What more sky time buys

Why the gain is a square root

Light arrives as separate photons, and they arrive at random, like raindrops on a paving slab. Count the drops that hit one slab in a second, then count again, and you get a different number each time. That unevenness is the grain you see in a single frame. It is not a fault in the camera. It is in the light.

When you add frames together, the galaxy's light adds up in step: twice the frames, twice the light. The random part adds up more slowly, because a chance high in one frame partly cancels a chance low in another. Twice the frames gives only 1.41 times the randomness, which is the square root of 2. Divide one by the other and the picture is 1.41 times cleaner for every doubling, and 2 times cleaner for every quadrupling. Astronomers call that ratio signal-to-noise, and the rule is written SNR ∝ √time.

Most of the noise does not come from the galaxy. The night sky glows, even at a site as dark as Rockwood, and the glow lands on every pixel. In your frames the faint outer arm of Andromeda is only 15 % as bright as the sky it sits on. The processing subtracts the sky's average level, but the sky's randomness stays behind. A long integration exists to wear that leftover randomness down.

The measurement

To measure noise without the galaxy getting in the way, I split each pile of frames into two random halves and stacked each half on its own. Both halves hold the same galaxy and the same stars, so subtracting one from the other removes everything real and leaves only noise. The dots below are the size of that leftover in two patches of empty sky.

What each added hour is worth

The same data on ordinary axes shows the catch. The curve climbs steeply and then leans over. It never goes flat, but it asks more for every step.

There is a useful way to picture a step you have not taken yet. Going from 10.9 hours to 25 hours multiplies the time by 2.3. Going from 4.8 hours to 10.9 hours multiplied it by the same amount, and you already own both of those stacks. So the improvement still to come on M 31 will look like the improvement below.

Where to look when you zoom in

Depth does not improve everything at once. Bright things arrive first and faint things queue behind them.

• Bright stars and the core are complete in one frame. More time adds nothing to them.
• Colour settles before detail does. The red, green and blue confetti in the short stacks is noise in the three colour channels, each disagreeing with the others. By about an hour the star colours are real: blue-white young stars, orange old ones.
• Dust lanes are dark detail on a bright ground, so they need the ground to be smooth. They separate from the grain at about 15 minutes and go on sharpening for hours.
• The outer arm is the true test. For the first several minutes that part of the frame holds stars and nothing else. The arm is there by one hour and has texture by four.
• Sharpness does not change. Star sizes are the same in every panel. Resolution is set by the 30 mm lens, the 3.7-arcsecond pixels and the steadiness of the air. Time buys depth, not resolution, though a clean deep stack does tolerate more sharpening afterwards.

The whole picture

Zoomed out, the story is different, and it is worth seeing why. Below, each depth is finished the way your pipeline finishes it: the sky is set to 10 % grey and the black point sits 2.8 times the noise below the sky. Because the rule is tied to the noise, a cleaner stack is automatically stretched harder.

Fainter stars

Stars give a second, independent measure. I counted the stars that stand at least five times above the noise in the two empty-sky test fields of Fig. 1.

The stars show one more thing: the sky is filling up. In the deep stack the two independent halves agree with each other, pixel by pixel, even in the patches I called empty. The agreement is about four-fifths the size of the remaining noise, and it leans toward the bright side. Noise cannot do that. Faint stars can. At eleven hours the dark sky between the stars you can see is already speckled with stars you cannot quite see. More time will turn some of that speckle into stars and uncover more behind it. In a rich field like this one, the background of a deep image never becomes perfectly smooth, and that is the sky's doing, not the camera's.

Not every hour is the same hour

The square-root rule assumes each minute is as good as the last. On a real sky that is not so. Because I measured the noise of every frame separately, I can price each night against the best one.

• Moonlight is the big one. On 27 October the Moon was 94 % lit and high in the sky. The sky glow was 13 times brighter than on a moonless night, and an hour was worth 8 minutes. The 27 minutes collected that night count for about 4 dark-sky minutes in the stack.
• Height matters steadily. Near your 35° limit the telescope looks through more air, which dims the galaxy and brightens the sky. A minute there is worth 71 % of a minute overhead.
• Dark nights still differ. The two new-moon nights, 10 and 11 November, were a little hazier than 5 November and caught M 31 lower, so they came in at 75 to 80 %.

Put together, M 31's 10.9 hours on the clock are worth 8.9 hours of best-night time. The campaign counts clock hours. The picture only knows the second number.

The same rule on your other three targets

I ran the identical measurement on the North America Nebula, the Pleiades and the Orion Nebula. Different objects, a different filter on one of them, different nights. The rule holds on all of them.

NGC 7000, the North America Nebula · 6.1 of 15 hours

This target is shot through the dual-band filter, which passes the red glow of hydrogen and the teal glow of oxygen and blocks most other light. In your frames the sky glow is about nine times weaker than in the unfiltered M 31 frames. That is why the bright ridge called the Cygnus Wall is brighter than the sky behind it, where Andromeda's outer arm was a sixth as bright. The nebula shows up sooner, but the rule for improving it is unchanged.

M 45, the Pleiades · 1.0 of 20 hours

The Pleiades have had one hour, and it shows. The blue haze is starlight scattered off dust, and around Merope it is about 30 % as bright as the sky. This is the target where your next hours will do the most. The 20-hour goal means 4.4 times today's signal-to-noise. For a sense of scale, that is the step from 3.5 minutes to today, shown below.

M 42, the Orion Nebula · 2.4 of 12 hours

Orion is bright enough to look finished in minutes, and that hides what depth adds here. Look at the dark dust in the third region of the figure. For the first ten minutes it is not in the picture at all. The 12-hour goal is 2.2 times today's signal-to-noise, the step from 29 minutes to today.

Progress, counted two ways

Because quality grows as the square root of time, you are always further along in quality than in hours. Half the hours is 71 % of the final signal-to-noise. A quarter of the hours is half of it.

What I would do with this

1. Judge M 31's goal with Fig. 4. The remaining 14 hours buy +51 %. Stopping at 15 hours would buy +17 %, and 20 hours +35 %. If the step in Fig. 4 looks worth it to you, keep 25. Your preference for one great image over several fair ones argues for keeping it, since the outer disc is exactly what the extra hours improve.
2. Expect the biggest visible change on M 45. An hour there adds 41 %. The same hour on M 31 adds 4.5 %. The campaign order has its own reasons, such as NGC 7000 leaving the evening sky, and I would not change it on this alone. Just know that the Pleiades picture you have today is the least finished of the four by a wide margin.
3. Keep broadband targets off bright-moon nights. Your moon rules already do this. The measurement backs them: under a 94 % Moon an hour on M 31 was worth 8 minutes.
4. Leave the 35° altitude limit alone unless another target is waiting. Low frames are worth 71 % of high ones. That is a discount, not a waste, and it beats an idle telescope. It only matters when a different target could use the same minutes higher up.
5. Trailed frames: measured, and left in. A by-product of this work: 6 % of the frames in the 12 November stacks show stars drawn into short lines, and about 7 in 10 of them are the first frame after a pause in the sequence, which is where the scope's dither moves fall. They pass the sharpness and roundness tests because the star's bright head is still round. The stacker's pixel rejection removes the bright part of each trail. What stays is a faint one-sided tail holding about 0.1 % of each star's light. Those frames still carry the galaxy and the nebula, so leaving them out costs about 2 % of signal-to-noise. A check that would leave out the 4.3 % of frames with a clear tail or a doubled star is drafted and tested. You decided on 12 November that 2 % of depth is not worth cleaner stars right now, so it stays parked and the stacks keep these frames.