Timbuctoo, and that the S-meter has been calibrated. The latter would rarely be a safe assumption, but we have to start somewhere. According to hamwaves.com, a reading of S3 equates to -109 dBm at 50 ohms, provided that the S-meter is ‘well-designed’ and has been calibrated for the high frequencies.4 But, what does that mean?

    dBm to Watts: First, observe that the letter ‘m’ in dBm stands for ‘milliwatt’. That’s easy enough. However, at this point it becomes necessary to introduce a swear word. If you already hang out with swear words, then this one is not particularly fearsome. It is the word ‘logarithm’. More precisely it is the common logarithm, where ‘common’ refers to base-10, the number base that is commonly used for counting things. The symbol dB (decibels) is nothing more than ten times the base-10 logarithm of a number. One of the shortcuts we learn in ham radio is that a 3dB power gain corresponds to doubling the power. Doubling, of course, is the same as multiplying by 2, and the base-10 logarithm of 2 is approximately 0.3, which multiplied by 10 makes 3 dB.

    Before losing track let’s return to the S-meter reading. Our signal was S3, or -109 dBm in Timbuctoo. Although these intermediate steps are probably explanation overkill, to be clear, -109 dB means that the logarithm of the power ratio is -10.9 (divide dB by 10). To work backwards from dBm to power, multiply 1 milliwatt (.001 Watts) by 10-10.9. As may be  expected, the result is a very small quantity. Expressed to an absurd number of decimal places the value works out to be .0000000000000125892541179416721... Watts. Believe it or not, there is a prefix for this miniscule amount of power. Rounded to 3 places, it is 12.6 femtowatts.

    The number of photons per second impinging upon the receiver is this small number divided by the previously computed energy of a photon at 7.15 MHz (Planck's constant multiplied by 7,150,000 Hz). Drum roll... The number is 2,657,283,732,002 (more or less). In words, it is two trillion, six hundred fifty-seven billion, two hundred eighty-three million, seven hundred thirty-two thousand and two photons per second. In contrast to the enormous number that was computed for the transmit power, this one is small enough to be dollars instead of photons! As dollars it would be a small fraction of the US national debt.5

    Photons per unit area per second: Counting energy packets that equate to an S-meter reading was messy to say the least. However, to continue in the spirit of glorifying big numbers, and expressing them to preposterous precision, the exercise can be carried one step further. The concept is called ‘photon flux’. Stay tuned.
* * *

    Here comes another swear word, ‘isotropic’. It is the ‘i’ in dBi, in case you happen to have seen that symbol. The term ‘isotropic’ means ‘same in every direction’. Ham radio antennas are not isotropic. On the other hand, the gain of a directional antenna may be conveniently expressed in dBi units, or decibels over isotropic. For example, the gain of a dipole (in free space) is 2.15 dBi. The dipole itself often serves as a comparison reference for expressing the gain of other antenna types. Decibels (logarithms) are additive, so if a Yagi antenna has 8 dB forward gain with respect to a dipole, it would be the same to say its forward gain is approximately 10 dBi. Bear in mind that real antennas are not ideal antennas. Real antennas are affected by their height above ground, and countless other installation-specific environmental factors.

    The word isotropic plays an important role in a concept called the ‘inverse square law’. I don’t want to make more of this than is obvious. If a substance emanates outwards from a point equally in all directions, then at any given distance the substance will be dispersed evenly on the surface of an imaginary sphere whose radius is said distance. In school we either deduced or memorized (and then quickly forgot) the formula for the surface area of a sphere 4πr2, where r stands for its radius. In the inverse square law, the letter d (distance) replaces r (radius) in this expression. Putting the inverse square law together with the count of photons at the source leads to the concept of photon flux, sometimes denoted by an uppercase Φ. Flux is simply the number of photons per second per unit area at a given distance from the source.

Photon flux formula    At this point it may be necessary to apologize for sneaking in a potentially frightening formula. But each of the symbols in the formula stands for something that has been explained already, P for power in Watts, h for Planck’s constant, f for frequency in Hz, and d for distance in whatever units you want. If distance is specified in meters, then area will be meters squared, and so forth. As illustrated here, the photon flux calculation assumes isotropy. For convenience the formula has been split into sub-expressions to emphasize the radiative source (left) and its inverse square diminution (right).6

    Effective area: Ham radio transmit antennas are not isotropic. And receive antennas don’t resemble a chunk from a spherical collecting surface. Moreover, who’s going to climb an antenna tower with a tape measure in hand, and what would they do with it if they did? As it happens, nobody has to climb the tower, or measure antenna parts because, according to this page, “the effective area of an antenna ... is the area of the idealized antenna that absorbs as much net power from the incoming wave as the actual antenna [does].” With a few lines of symbol manipulation that same page deduces the effective area of a (Hertzian) dipole7 to be three times the wavelength squared divided by 8π. In the spirit of fudging when expedient to do so, I will use this expression to compute the area of the make-believe receive antenna in Timbuctoo.

   The frequency 7.150 MHz falls in the 40 meter ham radio band. At this frequency the free-space wavelength is closer to 41.9 meters, but no-matter. Let’s say λ ≈ 40 meters. That should be close enough. Applying the effective area formula from the preceding paragraph we get approximately 190 m² for the area of the receive antenna. (Check me!) At this point it is time to let go the excessive precision of photon counts. Dividing 2,657 billion by 190 meters squared obtains a flux of approximately 14 billion photons per second per meter squared at the receiving antenna. Hmm. Now I’m thinking of even crazier calculations, like given the distance to Timbuctoo, how would the antenna’s measured gain compare to the isotropic flux calculation for the same distance. Of course it would not be necessary to use photon flux in the calculation, as the same comparison can be done with power in Watts. Big numbers on this page are just for fun.

Sine wave at 7.150 MHz    Wrap up and wind down: Waves are everywhere, especially in ham radio, long waves, medium waves, shortwaves, waves of voltage and current on the transmission line and antenna, electric and magnetic components of the RF wave, in-phase and quadrature to the SDR, and so on.

    Waves correspond to the natural way of thinking about radio, as we know it. They form a conceptualization that works, and has passed the test of time. Nevertheless it is fun to contemplate the magic of radio in a different way, to visualize billions and billions of infinitesimal beadies shooting forth, bouncing and penetrating and otherwise wending their way to remote and exotic places.



 

Projects Home


1. Chapter 1 in Sagan’s posthumous Billions & Billions... Sagan did intentionally pronounce the word with a hard ‘b’ to emphasize that he was not saying ‘millions’.
2. https://en.wikipedia.org/wiki/Schumann_resonances
3. https://www.varsitytutors.com/hotmath/hotmath_help/topics/big-and-small-numbers
4. S-meter calibration differs for VHF/UHF. For example, S9 at HF translates to -73 dBm at 50 ohms, while for VHF/UHF it is -93 dBm.
5. 8.4% at the time of this writing (https://www.usdebtclock.org/).
6. In the flux formula, frequency is sometimes expressed as the speed of light divided by wavelength (f = C/λ), which puts wavelength in the numerator and the constant C in the denominator.
7. Directivity = 3/2.




Project descriptions on this page are intended for entertainment only. The author makes no claim as to the accuracy or completeness of the information presented. In no event will the author be liable for any damages, lost effort, inability to carry out a similar project, or to reproduce a claimed result, or anything else relating to a decision to use the information on this page.