Integration time and signal-to-noise: the CCD equation, sub length and total hours

The CCD equation

The planning rules in this guide all come from one equation. Sheffield’s course notes on the CCD equation list four noise sources: the shot noise of the target, of the sky background and of the dark current, each the square root of its signal, and the read noise R, which has no square root because it carries no signal and does not grow with time. Independent noises add in quadrature, so with signals in electrons:

S/N = S_obj × t ÷ √(S_obj × t + n_pix × (S_sky × t + S_dark × t + R²))

Here S_obj is the target’s total rate, S_sky and S_dark are rates per pixel, and n_pix is the number of pixels the target covers. ESO’s exposure time calculators use the same structure, in equation 2.1 of their formula book. Sheffield adds that subtracting the mean sky and dark levels does not remove their shot noise.

Three regimes

Sheffield’s notes work out the limiting cases:

• Target-limited: S/N grows as √t; doubling the exposure gives 1.4 times the S/N.
• Background-limited: S/N also grows as √t, and for a given target it falls as the square root of the sky level.
• Read-noise-limited: S/N grows in proportion to t, because the noise no longer grows with time.

For faint targets, Olivier Hainaut’s ESO note on signal and noise treats the last two cases in stacking terms: when the sky dominates, S/N depends on the total time, and one 3000-second exposure equals ten 300-second exposures as long as each frame is sky-noise dominated. When the sky is faint, as in narrowband imaging where his example sky is a couple of electrons, read noise dominates and individual exposures should be as long as possible.

Worked numbers: what more S/N costs

In the shot-noise regimes, time scales with the square of the S/N you want (computed):

Target S/N gain	Time needed	Starting from 2 hours
1.5 ×	2.25 ×	4.5 h
2 ×	4 ×	8 h
3 ×	9 ×	18 h

Baader’s narrowband whitepaper states the same rule for frames, four for twice the S/N and sixteen for four times, and adds that read noise enters once for every frame, so exposures should be made as long as practical before adding more of them.

Sub length: when read noise matters

The equation also shows how much a given sub length costs (derived; not quoted from the sources). Take N subs of length t on a faint target. If B is the sky signal per pixel in one sub, in electrons, the noise variance per pixel is N × (B + R²) instead of the read-noise-free N × B. The fraction of S/N kept is:

S/N kept = 1 ÷ √(1 + R² ÷ B)   (time needed to make up the loss = 1 + R² ÷ B)

Using Hainaut’s example sky levels with his 3 e read noise, and then with published read noise for two cooled cameras (computed):

Case	Sky per sub B	Read noise R	S/N kept	Total time needed instead
Broadband, Hainaut’s example	10,000 e	3 e	99.96 %	× 1.0009
Narrowband, Hainaut’s example	2 e	3 e	42.6 %	× 5.5
Same sky, ZWO ASI2600MM Pro at 1.0 e	2 e	1.0 e	81.6 %	× 1.5
Same sky, ZWO ASI6200MM Pro at 0.86 e	2 e	0.86 e	85.4 %	× 1.37
Same sky, ASI2600MM Pro at 3.3 e	2 e	3.3 e	39.4 %	× 6.4

The read noise values are from ZWO’s ASI2600MM Pro manual (1.0–3.3 e, lower at high gain) and the ASI6200 product page (0.86 e). Hainaut’s sky levels describe ESO instruments, not a backyard; your own B is the background of a calibrated sub converted to electrons with the camera’s e-/ADU, as explained in the gain and read noise guide and the calibration frames guide. Once B is ten times R², 95.3 % of the S/N is kept.

Longer is not free. Hainaut lists reasons to split exposures, among them removing cosmic rays and improving the flat field, and QHYCCD’s QHY294 page notes that bright stars saturate more easily in long exposures. The narrowband guide covers why narrowband frames sit so close to the read-noise regime.

Total time, mosaics and storage

When the sky dominates, total time is what counts. That has a direct consequence for mosaics: each panel needs its own integration, so the mosaic planner’s panel count multiplies the hours. Four panels at four hours each is sixteen hours. Spreading the same four hours over four panels gives each panel one hour and half the S/N of a single four-hour field (computed, √(1/4) = 0.5).

Sub length also sets the file count. Four hours with an IMX571 camera, whose 16-bit FITS frame is 52.19 MB (computed, ignoring gaps between frames):

Sub length	Frames	Data from the camera
30 s	480	25.05 GB
180 s	80	4.18 GB
600 s	24	1.25 GB

The storage calculator adds calibration frames, gaps and processing copies. Hainaut also notes that n_pix scales with the seeing squared, so a star spread over more pixels collects more sky and read noise; the sampling calculator shows how many pixels your stars cover.