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    Re: Photographic lunars
    From: Frank Reed
    Date: 2009 Dec 15, 20:22 -0800

    Peter Monta, you wrote:
    "Interesting---I hadn't known of this possibility."

    Well, it's one of those things that I "invented". Of course, as is always the case, it turns out that other people have invented this idea before, and there are supposedly some folks trying to commercialize this system, but that's unconfirmed and until it appears on the market, it's not much better than a rumor.

    And you wrote:
    "A considerable advantage is that it's unaffected by platform acceleration, unlike anything that tries to measure the gravity vector."

    Yes, exactly! So you don't need an inertial system, and you don't need a visible horizon. It's interesting to note that the refraction up above 45 degrees is useless for this purpose since refraction shrinks all distances among stars and other celestial objects by very nearly the same proportional amount. So the change in angular distance between two stars 60 and 70 degrees high, both on the same azimuth, is just the same as the refractional change in angular distance between two stars both 85 degrees on opposite azimuths. We need that highly non-linear refraction down low to locate the vertical using refraction.

    "If my arithmetic is right, moving one arcminute of latitude on the Earth's surface would change the relative refraction between a near-zenith star and one at 15 degrees elevation by about 0.4 arcsecond. That's a small signal, but maybe reachable with some averaging."

    I'm thinking about this a little differently... Consider this scenario: I take a photograph of the sky where the center of the image is about five degrees high. There are a few dozen stars visible. We run it through an astrometric processing system. Because of the very non-linear nature of refraction at low altitudes, we can get a rather good estimate of the altitude of the center of the image. How many stars would it take to get an accuracy in that altitude of 1' of arc?

    And you wrote:
    "It seems like the overall refraction amplitude would not need to be
    known that well, since one would be comparing delta-refractions across
    the whole sky. So static variations from temperature and barometric
    pressure should drop out."

    Yes indeed. Not only would they drop out, but we get the combined value of the pressure/temperature factor as an output. So this system is also a very complicated weather instrument. ;-)

    You wrote:
    "Any variation over azimuth would be a problem, though."

    Yes, they would. Below 3 degrees altitude, this would be a serious problem since you're looking through atmosphere that's many miles away, so there would be some optimal altitude above the horizon where you get enough variation of refraction with altitude and also enough stability in refraction.

    "Multiple cameras would be fine, but another possibility is to use a single
    camera and split its aperture across different areas of the sky with
    mirrors on a stable lens-attachment, a la Hipparcos."

    Sure. That's for the engineers and the accountants to work out. :-)

    And you wrote:
    "So the upshot seems to be: with just a camera and software, one can
    recover UTC from the moon, and the zenith from refraction, and therefore
    both latitude and longitude! Starting with nothing---no UTC, no visible
    horizon even."

    Well, I don't think anyone is EVER going to lose UTC again, so I don't see any practical point in doing lunars here, except that a lunar observation can also give you a line of position at known UTC. It's not very accurate, but if you're taking hundreds of photographs, eventually it converges. This method, like the refraction game above, yields a celestial fix without requiring a visible horizon.

    You wrote:
    "I'd be curious as to which satellites are both bright and have orbits that will remain stable for a long time (years or decades, say). Needing up-to-date orbital elements would be one dependency it would be nice to avoid."

    Up-to-date orbital elements are available though. Even if you expect to be away from data sources (a.k.a. the Internet) for a couple of weeks, a great many satellites are high enough that their orbits can be extrapolated several weeks ahead. The principal error that arises over longer times is mean orbital longitude which translates into a smearing of the calculated position along the projected ground track of the satellite. Ideally, if the satellite's orbit is known exactly, you get a position fix from one photo. But due to this potential uncertainty in mean longitude, you get a finite-length line of position from a single photo. So you find a second satellite and get a fix by crossing LOPs as in traditional celestial navigation.

    And:
    "This almost rules out anything in LEO."

    There's a trade-off. Low satellites have more variable orbits, but being closer, they also yield better fixes in direct proportion to distance. Consider a satellite 300 miles high. That's high enough that atmospheric drag is calculable and predictable. If it's more or less straight up when we photograph, then just by reading off its RA and Dec to the nearest 0.1 minutes of arc from the star field around it, we get (after a simple calculation) our position accurate to about (300 miles)*0.1/3438 or about 50 feet. That's GPS accuracy! In reality, few satellites will have orbits known to that accuracy so if the satellite's orbit is out by 500 feet, which wouldn't be surprising, then so is your position. And of course, satellites are usually only visible in twilight unless they're more than ten times further away (and out of the Earth's shadow).

    "I guess with any satellite there's always the risk of maneuvering, but with several being tracked, an isolated maneuver could be detected and removed from the solution."

    Luckily (for this game at least), most satellites are defunct, so their orbits evolve in ways which are usually predictable except when they're low enough to experience variable atmospheric drag. The ISS, which is nice and bright of course, and easy to photograph, is about as low as practical so the atmosphere's always eroding its orbit which then implies that there are many orbital maneuvers to bring it back up. It's definitely of no use here except in the very short-term.

    You concluded:
    "Incidentally, I made a mistake with the first email regarding the error---it
    is actually almost three arcminutes"

    Ahh. That's more believable for an early trial. But there's plenty of accuracy available in this process. I'm a fan of the idea of using a longer exposure with a neutral density filter to dim the Moon on some sort of arm attached out in front of the camera. Like a sextant filter, the filter here would have to be free of prismatic distortions, but it doesn't have to be oriented exactly perpendicular to the line of sight since we're dealing with effectively parallel light rays from very distant objects. At least I think that's right...

    Fascinating topic, by the way. Though it stretches the limits of "traditional navigation", it's also one of the few ways that traditional navigation may remain in existence, albeit in an automated form, in years ahead.

    -FER

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