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Amsat-UK's Oscar News, 1994 Apr No. 106 p8-9

The Earth Moved


James Miller G3RUH

Moon Beacons

From time to time there appear proposals to put transponders or beacons on the Moon. What you never see is any system analysis. Here's a few thoughts. One of the claimed conveniences is that the Earth appears stationary from the Moon.

Well almost; after all, when viewed from Earth, the Moon appears to rock a little. This is called Moon's Optical Libration; because of this, some 60% of the Moon's surface is visible to us.

Conversely Earth, viewed from the Moon, appears not to be fixed in space, but glides around with a complex epicyclic motion.

Moon's Optical Libration

We can plot this motion quite easily. Refer to Figure 1. The large circle indicates scale, and has a radius of 10°. The black blob in the middle is the same size as the Earth.

 Overall Impulse

Figure 1. Moon's Optical Libration. This is the trajectory of the Earth as seen from an observer located on the Moon. The Earth appears to wander about with a monthly motion as indicated. Maximum excursion is as much as 9.5°.

The coordinates are ecliptic latitude and longitude. That is, the line across the middle is the ecliptic plane. The Sun glides across here from left to right, once a month, and when in the middle it's Full Moon. If the Moon's in the middle too, there's an eclipse of the Moon.

The average up/down latitude excursion is 5.13°, corresponding to the Moon's orbital inclination with respect to the ecliptic. A variety of other influences from the Sun add up to +/- 0.9° more to this.

The left/right longitude excursions are caused mainly by the Moon's orbital eccentricity. The major term is 2e radians, or 6.29°, where e = 0.055 is the orbit eccentricity. Interaction with the Sun contributes a little more, bringing the longitude peak movement to some 8°.

The combined latitude and longitude movement is a maximum of 9.5°, as can be seen at the "corners" of the plot.


All Moon tracking programs compute the Moon's ecliptic latitude and longitude, though it is not so obvious to the untutored eye.

Low precision formulae that are adequate for graphical work can be found at the end of section D of the Astronomical Almanac, published annually by USGPO and HMSO.

Moon Downlink

The maximum total excursion of 9.5° is the same as the beamwidth of a 5 wavelength diameter dish antenna. This has a gain of some 20 dbi, and represents an upper limit for an unsteered Moon-based antenna. However the higher the frequency used, the smaller mechanically is the antenna, which makes 2.4 or 5.6 GHz a good choice. Five wavelengths is 60 cm and 26 cm diameter respectively; quite small.

For a given TX e.i.r.p., signal strength received at Earth depends only on the mechanical size of the RX antenna; frequency is irrelevant [1]. Noise level however is not, and S-band (2.4 GHz) is a sensible downlink choice because very low noise performance is robustly obtainable "off the shelf".

An example, 1 watt transmitted from a 20 dbi gain dish on the Moon, received on a 1.2m dish at Earth with a system noise temperature of 100K results in a signal to noise ratio in 2.4 kHz bandwidth of 10.5 db. (Note that frequency matters not). This would support one rather noisy SSB voice signal. Alternatively it would carry an error-free 2400 bps binary PSK data transmission without coding, 9600 bps with modest coding [2].


1. Miller J.R.; Mode-S - Tomorrow's Downlink?,  Oscar News (GB)
     1992 Oct No. 97 p20-22.
     Also: Amsat-DL Journal Dec 1992 Nr4. Jg.19.
     Also: Amsat Journal (USA) 1992 September.
     Also: Amsat-VK Newsletter, No. 90, Sept 1992.
     Also: CQ-DL 9/93  1993 September, p.614-617.

2. Miller J.R.; Shannon, Coding and the Radio Amateur,  Oscar News (GB)
     1990 Feb No.  81 p.11-15.

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Created: 1994 Dec 09 -- Last modified: 2005 Oct 29