This article is from the Astronomy FAQ, by Joseph Lazio (jlazio@patriot.net) with numerous contributions by others.

By Al Aburto <aburto@nosc.mil>,

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.

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