A magnitude has to say which light

The magnitude scale is a logarithm of a received flux. The figure below draws the most familiar quantity built out of it — $B-V$ against temperature — twice, and only one of the two curves is a star.

The solid curve is arithmetic: Planck’s law integrated against a $B$ response and against a $V$ response, and the two results subtracted. The dots are the main sequence as photometry actually finds it, and they do not lie on the curve. At the Sun’s temperature the blackbody gives 0.446 where the sky gives 0.653, and at M0V it gives 0.995 against a measured 1.40.

The disagreement is the essay. It is not a defect in the quadrature and it is not small — a fifth of a magnitude at the Sun, against a photometric precision of a millimagnitude. The two lines touch at one temperature only, and they touch there because somebody decided they should.

A magnitude is not a property of a star #

The claim, stated so it can be argued with: a magnitude without a named band is not a number, and no care in the photometry repairs that.

Two observatories pointing at the same star through different glass record different magnitudes, and neither is wrong. The star has no single brightness for them to disagree about. Each records the light that got through its own filter, and a filter is a function of wavelength — so the datum is an integral, and two different integrals of one spectrum have no reason to be equal. That the scale also runs backwards and multiplies is an inconvenience of arithmetic, settled once by Pogson’s definition; that a magnitude belongs to a passband rather than to a star is not an inconvenience but what the quantity is.

The integral, written out #

A detector behind a filter of transmission $S(\lambda)$, looking at a source whose flux at Earth is $f_\lambda$, records

$$m_X = -2.5\log_{10}\!\int f_\lambda\, S_X(\lambda)\, \mathrm{d}\lambda + C_X,$$

where $C_X$ is a constant belonging to the band and to nothing else. A colour index is the difference of two such expressions: a ratio of two integrals with two constants left over, which is why an index carries a convention however carefully the star is measured. The response functions are worth looking at, because their widths decide how much of a spectrum any one number sees. A $V$ magnitude of the Sun therefore accounts for an eighth of the Sun’s output, with seven eighths of the light outside the measurement — and because the humps overlap, the same photons are counted in $B$ and again in $V$.

Zero is where somebody chose to put it #

The constants $C_X$ have to come from somewhere, and where they came from is a decision rather than a discovery. Johnson and Morgan fixed the $UBV$ system in 1953 by requiring the mean of six A0V stars to have $B-V = U-B = 0.00$. That single stipulation sets both constants and propagates into every colour index quoted since: an A0V star is defined to be colourless in every pair of bands, so all the numbers in the subject are measured from a star at about 9,600 K. Vega is the shorthand for that anchor, and it is a poor one — variable, dusty, seen nearly pole-on — but the convention is older than the objections.

Two objects radiating identical total power are ranked differently by different glass: the Sun delivers about half of Vega’s flux in $U$ and twice it in $I$, a factor of four in relative brightness bought by nothing but the filter. The differences between the five numbers are the only reason a colour index exists; the numbers themselves say almost nothing. The modern repair abandons the star: the AB system sets its zero point at a flat 3,631 janskys, so that a magnitude converts to a flux with no reference object at all. It has not displaced the older convention, because a century of catalogues is written in that one.

Change one filter and every number changes #

If a colour index were a property of a star, its value in one pair of bands would fix its value in another. It does not, and the machinery that drew the hero figure shows as much with one option changed.

The two curves differ in the more useful respect as well, which is slope. $B-V$ flattens above about 10,000 K, where both filters sit far out on the Rayleigh–Jeans tail and the ratio between them stops changing, while a redder pair keeps its slope at the cool end. Choosing a filter pair is a statement about which part of the Planck curve still has information in it, and a survey quoting one index has chosen a temperature range to be good at.

Which curve is the model, and which is the sky #

The gap in the hero figure runs one way almost everywhere: the blackbody is too blue, and cool stars are redder than any Planck curve by far more than the measurement error.

The reason is that light is not removed from a stellar spectrum evenly. A stellar spectrum’s dark lines crowd towards the blue, and in a cool star the crowding becomes a continuum of its own: titanium oxide and other molecules survive below about 4,000 K and take out broad swathes of the blue and green. The $B$ band is where that damage is worst, so a real M dwarf’s $B$ magnitude is fainter than a blackbody’s and its $B-V$ larger. At the hot end a smaller version of the same thing happens for a different reason — the Balmer jump at 365 nm sits inside the $U$ band and nowhere else. The two curves are therefore not two attempts at one quantity. One is what a body in thermal equilibrium would do; the other is what stars do, and the difference is an inventory of everything a photosphere is besides a temperature.

That also leaves the zero point looking less arbitrary than it was called above. With the constant fixed at 9,600 K the model and the sky agree exactly there and disagree on both sides — an interior best point, and the one feature of the arrangement that is not a matter of taste. The convention sits where the model it is not is least wrong, near A0 being where the molecules have gone and the Balmer jump has not yet arrived.

From one band to all of the light #

The quantity physics wants is a luminosity, and getting there means recovering the whole spectrum from one slice of it. The step is the bolometric correction, $\mathrm{BC} = M_{\rm bol} - M_V$, and it is a model wearing the clothes of a measurement.

The asymmetry is the practical content. Between the Sun and a B0V star the correction changes by 2.57 magnitudes, a factor of 10.6: treating a $V$ magnitude as a proxy for luminosity misstates a hot star’s output by an order of magnitude relative to the Sun’s, in the same direction every time — and those are the stars that set a young cluster’s mass and a galaxy’s ultraviolet output.

The interior maximum has a closed form, and it arrives from an unexpected direction. For a narrow band at $\lambda$ the admitted flux goes as $B_\lambda(T)$ while the total goes as $T^4$, so the ratio is stationary in temperature where

$$\frac{x e^{x}}{e^{x}-1} = 4, \qquad x = \frac{hc}{\lambda k T}.$$

That is Wien’s displacement condition with a 4 where Wien has a 5, and the 4 is the Stefan–Boltzmann exponent: the law of total power has walked into a question about one filter. Its root is $x^\ast = 3.92069$, giving $\lambda T = 3.670\times10^{6}$ nm K, or 6,660 K at the $V$ band’s 551 nm. The figure’s own maximum, computed with the real 88 nm bandwidth, is 6,724 K: the two agree to one per cent, and the difference is the width of a filter the closed form does not know exists.

What is actually measured, and through what glass #

The raw datum is a count of electrons behind one filter, and the reduction from counts to a calibrated magnitude is a chain of its own. What belongs here is the part the band is responsible for, which is the part no exposure time improves.

Johnson and Morgan built the system on an RCA 1P21 photomultiplier at McDonald Observatory, and the tube rather than any design decision is why the bands sit where they do: its sensitivity ran out towards the red, which is why $V$ is at 551 nm and not in the infrared, and the $U$ band’s blue edge is the atmosphere rather than the filter. The most-used photometric system in astronomy is a fossil of one 1950s vacuum tube and the air above west Texas — so a $U$ magnitude taken at sea level is not quite the quantity a mountain measures.

Everything downstream is a transformation. An observer’s filters are never quite the system’s, so instrumental magnitudes are regressed onto catalogue values for a field of standard stars, and the fit carries a colour term — a slope against the star’s own index, because a mismatched filter’s error depends on the spectrum it looks at. It works for stars resembling the standards and fails for anything else, leaving a systematic no repetition removes.

A colour index the dust has moved #

One quantity is built from nothing but the difference between two bands, because dust does not remove light evenly. The colour excess $E(B-V)$ is the observed index minus the intrinsic one — a difference of two differences, and so a number that survives when neither magnitude is trustworthy alone — and the ratio $R_V = A_V/E(B-V)$ converts it into a dimming. The law behind that vector is a separate argument; the reason it can be applied at all is this essay’s, because extinction is a different number in every band and ratios of indices are therefore fixed even when the amount of dust is not. Johnson’s reddening-free parameter $Q = (U-B) - 0.72(B-V)$ is that turned into an instrument: three magnitudes combined so that the reddening cancels and a property of the star is left.

The same question asked of a planet and of a galaxy #

A transiting planet’s radius is measured from the fraction of light it removes, and that fraction depends on wavelength, because a planetary atmosphere is opaque at some wavelengths and clear at others. So a planet has a different radius in every band, and the spectrum of radii is the atmosphere’s composition — what makes a magnitude band-dependent is, there, the whole measurement.

Galaxies are stranger. Their colours integrate whole stellar populations rather than one photosphere, and the distribution of galaxy colours is bimodal: two clumps with very little between them. That structure lives in a colour index and is invisible in any single magnitude — the same asymmetry the equal-brightness figure draws, with the numbers carrying little and the differences carrying the physics.

The escape, where available, is to measure the light and the angle instead: a bolometric flux with an angular diameter measured by interferometry gives a temperature with no filter system in the chain, for the few hundred stars near enough to resolve.

The band that is nearly all of them #

The systems described so far are narrow enough that a magnitude in one of them samples a defined slice of a spectrum. The most-used photometry in astronomy today is not like that, and the difference is worth setting out because it inverts several of the arguments above.

A satellite surveying the whole sky repeatedly has to record as many photons as possible from every star, so its main photometric band is made as wide as its optics allow — spanning from the near ultraviolet to the near infrared in one measurement, several times the width of a Johnson band. Two further bands split roughly the blue half and the red half of the same range, and the difference between them serves as a colour index.

Everything this essay has said about band-dependence becomes more severe rather than less. A very wide band integrates across the whole of a cool star’s molecular absorption and across the Balmer jump of a hot one, so the relation between such a magnitude and any narrower system’s depends strongly on the star’s spectrum — the colour term is not a small correction but a large and nonlinear one, and the transformation between the wide system and $V$ has residuals of tenths of a magnitude for red stars however many terms are fitted.

The repair is not a better transformation. It is to abandon the idea of transforming at all: the same satellite records low-resolution spectra for the same stars, so a magnitude in any system can be computed by integrating each star’s own measured spectrum against that system’s published response. That is synthetic photometry, and it does exactly what the definitional section of this essay says a magnitude is — it performs the integral rather than approximating it.

The consequence is a change in what a photometric system is for. Historically a system was a physical thing: a filter, a detector, a mountain, and a set of standard stars tying everything to them. Synthetic photometry makes a system a specification — a response curve published as a table — that anyone with a spectrum can evaluate, and the standard stars become a check rather than the definition.

The band-dependence has not gone away; what has gone is the need to have observed through the band in question. A magnitude still says which light, and the spectrum now says how much of it any named band would have collected.

What the picture cannot show #

The filters are Gaussians. A real $UBVRI$ set is not symmetric: $B$ and $V$ have steep blue edges and red tails set by the detector, Cousins’ $R$ and $I$ are broad and flat-topped, and $U$ has no blue edge of its own. The widths drawn are right and the shapes are not.

Both stars in the equal-brightness figure are blackbodies. The curve labelled as the Sun is a Planck curve at 5,772 K, so its $B-V$ is the model’s 0.446 rather than the measured 0.653. That 1.48-magnitude spread is a spread of two Planck curves; two real spectra would give a wider one, so the figure understates its own case.

The bolometric correction is not a measurement. No detector covers all wavelengths, so the total flux in that ratio is always computed — here from a Planck spectrum the hero figure has already shown to be wrong by tenths of a magnitude in the optical. Published corrections come from synthetic spectra, and their mutual disagreements at the cool end are the size of the effect.

One star, unreddened, unresolved and unmoving. An unresolved binary contributes two spectra and sits off any sequence; a fast rotator’s temperature varies across its own surface.

A colour is a temperature only where the curve is steep #

One practical consequence of the shape of the hero figure deserves separating out, because it decides which index a survey should use and is usually stated as folklore.

An index is useful as a thermometer in proportion to how fast it changes with temperature. Where the curve is steep, a small error in the measured colour is a small error in the inferred temperature; where it is flat, the same colour error is enormous.

$B-V$ is steep between about 4,000 and 8,000 K and flattens badly above 10,000, for the reason given earlier: both filters are then on the Rayleigh–Jeans side of the peak, where the spectrum’s shape barely changes with temperature. So $B-V$ is a fine thermometer for solar-type and cooler stars and a poor one for hot ones, and the standard replacement for the hot end is an index built from the ultraviolet, where the peak still is.

At the cool end the flattening happens for a different reason and to a different index. Below about 3,500 K the blue bands are collecting so little light, and so much of what they collect is chewed by molecular absorption, that the measurement noise dominates — so $B-V$ becomes useless not because the relation is flat but because the observable is unmeasurable. A red index takes over there, and the near-infrared ones keep their sensitivity to the bottom of the main sequence.

The result is that no single index is a thermometer across the range, and a survey quoting temperatures from photometry has chosen a temperature interval to be trusted in. The zero point is where the convention put it, and the useful range is where the arithmetic put it, and neither is a property of any star.

The bolometric correction depends on which band it corrects from, and the dependence is the clearest statement of what a passband costs.

That is why a survey of cool stars is done in the infrared and not because infrared detectors became cheap. The bolometric correction is the part of a luminosity that comes from a model rather than from a measurement, and the band that minimises it is the band that minimises the model’s contribution. A luminosity computed from V for an M dwarf is mostly extrapolation; one computed from K is mostly measurement.

The same logic runs the other way for hot stars, whose output peaks in the ultraviolet and is unobservable from the ground at all. There the correction from V is large and negative and depends sensitively on the temperature, so an O star’s luminosity carries an uncertainty that no amount of photometric precision reduces — the light being corrected for was never seen.

Between those two regimes sits a narrow range of temperatures, near the Sun’s, where V happens to catch most of the light and the correction is both small and insensitive. That the standard visual band works well for solar-type stars is not a coincidence and not a design achievement either; the band was defined by what the human eye and early photographic plates could do, and the human eye evolved under a G2 star.

Where the ladder goes next #

This rung establishes only that a magnitude belongs to a band. Above it sit the systems built to exploit that — the Strömgren narrow bands, where one index measures gravity and another metallicity — and then synthetic photometry, which computes a survey’s magnitudes by integrating a model spectrum against a published response.

The harder rung is the one after: turning a magnitude into a distance needs an absolute magnitude in the same band as the apparent one, and the systematic that has cost the distance scale most is not a mismeasured brightness but two bands assumed to be the same.