Astronomy & Celestial Mechanics
the sky, computed
Concept

Convolution — where it appears

An operation that smears one function by another, summing shifted copies of the first weighted by the second. A population's supernova rate is its star-formation history convolved with the delay-time distribution; an image is the sky convolved with the point-spread function.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous.

A planet radius is a stellar radius

A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

exoplanets · Planet composition A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism.

The iron clock has no single delay

The α-element knee is drawn as though Type Ia supernovae switched on a billion years after the stars that made them. Measured rates say otherwise — the delays are spread evenly over every decade from forty million years to a Hubble time, as a power law with no timescale in it — and a clock with no timescale bends where a clock with one would break.

galaxies · Chemical evolution 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 A disc that bends up into a halo only the telescope made. The surface brightness of a face-on exponential disc against radius in units of its scale length, in magnitudes per square arcsecond with the centre at 20, as it is (black, a straight line) and as a telescope sees it if 2% or 5% of each point's light is spread into faint wings falling as the inverse square of distance out to ten arcminutes. Scattered light from the bright centre lands in the outskirts, and once the disc has faded below it the profile stops falling and bends up into a shallow component that looks like a stellar halo: the observed profile rises three-quarters of a magnitude above the disc beyond 10.3 and 9.3 scale lengths, near 29.2 magnitudes per square arcsecond. At ten scale lengths the disc itself is at 30.9 and the image at 30.2 and 29.6.

A halo the telescope paints on every galaxy

A telescope puts almost all of a star's light into a dot and a few per cent of it into faint wings that reach arcminutes away. Around a galaxy those wings add up. The scattered light from the bright centre lands in the outskirts, and below about twenty-nine magnitudes per square arcsecond it outshines the galaxy's own — bending the profile up into a halo, or an edge-on disc out into a thick one, at the same depth around every galaxy whatever its size.

starlight · Photometric systems

Named alongside it

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

Alpha enhancementAlpha kneeChemical evolutionCosmic star formation historyDelay time distributionDouble degenerateExponential discFourier transformGravitational wavesLimb darkeningLine-broadeningMacroturbulence

All concepts