Interstellar Radio Propagation

Proxima Centauri: The Closest Star.
Credit & Copyright:  David Malin, UK Schmidt Telescope, DSS, AAO

It's a staple of Science Fiction, and an unquestioned fact of our modern age, that aliens could be listening to our radio and watching our TV broadcasts, as our signals race across the galaxy at the speed of light.  They could be studying our weaknesses, preparing their attack!  But really, is that possible?

 

I have long been fascinated by the possibility of finding life beyond our solar system, or of aliens finding us.  But rather than wishful thinking, scaremongering or falling for alien abduction tales, I'm far more interested in the realistic prospects of such a discovery.  So when Prof. Brian Cox threw down the gauntlet for listeners to BBC Radio 4's The Infinite Monkey Cage to carry out a fundamental but accessible calculation to illustrate the real likelihood of one form of contact, I was fascinated.

 

Episode 5 of Series 12 was broadcast on 3 August 2015, and I heard it several weeks later via the show's podcast feed.  The previous week's episode focussed on extra-terrestrial life and alien contact, but Episode 5 concentrated on speed, including land speed record attempts as well as the fundamental barrier in physics that is the speed of light.

 

If you want to download and listen to the episode yourself, at 38m 38s, presenter Robin Ince asks about radio signals leaking into space and Professor Danielle George, of University of Manchester, describes broadcast transmissions degrading in power with the inverse square law. Then Robin asks Brian Cox to calculate how far away through space their own radio broadcast would be detectable. Prof. Cox ad lib ponders the problem and then defines the listeners' challenge, which I summarise here:

"Suppose a 200kW transmitter broadcasts for 1 second at 198kHz, find the distance at which there remains one photon per square metre."

Now, we can argue the merits of this threshold, whether one photon per second per square metre is easy or unduly difficult for advanced aliens to detect, (and I shall return to this question).  But for now, let's solve the problem.

First, we need to know how many photons of radio energy are transmitted in one second.  Then we need to find the distance at which all these photons are spread out to one per square metre.

 

So let's do it...  First, let's define some parameters and constants:

Transmitted power,       P = 200kW

Frequency,                    f = 198kHz

Planck's constant,         h = 6.6x10^-34Js

 

As Prof. Cox helpfully reminded us, the energy of a photon is given by its frequency multiplied by Planck's constant, so

 

Photon energy,             E = hf

 

So each photon at 198kHz carries 198x10³ x 6.6x10^-34 = 1.3x10^-28J of energy.

 

And since a power of 200kW delivers precisely 200kJ of energy per second, in one second the transmitter delivers 200kJ of energy.

 

So we divide the energy transmitted by the energy per photon to find the number of photons transmitted.

 

Number of photons, N = 200x10³ / 1.3x10^-28 = 1.25x10³³ photons.

 

That's an awful lot of photons!  So now we need to spread these photons out over a sphere to the point where there's one square metre of area on the sphere for each photon.

 

The area of a sphere, A = 4πr² m², where r is the radius of the sphere.

 

So, a sphere with an area of 1.25x10³³ is given by the equation 1.25x10³³ = 4πr².

 

Rearranging this to find r gives, r = √( 1.25x10³³ / 4π ) = 1.0x10¹⁶ metres.

 

That's an unfeasibly large distance by human standards, but on the astronomical distance scale, it's almost exactly one light year!

 

So once the Radio 4 long wave signal broadcasting The Infinite Monkey Cage gets to a light year from Earth, it will comprise only one photon per square metre, per second.  And by Brian Cox's criterion, it will have degraded to the point of undetectability.

 

Now bear in mind that the nearest extra-solar star, pictured above, is Proxima Centauri which is 4.2 light years away.  And if that was conducive to intelligent life, which it is not, our signal would not make it a quarter of the way there.  So by this criterion, which is not unreasonable at all, we are to all intents and purposes radio silent to any alien life out there, as far as commercial broadcast transmissions are concerned.

 

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So now, how reasonable is this as a limit?  Can we find an argument which breaks this?

 

One photon per square metre per second seems like an arbitrary limit, why can't advanced aliens detect those? Well, as advanced as aliens might be, there has to be a signal to receive.  The bandwidth of an audio signal is a few kHz, which means that you'd need at least 5,000 samples per second to reconstruct the transmitted signal.  And that's not a technological limit, which advanced civilizations could surpass, it's a fundamental information limit.  Worse still, one photon carries no amplitude information, so unless the reconstruction is to be distorted beyond use, they would need to detect many photons per sample.  A good signal would use several hundred amplitude levels, but you could get away with perhaps 20 or so.  So now, to reconstruct a useful signal, you'd need 100,000 photons per second.  At just one light year, that would require a receiving antenna with an area of 100,000m², or a perfect dish with a diameter of 350 metres (1,200 feet).

 

But aliens have limitless capabilities, because... well, aliens! So they could build a 350 metre dish.  Well, perhaps.  But now consider that signal power drops with the square of distance, and dish area increases with the square of diameter.  So double the distance, double the dish diameter.  There are plenty of stars nearby, but to find one which can possibly be inhabited by life which could evolve to sufficient intelligence, we need to look tens of light years away. Say fifty light years.  So now they need a dish fifty times bigger, that's 18km (11 miles) across.  And to search their neighbourhood to a fifty light year radius, they'd need to steer that, and keep it adequately parabolic too.  Consider too that fifty light years is on the extreme edge of optimism for reasonable numbers to plug into the Drake Equation, and the probability of another technologically advanced lifeform existing within 50 light years from us is not zero, but it must be very, very low.

 

So what about higher powered transmitters?  The 200kW BBC Radio 4 long wave transmitter is fairly typical for its waveband.  The Europe 1 transmitter in Germany is about the most powerful long wave transmitter on the planet, pushing out 2000kW at 183kHz.  That'll increase range by about three times, to 3 light years.  In terms of astronomy, that half an order of magnitude  and makes little difference to the feasibility of being heard.  It increases Brian Cox's limit from a light year to three, still well short of Proxima Centauri.

 

How about other wavebands?  Our atmosphere only allows through certain wavebands.  Long wave will get through, short wave will not.  Above that, VHF radio and UHF TV transmissions can get through, but as frequency increases, so does the energy in each photon.  So at higher frequencies, the number of photons for the same power is proportionately less, and so the range to receive sufficient photons to reconstruct the transmitted signal reduces too.  The upshot is, signals at frequencies above long wave will be undetectable closer rather than farther out.

 

You've only considered omni-directional signals, how about directed beams?  Well, yeah.  If you're talking SETI listening to the equivalent of Arecibo, then that's a different question entirely.  What I'm talking about is our routine commercial broadcast transmissions.  Many of those are vaguely directed, particularly the higher frequency transmissions.  And those already suffer from worse propagation issues than long wave.  But it's true that a directed transmission is more powerful than an omni-directional one, (one which transmits equal power in all directions).  And although the power density increases in the transmitted direction, the area of sky covered reduces, reducing the likelihood of any receiver within range detecting the signal.  So directed broadcast transmissions don't help us.

 

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Finally, let me be clear: I'm not saying that it's impossible for our transmissions to be detected by alien civilisations, if they exist.  But what I am saying is that the above is a reasoned argument supported by calculation that it's very, very improbable that there could be any within receiving range.  It's just not as easy as E.T. sitting on a planet orbiting, say, Tau Ceti with his transistor radio, listening to Hancock's Half Hour or I Love Lucy.  If we're going to make contact with technological civilisations, we'll need a highly funded, planned and directed effort.  Trusting on radio broadcasts leaking into space isn't going to cut it.