Astronomy & Celestial Mechanics
the sky, computed
Generator

The line-profiles generator

Three profiles of equal equivalent width, 28.6 mÅ
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.

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.

7 essays call line-profiles. 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.

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 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. Starlight

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.

 A wind that writes its own terminal speed on the blue edge, at 2017 km/s. A P Cygni profile, computed by integrating a spherical wind on a grid rather than drawn. The wind accelerates outward as one minus the inverse radius to the power 0.8, reaching 2000 kilometres a second, and its density falls as the inverse square of the radius over its own speed; the horizontal axis is the Doppler shift in units of that terminal speed, with blue to the left. The column directly in front of the stellar disc is moving toward the observer at every speed from nearly nothing near the surface up to the terminal value far out, so it takes light out of the beam across that whole range and the absorption trough runs from the rest wavelength to the blue edge at -1.01 of the terminal speed. That edge is the measurement: no model of the star, its distance or its mass enters it, only the geometry of a column seen end-on. The emission comes from everywhere else, where the wind moves across the line of sight and can only add photons; it is Infinity times stronger to the red than to the blue, because the part of the shell receding directly away is hidden behind the star and its blueshifted counterpart is not. Two features, one shell, and between them a velocity and a mass-loss rate. Starlight

A speed read off an edge

A hot star drives material off itself at thousands of kilometres a second, and the profile that wind prints on the star's own spectrum has a sharp blue edge. That edge is the terminal velocity, measured with no model of the star, no distance, and no calibration — one of the few numbers in stellar astrophysics obtained from geometry alone.

 A hundredfold expansion is a four-order slowdown. Equatorial rotation speed against radius for three stars leaving the main sequence at 2, 10, 100 km/s, on the single assumption that nothing exerts a torque. Both axes are logarithmic. The specific angular momentum is held fixed along each track and checked at four radii rather than asserted, so the speed falls as the reciprocal of the radius and then faster, as the moment of inertia coefficient slides from 0.073 on the main sequence to 0.02 in a centrally condensed giant envelope. A star at 100 km/s crosses the 8 km/s line — the usual boundary for calling a giant a rapid rotator — at 45.6 solar radii, and everything larger is slower. That is why the observed giants are almost all under two kilometres a second, and why the one or two per cent that are not cannot be explained by anything the star did on its own: the angular momentum has to have been delivered, by a swallowed companion or a merger. The figure assumes no mass loss, which for the largest radii drawn is the weakest of its assumptions. Stars

A surface that slowed because the star grew

A star leaving the main sequence expands by a factor of a hundred. Conserve its angular momentum and its equator slows by four orders of magnitude, which is why every red giant is a slow rotator. One or two per cent are not, and a giant turning at fifty kilometres a second has to have been given the angular momentum by something else.

 Two effects that are blind in opposite directions. The sensitivity of two magnetic diagnostics against field strength, on a logarithmic axis spanning five decades. The Zeeman effect measures the line-of-sight component and adds it up along the path, so a field tangled into 100 independent cells inside one resolution element averages down by a factor of 10 and reports almost nothing — which is exactly the situation in a chromosphere or a turbulent cloud. The Hanle effect is a different device altogether: a field precesses the atom between absorption and re-emission, so a scattering line's polarisation is rotated and reduced, and the amount depends on how far the precession gets in one radiative lifetime. That makes it sensitive around 1 gauss for a 100-nanosecond level, and — the useful part — it does not care about sign, so a tangled field does not cancel. Above the crossing at 27.7 gauss the Hanle signal has saturated and carries no strength information, and the Zeeman effect is the instrument. Neither is a measurement of the field; each is a measurement of what the field did to something else. Starlight

Two instruments blind in opposite directions

The Zeeman effect measures the component of a field along the line of sight and adds it up, so a field tangled into a hundred cells reports a tenth of one cell's strength. The Hanle effect measures how far an atom precesses between absorbing and re-emitting, does not care about sign, and saturates just where the Zeeman effect becomes useful.

 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. Starlight

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.

 A bump that crosses the line from -41 to 25 km/s. The residual of a rotationally broadened line profile during a transit, drawn at five epochs and offset vertically. The planet covers a strip of the stellar disc whose radial velocity is the projected rotation at that position, so it removes light from one velocity and leaves a bump in the residual there. As the planet crosses, the bump travels across the profile — and where it starts and ends is set by the geometry of the chord. An orbit aligned with the star's equator gives a track symmetric about the line centre; this one, tilted by 30 degrees, runs from -41 to 25 kilometres a second and is not. The measurement is of a path rather than of a centroid, which is why it works on rapidly rotating stars where the velocity anomaly is swamped by the line's own width. Exoplanets

A shadow crossing a rotating line

A transiting planet hides a strip of a rotating star, and that strip has a definite velocity. So the planet removes light from one place in the line profile and leaves a bump there — a bump that travels across the line as the transit proceeds, tracing the path the planet took across the disc.

The whole library · All essays