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Gary, you wrote:
"OK, so you end up with a position defined by the lat-long of the spot on 
the surface of the earth directly between the spacecraft and the center of 
the earth (the spot that the spacecraft is directly above and where a person 
on the surface would measure 90� with his sextant to the spacecraft) and the 
radar distance. But is this useful for space navigation? how do you relate 
this to an inertial frame or sidereal frame or to determine if you are on 
course to the moon or mars?" 

Ok, first, here's how it has been done relatively recently:
 --Suppose you're designing software to navigate a space probe out beyond the 
orbit of Mars to visit multiple asteroids. The positions of the asteroids 
are known with relatively high accuracy from decades of Earth-based 
observations. So you load your computer with this almanac data. It consists 
of coordinates in space for every instant of time (actually polynomials 
which allow you to quickly reconstruct the position for any instant of 
time). You also have a database of every star brighter than magnitude 10.0 
giving their positions in angular terms, RA and Dec. Based on your probe's 
approximate position, call it "AP", the computer calculates where to aim the 
camera to see asteroid XYZ. It takes a photo of that region and matches it 
against the database of stars. If the AP isn't too far off, asteroid XYZ 
will be an interloper near the middle of the frame of the camera image. 
Comparing against the background of stars, we can "read off" the exact RA 
and Dec of the asteroid. Now since we know where the asteroid is in space 
(its Cartesian coordinates in 3d space) at the instant the photo was taken, 
we can draw a "ray" of position extending from that spot across the Solar 
System towards the spot opposite the calculated RA and Dec of the asteroid. 
See how that works?? If the asteroid is observed at RA=6h 00m 00s and 
Dec=10d 0' 0" South, then we must be on a ray emanating from the known x,y,z 
location of the asteroid and extending towards RA=18h 00m 00s and Dec=10d 0' 
0" North. We can do a little better and roughly estimate our location along 
the line by measuring the apparent magnitude of the asteroid, but this is 
only a rough estimate. Now we turn the digital camera platform and measure a 
second asteroid's position. That gives a second ray of position. Where those 
rays cross is where our space probe must be. If any time has elapsed, we can 
advance the previous line of position. Once the position is known, and the 
velocity checked, we can apply standard celestial mechanics and find out if 
we're on the right trajectory to reach our target. 

Next, suppose we're imagining doing this 40 years ago on a manned mission to 
the Moon.
 --Just so we're clear, they didn't use celestial. They used ground 
observations of position relayed to the spacecraft and inertial navigation 
on-board, primarily for orientation. The "sextant" onboard was relegated to 
a somewhat different, but still important role. It was used to check the 
alignment of the inertial navigation platform. That is, it was used for 
direction-finding, like a 3d astro-compass, rather than for 
position-finding. Nonetheless, the astronauts themselves insisted on a 
backup just in case all of their radio equipment failed and their inertial 
platform simultaneously became unreliable. You can see their thinking 
here... 'don't doom us to a nasty death just because the radio is dead!' So 
there were procedures prepared that could give them basic position finding 
capabilities using the onboard sextant, and these were tested briefly on 
Apollo 8, forty years ago (December 1968). The principle here is nearly the 
same as the case with the asteroids above except that the you use the Earth, 
Moon, and Sun as nearby references to get rays of position. Unlike 
asteroids, since these objects are very bright (and since computation 
capability was primitive back then) you can't just photograph them in front 
of a "starry background" and "read off" the RA and Dec of the object. 
Instead you measure angles to bright stars in roughly perpendicular 
directions. Each measured angle gives a cone of position with the apex of 
the cone at the center of the Earth (or Moon or Sun). Two cones emanating 
from the same center intersect in two rays. These are just the same as the 
rays of position above and mathematically equivalent to calculating the 
exact RA and Dec of the body we're measuring from. Our spacecraft must lie 
along one of these lines (and as in traditional cel nav, we can throw out 
one by knowing an approximate DR position). Then we measure some other 
angles off the limb of another celestial body. Where the two rays cross, 
that's where we must be. And what kind of observations are these??? Why of 
course! They're lunar distances (technically "lunar" only if they're 
measured from the limb of the Moon, but the principle is the point). You'll 
note that we have to correct for the semi-diameter of the object (Moon, 
Earth, or Sun). If we know our approximate distance from the object, we can 
tabulate these. Otherwise, we can measure them. 

Note that this sort of navigation also works right here on the surface of 
the Earth. Measure a pair of lunar distances and you can get a complete 
position fix (under the assumpion we're on the surface of the Earth) without 
any horizon reference at all. That's the "fix by lunar distances" that I 
described on the list back in the fall of 2006. 

 -FER 



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