Planck Black Body Radiation
9006006
9006006a
We shall assume that the alien planet orbits a star very much like our own Sun. The
graph 9006015 illustrates the general Planck radiation curve
for the Sun which peaks in the visible part of the spectrum. This Planck function peaks at
an energy density of about 1.7 W/m^{2}.nm. The energy in sunlight is approximately
1.4 kW/m^{2}. For communications analysis it is preferable to express the Planck
function in terms of frequency 9008034. The noise (energy
density in a 1 Hz bandwidth as a function of frequency) due to the Planck black body
radiation continuum from a solartype star, is given by:
2.pi.h.f^{3.}r^{2
}N_{pl} =  Watts/meter^{2}.Hz
c^{2}(e^{(hf/kT)}  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),
r = radius of star (6.96 x 10^{8} m),
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 = distance of receiver (10 light years = 9.461 x 10^{16} m).
9006006b
For the rest of this analysis, we use the background radiation power P_{b} as
a measure of the Planck radiation received.
P_{b} = N_{pl}.(f)A_{r}.B_{o}
where:
A_{r} = collecting area of the receiving telescope,
B_{o} = optical filter bandwidth.
For the parameter values above, the Planck spectral energy density at 656 nm is:
N_{pl} = 5.48 x 10^{24} W/m^{2}.Hz
Our Sun produces a Planck spectra energy density in the region of the Earth's orbit (R
= 1 A.U.) of 2.29 x 10^{12} W/m^{2}.Hz. If the Planck black body curve is
integrated over the visible and near visible range of wavelengths (B_{o} ~ 10^{15}
Hz), the total energy is about 1.39 kW/m^{2}. Again, this is the energy in
sunlight.
9006006c
Since the optical heterodyne receiver would normally detect energy only along one
plane of polarization, we can halve the calculated optical spectral energy density, viz.
N_{pl} = 2.74 x 10^{24} W/m^{2}.Hz
Ignoring any emission and Fraunhofer lines in the spectrum, the amount of this energy
collected by a 10 meter diameter telescope through a 1 Hz optical prefilter, and
delivered to the photodetector via a polarizer is 2.15 x 10^{22} W. Thus,
P_{b} = 2.15 x 10^{22} W
It is important to note here, that the action of a heterodyne receiver in
downconverting that part of the Planckian continuum situation above and below the local
oscillator frequency, is to double the noise spectral density in the Intermediate
Frequency spectrum. Thus for the purposes of determining the SignalToPlanck Ratio (SPR)
at the output of the receiver we take the Planckian spectral density as being 2N_{pl}.
Illustration 9008072 shows how the electrical energy of the
Planck continuum is doubled on being downconverted.
9006006d
Graphs 9008033, 9008005, 9008034, 9008006 show plots of the
above Planck function out to a range of 10,000 L.Y. Graphs 9008025
and 9008026 illustrate how the microwave and optical Planck
spectral density levels compare to the received signals. These, and other graphs, pertain
to the optical side of the photodetector, so the ratio between the signal and Planck noise
spectral density at the output of the heterodyne receivers should be increased by 3 dB to
reflect the Planck noise doubling effect.
The Planck levels for the 1.5 GHz microwave system assume that the Sun has an
effective radio brightness temperature of 136,000 K, not the normal 5778 K applicable to
visible and infrared wavelengths. Clearly, the SignalToPlanck Ratios (SPRs) for each of
the systems, 74 dB versus 69 dB, are very similar, even though the absolute signal and
noise levels are very different.
The Columbus Optical SETI Observatory
Copyright (c), 1990
