David Woolley <david@djwhome.demon.co.uk>

Representative results are presented in Tables 1 and 2.  The short
answer is
 (1) Detection of broadband signals from Earth such as AM radio, FM
     radio, and television picture and sound would be extremely
     difficult even at a fraction of a light-year distant from the
     Sun.  For example, a TV picture having 5 MHz of bandwidth and 5
     MWatts of power could not be detected beyond the solar system
     even with a radio telescope with 100 times the sensitivity of the
     305 meter diameter Arecibo telescope.

 (2) Detection of narrowband signals is more resonable out to
     thousands of light-years distance from the Sun depending on the
     transmitter's transmitting power and the receiving antenna size.
            
 (3) Instruments such as the Arecibo radio telescope could detect
     narrowband signals originating thousands of light-years from the
     Sun.
            
 (4) A well-designed 12 ft diameter amateur radio telescope could
     detect narrowband signals from 1 to 100 light-years distance
     assuming the transmitting power of the transmitter is in the
     terawatt range.

What follows is a basic example for the estimation of radio and
microwave detection ranges of interest to SETI.  Minimum signal
processing is assumed.  For example an FFT can be used in the
narrowband case and a bandpass filter in the broadband case (with
center frequency at the right place of course).  In addition it is
assumed that the bandwidth of the receiver (Br) is constrained such
that it is greater than or equal to the bandwidth of the transmitted
signal (Bt) (that is, Br >= Bt).

Assume a power Pt (watts) in bandwidth Bt (Hz) radiated isotropically.
At a distance of R (meters), this power will be uniformly distributed
(reduced) over a sphere of area: 4 * pi * R^2.  The amount of this
power received by an antenna of effective area Aer with bandwidth Br
(Hz), where Br >= Bt, is therefore:

  Pr = Aer * (Pt / (4 * pi * R^2))

If the transmitting antenna is directive (that is, most of the
available power is concentrated into a narrow beam) with power gain Gt
in the desired direction then:

  Pr = Aer * ((Pt * Gt) / (4 * pi * R^2))

The antenna gain G (Gt for transmitting antenna) is given by the
following expression.  (The receiving antenna has a similar expression
for its gain, but the receiving antenna's gain is not used explicitly
in the range equation.  Only the effective area, Aer, intercepting the
radiated energy at range R is required.)

  Gt = Aet * (4 * pi / (w^2)), where
     
       Aet = effective area of the transmitting antenna (m^2), and
         w = wavelength (m) the antenna is tuned to.
         f = c / w, where f is the frequency and c is the speed of light.
         c = 2.99792458E+08 (m/sec)
        pi = 3.141592654...

For an antenna (either transmiting or receiving) with circular apertures:

  Ae = <eta> * pi * d^2 / 4

        <eta>r = efficiency of the antenna,
        d = diameter (m) of the antenna.

The Nyquist noise, Pn, is given by:

  Pn = k * Tsys * Br, where
     
          k = Boltzmann's constant = 1.38054E-23 (joule/kelvin)
       Tsys = is the system temperature (kelvins), and
         Br = the receiver bandwidth (hertz).

The signal-to-noise ratio, snr, is given by:

  snr = Pr / Pn.

If we average the output for a time t, in order to reduce the variance
of the noise, then one can improve the snr by a factor of 
sqrt(Br * t). Thus:

  snr = Pr * sqrt(Br * t) / Pn.

The factor Br*t is called the "time bandwidth product," of the receive
processing in this case, which we'll designate as:

  twp = Br * t.

We'll designate the integration or averaging gain as:

  twc = sqrt(twp).

Integration of the data (which means: twp = Br * t > 1, or
t > (1 / Br) ) makes sense for unmodulated "CW" signals that are
relatively stable over time in a relatively stationary (steady) noise
field.  On the other hand, integration of the data does not make
sense for time-varying signals since this would distroy the
information content of the signal.  Thus for a modulated signal 
twp = Br * t = 1 is appropriate.

In any case the snr can be rewritten as:

snr = (Pt * Gt) * Aer * twc / (4 * pi * R^2 * Br * k * Tsys)

Pt * Gt is called the Effective Isotropic Radiated Power (EIRP) in
the transmitted signal of bandwidth Bt. So:

EIRP = Pt * Gt, and

snr = EIRP * Aer * twc / (4 * pi * R^2 * Br * k * Tsys)

This is a basic equation that one can use to estimate SETI detection
ranges. 

#######################################################################
# If Rl is the number of meters in a light year (9.46E+15 [m/LY]),    #
# then the detection range in light years is given by                 #
#                                                                     #
# R = sqrt[ EIRP * Aer * twc / (4 * pi * snr * Br * k * Tsys) ] / Rl  #
#                                                                     #
# If we wanted the range in Astronomical Units then replace Rl        #
# with Ra = 1.496E+11 (m/AU).                                         #
#######################################################################

Note that for maximum detection range (R) one would want the transmit
power (EIRP), the area of the receive antenna (Aer), and the time
bandwidth product (twp) to be as big as possible.  In addition one
would want the snr, the receiver bandwidth (Br), and thus transmit
signal bandwidth (Bt), and the receive system temperature (Tsys) to be
as small as possible.

(There is a minor technical complication here.  Interstellar space
contains a plasma.  Its effects on a propagating radio wave including
broadening the bandwidth of the signal.  This effect was first
calculated by Drake & Helou and later by Cordes & Lazio.  The
magnitude of the effect is direction, distance, and frequency
dependent, but for most lines of sight through the Milky Way a typical
value might be 0.1 Hz at a frequency of 1000 MHz.  Thus, bandwidths
much below this value are unnecessary because there will be few, if
any, signals with narrower bandwidths.)

Now we are in a position to carry out some simple estimates of
detection range.  These are shown in Table 1 for a variety of radio
transmitters.  We'll assume the receiver is similar to Arecibo, with
diameter dr = 305 m and an efficiency of 50% (<eta>r = 0.5).  We'll
assume snr = 25 is required for detection (The META project used a snr
of 27--33 and SETI@home uses 22; more refined signal processing might
yield increased detection ranges by a factor of 2 over those shown in
the Table 1.)  We'll also assume that twp = Br * Tr = 1.  An
"educated" guess for some of the parameter values, Tsys in particular,
was taken as indicated by the question marks in the table.  As a
reference note that Jupiter is 5.2 AU from the Sun and Pluto 39.4 AU,
while the nearest star to the Sun is 4.3 LY away.  Also any signal
attenuation due to the Earth's atmosphere and ionosphere have been
ignored; AM radio, for example, from Earth, is trapped within the
ionosphere.

The receive antenna area, Aer, is 

  Aer = <eta>r * pi * dr^2 / 4 = 36.5E3 m^2.

(Scientific notation is being used here; 1E1 = 10, 1E2 = 100, 1E3 =
1000, so 36.5E3 is 36.5 times 1000.)  Hence the detection range (light
years) becomes

  R = 3.07E-04 * sqrt[ EIRP / (Br * Tsys) ].

Table 1 Detection ranges of various EM emissions from Earth and the
          Pioneer spacecraft assuming a 305 meter diameter circular
          aperture receive antenna, similar to the Arecibo radio
          telescope. Assuming snr = 25, twp = Br * Tr = 1, <eta>r =
          0.5, and dr = 305 meters.
-------------+--------------+-----------+--------+--------+-----------+
Source       | Frequency    | Bandwidth | Tsys   | EIRP   | Detection |
             | Range        |    (Br)   |(Kelvin)|        | Range (R) |
-------------+--------------+-----------+--------+--------+-----------+
AM Radio     | 530-1605 kHz |  10   kHz | 68E6   | 100 KW |  0.007 AU |
-------------+--------------+-----------+--------+--------+-----------+
FM Radio     |  88-108  MHz | 150   kHz |  430   |   5 MW |    5.4 AU |
-------------+--------------+-----------+--------+--------+-----------+
UHF TV       | 470-806  MHz |   6   MHz |  50  ? |   5 MW |    2.5 AU |
Picture      |              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+
UHF TV       | 470-806  MHz |   0.1  Hz |  50  ? |   5 MW |    0.3 LY |
Carrier      |              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+
WSR-88D      |   2.8    GHz |  0.63 MHz |  40    |  32 GW |   0.01 LY |
Weather Radar|              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+
Arecibo      |   2.380  GHz |  0.1   Hz |  40    |  22 TW |    720 LY |
S-Band (CW)  |              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+
Arecibo      |   2.380  GHz |  0.1   Hz |  40    |   1 TW |    150 LY |
S-Band (CW)  |              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+
Arecibo      |   2.380  GHz |  0.1   Hz |  40    |   1 GW |      5 LY |
S-Band (CW)  |              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+
Pioneer 10   |   2.295  GHz |  1.0   Hz |  40    | 1.6 kW |    120 AU |
Carrier      |              |           |        |        |           |
-------------+--------------+-----------+--------+--------+-----------+

It should be apparent then from these results that the detection of AM
radio, FM radio, or TV pictures much beyond the orbit of Pluto will be
extremely difficult even for an Arecibo-like 305 meter diameter radio
telescope!  Even a 3000 meter diameter radio telescope could not
detect the "I Love Lucy" TV show (re-runs) at a distance of 0.01
Light-Years!

It is only the narrowband high intensity emissions from Earth
(narrowband radar generally) that will be detectable at significant
ranges (greater than 1 LY).  Perhaps they'll show up very much like
the narrowband, short duration, and non-repeating, signals observed by
our SETI telescopes.  Perhaps we should document all these
"non-repeating" detections very carefully to see if any long term
spatial detection patterns show up.

Another question to consider is what an Amateur SETI radio telescope
might achieve in terms of detection ranges using narrowband FFT
processing.  Detection ranges (LY) are given in Table 2 assuming a 12
ft (3.7 m) dish antenna operating at 1.42 GHz, for various FFT
binwidths (Br), Tsys, snr, time bandwidth products (twp = Br*t), and
EIRP values.  It appears from the table that effective amateur SETI
explorations can be conducted out beyond approximately 30 light years
provided the processing bandwidth is near the minimum (approximately
0.1 Hz), the system temperature is minimal (20 to 50 Degrees Kelvin),
and the EIRP of the source (transmitter) is greater than approximately
25 terawatts.


Table 2     Detection ranges (LY) for a 12 foot diameter amateur
            radio telescope SETI system, operating at 1.420 GHz.
                                 +-------------------------------+
                                 |             EIRP              |
                                 +-------+--------+------+-------+
                                 | 100TW |  25TW  |  1TW | 100GW |
-------+-------+----------+------+-------+--------+------+-------+
   Br  | Br*t  |   Tsys   | snr  |        Detection Range        |
  (Hz) |       | (kelvin) |      |             (LY)              |
-------+-------+----------+------+-------+--------+------+-------+
  0.1  |   2   |    50    |  25  |   28  |    17  |  3.4 |   1.1 |
-------+-------+----------+------+-------+--------+------+-------+
  0.1  |   1   |    50    |  25  |   20  |    12  |  2.4 |  0.76 |
-------+-------+----------+------+-------+--------+------+-------+
  0.5  |   2   |    50    |  25  |  12.7 |    6.4 |  1.3 |   0.4 |
-------+-------+----------+------+-------+--------+------+-------+
  0.5  |   1   |    50    |  25  |    9  |    4.5 |  0.9 |   0.3 |
-------+-------+----------+------+-------+--------+------+-------+
  0.1  |  20   |    50    |  25  |   90  |    54  |  11  |   3.4 |
-------+-------+----------+------+-------+--------+------+-------+
  1.0  | 200   |    50    |  25  |   90  |    54  |  11  |   3.4 |
-------+-------+----------+------+-------+--------+------+-------+


REFERENCES:
        Radio Astronomy, John D. Kraus, 2nd edition, Cygnus-Quasar
        Books, 1986, P.O. Box 85, Powell, Ohio, 43065.

        Radio Astronomy, J. L. Steinberg, J. Lequeux, McGraw-Hill
        Electronic Science Series, McGraw-Hill Book Company, Inc,
        1963.

        Project Cyclops, ISBN 0-9650707-0-0, Reprinted 1996, by the
        SETI League and SETI Institute.

        Extraterrestrial Civilizations, Problems of Interstellar
        Communication, S. A. Kaplan, editor, 1971, NASA TT F-631
        (TT 70-50081), page 88.