NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Navigation without Leap Seconds
From: Frank Reed
Date: 2008 Apr 18, 02:55 -0400
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. -FER --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---