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Hi Frank,

> Yes, that makes sense. I have wondered how much the "plate correction" model
> changes from one shot to the next with these handheld cameras. The detector
> array can be tilted slightly skewing the image in a variable way. If there
> are enough stars in the field, it can all be calculated. But if we have a
> short exposure and just a handful of stars, that may limit the accuracy.

I don't know how much the camera optics would flop around.  As you
imply, any fixed miscollimation, such as the detector plane not being
exactly normal to the optical axis, would be calibrated out.  Only
nonrigid motions would be problematic; one could test this with a few
images right-side-up then upside-down, so that gravity is pulling the
lens in different ways.

> Are you able to extract the so-called "plate constants" or "lens-distortion
> model" from the results when it's done?

Yes.  They are given in the WCS (world coordinate system) header in
the form of a 2D polynomial in pixel space (for the default SIP
representation).  They're ASCII floats in the header lines, annotated
with comments, so they're simple to extract.

> If we can get at the model derived
> from a field of stars, and if it's relatively stable, then we can turn a
> camera into an accurate sextant.

Right.  It would seem logical to take a number of
moderately-long-exposure tripod star shots, estimate distortion terms
for them, then take their average, to get a robust master estimate for
the camera's distortion at that zoom setting.  This can then be
supplied a priori to the images from the much sparser handheld shots,
to prevent overfitting.

Refraction might be a slight issue preventing a single fixed
distortion model from applying to all images.  Maybe the calibration
images should be taken near the zenith, and then when it comes time to
supply the a priori distortion to a target image, modify the zenithal
polynomial by applying refraction analytically, assuming a rough
altitude is available.  But with a small field of just a few degrees,
this correction is likely to be quite small.  In the limit of small
fields, it would just squish the images vertically a little more for
the larger zenith angles, which would be a small modification to the
first-order y term in the polynomial.  Probably all this can be
ignored at first.

> ... It's a fair start, but clearly if we could get at this
> distortion model without having to build new software for it, that would be
> terrific.

Sure.  SCAMP is another useful tool in this area:

http://www.astromatic.net/software/scamp

> I know you know this, Peter, but I think it's worth mentioning that the
> distance that counts here is the range to the satellite at the time of the
> photo (or visual observation).

Well, I may have forgotten it :-).  Certainly low-inclination objects
like HST are not going to have overhead passes in most of the USA or
Europe, so it is over-optimistic to use closest distance for it; but
polar orbiters like Iridium will have good passes everywhere.

> I don't know... The standard SGP/SDP models are awfully good! Plus they're
> tied very tightly to the TLE data. It's possible to do what you suggest, and
> people do run such integrations, of course, but mostly I've seen it done for
> critical cases like satellites near re-entry. Do you follow Seesat-L
> (http://satobs.org/seesat/)?

From what I can gather from googling a few papers on the topic, the
fancy propagators don't help as much as I would have expected.
Typical errors for 750km-ish objects seem to be in the vicinity of 1
km cross-track and a few tens of km along track after two weeks.
That's not too bad, but it does set a limit.  Using multiple objects
will help a bit I guess.

I followed Seesat-L brifely, but quite a few years ago.  They would
probably be the best community to advise on all this.  The new wrinkle
here would seem to be the use of handheld instruments.

> It's a good question: just what is the 'game' here? What kind of system are
> we describing? Something just for fun? Or something with real, practical
> value? I think there is practical value here, but it exists along a spectrum
> of options.

Well, it broadens one's thinking about navigation, which has value all
its own.  To give practical value, though, such systems will have to
be fully automated.  Enthusiasts willing to put in manual work will
already have many options for backup navigation, such as classical
horizon-based or lunar-based CN of course.

> ... We could build a highly
> accurate, automated system, but that takes money and time, and I think there
> are some groups working on these already.

Would you happen to have any pointers for these groups?

Also, what would be the best scheme with just binoculars?  A reticle
would allow a measurement of the closest approach to bright stars as
they pass by; but with no reticle, this estimate will be really rough
("20% of the field", or "40% between star A and star B"; many
arcminutes of error).

By the way, here's a refinement that may be starting to get a bit
ridiculous:  if GPS is available for part of the voyage, it can be
used to maintain the orbits.  These optical observations are pretty
good and will start to contribute useful orbital corrections after
only a few days (provided GPS position is available).  If and when GPS
is lost, the orbits will be in better shape than they would have been
otherwise.

Cheers,
Peter

   

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