The same star through two telescopes

The first rung of this ladder established that a magnitude has to say which light: a number without a band attached is meaningless, because the same star is brighter than another in $B$ and fainter in $V$.

The second thing a magnitude has to say is whose band, and that is a harder statement to make good on. A band is not defined by a name or by a central wavelength; it is defined by a response curve — the product of the filter’s transmission, the telescope’s mirror reflectivity, the detector’s quantum efficiency and the atmosphere above the site. No two observatories have the same product, and it follows that no two observatories measure the same magnitude for the same star.

Why the response is not the filter #

It is tempting to think of the filter as the band, with everything else a small correction. The arithmetic says otherwise.

A Johnson $V$ filter passes light from about 480 to 640 nanometres. Across that range a bare aluminium mirror varies in reflectivity by a few per cent — negligible, until it is squared for two mirrors and cubed for three. A thinned back-illuminated CCD’s quantum efficiency, however, runs from 60 per cent at 480 nanometres to 90 per cent at 600 and back to 70 at 700, and interference in the thin silicon layer superimposes a fringe pattern on top of that. The detector shapes the band as strongly as the filter does.

Above both sits the atmosphere. Rayleigh scattering removes blue light preferentially — a factor of four more at 400 nanometres than at 600 — and ozone absorbs across the middle of the visible band. So the effective response of a system changes with airmass, which means it changes over the course of a night, which means the band is a function of where the telescope is pointing.

The natural system #

Given all that, the honest position is that each observatory has its own natural system: the magnitudes its own equipment produces, on its own night, with no transformation applied. Those numbers are internally consistent and externally meaningless.

The bridge to the outside is a network of standard stars whose magnitudes on an agreed system have been determined by long campaigns. Johnson and Morgan established the original $UBV$ system in 1953 on ten stars; Landolt’s equatorial standards, published between 1973 and 2009, are the network most optical photometry is still tied to. An observer measures the standards in their own natural system, fits the transformation, and applies it.

The transformation is written as

$$V = v + \zeta_V + \varepsilon_V\,(b-v) + k_V X,$$

with $v$ and $b-v$ the instrumental values, $\zeta$ a zero point, $\varepsilon$ the colour term, $k$ the extinction coefficient and $X$ the airmass. The colour term is the whole subject of this essay: it is the slope of the line in the figure above, it is a property of the equipment rather than of the sky, and it is typically a few hundredths to a tenth of a magnitude per magnitude.

Why a linear term is enough, and when it is not #

The reason a one-parameter correction succeeds as well as it does is worth stating precisely, because it also predicts its failures.

To within a few per cent, a normal star’s optical spectrum is a blackbody modified by a smooth opacity — and the family of such spectra is one-dimensional. A single colour index locates the star in that family, and once located, the difference between two bands’ responses is determined. There is nothing left to know.

Fitting a straight line to the blackbody sequence in the figure above leaves 1.5 millimagnitudes of scatter across a factor of fourteen in temperature. That is far below the systematic floor of any real photometry, so linear is not merely adequate but generous; the second-order term is fitted only for the widest bands and the largest colour ranges.

The failure is a star whose spectrum is not in the family.

The residuals are not small. For M dwarfs the transformation between two $R$-band systems can be wrong by several hundredths of a magnitude, which is ten to thirty times the internal precision. For carbon stars, worse. For emission-line objects — cataclysmic variables, young stellar objects, active galactic nuclei — the concept of a transformation barely applies, because their spectra are not a family at all.

Metallicity supplies a subtler version of the same problem. A metal-poor star has weaker line blanketing in the ultraviolet, so it is brighter in $U$ than a solar-abundance star of the same temperature — the ultraviolet excess that has been used to estimate metallicities since the 1950s. That means the family is two-dimensional, and a one-colour transformation absorbs the second parameter into its residuals.

What this costs, and where it does not #

The practical consequence is a hard limit on all-sky photometry that has stood for decades.

Transforming to a standard system and comparing against measurements made elsewhere is reliable to about one per cent for ordinary stars, and one per cent is not a statistical limit. It is the accumulated cost of the transformation residuals, the standard network’s own internal errors, and the extinction correction. Doubling the exposure time does not improve it.

Differential photometry has no such limit, and the difference is instructive. Measuring a target against comparison stars in the same field, on the same image, through the same air means the response, the extinction and the zero point are common to both and cancel in the ratio. Nothing needs to be transformed, because nothing is being compared with anything elsewhere. The precision floor drops from one per cent to a millimagnitude or better.

The remaining residual, even differentially, is a colour term again: if the target and the comparison have different colours, then a change in airmass or transparency across the night affects them differently. That is why a transit observation chooses comparison stars matched in colour where it can, and why the residual second-order extinction is the largest systematic in ground-based transit photometry.

Two ways of being wrong, and only one of them shows #

There is a distinction worth drawing between the two errors a transformation can leave, because they behave completely differently in a survey.

A zero-point error shifts every star by the same amount. It is large, it is obvious, and it is removed by any comparison against a catalogue — and where it is not removed it produces an offset in a derived distance that is the same for every object, which is a systematic in the worst place but at least a legible one.

A colour-term error shifts stars by an amount proportional to their colour. It cannot be found by comparing mean magnitudes, because the mean is unaffected if the colour distribution is symmetric. It shows up only as a slope: fit the difference between two catalogues against colour, and the residual colour term is the slope of the fit. A survey that reports agreement in the mean has not tested its colour term at all, and the checks that do test it are the ones that involve comparing populations of different colours in the same field.

The system that removed the problem by redefining it #

The AB system, introduced by Oke and Gunn in 1983, takes a different route: it defines the zero point by a flat spectrum in $f_\nu$ rather than by a star.

$$m_{\rm AB} = -2.5\log_{10} f_\nu - 48.60,$$

with $f_\nu$ in erg s⁻¹ cm⁻² Hz⁻¹. An AB magnitude of 0 in any band means the same physical flux density, so an AB colour is directly a ratio of flux densities and needs no reference star at all.

That removes the zero-point problem entirely and leaves the response problem untouched, which is the distinction worth carrying. Two AB systems with different response curves still disagree about a star, and still need a colour term to reconcile. What AB removes is the historical accident by which $U-B = B-V = 0$ for Vega, and with it the awkwardness that Vega is a variable star with a debris disc that is not, in fact, a good absolute standard.

Synthetic photometry, and what it settles #

Everything above can be computed rather than fitted, provided two things are known: the system’s response curve as a function of wavelength, and the star’s spectrum.

$$f_{\rm band} = \frac{\int f_\lambda S(\lambda)\,\lambda\,\mathrm d\lambda}{\int S(\lambda)\,\lambda\,\mathrm d\lambda}$$

That integral, evaluated over a library of stellar spectra for two response curves, gives the transformation between them exactly — including its non-linearity and including the outliers. It is how the figure at the top of this essay was made, and it is how modern surveys derive their transformations rather than by fitting standard stars.

The catch is the response curve. Measuring $S(\lambda)$ for a real instrument to the accuracy required means measuring the filter, the mirrors, the detector and the atmosphere as installed, and the usual practice of multiplying together the manufacturers’ curves is good to a few per cent, which is not good enough. The systems whose responses are known best are the ones flown in space, where there is no atmosphere and the instrument has been characterised on the ground and does not change.

The part of the response that is weather #

For a ground-based system the response curve includes the atmosphere, and the atmosphere is not a fixed component.

Light reaching a detector has passed through a column of air whose length depends on the zenith angle. The standard bookkeeping is the airmass, $X \approx \sec z$, and the measured magnitude is corrected as

$$m_0 = m - k X,$$

with $k$ the extinction coefficient in that band. At a good site $k$ is about 0.15 magnitudes per airmass in $V$ and 0.55 in $U$, so a star observed at 60° from the zenith is dimmed by nearly a magnitude in the ultraviolet.

Two features of that correction make it a source of exactly the errors this essay is about.

The coefficient is not a constant. It depends on the aerosol content, the water vapour and the ozone above the site, all of which change from night to night and some of which change within a night. Photometry to a per cent requires measuring $k$ on the same night as the science, by observing standards over a range of airmass — which is why a classical photometric night’s observing plan is half calibration.

And it is itself colour-dependent. A broad band is not monochromatic, so extinction removes more of the blue end of the band than the red, which shifts the effective wavelength of the band as the airmass grows. A red star and a blue star observed through the same air are therefore extinguished by different amounts, and correcting for that requires a second-order term $k'(B-V)X$.

So the atmosphere changes the response curve as a function of both time and the star’s own colour, which is the same structure as the transformation problem itself, arriving from outside the instrument. That is the fundamental reason a space-based system can be characterised once and a ground-based one cannot.

Calibrating without standards #

The classical answer was a network of stars whose magnitudes had been measured carefully and published, against which everything else was compared. That network’s descendants are still in use, and the modern practice has largely replaced it with something structurally different.

The difficulty with a standard-star network is that it propagates its own errors outward. Every magnitude in a survey is tied, through a chain of comparisons, to a few hundred stars measured on a few hundred nights with one instrument, and any error in those propagates to everything. The network’s internal consistency is around one per cent, which for a survey aiming at better than that is a floor rather than a calibration.

Übercalibration removes the network. A wide-field survey observes overlapping regions repeatedly, and a star in an overlap is measured twice under different conditions. Requiring every such repeated measurement to agree gives an enormously overdetermined system of equations for the per-night zero points, the flat field and the extinction, which is solved globally over the whole survey at once.

The result is a photometric system that is internally consistent to a few thousandths of a magnitude across the sky, tied to nothing external. It is a relative calibration, so an absolute zero point still has to come from somewhere — usually from a handful of white dwarfs whose spectra are calculable from atmosphere models — but the relative part, which is what colours and distances depend on, no longer inherits anybody’s standard-star errors.

That is the shape of the modern answer to this essay’s problem: rather than transforming one system into another, measure everything with one instrument and solve for the system self-consistently.

What has not changed is the problem this essay began with. A self-consistent survey still has its own response curve, and comparing it with anybody else’s still needs a colour term — the calibration has moved from being a chain of stars to being a property of one instrument, and the instrument is still not the same as the next one.

The white dwarfs deserve a word, because they are an unusual kind of standard. A hot hydrogen-atmosphere white dwarf has a spectrum that can be computed rather than measured: its atmosphere is pure hydrogen, in local thermodynamic equilibrium, with a surface gravity high enough that the physics is simple and a temperature high enough that the opacity is dominated by processes that are well understood. So the shape of its spectral energy distribution follows from a model with two parameters, both of which are read off the same spectrum. The absolute calibration of modern photometry therefore rests on a handful of dead stars whose light is predicted rather than compared, which is a considerably more satisfactory foundation than a bright variable star with a disc around it.

The remaining gap is that a model atmosphere is still a model: it assumes a plane-parallel layer in equilibrium with a particular treatment of the line opacity, and the residual disagreements between independent calculations of the same white dwarf are at the per-cent level in flux. That is the current floor on absolute photometry, and it is a floor made of theory rather than of instrumentation — an unusual place for an observational limit to sit.

Where the model stops #

Three limits.

The response curve changes. Mirrors oxidise, filters age and shift, detectors are replaced, and the atmosphere above a site varies with the season. A colour term determined in 2010 is not the colour term in 2025, and a survey that spans a decade has to monitor it.

The transformation assumes the star is a star. Extended sources bring an additional problem the equations above do not contain: the response varies across the field, and a galaxy’s colour gradient interacts with that variation.

And a transformation determined from standards is only valid over the range the standards span. Landolt’s network is dominated by stars between $B-V = -0.3$ and $+1.5$; applying a linear term fitted there to an M dwarf at $B-V = 1.9$ is an extrapolation, and extrapolating a fit whose residuals are already structured is how a systematic becomes a discovery that is not one. The transformation’s whole difficulty is that a colour index is a proxy for a spectrum, and both halves of that are worth drawing at a second setting.

Where this ladder goes next #

This rung establishes that a photometric system is a response curve, that the difference between two of them is a function of colour, and that the function is one-dimensional only because stellar spectra nearly are.

Above it lies calibration proper: how a response curve is measured absolutely, using laboratory sources flown to the top of the atmosphere or white dwarfs whose spectra are calculable from first principles, and what the current one-per-cent floor on absolute stellar fluxes actually consists of.

Beside it lies the design question. A photometric system can be chosen to measure something — the Strömgren system’s narrow bands were placed to isolate the Balmer discontinuity and the metal lines, so its indices give surface gravity and metallicity rather than merely temperature. A band is an instrument for asking a question, and choosing where to put it is the same problem as choosing what to ask.