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Three shifts larger than the error bar, and two that cancel

An unresolved companion, a reddening and a metallicity each move a colour index by more than any modern photometer's precision. Two of them move it in opposite directions, so the three together can return the right temperature by cancellation rather than by accuracy.

Assumes Stellar colour and Extinction.

16 min read 6 figures Light is the only messengerSmall perturbations, large consequences

Colour photometry is the cheapest useful measurement in astronomy and it has a systematic floor that no exposure time reduces. Three effects move a colour index by more than a modern photometer’s precision, all three are properties of the star or of the path rather than of the instrument, and none is detectable in two magnitudes.

Two of them move it in opposite directions.

Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the V light reddens the index by 0.093 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5331 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.005 magnitudes against 0.405 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5744 K, which is 28 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.
Fig. 1 The blackbody B − V index against temperature with three systematic shifts marked at 5,772 K, all computed from the same Planck integrals the relation itself is. A 3,800 K companion contributing a quarter of the V light reddens the index by 0.093 magnitudes; 0.35 magnitudes of extinction adds 0.113; a metallicity of −1 dex subtracts 0.200. Read as temperatures the same star comes back at 5,331 K, 5,243 K and 7,021 K. All three together shift it by 0.005 magnitudes and return 5,744 K — within 28 K of the truth, by cancellation and not by accuracy.

The companion nobody resolved

A third of solar-type stars have a companion, and at a typical distance almost none of those companions is resolved. What a photometer measures is the sum of two spectra.

The composite colour is not the mean of the two colours. It is the colour of the summed flux, weighted by each star’s brightness in each band separately — and because the companion is cooler, it contributes a larger share of the V light than of the B light. The composite is therefore redder than the primary alone.

The size depends on the flux ratio and on the temperature difference, and it is largest for an intermediate case. An equal twin adds nothing, since two identical spectra sum to the same colour. A companion a hundred times fainter adds nothing measurable either. The maximum effect is for a companion contributing a tenth to half the light and several thousand kelvin cooler, which is a common arrangement.

A quarter of the V light from a 3,800 K star reddens B − V by 0.093 magnitudes, which reads as a temperature 441 K too low.

Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 4,800 K companion contributing 50 per cent of the V light reddens the index by 0.070 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5432 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by -0.017 magnitudes against 0.383 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5860 K, which is 88 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.
Fig. 2 A brighter and less dissimilar companion: half the V light from a 4,800 K star. The reddening is 0.070 magnitudes — smaller than the fainter and cooler companion of the first figure produced — because the effect depends on the difference between the two spectra’s shapes as well as on the flux ratio, and a companion only a thousand kelvin cooler has nearly the same colour as the primary. The shift is not monotone in either variable, which is why a survey cannot correct for binarity by assuming a typical companion.
Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,000 K companion contributing 10 per cent of the V light reddens the index by 0.061 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5475 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by -0.026 magnitudes against 0.374 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5910 K, which is 138 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.
Fig. 3 A tenth of the V light from a 3,000 K companion — a red dwarf beside a solar-type star, which is by far the commonest arrangement in the Galaxy. The shift is 0.061 magnitudes and reads as a temperature of 5,475 K against a true 5,772, which is 297 K too low. That is the case a survey actually faces: not a dramatic pair but a faint red secondary contributing a few per cent, present in a third of the sample, undetectable in imaging, and shifting every affected star’s temperature in the same direction.

The systematic nature of that shift is what makes binarity worse than it looks. An effect that scattered temperatures randomly would broaden a distribution and leave its mean alone; this one moves a third of the sample in one direction, so it shifts a cluster’s fitted isochrone, its inferred age and its inferred distance together.

The dust on the way

Interstellar dust removes blue light preferentially, so a star behind it is reddened. The amount is quantified as a colour excess E(B−V)=AV/RVE(B-V) = A_V/R_V, with RV≈3.1R_V \approx 3.1 for diffuse Galactic dust, and the definition makes the relation between extinction and reddening exact by construction.

Reddening is the only one of the three that is not a property of the star, and it is also the only one whose magnitude is unbounded. A star at the Galactic centre carries thirty magnitudes of visual extinction and is invisible in the optical entirely; a star a hundred parsecs away at high Galactic latitude carries a hundredth of a magnitude.

So the correction is large where it is hardest to estimate and negligible where it does not matter. Estimating it requires a three-dimensional dust map, or a spectrum, or a comparison against stars of known intrinsic colour — and the last of those is circular, since the intrinsic colour is what is being measured.

Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the V light reddens the index by 0.093 magnitudes, because the companion is relatively brighter in the redder band; 1.2 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.387 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -0.4 dex subtracts 0.080, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5331 K, 4278 K and 6214 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.400 magnitudes against 0.560 of combined magnitude — a cancellation of 0.160, which is not enough here to save the measurement — and return 4241 K, which is 1531 K from the truth and the cancellation has not rescued it. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.
Fig. 4 More dust and less metal deficiency: 1.2 magnitudes of extinction and a metallicity of −0.4 dex. The reddening now dominates and the three no longer cancel usefully — the net shift is 0.400 magnitudes against 0.560 of combined magnitude, and the recovered temperature is 4,241 K against a true 5,772, an error of 1,531 K. The cancellation in the first figure was a coincidence of magnitudes, not a mechanism, and nothing arranges for the three effects to be of comparable size.

The metals that eat one band

The third shift is the only one with a sign opposite to the other two, and it is the one least often remembered.

A star’s spectrum is not a blackbody; it is a continuum with thousands of absorption lines cut into it, and the lines are far denser in the blue than in the visual. Iron alone contributes tens of thousands of transitions between 350 and 500 nm, and their collective effect — line blanketing — removes a substantial fraction of the B band’s flux.

A metal-poor star has fewer of those lines, so it loses less of its B flux, so it is bluer than a metal-rich star of the same effective temperature.

A metallicity of −1 dex makes a star about 0.2 magnitudes bluer in B − V, which reads as a temperature of 7,021 K against a true 5,772 — 1,249 K too hot, and by far the largest of the three. That is a large error and it is in the opposite direction to both of the others.

The physical picture also explains why the effect is band-dependent. A colour index built from redder passbands — V − K, say — is far less sensitive to blanketing, because neither band is where the lines are. That is why infrared colours are preferred as temperature indicators for metal-poor stars, and why a temperature scale calibrated in the optical and applied in the infrared disagrees with itself.

Three shifts larger than the error bar, and two that cancel. The blackbody B − R index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the R light reddens the index by 0.135 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.280, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5378 K, 5439 K and 6811 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by -0.032 magnitudes against 0.528 of combined magnitude — a cancellation of 0.497, which is not enough here to save the measurement — and return 5873 K, which is 101 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.
Fig. 5 The same three shifts measured between B and R rather than B and V. Every shift is larger in magnitudes over the wider baseline — 0.135 for the companion, 0.113 for the dust, 0.280 for the metals — and the index’s own temperature sensitivity is larger too, so the three return a temperature 101 K from the truth rather than 28. A wider baseline is a better thermometer and not a cure: it improves the signal and the systematics together, because both are shifts in the same units on a scale that has stretched.

The size of each, against what a photometer can do

It is worth setting the three shifts beside the measurement precision they are competing with, because the comparison is what makes them systematics rather than noise.

Ground-based broad-band photometry reaches about 0.01 magnitudes in a colour index for a bright star, and space-based photometry does better by a factor of a few. Gaia’s colours are good to a few millimagnitudes for the brightest stars in its catalogue.

Against that, the three shifts computed here are 0.093, 0.113 and 0.200 magnitudes — between nine and twenty times the precision, and each of them an order of magnitude larger than anything the instrument contributes.

So a colour index is a quantity measured to one per cent and known to ten, and the gap between those two numbers is the whole subject here. Improving the photometry by another factor of ten changes nothing about it.

That ratio is also what decides how the measurement is reported. A published colour temperature carries a statistical error from the photometry and a systematic from the calibration, and the second is always the larger — usually by the factor computed above, and sometimes by more for a star in a direction where the dust map is poor.

Why the cancellation is the problem

It would be natural to read the near-cancellation in the first figure as good news, and it is the opposite.

If the three effects always cancelled, the measurement would be robust and nobody would need to know about them. If they never cancelled, each could be corrected in turn and the residual would be small.

What actually happens is that they cancel sometimes, depending on three quantities that are not known — and so a colour temperature that happens to be right and one that happens to be wrong are indistinguishable. A measurement whose error is sometimes zero and sometimes a thousand kelvin, with nothing to say which, is worse than one that is always three hundred kelvin out.

The degeneracy is structural rather than accidental. Three unknowns — temperature, reddening, metallicity — and one observable is an underdetermined system, and no amount of precision on the one observable improves it.

A reddened metal-poor star and an unreddened solar-metallicity one of lower temperature produce exactly the same colour index. There is no third number in two magnitudes to separate them.

A fourth shift that is not a shift

There is an effect often listed beside these three that belongs in a different category, and separating it clarifies what the other three are.

A star’s distance affects its apparent magnitude and not its colour, because both bands are dimmed by the same factor and the ratio is unchanged. A brightness is a distance only if something else is known, and a colour is not a distance at all.

That invariance is the reason colour is used as a temperature indicator in the first place, and it is worth noticing what breaks it. Reddening breaks it, because dust is wavelength-selective and its amount grows with path length — so a colour is weakly distance-dependent, through the dust, and only through the dust.

So of the three effects here, one is a property of the star, one of its companion, and one of the space between. The third is the only one that grows with distance, which is why it is the one that dominates for a survey reaching further out, and why deep surveys report colours corrected for reddening while shallow ones often do not bother.

What breaks it

The escapes are all of one kind: more observables.

A third band. Reddening, metallicity and temperature affect three colour indices differently, so with three indices the system is determined. That is the basis of every photometric metallicity estimate, and it works because the reddening law’s wavelength dependence differs from a blackbody’s.

A reddening-free index. A particular combination of three magnitudes, chosen so that the reddening vector’s projection onto it is zero, is unchanged by dust whatever the amount. The Strömgren system was designed around exactly that, and its β\beta index measures the hydrogen line strength — hence the temperature for hot stars — with no dust sensitivity at all.

A spectrum. Line strengths give the metallicity directly, the Balmer line profiles give the temperature and gravity, and the reddening comes out of the continuum shape. A spectrum resolves all three and costs a hundred times the photons — which is the trade every choice between photometry and spectroscopy comes down to, and why colour survives as a technique despite everything in this essay.

Or a different wavelength. In the infrared the extinction is a tenth of the optical value and the blanketing is small, so a J−KJ - K colour is nearly free of two of the three. It is a poorer thermometer intrinsically, for the reasons the classical limit makes clear, and the trade is usually worth it for a reddened star.

Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 3 per cent of the V light reddens the index by 0.013 magnitudes, because the companion is relatively brighter in the redder band; 0.05 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.016 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -0.05 dex subtracts 0.010, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5706 K, 5690 K and 5824 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.019 magnitudes against 0.039 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5675 K, which is 97 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap.
Fig. 6 The three effects reduced to what a nearby, single, solar-metallicity star carries: a hundredth of a magnitude of dust, a twentieth of a dex of metallicity, and a three per cent companion. The shifts are now 0.013, 0.016 and 0.010 magnitudes, all within a few times the photometric error, and the recovered temperature is 5,675 K — 97 K low, because the two reddening terms outweigh the one blueing term and there is no longer enough of anything to cancel. Even a well-chosen star carries a systematic comparable to its statistical error, which is the honest floor of the method.

What the corrections cost in practice

Each of the three has a standard correction and each correction has a standard failure.

Binarity is corrected statistically rather than individually. A survey’s binary fraction and companion mass-ratio distribution are known well enough to model the effect on a population, and not at all for a given star. So an individual temperature carries the full shift as an unmodelled error, and a population’s mean temperature carries a much smaller one.

Reddening is corrected from a dust map, three-dimensional where parallaxes are available. The maps are good to a few hundredths of a magnitude at high Galactic latitude and much worse in the plane, where the dust is patchy on scales smaller than the map’s resolution. The failure mode is a star behind a small dense cloud that the map has smoothed away.

Metallicity is corrected from a spectroscopic measurement where one exists and assumed where it does not — usually by assigning the mean metallicity of the population the star is believed to belong to, which is circular whenever the population membership was decided photometrically.

All three corrections therefore work well on the stars that need them least. A bright, nearby, spectroscopically observed single star has small shifts and good corrections; a faint distant one in the Galactic plane has large shifts and poor corrections, and it is the second kind a survey is mostly made of.

Two shifts that are exact and one that is modelled

Everything drawn is computed from Planck integrals through Gaussian approximations to the Johnson–Cousins passbands, and the three shifts are modelled rather than taken from data.

The companion’s effect is computed exactly, as the colour of the sum of two blackbodies at a stated flux ratio, and that computation involves no assumption beyond the two stars being blackbodies.

The reddening’s effect is exact by definition: E(B−V)=AV/RVE(B-V) = A_V/R_V is how the colour excess is defined, so adding AV/3.1A_V/3.1 to the index is not a model but a restatement.

The blanketing is the modelled one. Taking it as 0.2 magnitudes per dex in B − V is a linearisation of a relation that is not linear, depends on temperature, and differs between individual elements. Real calibrations are polynomial in temperature and metallicity together and disagree with each other at the few-hundredths level, which is the size of the effect being computed.

A fourth shift, a fifth, and a correlation the figures do not carry

The three effects are not independent in a real population. Metal-poor stars are old and belong to the halo, which is where the dust is thin; metal-rich stars are in the disc, where it is not. So a survey’s reddening and metallicity are anticorrelated, and the accidental cancellation drawn here is less accidental than it looks.

Surface gravity is a fourth shift and is not drawn. A giant and a dwarf of the same effective temperature have different colours, because the lower pressure in a giant’s atmosphere changes the continuous opacity and the line strengths. The effect is smaller than the three here for solar-type stars and comparable for cool ones.

And rotation is a fifth. A rapidly rotating star is oblate and its poles are hotter than its equator, so its colour depends on which way it is inclined — an effect with no counterpart in any of the others, since it makes one star have several colours.

Why the metallicity term is the strangest of the three

The blanketing shift deserves a second look, because unlike the other two it is a departure from the blackbody assumption rather than a contamination of it.

A companion adds a second blackbody. Dust multiplies by a transmission. Both leave the star itself a blackbody and alter what arrives.

Blanketing is different: it says the star was never a blackbody, and the amount by which it is not depends on its composition. The B band’s flux deficit relative to a blackbody is not a small perturbation — for a solar-metallicity G star it is tens of per cent — and what the metallicity shift measures is the change in an already-large deficit.

That has two consequences. The relation between colour and temperature has to be calibrated against real stars or model atmospheres rather than against Planck curves, because the blackbody relation is wrong even at the reference metallicity. And the blanketing correction is not additive in the way the figures draw it; it depends on the temperature as well, since the lines that do the blanketing are ionised away in hot stars and swamped by molecules in cool ones.

The linearisation used here is therefore the one genuinely modelled quantity in the essay, and the figure’s captions say so. The strength of a line is a competition between abundance and ionisation, and a coefficient in magnitudes per dex compresses that competition into one number.

The general shape of a degeneracy

The recurring statement is that an observable is a projection, and a projection of three quantities onto one line loses two of them irretrievably.

Nothing about the measurement can recover them. Improving the photometry improves the precision of the projection and does nothing about its rank. That is the same structure as an inclination that cancels out of a velocity amplitude, and as a transmission spectrum whose reference radius trades against its abundance.

The escape in each case is an additional observable with a different projection, and finding one is usually the harder half of the work. The reddening-free index is the most elegant instance in this subject: rather than measuring more and solving, it constructs a combination onto which the unwanted quantity projects to zero.

A quantity that projects to zero is better than a quantity that is measured, because it needs no correction and carries no error from one.

Still open: the temperature scale itself

The three shifts here are each corrected routinely and the corrections are believed at the level of tens of kelvin. What is not settled is the zero point they are corrected towards.

Converting a colour into an effective temperature requires either an empirical calibration against stars with directly measured angular diameters — of which there are a few hundred, all bright and nearby and mostly giants — or a theoretical one from model atmospheres. The two disagree by 50 to 150 K for cool stars, and the disagreement has not narrowed in twenty years.

That matters because a stellar temperature propagates into a radius through Stefan–Boltzmann, into a luminosity, into an age from isochrone fitting, and thence into the age of the oldest stars. A hundred kelvin at the bottom of that chain is a gigayear at the top.

From here the index stops being one number

Everything above has treated a colour as one number from two bands. A modern survey measures five, ten or a hundred bands, and the natural continuation is what changes when the index becomes a vector.

The answer is that the degeneracies described above become soluble in principle and remain hard in practice, because the extra bands are correlated — a star’s spectrum is smooth, so ten broad-band magnitudes contain far less than ten independent numbers, and how many they do contain is a measurable quantity nobody usually measures.

Where this goes next

Blackbody curves at 3000, 5800, 10000 K. Thermal 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. 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. The extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V. Dust makes everything look further away A star behind dust is fainter, so a distance taken from its brightness comes out too large. The same dust also makes it redder, and the reddening is measurable where the dimming is not — which is the only reason the correction can be applied at all. Two laws that are the peak and the area of one curve. Planck curves at 3,000 K, 5,772 K, 9,600 K on logarithmic axes, with each peak marked. Normalised by its own peak, Planck's law is a universal function of x = hc/λkT, and three exponents follow from that alone and are fitted here off the drawn curves rather than quoted: the peak wavelength goes as T^-1.000, which is Wien's displacement law with a constant of 2.897772 mm K obtained by solving 5(1 − e^(−x)) = x for x = 4.965114; the peak HEIGHT goes as T^5.000; and the area goes as T^4.000. The third is the first two multiplied. A peak five powers high on a curve one power narrow encloses four powers of area, so Stefan–Boltzmann is not an independent fact about radiation — it is Wien's law and the height of the peak, taken together. That is also why the two are worth having at once. A colour gives the temperature and a flux gives the luminosity, and L = 4πR²σT⁴ then gives a radius: for the Sun at 5,772 K receiving 1361 W/m² at 1.000 AU, the arithmetic returns 6.957·10⁸ m against a measured 6.957·10⁸. A thermometer alone cannot do that, because a colour is a ratio and a ratio has no size in it; the radius comes from the one law that is an absolute quantity rather than a shape. Two laws that are one curve read twice Wien's displacement and Stefan–Boltzmann are the peak and the integral of the same function. The peak is five powers high and one power narrow, so the area is four — and the fourth power that is taught as a separate law is the first two multiplied.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Binary starBlackbodyColour indexDegeneracyEffective temperatureInterstellar reddeningLine blanketingMetallicityPhotometric systemReddening