Astronomy & Celestial Mechanics
the sky, computed
Starlight

A scale that runs backwards, multiplies, and works

Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are.
11 min read 6 figures Light is the only messengerMeasuring the unreachable

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 magnitude scale, plottedThe logarithm of the received light against magnitude, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller.-20-100102030apparent magnitudethe Sunfull MoonVenus at its bestSiriusVega, by definitionthe naked-eye limita good amateur telescopea large ground-based surveythe deepest exposuresbrighter ←→ fainterfive magnitudes = ×100 in light
Fig. 1 The logarithm of received light against magnitude. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a factor of one hundred.

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:

m1−m2=−2.5log⁡10F1F2.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 1001/5=2.512100^{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 10410^4. And the range in the figure — from the Sun at −26.7-26.7 to the faintest galaxies at +31+31 — is 58 magnitudes, which is a factor of 2×10232 \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.

The magnitude scale, plottedThe logarithm of the received light against magnitude, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller.-202468apparent magnitudeSiriusVega, by definitionthe naked-eye limitbrighter ←→ fainterfive magnitudes = ×100 in light
Fig. 2 The naked-eye range alone: from Sirius to the limit of a dark sky. Eleven magnitudes, which is a factor of 25,000 — the entire dynamic range of unaided vision, and the whole of astronomy before 1609.

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 mm is what a detector records. The absolute magnitude MM 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 d2d^2, so the received flux falls as 1/d21/d^2, and putting that into Pogson’s relation gives the distance modulus:

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

Every distance measurement above the parallax ceiling is that equation rearranged. Establish MM somehow, measure mm, and dd follows.

The numbers are worth having. The Sun’s apparent magnitude is −26.7-26.7 and its absolute magnitude is +4.8+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 −1.46-1.46, and its absolute magnitude is +1.4+1.4: bright, but only 25 times the Sun’s output. Deneb is +1.25+1.25 apparent and about −8.4-8.4 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 distance ladder, and its overlapsThe reach of each distance technique on a logarithmic scale in parsecs. Each rung is calibrated where it overlaps the one below it, so an error low on the ladder propagates all the way to the top.10^-610^-410^-210^010^210^410^610^810^10radar rangingdirect: a timed echoparallaxgeometry, and nothing assumedspectroscopic parallaxassumes a star like the calibratorsCepheid variablesassumes the period–luminosity relation holdsTully–Fisherassumes rotation tracks luminositytype Ia supernovaeassumes a standard explosionHubble's lawassumes the expansion rate is knowndistance (parsecs)each rung is calibrated on the one belowan error at the bottom moves everything above it
Fig. 3 The distance techniques and their overlaps. Every rung above parallax works by establishing an absolute magnitude and comparing it with the apparent one — the distance modulus applied to a different kind of object at each scale.

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 101210^{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.

Blackbody curves at 3500, 5800, 12000 KThermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.visible050010001500200000.20.40.60.811.2wavelength (nm)3500 K, peak 828 nm5800 K, peak 500 nm12000 K, peak 242 nmeach curve scaled to its own peak
Fig. 4 Thermal emission at three temperatures. A filter samples one vertical slice of these curves, so what a magnitude records depends on both the star’s temperature and the filter — which is why colour indices exist and why bolometric corrections are needed.

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 Hertzsprung–Russell diagramLuminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉giantssupergiantswhite dwarfs1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units
Fig. 5 Luminosity against temperature. The vertical axis is an absolute magnitude with the sign flipped and the units changed — which is why every point on it required a distance before it could be placed.

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.

Parallax for a star at 1.3 parsecsThe same star observed from two ends of a baseline. The angle between the two sight lines is 0.77 arcseconds for a star 1.3 parsecs away — the definition of the parsec is the distance at which it would be exactly one.distant starsJanuaryJuly2 AU1.3 pc2p = 1.54″distance = 1 ÷ parallax in arcsecondsangles hugely exaggerated: the true angle here is a five-thousandth of a degree
Fig. 6 The geometric measurement at the bottom of it all. Absolute magnitudes exist because a few thousand stars are near enough for a triangle to be solved, and everything further is calibrated against them.

The two figures are the two halves of the same operation. A parallax gives dd; the distance modulus turns mm into MM; MM 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.

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.

Where this goes next

distant starsJanuaryJuly2 AU1.3 pc2p = 1.54″distance = 1 ÷ parallax in arcsecondsangles hugely exaggerated: the true angle here is a five-thousandth of a degree The triangle that reaches the stars, and stops Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it. visible050010001500200000.20.40.60.811.2wavelength (nm)3000 K, peak 966 nm5800 K, peak 500 nm10000 K, peak 290 nmeach curve scaled to its own peak Colour is a thermometer, and it reads across the galaxy A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy. R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉giantssupergiantswhite dwarfs1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units The diagram that sorted the stars, by plotting two things against each other Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy.