Wikipedia:Reference desk/Archives/Science/2024 August 25

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August 25

Arecibo message distance

The Arecibo message was broadcast towards M13, 25,000 light years away. Is there any chance that the message could be received that far away, or would it be long lost in noise? Bubba73 ^{You talkin' to me?} 04:00, 25 August 2024 (UTC)

Not only would the signal-to-noise ratio be minuscule, but due to its orbit around the galactic center the signal would have to be aimed at where the Messier 13 cluster will be 25k years from now. 136.54.237.174 (talk) 18:05, 25 August 2024 (UTC)

Now I'm curious: Do we know, then, what stars it's going to actually pass close by? -- Avocado (talk) 21:15, 26 August 2024 (UTC)

From the data given in the article, I arrive at a minimum diameter of the receiving antenna of 2 kilometers. The diameter of the Arecibo dish is $D=305$ meters, the wavelength of the signal is $\lambda =12.6$ cm. The beam divergence angle is then $\theta =1.22{\frac {\lambda }{D}}=5\times 10^{-4}$ radians. Because $\theta$ is very small, the solid angle is to a good approximation $\Omega =\pi \theta ^{2}$, the exact formula is $\Omega =2\pi \left[1-\cos(\theta )\right]$. The area of the beam after traveling a distance of r is then $\Omega r^{2}$. Then with the power of the beam of 405 kW, at a distance of $22.2\times 10^{3}$ lightyears, the flux of the signal will be $F=1.3\times 10^{-26}$ Watt/m^2 at M13. This signal can then be detected using one or multiple antennas. If the total area of the antennas is A, then the received power is F A. If we assume that the temperature of the antennas and receivers are T = 20 C = 293.15 K, then the noise power will be $P=kT\Delta f$ where $\Delta f$ is the bandwith, that in this case must be 10 Hz or larger, as this is the frequency shift used to modulate the signal. The signal power must be larger than the noise power. If we then equate F A to P and solve for A and then assume a single antenna is used, and put $A=\pi r^{2}$ then the diameter of the receiving dish is 2 r and if I didn't make any mistakes, this yields a minimum diameter of approximately 2 kilometers. Count Iblis (talk) 19:25, 25 August 2024 (UTC)

A 2 km dish is feasible, but will the signal get lost in the noise at that distance? Bubba73 ^{You talkin' to me?} 04:57, 26 August 2024 (UTC)

There's also the matter of integration time. Noise adds incoherently, signal adds (hopefully) coherently, so with a longer integration time, the signal may rise above the noise. In this case, the integration time is limited to no more than 100 ms by the 10 Hz bitrate. The difference between the 0 bit and the 1 bit was only one wave, so a longer integration time doesn't help to decode the signal, but it may still help to detect the carrier wave.

Beam size matters too. The wider the beam, the more noise from other sources like stars; the narrower the beam, the less likely those aliens pointed it well enough at Earth. PiusImpavidus (talk) 10:31, 26 August 2024 (UTC)

[1] says that the gain of Arecibo antenna at 2.38 GHz was 77 dBi, only 600 mdB short of Count Iblis's estimate from the physical diameter (an aperture efficiency × antenna efficiency of 87% if true). A receiver temperature of 20°C is a little pessimistic; usually the receiver would be cooled (it is not necessary to cool the antenna, assuming that it is low-loss). catslash (talk) 00:09, 28 August 2024 (UTC)

A Band-stop_filter can filter the noise between the valid symbols and thus enhance the signal noise radio over this. Or do I miss something there? 176.0.144.239 (talk) 19:30, 2 September 2024 (UTC)

Thanks for the informative replies. Bubba73 ^{You talkin' to me?} 04:47, 28 August 2024 (UTC)

Regarding the integration time: when there are two possible symbols (0 and 1) represented by two orthogonal signals of equal energy $E$ ( = received power × time) then the bit error rate is something like

$P_{\mathrm {err} }={\frac {1}{2}}\mathrm {erfc} \left({\sqrt {\frac {E}{2kT}}}\right)$

where $\mathrm {erfc} (x)$ is the complementary error function. This assumes (1) that it is known exactly what the two signals are - there is no random change in the phase between symbols, and (2) that the prior probability of each symbol is equal.

Without error-correcting codes it is impossible to reduce the error rate to zero, so it is necessary to decide what rate is acceptable before building the receiving antenna. catslash (talk)

[2] (Table 1: Legacy Arecibo Observatory planetary radar system.) says that the gain of Arecibo antenna at 2.38 GHz was 72.9 dBi, which seems more plausible. catslash (talk) 15:57, 1 September 2024 (UTC)