Astronomy & Celestial Mechanics
the sky, computed
Concept

Model atmosphere — where it appears

A calculation of how temperature, pressure and composition vary through a star's or planet's outer layers, and of the spectrum those layers emit. Temperatures, gravities and abundances read from spectra all pass through one.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A few per cent of diameter, hidden in the second lobe. Visibility against baseline in the natural variable πθB/λ, for stars with linear limb-darkening coefficients of 0, 0.3, 0.6, 0.9, each rescaled to the uniform disc that best fits its own first lobe. In the first lobe the four curves are within 0.72 per cent of one another; past the first null they differ by up to 4.2 per cent. That is the whole difficulty of measuring a stellar diameter. A limb-darkened star has a faint edge, so a uniform-disc fit returns a diameter 9.7 per cent too small at u = 0.9 and 2.5 per cent too small at u = 0.3 — and the information needed to tell which is in a region where the visibility is under five per cent and the calibration errors of a real interferometer are comparable to the signal. The correction from what is measured to what is wanted is taken from a model atmosphere, because the observation that would supply it is the hardest one there is. The fit here is done by least squares on the drawn curves rather than read from a conversion table, and the zero-coefficient case returns the uniform disc to 0.00 per cent, which is the fit checking itself.

A diameter that depends on a model atmosphere

An interferometer measures fringe visibilities and somebody fits a disc. A uniform disc and a limb-darkened one agree to within a per cent across the whole of the first lobe and differ by five in the second — where the visibility is under five per cent and the calibration errors are the same size.

starlight · Angular diameter A white dwarf cooling track in J−K and absolute V. The track a 0.6 solar-mass hydrogen-atmosphere white dwarf follows as it cools, absolute V magnitude against J−K colour, fainter downward. Dashed, a blackbody of the same radius, which slides steadily redward and fainter. Solid, with collision-induced absorption: the track reddens to J−K = 0.40 at 4250 K and absolute magnitude 16.1, then hooks back to the blue, reaching −0.86 at 3000 K and magnitude 18.1. Temperatures are marked along it. The oldest white dwarfs therefore sit in the same colour range as much younger, hotter ones and are told apart only by being fainter — which is why they were missed by surveys that selected by colour.

A colour that runs backwards at the end of cooling

A white dwarf reddens as it cools, like every other hot body — until its surface falls below about four thousand kelvin. Then pairs of hydrogen molecules, colliding in an atmosphere dense enough to crush them together, begin to absorb its infrared light, and the star turns blue again while it goes on getting fainter. The oldest white dwarfs in the Galaxy have the colours of young ones, and for years they were looked for in the wrong place.

stars · White dwarf cooling How steeply the ratio of all the light to one band rises with temperature. The exponent n in (bolometric flux ÷ band flux) ∝ Tⁿ, for a blackbody, against temperature, for bands at V (0.55 µm), J (1.25 µm), H (1.65 µm), K (2.2 µm). The bolometric flux rises as T⁴; the flux in a band rises as T on the long-wavelength side of the Planck peak and much faster on the short side, so n is the difference. For the Sun's temperature it is 2.33 in K, 1.70 in J and −0.56 in V. An error of one per cent in the modelled band flux becomes an error of 1/n per cent in the temperature: 0.43 per cent, 25 K, in K. In the visible the ratio barely depends on temperature and carries no information about it; in the infrared it does, and the temperature is read with its model error divided.

A temperature read against the light's long tail

A star's total light depends on its temperature to the fourth power; its light in the infrared depends on it barely more than linearly, because the infrared lies on the long, flat tail of the Planck curve. The ratio of the two cancels the star's size and distance and leaves a steep function of temperature — and the only model it needs is the infrared surface flux, whose errors are divided by two and a half on the way to the answer. That is the infrared flux method, one of the two foundations of the stellar temperature scale, and the other foundation disagrees with it by up to a hundred and fifty kelvin.

starlight · Stellar colour

Named alongside it

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

Angular diameterBlackbodyBolometric fluxEffective temperatureBaselineCollision induced absorptionColour indexCooling ageInfrared flux methodInterferometryLimb darkeningMolecular hydrogen

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