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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.

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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

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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

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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. 

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: 

[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

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.

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.   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 
= 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 
= 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.
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