De-rotation recovers 88 percent of a perfect stack
A 76 mm scope with no tracking and no finder, stacking 600 three-second frames, lands within 12 percent of the theoretical best a stack of that depth can do. De-rotating about the celestial pole recovers a 19.2x drop in noise where a perfect square-root-of-N stack would give 21.7x. The rest of the night's interest is in what the number costs to get: how close the field sits to the pole, and how long a star therefore dwells inside it.
Notes
The pole is 4.32 degrees from the field centre, bearing 54 degrees left of straight up, and it held all night — the fits from hour 00 and hour 04 agree. That matters more than it sounds. Dwell time inside a 0.69 x 0.52 degree field goes as the reciprocal of the distance to the pole, so every degree closer is worth real integration.
Over four nights the scope has been walked onto the pole: 13.38 degrees, then 7.66, then 4.32. At 13.38 degrees a star crossed the field in about 12 minutes; at 4.32 it dwells for 37 minutes. Push the field centre inside 0.28 degrees and a star stays put for the whole night. The stack depth that buys follows the same curve — 15x, then 27x, against a ceiling of 106x if the field were centred on the pole itself.
The whole night also renders as a single pole-centred polar strip, 1619 by 22561 pixels, spanning 149.2 degrees of position angle from 2978 frames at a median depth of 114. It holds 269 sources above 5 sigma. The strip works because in polar coordinates de-rotation is a pure shift along the position-angle axis, so a night's worth of rotation becomes one long translation rather than 2978 separate resamplings.
Focus is the one thing this rig cannot measure on a single frame, and that is worth stating plainly rather than quoting a number for. A real star here spreads about 13 sigma of detection over 5 or 6 pixels, so its brightest single pixel stands only 3 or 4 sigma above the noise. Anything that IS bright enough to fit per-pixel in one 3-second frame turns out to be a residual hot pixel or a cosmic ray — a fit restricted to single-frame sources above 20 sigma duly returned a sub-pixel 0.9 pixel "PSF" with elliptical shapes, which is a picture of the detector, not the optics. Depth is not a luxury on this instrument; it is the only way it can see its own focus. Fitting 2D Gaussians to stars standing at 130 to 270 sigma in the 600-frame stack gives 6.0 pixels FWHM, which at 1.925 arcsec per binned pixel is about 11.5 arcsec — and that is an upper bound, because any error in the pole enters the stack as smear.
Numbers
- Noise reduction from 600 de-rotated frames: 19.2x, against 21.7x for a
perfect square-root-of-N stack — 88 percent of ideal.
- Pole offset 4.32 degrees, bearing 54 degrees left of straight up; field
centre declination 85.7. Hour 00 and hour 04 fits agree.
- Four-night walk onto the pole: 13.38 -> 7.66 -> 4.32 degrees.
- Dwell inside the field, T = 9504 / distance in degrees seconds: about 12
minutes at 13.38 degrees, 37 minutes at 4.32, all night below 0.28.
- Stack depth available: 15x, then 27x, with a ceiling of 106x at the pole.
- Polar strip: 1619 x 22561 px, 149.2 degrees of position angle, 2978 frames,
median depth 114, 269 sources above 5 sigma.
- PSF 11.5 arcsec FWHM (6.0 px at 1.925 arcsec/px), IQR 5.7-8.3 px, from 2D
Gaussian fits to 130-270 sigma stars in the 600-frame stack. UPPER BOUND.
- Night total: 11,911 frames at 3 s.