A brightness is a distance only if something is known

A star’s brightness falls as the inverse square of its distance. Rearranged, that is a distance measurement: observe the brightness, know the intrinsic output, divide, take a square root.

In magnitudes it is a subtraction rather than a division, which is the reason the logarithmic scale survives:

$$m - M = 5\log_{10}\!\left(\frac{d}{10\ \text{pc}}\right).$$

The apparent magnitude $m$ is measured, to a hundredth of a magnitude routinely and to a thousandth with care. The distance $d$ is what is wanted. And $M$, the absolute magnitude — the magnitude the object would have at ten parsecs — has never been measured for any object outside the solar system, and never can be.

Everything in extragalactic astronomy rests on that asymmetry.

The one number that is not observable #

The relation has three quantities and it is worth being exact about the status of each.

$m$ is observed. A CCD counts photons; a comparison with a standard star in the same frame converts counts into a magnitude in a defined photometric band. This is one of the cleanest measurements in astronomy.

$d$ is wanted, and it is the quantity no instrument measures directly.

$M$ is inferred. It is defined as the apparent magnitude the object would have at exactly ten parsecs, and no object is at exactly ten parsecs. It can be computed for an object whose distance is already known — which is circular unless the distance came from somewhere else — and it can be predicted for a class of objects from a relation calibrated on members whose distances came from somewhere else.

That “somewhere else” is the rung below on the ladder, and this is precisely where the ladder structure enters. The distance modulus is not a method; it is the arithmetic that every method above parallax performs. What distinguishes the methods from each other is entirely how they obtain $M$.

Ten parsecs, and why it is arbitrary #

The choice of ten parsecs has no physical content. It was fixed by convention so that absolute magnitudes of ordinary stars come out as small numbers of either sign: the Sun’s is +4.83, Sirius’s is +1.43, a bright supergiant’s is around −7, a faint red dwarf’s around +16.

Any other reference distance would work and would shift every absolute magnitude by a constant. The only property that matters is that everybody uses the same one, which is the usual condition for a zero point.

There is one place where the arbitrariness leaks out and causes trouble. Because absolute magnitudes are defined with respect to a distance in parsecs, and parsecs are defined by the astronomical unit, and the astronomical unit was measured rather than defined until 2012, the zero point of the absolute magnitude scale was tied to a measured quantity for its whole history. That dependence is now gone, and the residual is a redefinition too small to notice.

The relation is steep in one direction and flat in the other #

The factor of five in front of the logarithm is the whole of the practical behaviour, and it cuts both ways.

Read as distance from magnitude, the relation is forgiving: a whole magnitude of error in $M$ gives a distance error of

$$\frac{\Delta d}{d} = 10^{\Delta M/5} - 1 = 0.585,$$

or 58%. Half a magnitude gives 26%, and a tenth gives 4.7%. To first order it is $0.46\Delta M$, and that number — 46% per magnitude — is worth memorising, because it converts every photometric uncertainty into a distance uncertainty in one step.

Read as magnitude from distance, it is unforgiving in the other direction: a factor of ten in distance is exactly five magnitudes, so an object a hundred times further away is twenty-five times fainter in flux and five magnitudes down the scale. That steepness is why the observable universe is dominated by nearby objects and why every flux-limited survey is a survey of the bright end of everything.

The asymmetry between the two readings is the reason standard candles have to be so uniform. It is not enough for a class of object to be bright; the class has to have a small intrinsic spread. Type Ia supernovae — white dwarfs at a mass limit — are used not because they are the brightest thing available but because, after a correction, their absolute magnitudes agree to about 0.1 magnitude — which is 5% in distance. Quasars are far brighter and are useless as candles, because their absolute magnitudes span six magnitudes with no way to predict where in that range a given one sits.

Extinction: the term that is always in the wrong direction #

The relation above is wrong for almost every object, because it assumes nothing intervenes.

Interstellar dust absorbs and scatters starlight, and the correct relation is

$$m - M = 5\log_{10}d - 5 + A,$$

where $A$ is the extinction in magnitudes along the line of sight. In the plane of the Milky Way $A_V$ is roughly 1 magnitude per kiloparsec, and toward the galactic centre it reaches 30 — a factor of $10^{12}$ in flux, which is why the centre of the galaxy is invisible in optical light.

Extinction is not a random error. It always makes an object look fainter, so it always makes it look further away, so neglecting it always overestimates a distance. Averaging over many objects does not help.

What saves the situation is that extinction is wavelength dependent — it is stronger in the blue — so a star’s colour is reddened as well as dimmed, and the two effects are related by a ratio that is roughly universal:

$$A_V \approx 3.1\,E(B-V),$$

with $E(B-V)$ the excess of the observed colour index over the intrinsic one. So measuring an object in two bands and comparing its colour with the colour a star of its type should have gives the reddening, and the reddening gives the extinction. The dust is measured by the way it distorts a spectrum rather than by the amount it removes, because the amount it removes is exactly what is unknown. That is a general trick worth carrying: an unwanted effect that is achromatic cannot be separated from the signal, and an unwanted effect that is chromatic can.

Two magnitudes that are not the same magnitude #

There is a second inference buried in the relation, and it is one that catches even careful work.

A magnitude is always a magnitude in a band. A $V$ magnitude counts the photons that get through a filter centred near 550 nm; a $K$ magnitude counts a different set entirely. The distance modulus is valid band by band — $m_V - M_V$ and $m_K - M_K$ both give the same distance, provided the extinction term is the one for that band — and mixing bands gives nonsense.

The quantity that would be band-independent is the bolometric magnitude, counting all the light at every wavelength. No detector measures it. It is obtained by applying a bolometric correction, $BC = M_{\text{bol}} - M_V$, which is computed from a model atmosphere: −0.19 for the Sun, about −4.1 for an O star whose output is mostly ultraviolet, about −2.5 for a cool M star radiating mostly in the infrared.

So the chain from a photon count to a luminosity runs: counts → magnitude in a band → absolute magnitude in that band, given a distance → bolometric magnitude, given a model atmosphere → luminosity, given a zero point. Two of those four steps are inferences from theory rather than measurements. This is why the near-infrared has taken over the upper rungs of the ladder. Extinction at 1.6 μm is a sixth of what it is in $V$, and bolometric corrections there vary far less between stellar types, so both of the inferences above become smaller. The cost is that infrared detectors are more expensive and the sky is brighter — a practical price paid for a systematic gain.

What was actually measured #

The Hyades cluster is the cleanest place to watch the whole apparatus operate, because its distance is known two ways and one of them uses no candle at all.

The moving-cluster method exploits the fact that a cluster’s stars share a velocity, so their proper motions converge on a point on the celestial sphere. Measuring that convergent point gives the angle between the cluster’s motion and the line of sight; measuring a radial velocity gives one component of the space velocity; and the two together with the proper motion give the distance geometrically. Applied to the Hyades in the twentieth century this gave 46 to 48 parsecs.

Hipparcos parallaxes in 1997 gave $46.34 \pm 0.27$ pc, and Gaia now gives $47.03 \pm 0.20$ pc from 250 members. The 1.5% disagreement between Hipparcos and Gaia was traced to a correlated systematic in the Hipparcos astrometry over small angular scales — a real error in the earlier data, found because two independent measurements existed.

With the distance fixed, the Hyades main sequence becomes a template of absolute magnitudes against colour, and that is what main-sequence fitting slides against. The chain is: parallax gives the Hyades distance; the Hyades distance gives absolute magnitudes; those absolute magnitudes let another cluster’s distance be read off its apparent magnitudes.

The precision of the last step is limited by something the relation does not contain. Two clusters of the same age and different metallicity have main sequences offset from each other by up to 0.5 magnitudes, because metals change a star’s opacity and therefore its radius and colour. So a distance from main-sequence fitting is only as good as the metallicity match, and this is the single largest systematic in the method — it is the same 0.5 magnitudes that the scatter figure above converts into 26%.

The generalisation: the standard-anything method #

The structure — measure something that falls with distance, know what it was at the source, divide — is not confined to brightness, and the variants are worth naming because each fixes a different weakness.

A standard ruler uses an angular size instead of a flux. The baryon acoustic oscillation scale in the galaxy distribution is a known physical length of about 150 megaparsecs, and its observed angular size gives a distance without any reference to luminosity, so extinction does not enter at all.

A standard siren uses a gravitational waveform. The amplitude of the wave from a merging binary depends on the masses and the distance, and the masses are readable from the frequency evolution of the same waveform — a period doing the work of a ruler — so the distance comes out with no calibration against anything. GW170817 gave $40^{+8}_{-14}$ Mpc this way.

A standard clock uses a timescale. Type II supernovae expand at a rate measurable from their spectra, giving a physical size to compare with an angular one.

Each is an attempt to remove the one thing the distance modulus cannot supply. And each has its own version of the same problem — the “known” quantity is known from theory or from a lower rung — which is why none of them has replaced the ladder and why the ones that come closest, sirens and rulers, are valued far beyond their current precision.

The bias that survives averaging #

There is a statistical trap in this relation that no amount of data removes, and it is worth a section because it is the reason a naive average of standard-candle distances is wrong.

Any survey is limited by how faint an object it can detect. Among a population of candles with intrinsic scatter, the ones detected at a given distance are preferentially the brighter members of the class — the faint ones fell below the limit. So the mean absolute magnitude of the detected sample is brighter than the mean absolute magnitude of the population, and using the population mean makes every distance too large.

That is Malmquist bias, and its size is $1.382\sigma^2$ magnitudes for a Gaussian scatter of $\sigma$ in a uniformly filled space. For $\sigma = 0.5$ that is 0.35 magnitudes, which the arithmetic above converts into a 17% distance error — systematic, one-signed, and larger for the more distant objects, so it also biases anything fitted against distance.

The bias grows with the intrinsic scatter squared, which is a second and independent reason to prefer a tight candle over a bright one. Halving the scatter divides the Malmquist bias by four.

It cannot be averaged away because it is not noise. It is corrected by modelling the selection — knowing the survey’s detection limit, the luminosity function of the class, and the spatial distribution — and every such correction is a place where an assumption enters a measurement that started as a photon count.

And a bias in the calibration itself #

There is a companion bias one rung lower, acting on the calibration rather than on the sample, and it has the same shape aimed at a different target.

Absolute magnitudes are calibrated on objects with measured parallaxes, and a distance is obtained by inverting the parallax. Inversion is not a linear operation, and the parallax carries a symmetric error while space does not: among the stars that could have produced a given measured parallax, more lie beyond the corresponding distance than within it, because the volume of a shell grows as the square of its radius. So the true distance of a star with a given measured parallax is on average larger than the inverted value, the star is intrinsically brighter than the calibration credits it with, and every absolute magnitude derived that way comes out too faint.

That is the Lutz–Kelker bias, and its size depends only on the fractional parallax error: negligible below about five per cent, a tenth of a magnitude at fifteen, and formally divergent as the error approaches the parallax itself. Which is also why a catalogue’s negative parallaxes cannot simply be thrown away — a negative measured parallax is a legitimate outcome for a distant star, and discarding those measurements selects the sample in exactly the direction the bias runs.

The modern treatment does not correct the inversion. It declines to perform it: the fit is done in parallax space with a prior on the spatial distribution, so the symmetric error stays attached to the quantity that actually has one, and a distance is a derived posterior rather than a reciprocal. That change of variable is the whole of the repair.

Where the model stops #

No cosmology. Beyond about 100 Mpc the relation needs replacing, because “distance” splits into several inequivalent quantities. The luminosity distance that appears in the flux relation and the angular diameter distance that appears in the size relation differ by a factor of $(1+z)^2$, and the modulus above is a small-redshift limit of the first.

No K-correction. A redshifted object is observed through a filter that samples a different part of its rest-frame spectrum, so its magnitude has to be corrected using an assumed spectrum before the relation applies.

Extinction is assumed grey within a band. The $A_V = 3.1E(B-V)$ ratio varies between about 2.5 and 5.5 along different sight lines, and using the average introduces an error that is systematic for any one object.

The figures draw a relation, and the measurement is a subtraction of two things one of which is not on the page. Every plot here shows $m - M$ against $d$ as if the modulus were a measurable quantity. It is not: it is the difference between a measurement and an inference, and no figure can show the inference, because it comes from a different object entirely.

The ladder from here #

Later rungs on this anchor: the magnitude system’s zero points, and the difference between Vega and AB magnitudes. Bolometric corrections, and the fact that no detector sees all the light. K-corrections. Luminosity distance against angular-diameter distance, and the four distances of cosmology. Malmquist bias, and why a flux-limited sample of standard candles is systematically too bright. Eddington bias. The photometric parallax method for individual stars.

There is a caution the whole essay comes down to, and it is best stated as a question to ask of any quoted distance: what was assumed about the object? For a parallax the answer is “nothing”. For everything else the answer is a class membership, a calibration, and a chain, and the number is only as good as the weakest of them.