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EJASA - Part 11

                                                                     Page 71

                                  APPENDIX A

                       THEORY AND SPECIMEN CALCULATIONS


    The Drake Equation:

    Fundamental to all SETI approaches is the belief that there are a
    reasonable number of technological civilizations out there who might be
    trying to communicate with us.

    The following formula for the number of technological civilizations in
    the galaxy is a modified form of the one devised in 1961 by Frank Drake
    [2-3] of Cornell (also President of the SETI Institute) and is known
    as the famous "Drake Equation": [13,25]

    N = R*.fp.ne.fl.fi.fc.L                                              (1)

    where  R* = number of stars in the Milky Way galaxy (400 X 10^9),
           fp = fraction of stars that have planetary systems (0.1),
           ne = average number of planets in such star systems that can
                support life (1),
           fl = fraction of planets on which life actually occurs (0.1),
           fi = fraction of such planets which intelligent life arises
                (0.01),
           fc = fraction of intelligent beings knowing how to communicate
                with other civilizations (0.1),
           L  = average lifetime (fraction of the age of its star) of such
                technical civilizations (0.001).


    Substituting what some might say are conservative values given in
    parentheses for the entire Milky Way galaxy:

                                   N = 4,000

    Thus, there could be a minimum 4,000 worlds for us to detect in our
    galaxy.  If there were only 4,000 technical civilizations within a
    galaxy that is 100,000 light years in diameter, then the probability of
    detecting ETI signals is likely to be small.  However, many SETI
    scientists and exobiologists give more optimistic values for these
    parameters, and thus yield higher values for N.  If fp, fl, fi, fc, and
    L are significantly higher, our galaxy would be teeming with
    intelligent technical civilizations.  If we assume that the average
    lifetime of a star is 10 billion years, then a value of L = 0.001
    implies that civilizations can last 10 million years.  Clearly, there
    is a substantial degree of uncertainty in the value of L.

    Within 1,000 light years of Sol there are 10 million stars, of which
    1 million are solar-type.  Thus, taking a more optimistic value for
    "N", the SETI community reasons that there is a significant chance of
    detecting an ETI signal if we "look" out to 1,000 light years, assuming
    of course, that we are tuned to the correct frequencies.  The issue of
    the correct frequencies to search is at the heart of this paper.


                                                                     Page 72

    Apparent Stellar and Signal Magnitudes:

    The relationship between Apparent Stellar Magnitude (m) [88-90] and the
    brightness or intensity of a solar-type star (or a laser operating at or
    near the peak of the photopic response) may be expressed in the form:

    m = -[19 + (2.5).log(Ir)]                                            (2)

    where Ir = received intensity (W/m^2).


    The threshold for unaided eye visibility (dark sky) is m = +6.  As
    mentioned above, this expression may also be used to estimate the
    approximate visibility of a laser, i.e., the apparent signal magnitude,
    if its wavelength is not too far removed from the peak of the low-
    intensity visual response at 500 nm.  Here are several intensities and
    corresponding magnitudes as a function of range R.  We note that the
    Sun's total output (EIRP) = 3.90 X 10^26 watts:

    At R = 1 A.U. (1.496 X 10^11 m):

                               Ir =  1.39 kW/m^2
                               m  = -26.8

    Thus the solar flux density at normal incidence just outside Earth's
    atmosphere is 1.39 kW/m^2.


    At R = 10 L.Y. (9.461 X 10^16 m):

                               Ir =  3.48 X 10^-9 W/m^2
                               m  = +2.2


    At R = 100 L.Y. (9.461 X 10^17 m):

                               Ir =  3.48 X 10^-11 W/m^2
                               m  = +7.2*


    At R = 1,000 L.Y. (9.461 X 10^18 m):

                               Ir =  3.48 X 10^-13 W/m^2
                               m  = +12.2*

    * Not visible to the unaided eye.


    In Table 2 (Page 22), Apparent Magnitudes are quoted for stars,
    extrasolar planets, and ETI transmitters on the basis of the visual
    brightness or intensity of each object acting alone.  Because the
    reason for quoting the Apparent Magnitudes is to demonstrate that
    relatively strong laser transmitters are still "visually" weak, the
    Apparent Magnitudes are only given for the visible wavelength.


                                                                     Page 73

    Planckian Starlight Background:

    For observations at night, the background Nb may be taken as the
    Planckian (black body) starlight continuum level (Npl). [88-90]  With
    no allowance for the Fraunhofer dark line absorption or bright line
    emission, the non-polarized spectral energy density is given by:

               2.PI.h.f^3r^2
    Npl = -----------------------  W/m^2.Hz                              (3)
          c^2[e^(h.f/k.T) - 1]R^2

    where  h  = Planck's constant (6.63 X 10^-34 J.s),
           c  = velocity of light (3 X 10^8 m/s),
           Wl = wavelength (656 nm),
           f  = frequency (c/Wl = 4.57 X 10^14 Hz),
           k  = Boltzmann's constant (1.38 X 10^-23 J/K),
           T  = temperature (5778 K),
           r  = radius of star (6.96 X 10^8 m),
           R  = distance of receiver (10 L.Y. = 9.461 X 10^16 m).


    At R= 1 A.U.:

                         Npl = 2.19 X 10^-12 W/m^2.Hz

    At R = 10 L.Y.:

                         Npl = 5.47 X 10^-24 W/m^2.Hz


    Full Width Half Maximum (FWHM) Angular Beamwidth:

    For the purposes of this part of the analysis, we have assumed a fully
    (uniformly) illuminated circular aperture and not a beam with a
    Gaussian intensity profile, as might be obtained from a laser with a
    single transverse TEMoo mode.  The diffraction limited half-power
    (-3dB) beamwidth is given by: [66,85]

                     (58.5).Wl
    FWHM Beamwidth = ---------  degrees                                  (4)
                         d

    where  Wl = wavelength,
           d  = diameter (aperture) of telescope.


    For d = 10 m (professional telescope) and Wl = 656 nm:

                      FWHM Beamwidth = 0.0138 arc seconds


    For d = 0.30 m (amateur telescope) and Wl = 656 nm:

                      FWHM Beamwidth = 0.461 arc seconds


                                                                     Page 74

    Full Width Half Maximum (FWHM) Diameter:

    The diffraction limited far-field half-power (-3 dB) beam diameter is
    given by:

                    (1.02).Wl.R
    FWHM Diameter = -----------  meters                                  (5)
                         d


    At R = 10 L.Y.:

                  FWHM Diameter = 6.33 X 10^9 m = 0.0423 A.U.


    Gaussian Beamwidth:

    If a laser is used to illuminate a transmitting telescope, and if the
    aperture is greater than 4wo, theory gives the far-field 1/e^2 beam
    diffraction angle as:

                         (115).Wl
    Gaussian Beamwidth = --------  degrees                               (6)
                          PI.wo

    where wo = the TEMoo mode waist radius of the Gaussian beam.


    For a compromise aperture diameter d = 2wo, where a little diffraction
    will occur and produce some sidelobe energy, the (1/e^2) diffraction
    angle of the main lobe of a 10-meter telescope is given by:

                    Gaussian Beamwidth = 0.0172 arc seconds


    The corresponding (1/e^2) Gaussian beam diameter at the target is:

                        Gaussian Diameter = 0.0527 A.U.


    This is not that different to the previous case for a fully-illuminated
    aperture (no amplitude taper apodization).


    Rayleigh Range:

    For a Gaussian beam, the Rayleigh or near-field range of a diffraction
    limited single or multi-aperture (array) telescope is given by:

          PI.wo^2
    Ray = -------                                                        (7)
            Wl




                                                                     Page 75

    At the Rayleigh range Ray, the beam diameter has expanded by a factor of
    1.414.  As the distance increases beyond the Rayleigh range, the beam
    diameter becomes proportional to distance, and the inverse square law
    applies to the beam intensity.


    Considering our 10-meter diameter transmitting telescope with a
    Gaussian beam, and a compromise aperture diameter d = 2wo.

    For wo = 5 m and Wl = 656 nm:

                              Ray = 1.2 X 10^8 m

                                  = 0.0008 A.U.


    Now consider an array that has a width of 10 km.

    For wo = 5 km and Wl = 656 nm:

                              Ray = 1.2 X 10^14 m

                                  = 800 A.U.


    Finally, consider a Mercury-size planetary phased-array as conjectured
    by Dr. John Rather. [56]

    For a wo = 2,439 km and Wl = 656 nm:

                              Ray = 2.8 X 10^19 m

                                  = 3,000 L.Y.


    With such a huge array, the inverse square law does not apply over
    considerable distances.  The Rayleigh range can stretch out over 3,000
    light years, so that the flux density is essentially undiminished by
    distance, accept for any interstellar absorption effects.  Of course,
    the implication that a pencil beam (celestial searchlight) some
    3,500 km in diameter, i.e, of planetary diameter, could be landed on a
    desired planet 10 lights years away, let alone 3,000 light years,
    somewhat stretches even this author's imagination!


    Polar Response:

    The Polar Response (PR) or Directivity of a transmitting or receiving
    telescope with a single fully illuminated circular aperture, with no
    amplitude taper (apodization), is given by: [85]

         [2.J1{(PI.d/Wl).sin(PHI)}]^2
    PR = ----------------------------                                    (8)
            [(PI.d/Wl).sin(PHI)]^2


                                                                     Page 76

    where  J1  = Bessel Function of the first kind,
           d   = diameter (aperture) of telescope,
           Wl  = wavelength,
           PHI = angular separation.


    For the 10-meter diameter telescope at 656 nm, the first sidelobe is
    located at 0.022 arc seconds from the main lobe, and the response is
    17.6 dB down.  The second sidelobe occurs at 0.036 arc seconds from the
    main lobe, and response is 23.8 dB down.


    In a diffraction limited space-based telescope system, where the angle
    PHI between the image of the transmitter and star is >= FWHM/2 (-3 dB
    half width half maximum), the Planckian suppression, ignoring
    scattering within the telescope, is given by:

                                             8
    Suppression Factor >= 10.Log[-------------------------]  dB          (9)
                                 PI.{(PI.d/Wl).sin(PHI)}^3


    Equ. 9 essentially shows that the suppression factor is inversely
    proportional to the telescope's aperture raised to the third power.
    For a transmitter at 10 light years, located 1 A.U. from its star, and
    centered on the main lobe of the receiver, the maximum angular
    separation of the star is 0.275 arcseconds.  Using the parameters for
    the 10-meter diameter 656 nm telescope which has a FWHM beamwidth of
    0.0138 arc seconds, we find that the condition PHI >= FWHM/2 is more
    than satisfied, and the minimum suppression factor for the Planckian
    starlight continuum is:

                              Suppression = 50 dB


    This value is added to the Signal-To-Planckian Ratio (SPR) to arrive at
    the effective SPR when a large telescope is diffraction limited, and
    viewing a nearby star system at right angles to the star's plane of
    ecliptic (Table 2, Line 23, Page 22).  The suppression factor can be
    larger than predicted by Equ. 9 (up to a limit set by scattering and
    secondary mirror diffraction) if the star's image happens to be situated
    in a response null.  However, scattering effects and non-ideal optics
    will set a limit to this suppression factor to between 40 and 50 dB.


    Antenna Gain:

    The gain of a uniformly illuminated antenna is given by: [5,71,85]

        4.PI.At
    G = -------                                                         (10)
         Wl^2

    where At = area of transmitting telescope mirror (78.5 m^2).


                                                                     Page 77

    For a 10-meter diameter telescope at 656 nm:

                               G = 2.3 X 10^15

                                 = 153.6 dB


    Effective Isotropic Radiated Power (EIRP):

    The Effective Isotropic Radiated Power [5,8,85] is given by:

    EIRP = G.Pt  Watts                                                  (11)

    where Pt = transmitter power (W).


    For Pt = 1 GW:

                            EIRP = 2.29 X 10^24 W


    Received Signal Intensity:

    The received signal intensity just outside Earth's atmosphere is:

           EIRP
    Ir = --------                                                       (12)
         4.PI.R^2

    where  EIRP = effective isotropic radiated power (W),
           R    = range (10 L.Y. = 9.461 X 10^16 m).


    At a range of ten light years, a 1 GW transmitter EIRP = 2.29 X 10^24 W
    produces an intensity (Ir) just outside our atmosphere of
    2.04 X 10^-11 W/m^2.  For a perfect space-based 10-meter diameter
    telescope, the received signal power (Pr) is 1.6 nW.


    Received Signal Power:

    From Equs. 10, 11, and 12, and because the receiving aperture area
    At = PI.D^2/4, we may write the "perfect" received signal for the
    symmetrical telescope system in the simple form:

             PI^2.D^4
    Pr = Pt.-----------                                                 (13)
            16.R^2.Wl^2


    It can be clearly seen from the above, that the received power is
    proportional to D^4 and inversely proportional to Wl^2.  Thus, beamed
    optical links, particularly those operating in the visible spectrum,
    have the potential for tremendous throughputs.


                                                                     Page 78

    A slightly simpler form of this expression has been used by Albert Betz
    in his recent CO2 paper. [57]  To a close approximation, Equ. 13 may be
    further simplified to:

               D^4
    Pr = Pt.--------                                                    (14)
            R^2.Wl^2


    A more conservative analysis for ground-based observatories, would take
    into account atmospheric transmission losses, aperture blocking, and
    spectrometer efficiency in the case of an incoherent receiver.  For a
    a ground-based telescope, the optical power reaching the photodetector
    is given by:

    Pr = Ir.Tr.Ae.Ar.SE                                                 (15)

    where  Ir = intensity just outside atmosphere (2.04 X 10^-11 W/m^2),
           Tr = atmospheric transmission (0.4 for visible, 0.6 for CO2),
           Ae = antenna efficiency (0.7),
           Ar = antenna aperture area (0.0707 m^2),
           SE = spectrometer efficiency (0.5).


    For a 30-cm diameter (12-inch) visible telescope, and the above
    parameter values (1 GW, 10 m transmitter, EIRP = 2.29 X 10^24 W,
    Ir = 2.04 X 10^-11 W/m^2), the received visible signal:

                        Prv = 2 X 10^-13 W (-127 dBW)

    For a 30-cm diameter (12-inch) CO2 telescope, and the above parameter
    values (1 GW, 10 m transmitter, EIRP = 8.78 X 10^21 W,
    Ir = 7.81 X 10^-14 W/m^2), the received infrared signal:

                        Pri = 1.2 X 10^-15 W (-149 dBW)


    Daylight Background:

    The sky background radiation power detected per pixel, is given by:

    Pb = (PI.THETA^2.Ae.Ar.SE/4).Bo.N(Wl)  W                            (16)

    where  THETA = diffraction limited beamwidth (5.34 X 10^-6 radians),
           Bo    = optical bandpass (0.143 nm),
           N(Wl) = spectral radiance (W/m^2.sr.nm).


    For the incoherent optical systems, the pixel has a diffraction limited
    field-of-view (FOV) corresponding to the Airy disk, i.e., (2.44)Wl/d
    radians, where Wl = wavelength, and d is the aperture diameter.  For
    coherent systems, a smaller FOV is employed; that corresponding to the
    FWHM response, i.e., (1.02)Wl/d radians.  The latter pixel size is
    smaller because of the requirement to reduce the amount of local-


                                                                     Page 79

    oscillator power that does not beat with the signal but only induces
    excess quantum shot-noise.


    At visible wavelengths:

    N(Wl) = 0.01 W/cm^2.sr.micron [71]
          = 0.1 W/m^2.sr.nm
    N(f)  = 1.43 X 10^-13 W/m^2.sr.Hz

    The daytime sky background for a 12" (30 cm) telescope at 656 nm (not
    allowing for atmospheric distortion effects) with an optical bandpass
    filter bandwidth Bo = 100 GHz (0.143 nm):

                        Pbv = 7.9 X 10^-15 W (-141 dBW)


    The background is about 14 dB (Prv - Pbv) below the signal from the
    1 GW transmitter which produces an EIRP = 2.29 X 10^24 W, and a flux of
    2.04 X 10^-11 W/m^2 at a range of 10 light years.  Thus, in this small
    photon-counting receiver, the fluctuation noise from the daylight
    background is 14 dB below that of the quantum shot-noise generated by
    the signal.  This has little effect on signal detectability.  If a
    polarizer is employed, Pb can be reduced by a further 3 dB.  For a
    perfect space-based 10 meter diameter visible telescope, the daylight
    spectral density is about 4 X 10^-26 W/Hz (Figure 3, Page 17).


    For infrared systems, the 300 K temperature of the atmosphere produces
    a black body peak at approximately 10,600 nm, with a spectral radiance
    given by:

    N(Wl) = 0.0002 W/cm^2.sr.micron [71]
          = 0.002 W/m^2.sr.nm
    N(f)  = 7.5 X 10^-13 W/m^2.sr.Hz

    The sky background for a cooled 12" (30 cm) telescope at 10,600 nm (not
    allowing for atmospheric distortion effects) with a cooled 0.35 percent
    optical bandpass filter bandwidth Bo = 100 GHz (37.5 nm):

                        Pbi = 1.1 X 10^-11 W (-110 dBW)


    For an EIRP = 8.78 X 10^21 W and Ir = 7.81 X 10^-14 W/m^2, the
    potential CO2 SNR is degraded by about 39 dB (Figure 6, Page 38)
    because the background noise is 39 dB -(Pri - Pbi) above the quantum
    shot noise.  The infrared graph of Figure 6 is plotted to the same
    scales as that of the Figure 8 (Page 44) visible graph, to make
    comparisons easier, and the pages may be flicked back and forth to show
    the differences more dramatically.  We can clearly see that the
    effective optical bandwidth must be substantially reduced if ETI signal
    detectability at 10.6 microns is not to be impaired.  Thus, only
    heterodyning receivers, with effective optical bandwidths measured in
    MHz and not GHz, are suitable for CO2 SETI within the atmosphere.


Continued


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