How many seconds you actually have before the stars start to smear

Thirty seconds of exposure feels like a gift. The app allows it, the tripod holds, and after the wait the screen shows a sky full of dots. Job done, or so it seems. Then you get home, open the file, zoom to one hundred per cent, and the dots are commas: all leaning the same way, all exactly the same length. The tripod never flinched and the focus was fine. The sky moved while the sensor was busy counting photons. Which leaves one question, and it has a numerical answer: how many seconds do you get before that happens?

Fifteen arcseconds every second

Earth completes one full turn with respect to the stars in 23 hours, 56 minutes and 4 seconds. That is the sidereal day, just under four minutes shorter than the solar one: in the meantime the planet has moved along its orbit and needs a little extra spin to put the Sun back where it was yesterday. Spread that rotation over a circle of 1,296,000 arcseconds and you get the speed at which the sky slides past a stationary lens: 15.04 arcseconds per second of time.

That figure belongs to the celestial equator. Closer to the pole, stars trace tighter circles in the same interval, so they slow down, and the factor is the cosine of declination. A star in Orion’s Belt sits at declination zero and runs at full speed, which makes it your worst case. Vega, at nearly thirty-nine degrees, hands you a good twenty per cent of extra margin. Polaris, barely a degree from the celestial pole, crawls almost eighty times slower than Orion. Anyone shooting the Plough works under different rules from anyone chasing the Milky Way low in the south, and hardly anybody notices.

Your pixel is seventy arcseconds wide

Knowing how fast the sky moves is useless until you know how much sky one pixel covers. The maths starts from the field of view. Apple lists the main camera on its Pro models at a 24 mm equivalent focal length and f/1.78, the ultra wide at 13 mm and f/2.2, the 5x telephoto at 120 mm, f/2.8 and a twenty-degree field. Combine those with the frame geometry, 4032 by 3024 pixels for a 12-megapixel file, and the scale falls out: at the centre of the frame, one pixel of the main camera covers roughly 74 arcseconds of sky, one pixel of the ultra wide around 136, one pixel of the telephoto just under 15.

The crossing time is then a division. On the celestial equator a star traverses one main-camera pixel in five seconds, one ultra-wide pixel in nine, one telephoto pixel in less than one. A single second. Anyone who has aimed the tele at the Pleiades and held the shutter open for a few seconds already knows how that ends; now they know why. Shoot at the full 48 megapixels and every pixel halves in width, so all of those numbers halve with it.

The 500 rule quietly allows you four pixels of trail

The rule of thumb passed down among astrophotographers says to divide 500 by the equivalent focal length and use the result as your maximum exposure. On the 24 mm main camera that is 21 seconds, on the ultra wide 38, on the telephoto a shade over four. Line those up against the crossing times above and a coincidence appears that is not a coincidence at all: in all three cases the 500 rule grants a little more than four pixels of trail. The ratio is fixed by construction, since both the allowance and the pixel scale scale with focal length, and on a 48-megapixel file those four pixels become more than eight.

The rule was born in the film era, calibrated on coarse grain and prints viewed from a sensible distance. Four pixels of smear on a 35 mm negative blown up to 8 by 10 were invisible to everyone. On a retina display, with two fingers pulling the image apart down to the individual photosite, they are obvious. The NPF formula, worked out by Frédéric Michaud for the Société astronomique de France, tries to be more honest by bringing in pixel pitch and aperture: it divides by the true focal length the sum of 35 times the f-number and 30 times the pixel pitch in micrometres. On phone sensors, where the pitch is tiny and the aperture is wide, the aperture term dominates the answer almost completely. That is not a flaw in the formula. It is a clue.

A star is never a point, even when it holds still

A lens does not concentrate a point source into a point. Diffraction spreads it into a small disc whose diameter depends only on the aperture and the wavelength. At around 550 nanometres, where the eye is most sensitive, the arithmetic gives roughly 2.4 micrometres across at f/1.78, three at f/2.2, close to four at f/2.8. Now look at the size of the photosites. Samsung quotes 0.64-micrometre pixels for its ISOCELL JN1 sensor, merged four at a time into 1.28-micrometre virtual pixels when light runs short. The diffraction disc alone spans two or three of them. Add the residual aberrations of a lens a few millimetres long and a perfectly stationary star still occupies a small handful of pixels.

From which follows the most useful line in this piece: demanding that a star move less than a whole pixel is wasted severity, because that pixel of trail hides inside a disc already twice as wide. Allow a trail about as long as the disc, say two binned pixels, and the practical numbers become livable. Ten seconds on the 24 mm main camera, eighteen on the ultra wide, two on the telephoto. If you want a formula to carry in your pocket, for a 12-megapixel file divide 235 by the equivalent focal length, then divide again by the cosine of your target’s declination. Call it your own 500 rule, recomputed for the pixel density you actually own. At full resolution the number to divide is 118.

The test that beats any formula

Formulas give you the starting point; your particular lens delivers the verdict. One clear evening, ten minutes, and the question is settled for good. Point at a patch of sky near the celestial equator (Orion’s Belt in winter, the head of Aquila in summer), lock focus and exposure, then shoot the identical framing at 2, 4, 8, 15 and 30 seconds, changing nothing but the time. At home, on a proper monitor, zoom to one hundred per cent on a bright star near the middle of the frame and walk down the ladder. The moment a dot becomes a comma is unmistakable, and it always arrives sooner than you expected.

One caution about where to look: the centre. Phone optics distort stars at the edges all by themselves, and anyone hunting for trails in the corners ends up blaming Earth’s rotation for the lens’s own sins. If every star in the frame is equally round and equally bloated, the culprit is something else entirely, and it is called focus. The signatures differ, and it pays to learn to tell them apart before you convict the sky.

The ceiling applies to one frame, not to the night

Here the whole thing tips in your favour. The limit you have just worked out applies to a single exposure. How long the session runs is a separate matter, and nobody is capping that. A hundred frames at eight seconds gives you thirteen minutes of collected light with clean stars, and the software aligns them one by one, tracking the rotation you cannot track yourself. That is precisely what live stacking is for, and it is why a phone on a motionless tripod can reach places a single exposure never will.

The trade has a price, small but real. Every sensor readout adds its own dose of electronic noise, so chopping the same half hour into ever shorter frames means paying that toll more times over. The sweet spot sits in the middle: the longest exposure your test ladder approved, repeated until the battery or your patience gives out. And if the target allows it, head north. In the circumpolar sky the cosine works for you, and the same lens that grants eight seconds on the celestial equator hands you twenty near Polaris. The sky slows down for nobody, but it does choose who gets a discount.