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angular speed of celestial objects

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#1 Nicole Sharp

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Posted 30 July 2020 - 02:16 PM

Is there a way to compute or approximate the apparent angular speed of celestial objects to determine the maximum usable photographic exposure time (from a nontracking mount) without visible field rotation?  I know the maximum angular speed for equatorial motion is approximately 360 degrees per day (15 degrees per hour, 15 arcminutes per minute, or 15 arcseconds per second) but stars near the celestial north pole (such as Polaris) appear to move much more slowly than that.

 



#2 TOMDEY

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Posted 30 July 2020 - 02:32 PM

It's right around 15 arc-min per minute, times (slowed down by) the cosine of the declination. That's all there is to it!    Tom


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#3 Nicole Sharp

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Posted 30 July 2020 - 03:09 PM

It's right around 15 arc-min per minute, times (slowed down by) the cosine of the declination. That's all there is to it!    Tom

Is that still too fast for Polaris?  Polaris has a declination of 89.3 degrees (0.739 degrees in declination from the North Celestial Pole).  From 39.7 degrees north in latitude, I was able to get a 30-second exposure of Polaris at 1000 mm of focal length (Canon APS-C sensor size of 22.3*14.9 mm^2 and sensor resolution of 6000*4000 pixels) from a manual altazimuth mount (no tracking) with practically no visible field rotation.  For an object moving with an apparent angular speed of 15.1 arcseconds per second, the maximum usable exposure time without visible field rotation (for Canon APS-C at 1000 mm) should be 0.0508 seconds.  The cosine of 89.3 degrees is 0.0128, which would give a predicted angular speed of Polaris as 0.194 arcseconds per second (0.194 arcminutes per minute).  At an angular speed of 0.194 arcseconds per second and a camera resolution of 0.767 arcseconds per pixel (undersampled for 90 mm of aperture), the maximum usable exposure time without visible field rotation (at 1000 mm for Canon APS-C) should be 3.96 seconds.  Zooming in to 200% with an exposure of 7.58 times the maximum value of 3.96 seconds, I think I see maybe a hint of field rotation with stars being slightly elliptical instead of circles, but it's barely noticeable.

 

https://www.nicolesh...8_jpg.jpeg.html

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Edited by Nicole Sharp, 30 July 2020 - 03:38 PM.


#4 spereira

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Posted 30 July 2020 - 04:42 PM

Moving to Science, Astronomy & Space Exploration.

 

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#5 Michael Covington

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Posted 30 July 2020 - 05:11 PM

You may be looking at a lens aberration.  Polaris is not exempt from the laws of spherical trigonometry.



#6 rkinnett

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Posted 30 July 2020 - 05:18 PM

Hey Nicole, your math looks right to me.  However, another way to look at is that, given your 0.77 arcsec/pix image scale, Polaris will have moved roughly 7 pixel widths, which is well below the FWHM of Polaris.  That said, celestial north is just to the left of this crop, so stars near the upper and lower right corners will have moved the most.  The stars in the lower right corner are clearly elongated, but does it add up? 

 

The bottom right star is SAO 548, at 88.78 deg declination.  It should have traveled about 12.5 pixel widths across your sensor.  It's clearly elongated, but doesn't look like 12 pixels of smear.  You would have to check the full res image.  Also, bear in mind that the brightness of each pixel within a star trail is the integration of intensity at that point over the duration of the exposure.  Pixels toward the center of the arc will have been exposed the that star's light longer than those near the ends.  You might consider the expected star trail length to be the traverse length (12px in this case), minus the FWHM of the star, which appears to be several pixels in this case, leaving you with about a half dozen pixels of apparent smear.  That seems to get you in the right ballpark.

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Edited by rkinnett, 30 July 2020 - 05:22 PM.

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#7 Nicole Sharp

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Posted 30 July 2020 - 05:24 PM

You may be looking at a lens aberration.  Polaris is not exempt from the laws of spherical trigonometry.

 

It's a 1000/90 (f/11) Maksutov-Cassegrain, so should have a flat field with minimal aberrations (less than an f/10 Schmidt-Cassegrain).  With a 30-second exposure on a manual altazimuth mount (no tracking), I was able to identify stars down to magnitude +14, so pretty happy about that for such a small aperture on a nontracking mount.

 

https://www.nicolesh.../07-29/1a3.html


Edited by Nicole Sharp, 30 July 2020 - 05:26 PM.

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#8 Nicole Sharp

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Posted 30 July 2020 - 05:30 PM

Hey Nicole, your math looks right to me.  However, another way to look at is that, given your 0.77 arcsec/pix image scale, Polaris will have moved roughly 7 pixel widths, which is well below the FWHM of Polaris.  That said, celestial north is just to the left of this crop, so stars near the upper and lower right corners will have moved the most.  The stars in the lower right corner are clearly elongated, but does it add up? 

 

The bottom right star is SAO 548, at 88.78 deg declination.  It should have traveled about 12.5 pixel widths across your sensor.  It's clearly elongated, but doesn't look like 12 pixels of smear.  You would have to check the full res image.  Also, bear in mind that the brightness of each pixel within a star trail is the integration of intensity at that point over the duration of the exposure.  Pixels toward the center of the arc will have been exposed the that star's light longer than those near the ends.  You might consider the expected star trail length to be the traverse length (12px in this case), minus the FWHM of the star, which appears to be several pixels in this case, leaving you with about a half dozen pixels of apparent smear.  That seems to get you in the right ballpark.

Thank you for that analysis.  I didn't compute the number of pixels of trailing (I was just looking for visual evidence of trailing), but that makes more sense now.  Polaris is indeed no exception.  "Usable exposure time" of course can allow a small amount of trailing, as long as stars are not smeared enough that they are no longer identifiable.  I was trying to determine the limiting magnitude for 1000/90 without a tracking mount, and got down to magnitude +14 with a 30-second exposure.  I think I could get even longer exposures maybe with identifiable stars, but now that I can compute the angular speed as being proportional to the cosine of declination, I can estimate in advance how long the star trails will be.


Edited by Nicole Sharp, 30 July 2020 - 05:37 PM.

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