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Laser Visibility at Intensities Suitable forInterstellar Communications -TheoryRadobs 09Preface: This document (RADOBS.9) is in support of the discussion document
(RADOBS.8).
Handy conversions:
1 Astronomical Unit (A.U.) = 1.496 X 10^11 m
1 Light Year (L.Y.) = 9.461 X 10^15 m
1 Light Year (L.Y.) = 63,242 A.U.
1 Parsec (psc) = 3.26 L.Y.
The relationship between Apparent Stellar Magnitude (m) and the brightness
or intensity of a star may be expressed in the form:
m = -[19 + (2.5).log(I)] (1)
where I = received intensity (W/m^2). The threshold for naked-eye
visibility is m = +6.
The Sun's total output (EIRP) = 3.90 X 10^26 watts. Here are several
intensities and corresponding magnitudes as a function of range R:
At R = 1 A.U. (1.496 X 10^11 m):
I = 1.39 kW/m^2
m = -26.8
At R = 10 L.Y. (9.461 X 10^16 m):
I = 3.48 X 10-^9 W/m^2
m = +2.2
At R = 100 L.Y. (9.461 X 10^17 m):
I = 3.48 X 10^-11 W/m^2
m = +7.2*
At R = 1000 L.Y. (9.461 X 10^18 m):
I = 3.48 X 10^-13 W/m^2
m = +12.2*
* Not visible to the naked eye.
----------------------------
With no allowance for the Fraunhofer dark line absorption, the Planckian
(black body) starlight continuum level (spectral energy density) is given
by:
2.PI.h.f^3r^2
Npl = ----------------------- W/m^2.Hz (2)
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
----------------------------
For the purposes of this analysis we shall assume a fully (uniformly)
illuminated aperture and not a beam with a Gaussian intensity profile, as
might be obtained from a TEMoo single-mode laser.
The diffraction limited half-power (-3 dB) beamwidth is given by:
(57.3).Wl
THETA = --------- degrees (3)
d
where d = diameter of telescope (10 m).
THETA = 0.0135 arc seconds
----------------------------
The half-power (-3 dB) beam diameter is given by:
(1.22).Wl.R
D = ----------- meters (4)
d
At R = 1 A.U.:
D = 12 km
At R = 10 L.Y.:
D = 7.57 X 10^9 m = 0.0506 A.U.
----------------------------
The gain of an antenna is given by:
4.PI.At
G = ------- (5)
Wl^2
where At = area of transmitting telescope mirror (78.5 m^2).
G = 153.6 dB
----------------------------
For a mean transmitted power Pt, the Effective Isotropic Radiated
Power (EIRP) is given by:
EIRP = G.Pt Watts (6)
For Pt = 1 GW:
EIRP = 2.29 X 10^24 W
----------------------------
The intensity of the beam at the receiver is given by:
EIRP
I = -------- W/m^2 (7)
4.PI.R^2
At R = 1 A.U.:
I = 8.1 W/m^2
At R = 10 L.Y.:
I = 2.04 X 10^-11 W/m^2
----------------------------
The signal power received by a telescope is given by:
Ps = I.Ar Watts (8)
where Ar = area of receiving telescope mirror (78.5 m^2).
At R = 1 A.U.
Ps = 0.64 kW Phew!!
At R = 10 L.Y.:
Ps = 1.60 X 10^-9 W
----------------------------
The effective system noise temperature of an optical receiver may be
expressed in the form:
h.f
Teff = ----- K (9)
eta.k
where eta = photodetector quantum efficiency (0.5).
Teff = 43,900 K
----------------------------
The Carrier-To-Noise Ratio in a perfect shot (quantum) noise limited optical
heterodyne system is given by:
eta.Ps
CNR = ------ (10)
hfB
where Ps = received optical power (1.6 nW),
B = Intermediate Frequency bandwidth (30 MHz).
At R = 10 L.Y.:
CNR = 19 dB
There was no need to do a calculation for R = 1 A.U. since the optical
receiver went up in smoke! Hence, CNR effectively equal to zero!
December 23, 1990
RADOBS.09
BBOARD No. 277
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* Dr. Stuart A. Kingsley Copyright (c), 1990 *
* AMIEE, SMIEEE *
* Consultant "Where No Photon Has Gone Before" *
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