A scale that runs backwards, multiplies, and works

Astronomy’s brightness scale is indefensible on every count except one. It runs backwards, so that brighter objects have smaller numbers and the brightest have negative ones. It is logarithmic, so equal steps mean equal ratios. Its zero point is an arbitrary star. And it comes in a dozen flavours depending on which filter was used.

It survives because it was right about the important thing by accident. Hipparchus sorted the visible stars into six grades by eye, and the eye responds to ratios rather than differences — so the scale he produced was logarithmic before anyone knew what a logarithm was.

The accident deserves a moment, because it is the reason the scale was worth keeping rather than merely inconvenient to replace. A sensory system that responded linearly to intensity would be useless across the range light actually varies over: the difference between starlight and noon sunlight is a factor of about $10^{9}$, and no linear detector spans that. Biological detectors compress, and compression means the response goes as the logarithm of the stimulus over most of the usable range. Sorting stars into “grades” therefore produces steps of roughly constant ratio whether or not the sorter has any idea that is what is happening. Hipparchus was not being clever. He was reporting what a logarithmic instrument told him, and the instrument was his retina.

Making the old scale exact #

By the nineteenth century photometry was precise enough to check Hipparchus’s grades, and Pogson noticed in 1856 that a first-magnitude star was almost exactly a hundred times brighter than a sixth-magnitude one. Rather than replace the scale, he defined it to be so:

$$m_1 - m_2 = -2.5\log_{10}\frac{F_1}{F_2}.$$

Five magnitudes is a factor of 100 by definition; one magnitude is $100^{1/5} = 2.512$. The minus sign preserves the inversion, and the 2.5 is not a round number chosen for elegance but the consequence of insisting on the factor of 100.

Some useful landmarks fall out. A difference of 1 magnitude is a factor of 2.5; 2.5 magnitudes is a factor of 10; 10 magnitudes is $10^4$. And the range in the figure — from the Sun at $-26.7$ to the faintest galaxies at $+31$ — is 58 magnitudes, which is a factor of $2 \times 10^{23}$ in received light. One scale, running from the brightest object in the sky to the faintest ever recorded, over twenty-three orders of magnitude.

Apparent, absolute, and the difference between them #

A magnitude as observed says nothing about a star. It confounds how much light the star emits with how far away it is, and those are independent quantities.

The apparent magnitude $m$ is what a detector records. The absolute magnitude $M$ is defined as the apparent magnitude the object would have at exactly 10 parsecs — a stipulated distance that removes the distance and leaves the star.

The two are related by the inverse-square law. Light spreads over a sphere whose area grows as $d^2$, so the received flux falls as $1/d^2$, and putting that into Pogson’s relation gives the distance modulus:

$$m - M = 5\log_{10}(d/10\,\text{pc}).$$

Every distance measurement above the parallax ceiling is that equation rearranged. Establish $M$ somehow, measure $m$, and $d$ follows.

The numbers are worth having. The Sun’s apparent magnitude is $-26.7$ and its absolute magnitude is $+4.8$ — at 10 parsecs it would be an unremarkable naked-eye star, invisible from any city. Sirius appears brightest of all the night stars at $-1.46$, and its absolute magnitude is $+1.4$: bright, but only 25 times the Sun’s output. Deneb is $+1.25$ apparent and about $-8.4$ absolute, which makes it roughly 200,000 times the Sun and one of the most luminous stars visible.

Sirius and Deneb look almost equally bright. One is 2.6 parsecs away and the other about 800. The night sky is a projection in which distance and luminosity have been multiplied together and cannot be separated by looking.

The standard candle, and its assumption #

An object whose absolute magnitude is known independently is a standard candle, and the whole extragalactic distance scale is built from a small number of them.

Cepheid variables pulsate with periods from days to months, and Henrietta Leavitt found in 1912 that the period predicts the luminosity. Her insight was to study Cepheids in the Small Magellanic Cloud, where all of them are at essentially the same distance, so their apparent magnitudes could be compared without knowing that distance. The relation appeared immediately; calibrating it — turning a relative relation into an absolute one — required exactly one Cepheid with a geometric distance, and that took another decade.

Type Ia supernovae are the top rung. A white dwarf pushed over its stability limit detonates with a fairly repeatable output, and the small remaining spread correlates with how fast the light curve fades — a correction that turns a rough candle into a good one. They reach billions of parsecs, and the measurement of their brightness against redshift is what found the acceleration of the expansion.

Every one of these is an assumption dressed as a measurement. The assumption is that the object at great distance is the same kind of object as the calibrators nearby, and it is not always true: Cepheids’ luminosities depend on metallicity, and distant galaxies have different compositions from the Milky Way.

What is actually being counted #

“Brightness” is not a single quantity, and the ambiguity is the source of most of the scale’s complications.

A detector counts photons arriving in some range of wavelengths, set by a filter. The standard system uses U, B and V — ultraviolet, blue and visual — and a star’s magnitude differs between them by an amount that depends on its colour, which is its temperature. Quoting a magnitude without a filter is meaningless.

The bolometric magnitude integrates across all wavelengths, and it is the physically meaningful one — total energy output. It is also the one that cannot be measured directly, since no detector covers everything and the atmosphere blocks much of the spectrum. Getting from an observed V magnitude to a bolometric one requires a correction that depends on the star’s temperature, and the correction is large for the hottest and coolest stars, which emit most of their light outside the visible band.

The zero point is arbitrary and historical: Vega was defined as magnitude 0 in every filter, which is why a star’s colour indices are all zero if it happens to resemble Vega. Modern systems use a defined flux instead, because Vega turned out to be variable, surrounded by dust, and viewed nearly pole-on.

The other things that dim a star #

The distance modulus assumes light travels unimpeded. It does not.

Interstellar dust absorbs and scatters starlight, and a star seen through it appears fainter than its distance implies — which, applied naively, makes it seem further away. Extinction in the plane of the galaxy can reach several magnitudes per kiloparsec, and toward the galactic centre it is around 30 magnitudes in visible light, a factor of $10^{12}$. The centre of the galaxy is not visible at all in optical light, and everything known about it comes from the infrared and radio, where the dust is transparent.

Dust also reddens: it scatters blue light more effectively than red, so a reddened star looks cooler than it is. That is a nuisance and also the correction’s own solution, because comparing the observed colour with the colour expected from the spectral type measures the reddening, and the reddening gives the extinction. A contamination that carries its own diagnostic.

What was actually measured #

A magnitude looks like a reading. It is a reduction, and the steps between the photons and the number are where the entire difficulty lives.

Start with what a detector produces: a count of electrons in each pixel over an exposure. Turning that into a flux requires subtracting the bias level and the dark current, dividing by a flat field that records each pixel’s individual sensitivity, summing the counts inside an aperture large enough to catch the star’s wings but small enough to exclude its neighbours, and subtracting an estimate of the sky background measured in an annulus around it. Each of those steps has a systematic attached, and the aperture choice alone moves the answer by several hundredths of a magnitude if it is made carelessly.

Then the atmosphere. Every photon has passed through a column of air whose thickness depends on how far the target is from the zenith, and the absorption is exponential in that thickness. The standard treatment is to observe the same field repeatedly as it rises and sets, plot instrumental magnitude against airmass — the column depth relative to the zenith, which is close to $\sec z$ — and extrapolate the straight line back to zero airmass. The intercept is the magnitude above the atmosphere. That extrapolation is doing real work: at a good site the extinction coefficient in V is about 0.15 magnitudes per airmass, so a target observed at 45° from the zenith is being corrected by about a fifth of a magnitude.

Finally the zero point, which is not a physical constant but a network. A field of stars whose magnitudes have been established by repeated careful measurement is observed alongside the target; the offset between the instrumental and catalogue values for the standards is applied to the target. The catalogue in question is a chain of such measurements going back through several generations of photometry, and its internal consistency, rather than any absolute calibration, is what “magnitude 12.34” ultimately means.

The precision achieved is worth stating against that background. Ground-based differential photometry — measuring a target against a comparison star in the same frame, so that the atmosphere and the zero point largely cancel — routinely reaches a millimagnitude, a tenth of a percent in flux. Gaia’s broad-band photometry reaches about 0.3 millimagnitudes for bright sources. Absolute photometry, tied to a physical flux rather than to other stars, is good to perhaps 1%, which is a factor of thirty worse. The difference between those two numbers is the difference between comparing and measuring, and almost every result in the subject rests on the cheaper of the two.

Where the numbers land #

A magnitude on its own is a number. Paired with a distance it becomes a position on the diagram that organises the whole subject. The diagram is a plot of absolute magnitude against colour, drawn in physical units. Its horizontal axis is cheap and its vertical axis is expensive, and the expense is entirely the distance modulus. The two figures are the two halves of the same operation. A parallax gives $d$; the distance modulus turns $m$ into $M$; $M$ places the star on the diagram. Break the first step and the other two produce a confident and wrong answer, which is exactly what happened to the globular-cluster ages before the distance scale was revised.

The version that does not care how far away anything is #

There is one derived quantity in photometry that behaves so unlike the rest that it is worth the heading, and it is the generalisation the scale points toward.

Surface brightness is the magnitude per square arcsecond of an extended object — a galaxy, a nebula, the night sky itself. Its odd property is that it does not depend on distance at all. Move a galaxy twice as far away and the light received falls by a factor of four, exactly as the inverse-square law requires; but its angular area also falls by a factor of four, so the light per unit solid angle is unchanged. The two factors cancel identically.

This is why a nebula does not get brighter in a larger telescope, only bigger and better resolved, and why no eyepiece arrangement can make a faint galaxy stand out against the sky background — the galaxy and the sky are both surface brightnesses, and their ratio is fixed by physics rather than by optics. It is also why the darkest available sky matters more than aperture for that class of object, a fact amateur observers rediscover at some expense.

The surprising part is what the cancellation is sensitive to. It holds exactly in a static, Euclidean universe. In an expanding one it fails, and it fails in a specific and calculable way: surface brightness falls as $(1+z)^{-4}$, one factor of $(1+z)$ each from photon energy loss, arrival-rate dilation, and two from the angular-size relation. That prediction is the Tolman surface brightness test, proposed in 1930 as a way to distinguish an expanding universe from a static one in which redshift had some other cause. The measurement is hard, because galaxies at high redshift are also younger and intrinsically different, but it has been carried out, and the fourth power is there.

A scale invented to rank naked-eye stars by grade turns out, in one derived form, to be a test of whether space is expanding. That is not a connection anyone designed.

The things that are not stars #

The scale was built for point sources of constant brightness, and it is routinely applied to things that are neither. The conventions that result are worth knowing, because they are places where a number carries an assumption that is not stated.

A meteor is quoted at the magnitude of the point source it would take to deliver the same light — which is a sensible convention and an awkward one, because a meteor is a moving streak and the eye integrates a moving source differently from a stationary one. Two observers reporting the same meteor routinely differ by a magnitude, and the standard visual meteor catalogues are built on estimates of that quality.

A satellite is quoted the same way, and here there is a further complication: the brightness depends on the phase angle between the Sun, the satellite and the observer, so a single magnitude means nothing without the geometry. The convention for constellations is to quote the magnitude at a standard range and phase, which makes the numbers comparable and not directly usable.

A comet is the hardest case, because it is neither a point nor a uniform patch. Its total magnitude includes a coma whose apparent extent depends on the aperture used and the sky brightness, so published comet magnitudes from different observers differ systematically and by amounts that swamp the physical variation.

The common thread is that a magnitude is a statement about a flux, and a flux is well defined only once it is decided what counts as the object. For a star the decision is free; for anything with an edge that fades rather than ends, the decision is the measurement, and the scale’s apparent precision hides it completely.

Where the model stops #

Point sources. Extended objects — galaxies, nebulae — have a total magnitude that depends on how much of the faint outskirts is counted, which makes published values method-dependent at the tenth-of-a-magnitude level. A star, at any distance parallax can reach, is a point.

Constant sources. Many stars vary. Some of that variation is the measurement — Cepheids, and the eclipsing binaries that give stellar masses — and some is noise.

No absorption. Extinction, as above, and it is the largest systematic in the whole field.

One wavelength band. Bolometric corrections, as above.

The figures have a specific limitation. Plotting a straight line on log axes makes the relation look trivially simple, and it hides the fact that the measurement is not. Getting a magnitude to a hundredth requires flat-fielding, atmospheric extinction correction, aperture photometry and comparison standards, and the difference between a 0.1-magnitude measurement and a 0.01-magnitude one is the difference between a detection and a null result in most of modern photometry.

The ladder from here #

Later rungs: Pogson’s relation derived, and the eye’s logarithmic response. Filter systems and colour indices. Bolometric corrections. Extinction and reddening, and the reddening-free indices built to dodge them. The distance modulus applied to clusters — main-sequence fitting. Cepheids and the period–luminosity relation. Type Ia supernovae and the light-curve corrections. Surface brightness, which is distance-independent and therefore strange. And the AB magnitude system, which finally replaced Vega with a number.

Hipparchus’s six grades were a convenience for a catalogue. They are still in use, negative numbers and all, twenty-one centuries later — the oldest surviving measurement scale in any science.