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    Navigation with Moon images
    From: Peter Monta
    Date: 2020 Aug 23, 19:33 -0700
    Well, I've been circling back to navigation with camera images of the Moon.  I regard it as unfinished business.

    To get the Moon's position on the sky, you need either:

    - classical lunar distances with a sextant and a bright star (or several stars if you want declination also)
    - an image with Moon and stars

    The difficulty with the image, at least in past trials, has been an overly bright Moon and the need for a reasonably long exposure to show enough stars to serve as a frame of reference.  A neutral-density filter takes care of the Moon, but the long exposure directly conflicts with the need for stability when handholding the camera.

    I've recently taken some images that seem to be showing good performance, with right ascension and declination good to about 4 arcseconds standard error, single image.  That's 12 seconds in UT (presuming local sidereal time is known) and 8 km (~4 nautical miles) in latitude.  Averaging may improve this, and indeed you've got all night to keep pressing the shutter button.

    The new tricks are:

    - handheld photos, exposure time 1/10 second, using a monopod
    - automatic estimate of the center of the Moon from the image

    I don't think a monopod is cheating.  It seems to me that on board a ship, you're always sitting or standing on a solid surface, though of course that surface is pitching, rolling, and yawing in a noiselike way with some amplitude and timescale.  A monopod touching that surface gives the camera enough extra stability that 1/10 second seems to be viable, even without image stabilization.  Most of my images have trails less than one pixel (20 arcseconds); a few have trails of three or four pixels, but I throw those away.  (This throwing-away can be automated by looking at the quality-control numbers from the astrometry.)  I suppose this means I'm managing to keep camera body rates below 200 arcseconds per second by watching the Moon's image in the viewfinder and trying to calm it down.  I'm using a shutter self-timer delay of 2 seconds and electronic first-curtain shutter (EFCS) to eliminate shutter mechanical transients (and mirror slap in a DSLR).

    An example image is attached.  Because of the contrast-stretching you can see the "black hole" from the penumbra and umbra of the neutral-density filter suspended ~40 cm in front of the lens.  (The umbra is a generous 2.5 degrees in diameter.)  Outside of this hole the stars are unobstructed.  The Moon is not actually saturated, though it might look that way on this scaled view.

    The scheme for estimating the Moon's center uses some priors:  the Moon's rough position from dead reckoning, the vector to the Sun to establish the orientation of the Moon's limb, and the radius of the Moon in pixels.  The goal is to maximize the contrast between the limb's brightness at radius r and at radius r+dr, integrated over the 180 degrees of available limb (though with a thin crescent Moon, the horns are not contributing much and the effective limb is more like 140 degrees or 120 degrees).  The optimizer seems stable.  The points along the limb at the various radii are resampled with a cubic spline, and a dr of 0.5 pixel seems to work well.

    Once the Moon's center is available, it is mapped to celestial coordinates using the frame obtained from the surrounding stars.  Typically there are 15 to 30 star detections in the 10x15 degree field, depending on galactic latitude and threshold SNR.  (The stars range down to about mag 7.0.)  Finally, a comparison is made to the Moon's apparent position as calculated by Skyfield from the JPL numerical ephemeris and the topocentric position.  Currently, effects like irregularity of the limb's topography and any offset between center-of-mass and center-of-figure are ignored.

    Anyway, the plan is to clean all this up, then enjoy some effortless pushbutton horizonless navigation.

    The preliminary (very preliminary) code is on github in this repository:

    https://github.com/pmonta/lunar-astrometry

    Also attached is the positioning performance of 10 images taken on August 8.  Declination is looking really good (maybe because of the symmetry).  There seems to be a bias of about 5 arcseconds in right ascension, which I've yet to understand or fix.

    Cheers,
    Peter

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