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Bruce Hamilton, you wrote:
"This is the one place in the world I can ask this question and not get
laughed at, and probably get an answer."

LOL. I'm definitely laughing, but laughing with you!

You asked:
"Has any one worked out tables for celestial navigation on Mars?"

Well, let's see... The stars are in the same positions, so no problem there.
You would have to adjust the annual precession for "Martian" values, and
there is essentially no nutation. The annual aberration would be nearly the
same, though smaller in magnitude (about 16 arcseconds instead of 20, and
somewhat variable)  The standard, widely available JPL ephemeride solutions
are accurate enough in 3d space that we could easily compute the SHA and Dec
of the Sun and the navigational planets (Venus, EARTH, Jupiter, and Saturn).
The variability in delta-T that we have on Earth would be much smaller on
Mars.

Of course, the spherical triangle math doesn't change, so you could use HO
229, if you want. One small thing to keep in mind: in terms of linear
distances, celestial navigation is twice as accurate on Mars because one
minute of arc difference in a star's altitude corresponds to about 3000 feet
on Mars instead of one nautical mile as on Earth (if you want, you could
define a "Martian nautical mile" with that shorter length). This is a
specific case of the general principle that the accuracy of celestial
navigation (by altitudes from the horizon, or equivalently zenith distances
from the vertical) decreases as the radius of curvature of the surface
increases. So it's less accurate on Earth, more accurate on Mars, and really
accurate (in terms of distance on the ground per minute of arc change in
altitude) on a small spherical asteroid. Similarly, on one planet, if the
local radius of curvature is greater, which implies a flatter portion of the
planet, then the accuracy of celestial goes down. Note that the "surface"
whose curvature is being measured is really the surface of the geoid. If you
have an asteroid shaped something like a diverging lense, concave on both
sides, the geoid would be flat on both sides, and then celestial
observations could not distinguish positions at all (except by small changes
in parallax of objects which are nearby).

The density of the Martian air is less than 1% of that on the Earth, so
above about ten degrees altitude you could ignore refraction for standard
sextant observations (I say "about" because of the difference in atmospheric
composition).

Finally, we get to Phobos and Deimos. Lunar distance observations would give
quite accurate time on Mars, though in some latitudes the moons are always
below the horizon. The positions of the Martian moons are known to high
accuracy, but this is a case where ultra-high accuracy would pay back great
benefits. The moons potentially offer the easiest and most accurate method
of finding one's position on Mars. Since many bright stars can be seen in
daylight, this could be done without a sextant. You would photograph Phobos
or Deimos among the background stars, measure its RA and Dec and then from
its known 3d position in space (assuming the Universal Time is already
known), you would draw a ray which would intersect the surface of Mars at
your location. That would give you a very accurate latitude and longitude on
Mars. So we would essentially be using lunar distances to get a position fix
without any need to determine a local vertical... where have I heard that
before???

For the foreseeable future, on Mars, it seems to me that "air traffic
control" is probably a better approach to position-finding. In other words,
you emit a signal trackable from orbit, and "they" keep track of your
position and beam it back to you. We've seen some nice examples of this in
the past few days as the Mars Reconnaissance Orbiter has photographed the
Phoenix lander descending under its parachute (SPECTACULAR! Coolest thing
I've seen in months!) and then its landing site complete with ancillary
components. And from that we know the exact position of Phoenix on Mars.

 -FER



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