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    Re: Navigation without Leap Seconds
    From: Frank Reed
    Date: 2008 Apr 18, 02:55 -0400

    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. 
    Navigation List archive: www.fer3.com/arc
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