Project Cyclops, A Design... - Department of Earth and Planetary ...

islessfor shortranges, increases asthefifth powerof
range(thepowerperantennaincreases asR 2 and m
increases as R3), and so intersects the power required for
an omnidirectional beacon. The line labeled ZA t = A r is
the locus for which the total transmitting antenna area
equals the receiving antenna area of the assumed
receiving system. The curves assume that m is equal to
the number of F, G, and K stars within the indicated
ranges; m = R31440. The scale of m is shown along the
top.
I0 9
IO 8
ASSUMEDRECEIVER:
m=NUMBER OF ANTENNAS=NUMBER OF
F,, G, K STAR S
I I01 I0 2 10 3 I0 4 105 106
i i i I
ANTENNADIAMETER 3kin
NOISE TEMPERATURE20* K
BANDWIDTH
O,t HZ
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We conclude that long-range beacons will be omnidirectional
Power
and high powered.
Level of Beacons
How much power might we expect in an omnidirectional
beacon? If we make the assumption that all races
end up transmitting omnidirectional beacons as well as
searching with directive receivers, then the proper
beacon power is that which minimizes the total system
cost for all races involved. The total system cost will
increase as the search is carried to greater ranges, but at
any assumed range limit, it can be minimized.
The efficacy of a beacon-search-receiver combination
is proportional to PtAr where Pt is the beacon power
and A r is the receiving antenna area. At both ends these
factors represent the most costly items. Any race deciding
to transmit as well as receive will decide that the
other races will also be doing both for similar reasons,
and that all would have the goal of minimizing the total
system cost. The total cost includes many terms but a
dominating one will be
rr
m=m
O
ft.
laJ
I---
Q
IO 7
OMNIDIRECTIONAL
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where
C = KpP t + KAA r (22)
iv"
_I0 6
I,-
0
I--
10,5
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I0 4
I IO I I0 2 IO 3
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RANGE LIMIT, lighl years
Figure 6-6. Beacon power versus range.
We see that if we had a receiving array of a thousand
100-m dishes, we might use it as a beacon to illuminate
the stars out to about 76 light-years (or perhaps a
narrower spectral class, say G stars, to a greater range)
but beyond 100 to 200 light-years we would almost
certainly construct a separate omnidirectional beacon.
Kp = cost per unit power capacity of transmitter
K A = cost per unit area of antenna.
If the product PtAr is held constant, C will be a
minimum when
KpP t = KAA r (23)
Nuclear power costs about $300/kW of generating
capacity. The conversion of this power to monochromatic
microwave power costs about $1200/kW. Let
us therefore take Kp = $2000/kW. If we are willing to
spend KAA r = $4X 109 for antennas we should also be
willing to spend this same amount for beacons, which
makes Pt = 2X 106 kW or 2000 MW. These figures are
technology dependent, but it appears not unreasonable
to expect beacon powers in excess of 1000 MW.
Other Properties of Beacons
In addition to having high powers, beacons would
have other properties that make them easy to detect:
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