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I am trying to program the calculation of the declination of the stars, based on their apparent movement through a series of astrophotographs, purely algebraically, without using a solver based on the star catalog.

I'm a beginner in the field, so please forgive me if I'm not familiar with some well-known fact or source of information.

The main reason why I don't use some of the solvers, like Astrometry.net and similar (forks), is of an educational nature, because I definitely need equations of apparent motion, for feedback during acquisition. An additional contribution to that decision is the lack of Android (where app is going to run) libraries for Astrometry.net and its forks.

Let me first list and describe the data I intend to use for the calculation:

  • latitude, longitude of the observer
  • a set of frame extractions through the time period (451 x/y catalogs with approx. 90 sources, every 8 seconds, one hour in total), connected in continuous tracks of source movement
  • time of each catalog/frame acquisition (possible transformation into JD, ERA...)

Data that I do not have and that will gradually need to be calculated:

  • scale and rotation
  • Ra0 and Dec0, the position of the optical axis on the tangential plane
  • polynomial of optical distortions

I am aware that to solve the field completely, I need the missing data enumerated, but for the beginning I intend to use what I have and iteratively calculate the missing data one by one. Initially within a relatively small area in the middle of the frame, where optical distortions are relatively low.

My first question is, is there a parametric equivalent, in standard coordinates (plus scale, shift and rotation terms), of a general two variable quadratic conic section equation?

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

My idea was, by hyperbola or ellipse, curve fitting, to yield those six coefficients, then to translate them to standard coordinates?

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  • $\begingroup$ At most typical scales the motion would likely be indishtinguishable from a straight line. Especially if you allow errors for atmospheric and optical abberations. $\endgroup$
    – Greg Miller
    Commented May 9 at 22:39
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    $\begingroup$ You haven't provided enough information to understand what you know, personally, and what kind of inputs you would have. Do you know that the declination of most stars doesn't change much from year to year? Do you know the altitude and azimuth of points in your astrophotographs? Is the X axis in them the same as the local horizontal? You aren't "bothering" anyone. There are lots of questions on other of our sites routinely closed because they're looking for homework answers, are hopeless, or closed with a warning that OP moght be electrocuted or set his/her home on fire. Yours isn't one. $\endgroup$
    – stretch
    Commented May 9 at 22:45
  • $\begingroup$ @GregMiller As apparent motion curves approach equatorial plane, they become more straight, indeed. For cameras with wide FOV (68.1x53.6 degrees in this case), within hundred of pixels from equatorial plane, hyperbolas and ellipses becomes quite bumpy toward equator plane. Yes data are noisy, but easy least square fitted even with general form of cone section equation. Parametric equation, would provide unique solution. $\endgroup$
    – Danko Dnevic
    Commented May 10 at 1:14
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    $\begingroup$ @uhoh General form of "two variable quadratic conic section equation" is written just below section you quoted, and describes curves of apparent stars motion on tangential plane, that we are talking about: hyperbolas and ellipses (straight lines, parabolas and circles are also theoretically possible ). $\endgroup$
    – Danko Dnevic
    Commented May 10 at 1:22
  • $\begingroup$ @stretch Thanks on your response. Yes, I know that stars declination (including its Ra), doesn't change much, or at all (if that change is far below resolution of wide angle camera). Azimuth and Elevation of objects in the frame are unknown. Otherwise, I wouldn't ask these questions, I would just calculate Az and El from Ra, Dec, Lat, Lon and JD. Camera has unknown rotation and translation (current position of optical axis on the celestial plane). Latitude, Longitude and Time are well known values. Maybe rough estimation of focal distance and pixel size might be acquired also. $\endgroup$
    – Danko Dnevic
    Commented May 10 at 1:41

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