Astronomy & Celestial Mechanics
the sky, computed
Concept

Surface gravity — where it appears

The gravitational acceleration at a star's photosphere, spanning eight decades across the Hertzsprung–Russell diagram and read off a line's wings. It is read from a line's wings rather than from anything geometric, and it is what turns a spectral type into a luminosity and therefore into a distance.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Three profiles of equal equivalent width, 28.6 mÅ. Three absorption profiles with the same equivalent width — 28.6 milliångström, 1.72 km/s at 500 nm, matched to better than 0.1% by root-finding over the quadrature — differing in nothing but shape. Left: the cores, on a common velocity axis. Right: the same three normalised to their own half widths, on a logarithmic depth scale. The thermal profile is a Gaussian set by Fe's mass at 6000 K, 1.34 km/s; the collisional one a Lorentzian of γ = 2.28e-3 nm; the rotational one the classical kernel of a disc turning at v sin i = 1.73 km/s, which is exactly zero beyond 1.28 half widths and is the only one of the three with an edge. Matching the areas does not match the widths: the half widths are 1.11 km/s (thermal), 1.35 km/s (rotational), 0.68 km/s (collisional), a factor of 1.98 between the widest and the narrowest. At three half widths the collisional wing is 49 times the thermal one and at five it is 1.27·10⁶ times — six decades, which is why a line's shape stays diagnostic long after its width has stopped being so. 11% of the Lorentzian's own equivalent width lies beyond the right-hand panel's edge and is not drawn anywhere.

The same width for three different reasons

Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another.

starlight · Line formation A strong line at three gravities, 6000 K. Left, one strong absorption line of Fe computed at three surface gravities and drawn on the same wavelength scale — no normalisation to each profile's own width, which is exactly what the comparison is about. The thermal core is identical in all three, because the Doppler width is 0.00223 nm at 6000 K whatever the star's size. The wings are not: collisional damping is proportional to the density of perturbers, that density is proportional to the gas pressure, and in a grey atmosphere the pressure at the photosphere is proportional to g — so log g = 4.4 carries a damping parameter 794 times that of log g = 1.5. Right, the width at a tenth of the core depth against gravity, measured off those curves: a slope of 0.499, against the half a Lorentzian wing forces. That is the second dimension of a spectral classification. The first is the temperature, read from which lines are present; this one is the pressure, read from how wide they are, and it is the whole reason a spectrum can be turned into an absolute magnitude and then into a distance.

The second thing a spectrum says

Two stars of the same colour can differ in luminosity by ten magnitudes, and the difference shows in the widths of their lines rather than in which lines are present. That width is a pressure, the pressure is a gravity, and the gravity is a distance.

starlight · Spectra 14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here.

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

stars · Asteroseismology Two diagnostics, two bands, and a crossing to 49 kelvin. The plane of effective temperature against surface gravity, with the constraints from two spectroscopic diagnostics drawn as bands. The wings of a hydrogen line are broadened by collisions, so they respond steeply to the gravity and weakly to the temperature: a narrow, steep band. An ionisation balance — requiring that the same element give the same abundance from its neutral and its singly ionised lines — responds to both, and its band is much shallower. Neither diagnostic determines either quantity on its own. Where the two cross is the answer, and the size of the crossing region is set by the band widths divided by the difference of the slopes — so two diagnostics that respond similarly give a long, thin, nearly useless error region however precise each one is. Choosing diagnostics that disagree in their sensitivities is the whole of the art.

A temperature and a gravity that trade against each other

A stellar spectrum contains the star's temperature, its surface gravity and its composition, and no single feature in it contains only one of the three. Every diagnostic is a band in the parameter plane rather than a point, and the answer is where the bands cross — which makes choosing diagnostics that disagree in their sensitivities the whole of the art.

starlight · Spectra 742 kelvin between a dwarf and a supergiant, at the same ionisation. The fraction of calcium still neutral against temperature, for three surface gravities: a dwarf, a giant and a supergiant. The three curves are the same curve slid sideways. The Saha equation carries the electron pressure in its denominator, and the photospheric pressure follows the surface gravity as its square root — hydrostatic equilibrium gives a gas pressure of order g over the opacity, and the opacity in a cool star is set by the electrons themselves. So the pressure runs from 19 newtons a square metre in the dwarf down to 0.48 in the supergiant, a factor of 40, and the half-ionisation point moves from 4427 kelvin to 3685 — 742 kelvin apart. At fixed temperature the ionisation ratio goes as g^-0.50, exactly the square root the algebra requires. A star's luminosity class is a measurement of the density of its photosphere, read off which stage of an element the lines belong to.

A luminosity class is a density measurement

The Saha equation has an electron pressure in its denominator, and a supergiant's photosphere is forty times less dense than a dwarf's at the same temperature. So the same element sits in different ionisation stages in the two, the spectrum says which, and the second axis of stellar classification is a barometer.

starlight · Ionisation Two horizons, one formula, and a factor of two. Horizon temperature against horizon radius, on logarithmic axes, for the two kinds of horizon this collection has. The upper line is a de Sitter horizon at T = ħc/2πk_BR and the lower is a Schwarzschild horizon at T = ħc/4πk_BR — the same expression with the same constants, differing by exactly two, and both falling as one over the radius so that a bigger horizon is a colder one. The cosmological horizon today has a radius of 4,451 megaparsecs and a temperature of 2.65e-30 K, which is thirty orders of magnitude below the microwave background and will never be measured by anything. A solar-mass black hole sits at 6.2e-8 K, and a black hole as cold as the sky would weigh 2.32e+22 solar masses — of the order of the mass inside the observable universe, which is not a coincidence, since both numbers are c³/GH up to factors of order one. The factor of two between the two lines is the one place the analogy is not exact, and it is not a convention: it comes from the periodicity of the Euclidean time coordinate, which is 8πGM/c³ for a black hole and 2π/H for de Sitter space. The entropies, by contrast, agree exactly.

Two horizons that differ only in who is inside

A black hole's horizon and the cosmological one share an entropy formula exactly and differ in temperature by precisely a factor of two. The factor of two is the whole of the difference, and what it encodes is which side of the surface the observer stands on.

cosmology · Horizons The width of an emission core against a star's luminosity. Absolute visual magnitude against the logarithm of the width of the calcium K emission core in km/s, for the Wilson–Bappu relation Mᵥ = 27.59 − 14.94 log W₀ (line), with 120 illustrative stars scattered about it by 0.5 magnitudes, the relation's typical scatter, and the Sun marked at about 33 km/s and Mᵥ = 4.83. Over a range of fifteen magnitudes in luminosity — from dwarfs fainter than the Sun to supergiants a million times brighter — the logarithm of one width predicts the magnitude to half a magnitude, whatever the star's temperature. A factor of two in width is four and a half magnitudes in brightness, sixty times in luminosity.

A width that knows how bright the star is

In the dark trough of a cool star's calcium K line sits a small bright emission core made in its chromosphere. In 1957 Olin Wilson and Vainu Bappu found that the core's width predicts the star's absolute magnitude over fifteen magnitudes — from dwarfs to supergiants, whatever the star's temperature and however active its chromosphere — and the simplest physical account of why reproduces the relation's existence and misses its slope by half.

starlight · Spectra Where the photons stop mattering and the star starts. Noise in a 13-hour window — the length of an Earth-like planet's transit across a Sun-like star — against the star's brightness, for a space telescope that reaches 17 parts per million in 6.5 hours on a twelfth-magnitude star. Photon noise falls by a factor of ten for every five magnitudes of brightness; the star's granulation, averaged over the window, stays at 5.1 parts per million whatever the brightness. The two are equal at magnitude 10.1; brighter than that, the star is the floor. The dashed line is an Earth's transit depth across the Sun, 84 parts per million: a single transit on a twelfth-magnitude star is detected at 6.4 standard deviations, and on the brightest stars at no better than 17, set by the star alone — and supergranulation, spots and faculae add more on these timescales.

A floor the star sets for itself

Above the atmosphere, with comparison stars and a big enough telescope, every noise in a measurement of a star's brightness can be beaten down except one: the star's own surface, a mosaic of convective cells each brighter or fainter than the rest. That flicker does not average away against anything, and it is larger for stars that are bigger and puffier than the Sun. For the brightest stars, an Earth's transit is limited by the star, not the photons — and the stars Kepler watched turned out twice as noisy as the Sun.

starlight · Photon noise

Named alongside it

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

Damping wingsPressure broadeningAbsolute magnitudeConvectionEquivalent widthGranulationHydrostatic equilibriumIonisation balanceLuminosity classMK classificationSaha equationSpectral classification

All concepts