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Redden the integrand, not the answer
Two published extinction laws, a star behind a dust screen, and the arithmetic that separates a correct magnitude from a plausible one.
Chromaticity only, normalised to a peak of 1 — this shows the colour SHIFT, not the dimming. Extinction does both; a swatch can only carry one.
The error this page exists to make visible
Extinction is wavelength-dependent, so it has to be applied inside the passband integral: multiply the spectrum by 10−0.4A(λ) at every λ, then integrate. The tempting shortcut is to integrate the clean spectrum, then dim the result by A(λeff) — one number instead of a curve.
How wrong is it? Measured here, for a 20 000 K star at RV = 3.1 under G23, in magnitudes:
| Band | AV=1 | 3 | 5 | 10 |
|---|---|---|---|---|
| U | 0.013 | 0.014 | −0.015 | −0.193 |
| B | 0.002 | −0.021 | −0.083 | −0.417 |
| V | 0.018 | 0.036 | 0.029 | −0.106 |
| R | 0.012 | 0.018 | −0.003 | −0.172 |
| I | 0.007 | 0.014 | 0.014 | −0.019 |
| J | 0.005 | 0.013 | 0.019 | 0.025 |
Two things in that grid matter more than the headline size. FIRST, it is already at the level of good photometry at AV = 1 — 0.018 mag in V, 0.013 in U, 0.012 in R. Only B, at 0.002, is genuinely negligible there, so a worked example that quotes B alone flatters the shortcut. SECOND, the errors do not share a sign: at AV = 10, B is −0.417 while J is +0.025. An error that dimmed every band equally would be absorbed by a distance or a normalisation and cost nothing. This one does not — it moves the bands against each other, so it lands directly in the colours, which is what an extinction is usually being measured from in the first place.
So this is not a mistake that announces itself. Drag AV and watch the last column: the error grows faster than linearly, because the more the band is reddened the further its effective wavelength slides from the one the shortcut assumed. Cells past 0.01 mag — roughly good ground-based photometric precision, the level at which the error stops hiding inside the error bars — are marked.
| Band | λeff | no dust | reddened correctly | shortcut | error |
|---|
Where AV comes from, and where it doesn't
On this page it comes from a slider. That is honest for a bench and dishonest for a result: a real AV is a line integral of dust density along the sightline, so it differs star by star and is the mechanism behind differential reddening across a young cluster. The seam is already built for it — attenuation enters per star — but nothing computes it from the gas yet.
Note the two laws disagree outside the optical, and CCM89 returns nothing at all beyond its published domain rather than extrapolating. That is deliberate: a curve quietly extended past where it was fit is a fabricated measurement.