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Paul Morris
As seen from a given point on the moon, Earth will wobble in a box that measures plus or minus 7.7 degrees east to west (longitudinal libration), and 6.7 degrees north to south. The longitudinal libration is due to the elliptical orbit of the moon about the earth, and the latitudinal libration is due to the inclination of the moon's axis to the plane of its orbit.
Lunar equator is tilted 1 deg 32 min from the ecliptic
Lunar equator is tilted 6 deg 41 min from its orbital plane
Orbital plane is titled 5 deg 9 min from ecliptic
(these figures add up to make the first numbers)
To put this into perspective: Earth subtends an angle of about 2 degrees in the lunar sky, so it's moving roughly 3 times its diameter away from a mean point. This can be easily modeled for any point on the lunar surface. See the essay on viewing Earth from Angus Bay for more details and some illustrations of Earth's path through the lunar sky.
-7.7 North +7.7
-------+------- -- +6.7 deg
W | () | E
e | mean | a
s + + + s
t | Earth | t
| |
-------+------- -- -6.7 deg
South
Earth subtends an angle of about 1.9 degrees, as seen from the moon. So if you imagine a circle formed by the parentheses in the diagram above, it's approximately to scale. Earth will slowly move within this box over a 28-day cycle. The box will be tilted depending on your latitude on the moon. At the equator, it's rolled over 90 degrees. At the poles, it's as I've shown it here. At other lunar latitudes, the angle of tilt will be ( 90 - latitude ) degrees.
To keep an antenna bore-sighted on Earth, it would have to slew the rate of about 1/2 degree per day. We'd want the antenna to cover the whole 2 degrees of Earth to allow for continuous coverage from horizon to horizon. This is not a problem for most radio frequencies and antennas of interest, which have beam widths much wider than 8 degrees. Of course, people at a given location on Earth would be able to receive the signal for half of each Earth day.
Sometimes we'd get prime time in a given city; sometimes not. Programming could be adjusted for the cities along the strip of longitudes that will receive the signal in prime time. Advertisers in Chicago would pay more for a commercial spot at 7 PM than at 3 AM. That's OK; when it's 3 AM in Chicago it's 7 PM in Sydney, so our programming will be about what's up at the Opera House instead of how well the Cubs could play baseball on the moon even though people in both of those cities would be receiving the signal.
Note that this situation, with the moon halfway between Sydney and Chicago when it's 7 PM in Sydney, occurs at a specfic phase of the moon. Moon aficianados on Earth will learn to tell the phase of the moon based on what kind of programming is coming to them, and vice versa.
For most radio frequencies it will simply be necessary to point an antenna at the 8 degree region of the lunar sky where the earth can be found. For laser communication and very high radio frequencies it will be necessary to track the motion of the Earth. The easiest solution to tracking Earth would probably be to calculate its position, but if we want a stand-alone antenna guidance system to point an antenna, we could use a radio direction finder. Earth is radio-noisy across all the bands, so it should be very easy to close the guidance loop on radio signals.
Infrared should work, too. Earth is a very hot body against a very cold sky except during a terrestrial eclipse.
During an eclipse, it would be easy to find the limb of the Earth occulting the disk of the sun up to totality. During totality, use the camera to find the body occulting the solar corona. I'll bet Earth looks really pretty during a total eclipse, with the sunlight refracting around the atmosphere to form a ring of fire. The Luna City Hotel will likely be booked up decades in advance for those events.
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