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Far away, across the vastness of space, lies an alien civilization, who are pretty similar to us. They've got ice cream, peer to peer networking, and photography. More importantly, they do lots of their communication with radio waves, just like us. However, we don't know that they're there, because they are simply too far away.

If our aliens have colonized a few planets, but each has a total power output within an order of magnitude of the power output of Earth, how far from Earth would they need to be in order for us to have not noticed them yet? They have yet to start building Dyson spheres or harvesting solar energy en masse via satellite, so we won't be able to use their technology blocking out sunlight as a means of detection.

The aliens have been using modern technology for long enough that any transmissions they've made have reached us, and they aren't doing anything to either hide their presence or to advertise their location. Their technology at the point at which we are observing them (so offset by a number of years equal to the distance to their planet in terms of light years) is all near-future technology.

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    $\begingroup$ With sufficiently advanced technology, they could be in our fields, molesting our cows, and we would not be any wiser. $\endgroup$ – Serban Tanasa Apr 6 '16 at 18:05
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    $\begingroup$ Ok, added a bit about their technology. They're developed tehnology long enough ago that their transmissions have reached Earth and they aren't doing anything to hide it. $\endgroup$ – ckersch Apr 6 '16 at 19:18
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    $\begingroup$ About as far as they are now, I would guess. $\endgroup$ – Mystagogue Apr 6 '16 at 21:32
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    $\begingroup$ If their technology wasn't too advanced beyond ours, we wouldn't detect their radio signals unless they deliberately aimed a focused high-powered beam at us (as we did once), but there's the possibility that near-future telescopes designed to study exoplanets could detect either the illumination on the night side of their planet or the signature of industrial pollution from spectroscopic analysis of the atmosphere, see numbers 7 and 11 on this list. $\endgroup$ – Hypnosifl Apr 6 '16 at 21:46
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    $\begingroup$ Actually, if they are slightly more advanced than us, they will be harder to detect, because more efficient transmission methods are less likely to make it to another star. Or as Randall Munroe said, "If your TV signals are getting to another star, you’re losing money." $\endgroup$ – browly Apr 7 '16 at 16:11
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For the TL;DR, see the bottom of the answer.

We can approach answering this by considering how sensitive our most sensitive receivers on Earth are, how much antenna gain we can muster, how much power we can muster, and how much power needs to be transmitted for us to be able to detect the signal at interstellar distances. For simplicity, I'll just ignore the cosmic background radiation. In effect, this answer establishes an upper bound on how far away from us a civilization similar to us could be and we would be able to detect them.

The way to approach that is to construct a link budget for the transmission system and distance in question. The first thing we need for that is the equation for free space path loss, which is $$ 20 \times \log_{10}\left( 4 \pi \frac{d}{\lambda{}} \right) \approx -22 - 20 \times \log_{10}{\left(\frac{d}{\lambda}\right)} $$ where $d$ is the distance and $\lambda$ is the wavelength (where $\lambda = \frac{c}{f}$, where in turn $c$ is the speed of light in the medium and $f$ is the frequency). When $d$ and $\lambda$ are in the same units, the resultant value is the path loss in decibel (dB). Notice that the path loss scales with the distance in terms of wavelengths, so if you double the frequency (halve the wavelength) and halve the physical distance, the path loss is identical. On Earth there are other propagation modes as well (ionospheric, parasitic radiation, reflection, scatter, ...) that make calculating path loss far more complicated; however, for anything between different celestial bodies, this is the go-to equation for approximating path loss. It is only an approximation because it does not take into account for example losses in the interstellar medium.

For a reasonable frequency of interest, 3 GHz, the wavelength is 10 cm.

1 lightyear is, quite conveniently for our purposes, about $9.461 \times 10^{17} \approx 10^{18}$ cm.

$\frac{d}{\lambda} = \frac{10^{18}}{10} = 10^{17}$ so we calculate $-22 - 20 \times \log_{10}{\left(10^{17}\right)} = -22 - \left( 20 \times 17 \right) = -22 - 340 = -362$. Over 1 lightyear, our path loss is approximately 362 dB. For a more realistic example, let's take Proxima Centauri at about 4.25 lightyears from Earth; that gives us a total path loss of $-22 - 20 \times \log_{10}{\left(4.25 \times 10^{17} \right)} \approx 395$ dB.

(If desired, substitute the distance between the worlds of interest and an appropriate frequency in your case and recalculate the path loss. If you change the frequency, don't forget to recalculate the antenna gain below.)

Well, it's good that we now have a number, but what does that number mean?

The total worldwide electricity production in 2008 was 20,261 TWh, or if this were continuous (it probably wasn't) about 2.31 TW. Let's say we could somehow channel all of this into a transmission at 3 GHz (we almost certainly can't; even if we wanted to, there are efficiency limitations in real-world radio amplifiers, and with a really impure waveform, we might only be able to get maybe 90% efficiency in an amplifier which means we need to set our hair dryer to its 231 GW setting). The remaining about 2 TW is about +153 dBm. (If you compare this to the Arecibo observatory, you'll notice that the figure quoted by Wikipedia is an order of magnitude higher at 20 TW on the very similar frequency 2380 MHz. However, that's EIRP, which adjusts for antenna gain, while we are talking raw transmitter output power here. We'll get to antenna gain in a minute.)

A really good receiver, including a low-noise amplifier, might be able to detect a signal that measures something like -200 dBm at the antenna feedpoint terminals. The exact value varies with the receiver design and desired transmission rate, and I'm not completely sure what the state of the art actually is, but -200 dBm is likely close enough for our purposes particularly if the purpose is to simply detect the presence of the signal.

The really nice part about working with these numbers in dB relative to something (such as dBm, which is decibels relative to a milliwatt, or dB, which is just a ratio) is that we can simply add the numbers. If we feed that +153 dBm signal into the -200 dBm noise floor receiver, we have a margin of 353 dB for as long as we don't blow out the receiver circuitry (which would happen pretty quickly, but let's ignore that for a second).

The gain of a parabolic antenna (which Arecibo isn't, really; Arecibo is a spherical, not parabolic, reflector) is $$ G = \frac{4 \pi A f^2}{c^2} e_A $$ where $A$ is the reflector area in square meters, $f$ is the operating frequency in Hz, $c$ is the speed of light in m/s, $e_A$ is the aperture efficiency (defined as the ratio of effective aperture to physical aperture, or $\frac{A_e}{A_p}$), and $G$ comes out as a multiplication factor describing the antenna gain over an isotropic antenna (an antenna that has exactly the same sensitivity in all directions; no real-world antennas have such a radiation pattern, they are always somewhat directional). A good parabolic antenna might have an aperture efficiency in the 0.8 range, and I've seen (but can't seem to find again) a mention of 0.55 for poor ones. Because this is about interstellar communications, we'll use the best antenna we can reasonably muster, so let's call the aperture efficiency 0.8.

To give you an idea of its size, the Arecibo dish is 305 meters in diameter and weighs 900,000 kg. It's the largest single-aperture telescope in the world.

At 3 GHz, a 305 meter diameter parabolic dish with an aperture efficiency of 0.8 has a gain of $$ G = \frac{\left( 4 \pi \left( \pi \left( \frac{305}{2} \right) ^2 \right) \right) \times \left( 3 \times 10^9 \right) ^2}{\left( 3 \times 10^8 \right) ^2} \times 0.8 \approx 91811992 \approx 80 \text{ dB} $$ when compared to an isotropic antenna (EIRP gain). If we assume two Arecibo dishes pointed directly at each other, we can add another two times 80 dB of antenna gain to our link budget, so we get a bonus 160 dB for only a tiny difficulty in aiming (as anyone who has tried to aim a satellite dish can attest to; a household satellite dish has gain far lower than 80 dB at its frequency of interest). Here we can also see that even an antenna with a 1.0 (best theoretically possible) aperture efficiency wouldn't increase our gain by very much; such an upgrade would gain us another almost exactly 1 dB on either end.

So, to summarize, we are putting out +153 dBm, gain 80 dB in antenna gain, lose 395 dB along the way, gain another 80 dB in antenna gain, and need at least -200 dBm after all that for the signal to be detectable. Luckily, we are now at -82 dBm, so given these assumptions, the signal is well within the range of detectable. (In fact, I think my amateur radio transceiver could pick it up without much trouble, given an appropriate frequency downconverter. Of course, and perhaps thankfully for peace in the neighborhood, I don't have the Arecibo dish in my back yard.)

However, those are quite some assumptions that we are making in order to reach this conclusion. Basically, what we are doing is pouring the whole world's electricity production into the biggest radio antenna we can muster, expect the receiver to have an antenna just as large and that they are pointing it directly at us and are listening to just the right frequency at just the right time. Remove any one of these, and the signal goes from trivially detectable to anywhere between difficult and not a chance. The problem is exacerbated if we want to not just detect the presence of the signal, but also understand its content, at which point we start looking at noise over a larger frequency span and ultimately the Shannon-Hartley theorem, which gives a theoretical limit for the transmission rate of a communications channel of a given bandwidth and signal-to-noise ratio. Our terrestrial systems aren't really designed to be decoded at interstellar distances because, as pointed out in a comment to the question, companies like television networks and cell phone providers aren't really interested in Earth-bound investments in their Proxima Centauri viewership and customers.

For a real world comparison, SETI Sensitivity: Calibrating on a Wow! Signal from the SETI League indicates that the Wow! signal was received (in 1977, on 1420 MHz) on equipment that had a noise floor of -138.6 dBm (we are almost certainly doing better than this today) plus 55.3 dBi (dB over isotropic, or EIRP gain) antenna gain. Even if we would use such a receiver rather than our postulated -200 dBm receiver, but still use the Arecibo dish on 3 GHz, we "only" lose about 61 dB compared to the calculation above so we still have a margin of over 20 dB to the noise floor, which is quite decent and is going to stand out in any signal strength plot. (The Wow! signal peaked at about 30 times the ambient noise, equivalent to an about 15 dB margin. 20 dB means that the signal is 100 times the strength of the noise.)

As of September 2016, the Chinese are putting the finishing touches on what has been termed the five hundred meter aperture spherical radio telescope, or FAST for short, nicknamed Tianyan (天眼). While Arecibo has a 305 meter diameter spherical cap reflector, FAST has a 520 meter diameter spherical cap reflector (Five-Hundred Meter Aperture Spherical Radio Telescope (FAST) Cable-Suspended Robot Model and Comparison with the Arecibo Observatory, Ohio University, section 1) of which 300 meters is illuminated at any one time (The Five-Hundred-Meter Aperture Spherical Radio Telescope (FAST) Project, Rendong Nan et al, arXiv:1105.3794, doi:10.1142/S0218271811019335, PDF page 4 in the arXiv version). As such, it does not have significantly different properties as an antenna as compared to the Arecibo main reflector in situations where either can be used.

Scientific treatment of reception of incidental transmissions

(By "incidental", above, I am referring to those transmissions not actually aimed into space for the explicit purpose of being detected by an alien civilization.)

It appears that this has actually be discussed in proper scientific fora. For example, Rob Jeffries' answer on Astronomy SE on how we would detect an Earth doppelganger planet quotes Cullers et al. (2000) as stating that

Typical signals, as opposed to our strongest signals fall below the detection threshold of most surveys, even if the signal were to originate from the nearest star

and Tarter (2001) as stating that

At current levels of sensitivity, targeted microwave searches could detect the equivalent power of strong TV transmitters at a distance of 1 light year (within which there are no other stars)

In other words, an alien transmission would need to be far stronger than a current, powerful Earth TV transmission, for us to be able to detect it with currently available equipment. There aren't a lot of transmissions that would meet this criteria.

TL;DR:

To detect the radio transmissions of a civilization outside of the solar system is absolutely possible, but realistically does take deliberate effort, helpfully at both ends. We won't be picking up anyone's cordless phone. Nor will we be picking up anything like our own cell phone networks, nor likely our Earth-bound point-to-point radio links. We might be able to detect the presence of an EM radiation spectrum that does not match either the cosmic background radiation or what you would expect from natural processes in a solar system. However, if they aim a powerful transmitter in our direction for some reason, and we happen to be listening at just the right moment on just the right frequency, then we probably would detect it.

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    $\begingroup$ I understood some of those words you've used. $\endgroup$ – MKII Apr 7 '16 at 9:36
  • $\begingroup$ I think I remember reading that Arecibo could talk to its twin anywhere in our galaxy. So if we knew where & when to point the dish we could send & receive to anywhere in the galaxy. $\endgroup$ – Jim2B Apr 7 '16 at 23:35
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    $\begingroup$ @Jim2B - It's easy enough to plug some bigger numbers into the free space loss equation and see what happens. Turns out, we drop below the -200 dBm level at roughly $4 \times 10^{25}$ cm, or $4 \times 10^7$ light years. Since the Milky Way is roughly $10^5$ light years across, it's very true (for that wavelength). Of course, aiming is a huge problem. $\endgroup$ – Bobson Apr 8 '16 at 3:25
  • $\begingroup$ @Jim2B In addition to Bobson's excellent response to your comment, you'll notice that in my calculations I already posited one Arecibo dish at each end of the radio link (and mentioned the little detail of aiming). $\endgroup$ – a CVn Apr 8 '16 at 7:35
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    $\begingroup$ tl;dr for those who don't like math: Unless they're aiming a powerful transmitter right at us and we're also listening on the same frequency in that exact direction the chances of us noticing someone one solar system away are pretty nil, as far as radios go. $\endgroup$ – thanby Apr 8 '16 at 12:26
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We wouldn't notice a terrestrial civilisation at Alpha Centauri, the closest star. Radio and TV transmissions attenuate too quickly.

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    $\begingroup$ By analogy, when you drop a pebble in a pond the ripples spread out all over. But drop a boulder in Lake Ontario in Toronto, even on the calmest day, the people in Rochester, New York will never notice. Space is even bigger than Lake Ontario. $\endgroup$ – cobaltduck Apr 6 '16 at 18:08
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    $\begingroup$ By further analogy, drop a pebble into the pond during a hailstorm $\endgroup$ – Kys Apr 6 '16 at 18:20
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    $\begingroup$ @cobaltduck "Space is even bigger than Lake Ontario" citation needed? $\endgroup$ – Walt Apr 6 '16 at 21:16
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    $\begingroup$ @Amadeus9 Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. $\endgroup$ – MikeTheLiar Apr 6 '16 at 21:28
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    $\begingroup$ There's a pretty detailed analysis of the problem which confirms what you're saying on this page. Also see this article for a shorter summary. $\endgroup$ – Hypnosifl Apr 6 '16 at 21:37
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Here is a press release:

Unprecedented scope

The program will include a survey of the 1,000,000 closest stars to Earth. It will scan the center of our galaxy and the entire galactic plane. Beyond the Milky Way, it will listen for messages from the 100 closest galaxies. The telescopes used are exquisitely sensitive to long-distance signals, even of low or moderate power:

  • If a civilization based around one of the 1,000 nearest stars transmits to us with the power of common aircraft radar, Breakthrough Listen telescopes could detect it.

  • If a civilization transmits from the center of the Milky Way, with any more than 12 times the output of interplanetary radars we use to probe the Solar System, Breakthrough Listen telescopes could detect it.

  • From a nearby star (25 trillion miles away), Breakthrough Listen’s optical search could detect a 100-watt laser (energy output of normal household light bulb).

Since this project has funding to listen to signals from other galaxies, I presume the reasoning is sound in thinking that such a signal would be physically possible and a reasonable thing for an advanced civilization to emit.

I think the live talk (SETI siminar colloquium) indicated that it would be able to detect us at 200 light years.

Insterstellar Laser Communication

I can't find the talk again, but I recall it being shown that a laser "only" 10× more powerful than one being worked on would provide for practical insterstellar communication with 8 meter mirrors on each end.

The interesting thing here is the use of pulse lasers. The intensity of a pulse is far higher than the average power consumption. And, even though the brightness at the detector is lower than the background, the specific pulse cadence can still be detected.

This requires looking for such pulses if communication has not yet been established, and is highly directional.

Don't pay for the transmitter

In The Hercules Text, the aliens used a shutter to form a signal using a visible pulsar. Look at the "starshield" designs for a near-future telescope to image extra-solar planets. If you could get something in the line of sight between a target civilization and a pulsar or other star visible to them, it would not need to be very large to be effective.

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  • $\begingroup$ The cited text omits the issue of directionality, as referenced in the question and the main answer. For example, to use a pulsed laser we'd need both the sender and the detector to be aimed very precisely, and to share information about the pulse frequency to aid detection below the noise level. Those might be feasible for deliberate point to point communication,but the task of detecting an unknown signal are much larger. $\endgroup$ – Zeph Apr 12 '17 at 18:12
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Radio and television signals from Earth would have degraded sufficiently due to both the inverse square law and being absorbed by the interstellar medium that they are effectively impossible to make out from the interstellar noise. Various sites I've looked at seem to think that the effective range of Earthly radio signals would be only about a few light years. Perhaps worse, since we are switching over to much more efficient means of transmission (digital, fibre optic cable, satellite downlinks) the amount of radiated energy is much less than before.

The hypothetical aliens might be able to see a blast of analogue radio noise coming from Earth starting in the 1940's and tapering off in about the 1970's, but tis would register more as a spike of energy rather than coherent signals, and alien scientists would have a hard time determining what really caused this.

So unless the aliens are actively sending signals with high power transmitters and focused in relatively narrow beams at the Solar System, we would probably not be able to detect anyone farther than a few light years away.

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The time since they have radio, in years, is the minimal distance in lightyears.

Simply their transmissions, traveling only at the speed of light, did not propagate here yet.

So not that far, in a galactic scale. 200 lightyears could be plausible.

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  • $\begingroup$ Depends on what you mean by notice, but I'd say ~125 years because that's all the longer we've been listening or looking. $\endgroup$ – lonstar Apr 13 '16 at 19:33
  • $\begingroup$ I don't think it matters how long we've been listening. If we did not detect them today, there's probably no detectable signal reaching us today. $\endgroup$ – Emilio M Bumachar Apr 14 '16 at 19:35
  • $\begingroup$ We didn't see planets around other stars until about 15 years or so ago... $\endgroup$ – lonstar Apr 15 '16 at 1:09
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Depends on what you mean by notice, but I'd say ~125 years because that's all the longer we've been listening or looking with instruments powerful enough to see anything worth seeing.

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  • $\begingroup$ The author of the question implied that observations could be taking place in the future, not necessarily today. $\endgroup$ – HDE 226868 Apr 13 '16 at 20:52

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