Astronomy & Celestial Mechanics
the sky, computed
Concept

Rotational broadening — where it appears

The widening of a star's spectral lines by its rotation, as light from the approaching limb is blueshifted and from the receding limb redshifted. It keeps a line's equivalent width while making it shallower, and so blunts every velocity measurement.

Named by 4 essays across one field — 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 Where the velocity information in a spectrum lies. Above: a 600 km/s stretch of two synthetic spectra with the same continuum of 10000 photons per pixel at a resolving power of 100,000 — a cool, slowly rotating star, with 12 lines per nanometre, and a hot star rotating at 25 km/s, with 3 lines per nanometre. Below: the velocity information each pixel carries, (dA/dv)²/A, which is zero in the continuum and at line centres and largest on the steep flanks of deep, narrow lines. Summed over 100 nanometres of spectrum, the first star measures its velocity to 1.61 m/s and the second to 39.17 m/s — 24 times worse from the same number of photons. The photons that measure a velocity are the ones on the sides of lines; the continuum, however bright, contributes nothing.

The photons on the sides of the lines

A spectrum measures a star's velocity only through the slopes of its absorption lines. The continuum, however bright, carries no information at all, and a star with many deep, narrow lines gives a velocity twenty times more precise than a hot, fast-rotating one from the same number of photons.

starlight · The Doppler effect A spot's mark on a line, and how much of it a shift explains. Above: the disc-averaged profile of one absorption line from a star rotating at 5 km/s, observed at a resolution of 100,000, without (dashed) and with (solid) a dark spot 8° in radius at latitude 20°, 70 per cent darker than the surrounding surface, on the receding side of the disc. Below, magnified 185 times: the change the spot makes (blue), the part of it a pure Doppler shift of the whole line would make (orange, a shift of −36.3 m/s), and what is left over (green). The spot removes light from the velocity at which its own surface is moving, putting a bump into the line where it was darkest. Most of the change, 92 per cent of its square, looks like a shift; 8 per cent does not, and that remainder is the only part of the spot's signal a spectrum could tell apart from a planet.

A spot a slow star hides inside its velocity

A dark spot on a rotating star removes light from one part of each spectral line, and the line's centre seems to move: a velocity of metres per second that no planet caused. In principle the spot betrays itself, because it changes the line's shape and a planet does not. In practice the betrayal is a few per cent of the signal on a star that rotates as slowly as the Sun, and seeing it takes a signal-to-noise of hundreds per pixel across thousands of lines.

starlight · The Doppler effect 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 darkeningLine-broadeningMacroturbulenceRadial-velocitySpectral resolutionConvective blueshiftConvolutionCross-correlation functionDamping wingsDoppler broadeningDoppler imagingEquivalent width

All concepts