Magnitude 3.4 and invisible: the number the catalogue leaves out

Andromeda is magnitude 3.4. Pin that number on a star and you get something anyone can pick out by eye from a small town back yard. Point your phone the right way, ten seconds of exposure, look at the screen: dozens of stars, plenty of them fainter than 3.4, and no galaxy anywhere. The catalogue did not lie to you. It answered a question you were not actually asking.

One magnitude, spread across three degrees

Total magnitude tells you how much light arrives, added up over the whole object. The quantity was invented for stars. A star is a point source by definition, so its light lands in a handful of pixels and stays bundled there no matter what the sky around it does. An extended object breaks that arrangement, because the same amount of light has to cover the patch of sky the object occupies, and every pixel on the sensor gets only its own share.

The number you want is surface brightness, quoted in magnitudes per square arcsecond. The definition is literal: surface brightness 20 means each one arcsecond square of sky puts out, on average, as much light as a magnitude 20 star. Going from total magnitude to surface brightness is a single addition, and what you add is 2.5 times the logarithm of the area in square arcseconds.

Try it on Andromeda. NASA describes the galaxy as six times the apparent diameter of the full Moon, so roughly three degrees, and the catalogue ellipse narrows to about a degree on the short axis: a shade over thirty million square arcseconds. The correction comes to 18.7 magnitudes, and the mean surface brightness lands at 22.1. The Triangulum galaxy, magnitude 5.7 spread over 73 by 45 arcminutes, works out to 23.1. That is the same figure you get by converting the 14.2 magnitudes per square arcminute the catalogues publish for it.

The sky background plays in the same units

Those two numbers only mean something next to the sky you are shooting under, which happens to be measured on the same scale. On a moonless night, well away from the Milky Way and the zodiacal light, the natural background sits near 22.0 magnitudes per square arcsecond in the V band. That is a physical floor. The upper atmosphere glows on its own account and there is nowhere on Earth where it stops: Sky Quality Meter readings put a pristine site at 22.0, open countryside near 21.5, suburbs around 20, and a city centre below 18.

Now read the earlier figures again. Averaged across its full extent, Andromeda is as faint as the darkest sky you will ever stand under. Triangulum is a magnitude worse, which means its surface emits less light than the natural background does. Always, everywhere, including a flawless night at two thousand metres. No amount of gear moves that comparison.

Contrast takes one line of arithmetic: ten to the power of minus zero point four times the difference between the two surface brightnesses. Under a rural sky at 21.5, Andromeda adds 57 per cent on top of the background, which is exactly why it shows so readily. From a suburb at 20.0 it adds 15. Under a city centre at 18.5 it drops to 4 per cent, buried in a background nearly thirty times stronger, and digging it out takes hours where minutes used to do.

That is the whole reason light pollution kills nebulae and shrugs at stars. A star keeps its light packed into a few pixels and goes on beating the background even after the background climbs three magnitudes. An extended object fights the sky pixel by pixel, on level terms, and loses. The Bortle scale puts this in words; surface brightness writes it as a subtraction you can do in the car park.

Averages lie as freely as magnitudes

There is a trap inside the trap. The figure worked out above is a mean, and no galaxy is uniform. Andromeda’s core is far brighter per unit area than that 22.1 suggests, its outer arms far dimmer. In practice the bright middle turns up in thirty seconds from a suburban garden, while the three degree disc you see in other people’s pictures needs a different sky and a different number of hours.

The Orion Nebula pushes the same confusion to its limit. Catalogues list it at fourth magnitude across nearly a degree, but almost all of that light sits in the Trapezium region, a few arcminutes wide: it appears in seconds and clips if you keep pushing, while the outer wings stay down in the noise. Two catalogue numbers describe an average the object itself rarely honours.

Why the aperture matters and the magnification doesn’t

Exactly one parameter governs the brightness of the image forming on your sensor: the f-number. For an ideal lens, illuminance in the focal plane goes as the inverse square of the f-number, and focal length stays out of the equation. Change focal length at a fixed f-number and you change image scale while leaving light per unit area untouched.

On a phone the consequence is blunt. Apple specifies ƒ/1.78 for the iPhone 16 Pro main camera and ƒ/2.2 for the Ultra Wide: square them, take the ratio, and the main camera delivers one and a half times the light per unit of sensor area at the same exposure and gain. For deep sky work the lens choice is settled before you start. The telephoto, usually the slowest of the set, is the worst candidate on the phone despite all the reach it advertises.

The other half of the physics is harder to swallow. No optical system can raise the surface brightness of an extended source: that is conservation of étendue, a theorem, not a manufacturing limit. Magnify an extended object and its light spreads across more pixels, dropping its brightness in the image. Digital zoom adds nothing, cropping adds nothing either. Both enlarge the same signal-to-noise ratio.

What actually fits into one night

Once you think in surface brightness, the target list reorders itself. Open clusters (the Pleiades, the Beehive, the Perseus Double Cluster) are made of point sources and behave like stars, so they turn up under mediocre skies in a handful of frames. Compact nebulae and galaxy cores occupy the middle ground. Sprawling galaxies and large faint nebulae demand dark sky before they demand exposure time.

Integration is still the weapon you hold. Stacking N frames raises signal to noise by the square root of N, and from a suburb two hundred frames recover a good share of that 15 per cent. What the hours cannot recover is the sky itself. If the background sits at 18.5, every single frame carries the same wall of light with it, and the square root of N works on the noise in that wall, never on the wall.

Before you load the car, look up your target’s surface brightness and subtract the surface brightness of the site you plan to drive to. If the difference comes out a couple of magnitudes negative, you have already imagined the whole session.