Astronomy & Celestial Mechanics
the sky, computed
Concept

Line blanketing — where it appears

The cumulative depression of a spectrum by a forest of weak absorption lines, which removes flux from one region and redistributes it to another. It lowers the apparent continuum in crowded spectra, and it is strongest in cool, metal-rich stars.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Two means of one opacity, 75 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 6 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 75 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.76 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.13, and then turns up sharply, slope 0.99, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps.

A mean dominated by the gaps

The opacity in the equation of radiative transport is not an average of the opacity. It is a harmonic average weighted by the temperature derivative of the Planck function, which makes it a measurement of the transparent windows between the lines rather than of the lines — and that single fact decides what adding a million spectral lines to a table does.

starlight · Opacity A continuum drawn 4.3 per cent below the real one. A short stretch of spectrum with one strong line in it and 150 weak ones scattered across the same interval. The upper dashed line is the true continuum — the flux the star would emit with no lines at all — and it is not observable. The lower one is what a fit through the highest points of the spectrum returns, which is 4.3 per cent lower, because the weak lines have depressed the gaps between the strong ones. Measuring the strong line's equivalent width against the apparent continuum instead of the real one makes it 11.2 per cent too small. The error has a sign, it is worse in spectra with more lines, and it therefore correlates with metallicity — which is exactly the quantity being measured.

A continuum that was never observed

An equivalent width is an area measured relative to the continuum, and the continuum is not in the data. It is drawn — a curve through the highest points of the spectrum — and in any spectrum with many weak lines those highest points are already below the true continuum, because the weak lines have eaten the gaps.

starlight · Line formation 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.

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.

starlight · Stellar colour

Named alongside it

The objects these essays reach for when they reach for this one.

MetallicityAbundance analysisBinary starBlackbodyBound bound absorptionColour indexContinuum normalisationCurve of growthDegeneracyDifferential analysisEffective temperatureEquivalent width

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