Astronomy & Celestial Mechanics
the sky, computed
Concept

Line-broadening — where it appears

The widening of a spectral line beyond its natural width, by thermal motion, by collisions, by rotation or by turbulence. Each mechanism has its own profile shape, which is what allows several to be separated from a single well-resolved line.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

The same spectrum, at rest and at 900 km/s. A set of absorption lines at rest and shifted by a radial velocity of 900 km/s. The displacement is proportional to wavelength, so the reddest line here moves 2.06 nm and the bluest 1.18 nm — which is why the measured quantity is the ratio Δλ/λ and not a distance.

A shift in a line is a speedometer, and it works at any distance

A spectral line has a wavelength fixed by physics. Measuring where it actually arrives gives the source's speed toward or away — and the measurement does not degrade with distance.

starlight · The Doppler effect 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 27.5 m/s velocity the star does not have. Left: a rotating stellar disc, approaching on one side and receding on the other, with the chord a planet of 0.1 stellar radii takes across it at impact parameter 0.5 and a sky-projected obliquity of 0°. Right: the apparent radial velocity that results, computed by covering the disc cell by cell — the flux hidden at each phase and its mean line-of-sight velocity — rather than from a fitted formula. The star's centre of mass does not move at any point in this: the anomaly is entirely a statement about which parts of the line profile are missing. Its amplitude is 27.5 m/s at v sin i = 4.5 km/s, and the two numbers are related by the depth of the transit, since blocking a fraction f of light of mean velocity v shifts a flux-weighted centroid by f·v. The curve is antisymmetric about mid-transit to 0.00% of its own amplitude, which is what an aligned transit gives: equal time on the blue half and the red. Limb darkening is included at u = 0.6, and it matters: it weights the hidden light towards the centre of the disc, where the rotation velocity is smallest.

A velocity measured from a shape

A transiting planet hides part of a rotating disc, so the star's line profile loses a slice at one velocity and its fitted centroid moves. The star has not moved at all — and the lopsidedness of that motion is the whole measurement of whether the orbit lies in the star's own equatorial plane.

exoplanets · Spin–orbit alignment Mass from a velocity dispersion, across nine decades. The virial estimate M = 5 σ²R_e/G, drawn for half-light radii from 0.2 to 20 kpc, with four systems placed on it: M32 at σ = 75 km/s and R_e = 0.1 kpc, giving 6.5e+8 M☉; the Milky Way's bulge at σ = 105 km/s and R_e = 1 kpc, giving 1.3e+10 M☉; M87 at σ = 320 km/s and R_e = 7 kpc, giving 8.3e+11 M☉; the Coma cluster at σ = 1000 km/s and R_e = 1500 kpc, giving 1.7e+15 M☉. The same expression spans from a dwarf elliptical to a cluster of galaxies — nine decades of mass — because the virial theorem does not care what the moving objects are. In the last case they are whole galaxies, and the estimate exceeds everything visible in the cluster by a factor of about fifty, which is where the problem was first noticed.

A galaxy held up by disorder

An elliptical galaxy barely rotates. What holds it up against its own gravity is the randomness of its stars' motions, and the virial theorem turns that randomness into a mass by exactly the reasoning that turns a rotation speed into one.

galaxies · Velocity dispersion The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone.

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

exoplanets · Spin–orbit alignment One planet's helium line, around four kinds of star. The excess absorption at the centre of the 1083 nm helium line during transit, against the mass-loss rate, for one sub-Neptune — 26 Earth masses, 4.4 Earth radii, 0.0525 AU — losing an isothermal wind at 8,000 K, around four representative stellar spectra. Both scales logarithmic. The same outflow absorbs very differently: at 1 × 10¹⁰ g s⁻¹ the depth is 0.06%, 0.14%, 0.35%, 0.40% around an F star, a Sun-like star, an active K dwarf, an active M dwarf. Read the other way, a measured depth of 1% means 1.1 × 10¹¹ g s⁻¹ around an F star, 6.3 × 10¹⁰ g s⁻¹ around a Sun-like star, 3.0 × 10¹⁰ g s⁻¹ around an active K dwarf, 2.9 × 10¹⁰ g s⁻¹ around an active M dwarf — a factor of 4 between the extremes, from the star alone.

A helium line that weighs the star first

The metastable helium line at 1083 nanometres made the escape of a planet's atmosphere measurable from the ground. But the atoms it sees exist only in a balance between the star's extreme-ultraviolet light, which fills the level, and its mid-ultraviolet, which empties it — so the same outflow is six times deeper around an active K dwarf than around an F star, one depth allows a fifty-fold range of mass-loss rates across the wind's possible temperatures, and a rate inferred from it inherits a share of every error in a spectrum that cannot be observed.

exoplanets · Atmospheric escape The same two lines, seen as frequencies. The amplitude of the Fourier transform of the two line profiles of the previous comparison, on a logarithmic scale, against frequency in cycles per kilometre per second. At low frequency — the overall width — they agree, as the profiles did. At higher frequency they separate completely: the first star's transform falls to zero at 0.110 cycles per km/s and the second's at 0.132, because the rotation kernel's transform vanishes at 0.660/v sin i and no Gaussian-like broadening has a zero anywhere. The horizontal dashed line marks a noise level of 0.0005 in the transform, which a high signal-to-noise spectrum averaged over several lines reaches; the dip and the side lobe beyond it rise above it for both stars, so the rotation of each is measured by where its transform dips, independent of the macroturbulence. In a noisier spectrum the second star's weaker side lobe would be lost first.

A rotation found where the transform goes dark

A star's rotation and the turbulent flows on its surface both widen its spectral lines, and a profile fitted for one can be refitted for the other with almost no change. Convolution becomes multiplication in Fourier space, though, and the transform of a rotating disc has zeros — frequencies where it is exactly nothing — while the transforms of turbulence and of the spectrograph have none. Where the transform of a line goes dark is a measurement of rotation that turbulence cannot fake.

starlight · Line formation

Named alongside it

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

Limb darkeningStellar rotationMacroturbulenceRadial-velocityThe Rossiter–McLaughlin effectRotational broadeningAnisotropyAsteroseismologyAtmospheric escapeBlack holeConvolutionDamping wings

All concepts