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    Re: Space sextants
    From: Frank Reed
    Date: 2016 Jan 24, 13:31 -0800

    David Pike, you asked:
    "Please can someone tell me which unmanned probe and provide any updates since 2010?"

    It was an unusual spacecraft called "Deep Space 1". This was 50% exploration and 50% technology demonstrator. It tested a system called "AutoNav" which worked fine in practice as an autonomous navigation system, but basically it was a trivial test. It worked as expected, yes, and there were no surprises. Why did we do this again?

    In my message in 2010, I wrote: 
    "A digital camera beats a sextant in space any day".

    And David, you asked:
    "Please can someone explain to someone not particularly clued up on the inners of a digital camera or lens optics how a digital camera can be used for celestial navigation? Also can this be done in real time or almost real time?"

    Navigation in space is easy if you have enough data. You break the universe into two sets of objects:

    • Extremely distant objects that don't move and qualify as a "fixed background",
    • Relatively nearby objects that show significant parallax and serve as "lighthouses".

    For example, here on the Earth I can look at a dozen-odd first magnitude stars as a background and use the Moon as a nearby lighthouse. Then as I travel around on the Earth's surface, I can measure the Moon's angular distance from the bright stars with a sextant. It moves, of course, over time, but we can ignore that for now. The Moon's position relative to the background stars also moves because of simple parallax. As I travel from the Earth's Arctic to the Antarctic, the Moon will move about two degrees across the sky (because a ratio of 8,000miles/238,000miles is angle of nearly 2 degrees). By measuring the Moon's angular position relative to the bright stars, I can get a position fix on the Earth. The great difference in brightness between the Moon and the other stars makes a sextant a good choice for pinning down the Moon's position and mostly prevents us from using a camera for this purpose.

    The Moon is an obvious traditional choice for a nearby "lighthouse", but a much better choice would be artificial satellites. Since they're much closer they display much higher parallax. And since their brightnesses are comparable to the fixed stars, we can now use a camera to observe both the satellites as foreground "lighthouses" and the fixed stars as our "background". If we have positional data on the satellites accurate to a hundred meters or less (we do), then typically we can find our position on the ground to about the same level of accuracy. And even one satellite is enough if its position is accurate in time as well as location. When you see a satellite, like the International Space Station, graze past some known star, let's say Regulus just to be specific, at an accurately recorded instant of time, then based on the satellite's known ephemeris, we can draw a ray from the direction of Regulus through the satellite's location, and extend it to the Earth's surface. Assuming we know we are on the ground, that completely fixes our position. A single photo by a digital camera, properly processed, can do that in a flash.

    Now we go flying about in the asteroid belt, like that spacecraft "Deep Space 1". We can continue using a specific subset of the fixed stars as our background, and for foreground objects we can use a catalog of the brighter asteroids in the main belt. Since we will be sailing among them, their positions will change significantly as we travel. And just as with the case of satellites seen from the surface of the Earth, a single photo showing an relatively nearby asteroid against a background of distant stars can be easily processed to yield a position "ray" extending from the point in the heavens directly "behind" the asteroid in the photo. We're on that ray. And of course we get a position fix by adding two or more other asteroid photos to the analysis.

    In terms of processing the image, the background stars provide a reference that allows us to overlay coordinates, for example right ascension and declination, on top of the photo. This converts pixel x,y coordinates to astronomical coordinates. Then we read off the x,y coordinates of the nearby asteroid and invert the coordinate transformation. The math behind this was worked out a century ago for determining coordinates of astronomical objects on old photographic glass plates, and the transformation coefficients are still usually known as "plate constants".

    Note that this is really much more like coastal piloting than traditional celestial navigation which is so dependent on the gravitational field of the Earth and the local vertical. In this form of navigation, we look at a nearby object and see how it sits against a background. This is entirely analogous to deducing a small boat's position by seeing a lighthouse on a rock appearing directly in front of a distant landmark. And note, too, that like the coastal piloting case, the accuracy depends on the distance to the "nearby" lighthouse and also depends on the accuracy of the available charts. For space navigation, you need exceptionally accurate three-dimensional ephemerides of nearby planets and asteroids and you also need a sample of such objects that are relatively close to your track through space. 

    And yes, it's real-time by human standards. That's purely a question of processing power and even fifteen years ago, when Deep Space 1 was flying it was nearly real-time. Today a common handheld computer (a.k.a. "smartphone") could do the work in seconds, and a dedicated navigation processor should be able to keep a continuous position fix with millisecond delays at most.

    There's a different phase to this space navigation when you reach your target. Since we're usually talking about exploration missions, it's likely that the position of the target object is somewhat uncertain. You don't want to crash or zip right by, so in the final phase of navigation, you have to switch over to relative navigation (even more like coastal piloting). Here you're worried about your position relative to the final destination, but not so interested in your absolute position relative to the Earth and other known bodies.

    Hope that helps.

    Frank Reed

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