Planck Black Body Radiation

9006-006

9006-006a

We shall assume that the alien planet orbits a star very much like our own Sun. The graph 9006-015 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/m².nm. The energy in sunlight is approximately 1.4 kW/m². For communications analysis it is preferable to express the Planck function in terms of frequency 9008-034. The noise (energy density in a 1 Hz bandwidth as a function of frequency) due to the Planck black body radiation continuum from a solar-type star, is given by:

           2.pi.h.f3.r2
Npl  =  ------------------  Watts/meter2.Hz
        c2(e(hf/kT) - 1).R2

where:

h = Planck's constant (6.63 x 10^-34 J.s),
c = velocity of light (3 x 10⁸ m/s),
r = radius of star (6.96 x 10⁸ m),
Wl = wavelength (656 nm),
f = frequency (c/Wl = 4.57 x 10¹⁴ 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 10¹⁶ m).

9006-006b

For the rest of this analysis, we use the background radiation power P_b as a measure of the Planck radiation received.

Pb  =  Npl.(f)Ar.Bo

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/m².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/m².Hz. If the Planck black body curve is integrated over the visible and near visible range of wavelengths (B_o ~ 10¹⁵ Hz), the total energy is about 1.39 kW/m². Again, this is the energy in sunlight.

9006-006c

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/m².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 pre-filter, 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 Signal-To-Planck Ratio (SPR) at the output of the receiver we take the Planckian spectral density as being 2N_pl. Illustration 9008-072 shows how the electrical energy of the Planck continuum is doubled on being downconverted.

9006-006d

Graphs 9008-033, 9008-005, 9008-034, 9008-006 show plots of the above Planck function out to a range of 10,000 L.Y. Graphs 9008-025 and 9008-026 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 Signal-To-Planck 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