Re: ATM using DSC for polar alignment...

[Aplanatic@aol.com]

Benoit Schillings wrote:

> I was wondering of different ways I could use my digital setting 
>  circles to
>  speed up the polar alignment of my German Equatorial mount (MI-250)...
>  
>  I have 20000 steps in RA and 8000 steps in DEC... pretty good accuracy
>  and I'm looking for ideas on algorithms which would tell me how far I 
>  am
>  in azimuth and elevation based on the pointing of a few stars around 
>  the sky.

Your dec resolution is 2.7 arcminutes.  For the most demanding photographs, 
say 3 hours at +55 degrees declination, the required accuracy (depending on 
the sensor size and whether off-axis guiding is employed, etc) is 2 to 4 
arcminutes to reduce field rotation to something insubstantial.  So, the 
scheme has a chance.  The error terms are the following, in rough order of 
importance:

-- Optical axis non-orthogonality to declination axis.
-- Declination axis non-orthogonality to RA axis.
-- Optical axis droop, zenith to horizon.
-- Refraction, a known quantify not related to mechanical errors but very 
important to include in the solution, even if stars around 20 degrees 
elevation are used.  An approximate user location and time must then be 
entered to find local stellar elevations.

>  Seems to me that this would be possible (and pretty accurate since I 
>  would use
>  4 or 5 stars therefore averaging the encoder limited resolution ).
>   I just wonder if someone looked into that before diving back into
>  spherical trigonometry (yeak).

It turns out that four stars are enough to solve for all the variables, which 
include the three mechanical unknowns above, plus the following:

Polar axis east pointing error.
Polar axis north pointing error.
RA encoder zero.
DEC encoder zero.
  
There are a total of seven variables and eight measurements from four stars.  
So, the solution is slightly overdetermined, not a problem.  It turns out 
that five stars are better yet.  The solution is very sensitive to the DEC 
encoder zero, so it is also a good idea to have a known stop position in 
decliination to improve the accuracy of the solution.  This stop need only be 
accurate to about 1/2 degree, and the stop is not necessary at all if 
somewhat reduced accuracy is OK.

About a year ago I worked out the complete algorithm for finding, through an 
iterative procedure, the above variables for mount errors, polar axis 
pointing direction, encoder zero positions and the local hour angle (which 
appears in the full analysis as another independent variable).  For polar 
axis pointing the local hour angle is not important and can be removed, as I 
recall.  The iterative procedure can be replaced by a deterministic set of 
equations based upon perturbation analysis (small errors) with slightly 
reduced accuracy but then the declination stop becomes more important.  

As a note, I believe that the program TPOINT does all of this and more but I 
have never used it and don't remember the URL for it.  Nor do I know if it is 
easily used or not.  Kevin McCarthy had posted some information about it, I 
think.  Still out there, Kevin?

As another note, if you have no mount errors at all (unlikely) then the 
two-star or three-star calibration algorithm given by Taki and used in most 
DSCs will give you the polar axis pointing errors in a very simple manner.

If you would like to discuss this further, I would be pleased.

Dave Rowe
Torrance, CA
Medium Format Astrophotography:
http://members.aol.com/aplanatic/photos/astro.html