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Page updated January 20, 2008

### Calculation of the Expected Lunar Echo Receive Signal Strength

When a radio signal is transmitted from an earth station, scattered from the moon's surface and received back on the earth's surface, the propagation can be described by the bistatic radar equation where the received signal $P$r is given by :

in this equation:

$P$t is the transmitted power in Watts. The transmitter power of the HAARP facility is 3.6 X $106$ Watts.

$G$t is the transmitter antenna gain. For the frequency used in this experiment, the HAARP transmitting antenna has a gain of 735 (28.6 dB).

$G$r is the gain of the receiving antenna over isotropic. For a dipole antenna, the gain is approximately 1.5.

σ is the scattering cross section of the moon. We use the cross section for an isotropic sphere modified by the moon's albedo which at these frequencies is approximately 0.15. The value of σ used for this calculation is 1.4 x $1012$ square meters.

λ is the wavelength of the frequency used. For this experiment, the wavelength is 42 meters.

R is the distance from the earth to the moon. For this calculation, we have taken the distance from the moon to the transmitter to be approximately the same as the distance from the moon to the receiving antenna and equal to approximately 370,000 km.

#### Results of the calculation

Using the equation above, the power delivered at the terminals of a resonant 40 meter dipole antenna is 2.5 X $10-13$ Watts or about -96 dBm. This is equivalent to about 4.4 microvolts across 70 ohms. Typically, the S meters in commercial communication receivers are calibrated such that S-9 corresponds to an input signal of 50 microvolts and each succesively lower S unit represents a decrease of about 5-6 dB in voltage. Therefore, the calculated (predicted) level of the received lunar echo is equivalent to about an S-5 signal strength.

#### Other considerations

The calculation above is for an ideal case. Here are some other potential sources of loss that would affect the signal level for the lunar echo.
1. The height of the antenna used for reception. Antennas are generally mounted as high as possible to maximize the radiation and reception of signals at low elevation angles. However, a horizontal antenna mounted 0.5 wavelength above ground (21 meters at this frequency) would have an overhead null in its pattern. Thus, an antenna that performs well for long distance communication, may not work as well when the (lunar echo) signal is arriving from a high angle.
2. The HAARP antenna beam was repositioned to track the moon during the experiment in 3 minute steps. Although the experiment took place at the low point of the solar cycle and during evening hours when ionization could be expected to be low, there could still be some ionospheric absorption and refraction possible. Absorption would cause some small decrease in the signal both going toward the moon and returning. (The HAARP riometer indicated no absorption duuring the 19 January test and a small amount during the 20 January test.) Ionospheric refraction, if present, could have affected the exact positioning of the maximum in the HAARP transmitted signal as it passed beyond the ionosphere.
3. A planar antenna such as the HAARP array, loses gain as the beam is positioned off the vertical. For this experiment, the moon was at a zenith angle of about 35 degrees as seen from HAARP. The additional loss in gain of a few dB was not included in the calculation.
4. The moon's distance from the earth varied during the test. This resulted in a small but measureable Doppler shift on the lunar echo. The amount of this frequency shift was as much as 7 Hz positive or negative, depending on the specific time during the experiment. Several of the reports that have been sent to us have reported this Doppler shift.

#### References

Reference Data for Radio Engineers, Howard W. Sams & Co., Indianapolis, IN, 1977.
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