I have been wanting to write this for a long time. I will create a web-site
for this at some time, but for now, I'd just like to get your feedback.

There have been many questions in the newsgroup as to how far away our
earthly transmissions are detectable. Also the seti FAQ page deals with this
subject, and concludes that a seti@home - style receiver system cannot
detect much more than just the very strongest signals from earth if located
far outside our solar system.

In this posting, I want to put some formula's behind this, and basically
analyze the following question :

If an ET transmitter is designed to reach a receiver at a certain range, how
far does that range need to be so that we can detect it here on earth, say
100LYs away from the transmitter.

Basically, I split the sort of ET signals into two target applications :
  (1) point-to-point : this would be any signal intended for a receiver of
which the location is known (to ET)
  (2) radar : this would be any signal intended to detect a passive object
that will reflect some of the power, so that it can be detected in a
receiver (located close to the transmitter).

In the analysis below, I conclude that it is impossible for us to detect an
ET transmission which was intended for ET's planet, or close to the planet.

The only application that we have a chance of detecting is RADAR.
More specifically, we can only detect ET radar which attempted to detect
asteroids or other 'near-ET-planet' objects. Even if, and maybe especially
if, ET has a technology more advanced than ours.

All this might not come as a surprise to anyone, but I feel that it is
important to understand what sort of signals we are likely to receive (apart
from beacons, which apparently are not present around the water-hole). Also
the formula's might be of help to some of you determining the likelihood
that we will detect signals from extra-terrestrial origin. Also, it would
make sense to consider this analysis if we would start looking for 'leakage'
signals from ET.

Another conclusion is that if we would have a square-mile antenna for SETI,
this would not change much about the sort of signals we can expect to
detect. There will be more radar signals, but it would still be very
unlikely that we would detect any point-to-point ET communication signals.

As a side-note : IF there are ETI's out there within a 100LY range, AND they
use asteroid or high-resolution radar, then we WILL be able to detect these
radar signals... That puts the recent discarding of signal SHGb02+14a (which
can easily be a radar signal) in a whole new light....

Take this for what it's worth. Your comments are appreciated..

Rob


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Point-to-point transmissions :

Define :
   R  : range to the intended target receiver (meter)
   rt : effective radius of the transmitter antenna used (meter)
   rr : effective radius of the receiver antenna used (meter)
   lambda   : wavelength (c/f) used (meter)
   P : line power of the transmitter (Watt)
   SNR : signal/noise ratio in the receiver
   BW : bandwidth used in the receiver (Hz)
   kT : Bolzmann constant times system temperature
   pi : 3.141...

The following formula is basically the one from the seti FAQ page :

(1)    SNR = P * (2*pi*rt / lambda) * (rr / (4*R^2)) * 1/(BW*kT)


Radar :
   ro : effective radius of the object (or resolution) that the radar
attempt to detect

(2)    SNR = P * (2*pi*rt / lambda) * ( ro^2 / (4*R^2) ) * (rr / (4*R^2)) *
1/(BW*kT)

See here the R^4 drop-off of radar... This is due to reflection at the
target object.
The factor ( ro^2 / (4*R^2) ) might require some explanation.
I am assuming here that the object that the radar attempts to detect is
reflecting the power that it receives omnidirectionally. Essentially, if 10W
of power would hit the object, I assume that it absorbs 5Watt and reflects
the other 5 in the 180degree cone back to the direction of the transmitter.
An ideal spherical, metal object would create a factor 2*(ro^2 / 4R^2)
because it would reflect all incoming power back into the 180degree cone it
came from.
An ideal mirror has a totally different factor : pi*ro^2, because it
reflects all incoming power back. That power obviously would only be
directed back to the receiver if the mirror is in one exact position.
This (mirror) would destroy the R^4 drop-off which is typical for radar.
That is why stealth fighters have very 'flat' surfaces. Any incoming radar
signals will be reflected to different directions than the original incoming
radar signal, so nothing gets detected in the receiver (which is typically
at the same location as the transmitter).
Any reflection factor should have a 1/R^2 factor in it, so the factor
ro^2/(4*R^2) seems reasonable for radar, and if you want to, you can
multiply that by a factor (alpha) between 0 and 1, for different materials
of the target object...


Important for the next deduction are not the absolute numbers, but the ratio
of them :

Radar signals must be a factor 4* (R/ro)^2 stronger than any point-to-point
transmission, to obtain the same SNR for the same target range.

So what does this mean for us as external, far-away, eavesdropping observers
?
It means that radar signals can be detected a factor 2*R/ro further than
point-to-point transmissions.
That is in fact a very rough estimate, and requires more detailed analysis
(further down), but roughly this is correct.

How big is R/ro ?
Lets look at some radar applications on earth. I don't know much about the
specifications of various radar applications, but the following numbers seem
to make sense :

  - air traffic surveillance radar : 1m resolution at 100km range : R/ro =
100,000
  - Arecibo asteroid radar : 100km resolution at 10 light-minutes :
1.8*10^11 / 1*10^5 = 1,8M.

I'm not sure if these numbers are true and accurate, but it is clear that
radar systems would actually penetrate 100k to 1M times deeper into space
than do transmissions which are intended for a certain defined target
(point-to-point communication).
In other words, they are a factor 1*10^10 to 1*10^12 times stronger in power
(R/ro)^2)...

----


Now lets ask another question, and see if we can come up with a reasonable
answer :

If we, at say 100LYs distance, are listening to such ET signals, what would
the target range have to be of either a ET point-to-point transmission or a
ET radar transmission, so that we can still detect it ?

Obviously, if we want to achieve the SNR that ET intended for its
application, and we are using a similar receiving antenna, and receiver
sensitivity, then the range would need to be 100LYs. That would only be true
for beacons, but I do not want to discuss that here in this posting.

So where can we achieve advantage over ET's receiver, so that we can detect
her signal further than the intended range ? Here are some ideas with
ballpark estimates :

  a) We can settle with a lower SNR ration. Most applications require 30dB
or so SNR, and we might settle for just 10db, so we can gain 20dB here
(power factor 100, distance factor 10).

  b) We can use a bigger receiving antenna. The power factor we can gain is
(r/rr)^2, where r would be our antenna radius, and rr is ETs receiving
antenna for the application. Arecibo has a 100m effective radius (correct me
if I'm wrong), which can easily be 10x larger than what ET is using. So we
might gain 20db with this. A square-mile receiver (as is in the planning)
might add another 15db to this.

  c) We can integrate the signal we receive. Here it gets complex, but
important. Seti@home uses very narrowband (0.1Hz?) 'channels' and integrates
any signal with this. Integration typically enhances a signal by a factor
sqrt(BW*t) in SNR (power factor). t (time) must be restricted for us to
about 10sec (0.1Hz minimum bandwidth) or so, since any strongly beamed ET
transmission not intended for us would not be aimed at earth much longer
than that. Bandwidth could be small (for high-resolution asteroid radar), or
very wide (for broadband communication transmissions). Overall. I think that
a BW*t factor of 10 (for very narrowband, high-resolution radar or
communication) to 1G (for 100MHz broadband communication) is reasonable to
assume. That means that we can gain 5-35db with integration.

   d) We can improve the sensitivity of the receiver (system temperature).
Assuming that ET is not stupid, we cannot gain much here. 100K is reasonable
for any application, and thus the max we can gain here is about 5dB or so
(for a 30K receiver).

Overall, we should be able to obtain 50db (for narrowband transmissions) to
80dB (for broadband) improvement of SNR, with a seti@home style receiver
system on Arecibo. That translates to a distance-factor of about 300-10K.
That's not a lot, compared to the distance factor of 100K-1M of radar versus
point-to-point communication....

So, to answer the question : The target range for ET transmissions,
detectable at seti@home at 100LYs would be the following :
    - factor 100LYs / 10K = 0.01 LY = 650 AU for broadband point-to-point
communication
    - factor (100LYs / 300) / 1M = 0.17 Light-minutes = 3*10^6 km. for
narrowband radar.
    - factor (100LYs / 10K) / 1M = 94*10^3 km for broadband radar

Broadband radar is not really an option to consider, since for radar, the
resolution can be much better improved by narrowing the bandwidth. But who
knows what ET thinks about this...

So what conclusions can we draw from this ?
Seems that it is out of the question that we would ever detect any ET
point-to-point communication signal, since it would have to be targeting
something or somebody far out into or outside of ET's star system.
However, we DO have a chance to detect radar signals from ET. There is not
much of a chance to detect surveillance radar, targeting ET's terrestrial
objects, but there IS a chance to detect ET's narrowband asteroid radar, or
even high-resolution (military?) radar for ET's planet's close-proximity
(incoming missiles?) analysis. And there is a minute chance to find some
broadband radar signals from ET which were intended for objects on or near
ET's planet.

Interesting to note here, is that a square-mile antenna would not change
much about this analysis. The target range of ET transmissions would reduce
by a factor of 10 at most, but that still requires an absurdly large target
range for point-to-point transmissions (in the range of 65 AU). Only an
interplanetary radio-hub would become detectable, but still very faintly. So
overall, even if we increase the size of the receiver from Arecibo to a
square-mile antenna, we still would only be able to detect ET radar
signals.....

It might be time to start listening for radar signals...