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Perhaps the easiest way to answer this question is to quote from the sci.astro FAQ:

Representative results are presented in [the table]. The short answer is

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.

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.

Instruments such as the Arecibo radio telescope could detect narrowband signals originating thousands of light-years from the Sun.

[...]

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 = 0.5`

, and`dr`

= 305 meters.

Source Frequency Bandwidth Tsys EIRP Detection Range (Br) (Kelvin) Range (R) AM Radio 530--1605 kHz 10 kHz 68 x 10 ^{6}100 KW 0.007 AU FM Radio 88--108 MHz 150 kHz 430 5 MW 5.4 AU UHF TV Picture 470--806 MHz 6 MHz 50? 5 MW 2.5 AU UHF TV Carrier 470--806 MHz 0.1 Hz 50? 5 MW 0.3 LY WSR-88D Weather Radar 2.8 GHz 0.63 MHz 40 32 GW 0.01 LY Arecibo S-band (CW) 2.380 GHz 0.1 Hz 40 22 TW 720 light years Pioneer 10 Carrier 2.295 GHz 1.0 Hz 40 1.6 kW 120 AU 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`

R2. 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}), w = wavelength (m) the antenna is tuned to, f = c / w, where f is the frequency and c is the speed of light (c = 2.99792458 x 10^{8 m/s), and pi = 3.141592654... }For an antenna (either transmiting or receiving) with circular apertures:

Ae = eta * pi * d^{2}/ 4with eta = efficiency of the antenna and 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 (Kelvin), 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)The quantity 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 leads to the following basic equation that one can use to estimate SETI detection ranges:

If Rl is the number of meters in a light year (9.46 x 10

^{15}[m/LY]), then the detection range in light years is given byR = sqrt[ EIRP * Aer * twc / (4 * pi * snr * Br * k * Tsys) ] / RlIf we wanted the range in Astronomical Units then replace Rl with Ra = 1.496 x 10

^{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 = 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 * pi * dr^{2 / 4 = 36.5E3 m2. }(...) Hence the detection range (light years) becomes

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

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