Astronomy & Celestial Mechanics
the sky, computed
Generator

The doppler-lines generator

The same spectrum, at rest and at 900 km/s
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 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.

9 essays call doppler-lines. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Starlight

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.

 The curve of growth. The equivalent width of an absorption line against the number of absorbers along the sight line, both logarithmic, computed by integrating a Voigt profile with damping parameter 0.005. Three regimes: the width grows in proportion to the abundance while the line is weak, then almost not at all for two decades once the core saturates, then as the square root once the damping wings dominate. A line measured in the middle stretch carries almost no information about the abundance, and most strong lines in a stellar spectrum are there. Starlight

How much of a line is not in its depth

An absorption line stops getting deeper long before it stops getting stronger. What keeps growing is its area, and the way the area depends on the number of absorbers has three distinct regimes over five decades — one of which carries almost no information at all.

 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. Starlight

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.

 A 28.7 km/s correction, and a 12.5 m/s planet underneath it. Two years of radial velocities of a star at ecliptic latitude 12°, orbited by a companion whose reflex semi-amplitude is 12.5 m/s — the Sun's own, from Jupiter. The upper panel is what the spectrograph measures: the Earth's motion about the barycentre of the solar system, amplitude 28.72 km/s, which is V⊕ cos β to a fraction of a per cent. The planet is in that curve and is 2,297 times smaller than it, which is a line thinner than the stroke it is drawn with. The lower panel is the same data after the correction, and the correction is not a fit: it is computed from an ephemeris, the observatory's position on a rotating deformable Earth, and the star's own coordinates and proper motion. To leave a centimetre a second it has to be right to one part in 2.9·10⁶ — the light-travel time across the Earth's orbit, the relativistic terms, and the fact that the star moves are all inside that budget. What remains is the planet, at 4333 days, and a scatter of 1.2 m/s that is the star rather than the instrument. Starlight

The metre per second that is not the star

A line shift is a speedometer, and the speed it reads is mostly the observer's. Getting to a metre a second means removing thirty kilometres of the Earth's own motion to a part in three million, and then confronting a floor that is the star's own surface rather than the instrument.

 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. Galaxies

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.

 4000 lines averaged into one profile, and 2.3 m/s out of it. A cross-correlation function: the average absorption profile obtained by shifting a mask of 4000 line positions across a spectrum and summing what falls under it. The faint curves behind are individual lines, each with its own depth, its own width and its own small offset; the heavy curve is what averaging them produces. The velocity is the position of the peak, and its precision is the width divided by the contrast, the signal-to-noise and the square root of the number of lines — 2.3 metres a second here. Nothing about this construction is a measurement of any one line. It is a measurement of where a weighted average of thousands of them sits, and the weights are a choice: a mask built for one spectral type applied to another weights the disagreement between the lines differently, and moves the peak. Starlight

A velocity that is an average of lines that disagree

A radial velocity measured to a metre a second is not measured from a line. It is the position of the peak of a cross-correlation against a mask of thousands of lines, and those lines do not agree with each other by hundreds of metres a second — because each one forms at a different depth in an atmosphere that is boiling.

 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. Starlight

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.

 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. Starlight

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.

 Lines from the lit face follow the star that cannot be seen. Radial velocity against orbital phase, over two orbits, for a binary like Sco X-1 — period 0.787 days, a 1.4 M☉ neutron star, a donor of 0.42 M☉, inclination 40°. The donor itself is drowned by the light of the accretion disc and is not seen. The narrow components of the Bowen blend, emitted by the donor's face lit by X-rays, move in phase with the donor but with 109 km/s rather than the 139 km/s of its centre, because the lit face lies on the side nearer the neutron star. The neutron star moves in antiphase at 42 km/s, a motion the broad lines of the disc track only roughly. Measuring the narrow lines' amplitude puts a lower limit on the donor's velocity. Starlight

A wavelength coincidence that weighs a neutron star

An ionised helium atom and a twice-ionised oxygen atom happen to have lines a sixtieth of an ångström apart in the far ultraviolet, close enough that one absorbs the other's light. The oxygen then glows in a cascade of visible lines, and so, by a second coincidence, does nitrogen. In a bright X-ray binary that glow comes from the face of the companion star turned towards the X-rays — a star otherwise lost in the light of its accretion disc — and its Doppler shifts trace the companion's orbit, weighing the neutron star it circles.

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