The collection

Every essay — page 40

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence. Essays 781–800 of 909.

Starlight

Distance, brightness and colour: what can be measured when nothing can be visited.

A calibration chosen by the measurement it calibrates

A standard candle is calibrated on stars with good parallaxes, and "good" is decided by the parallax itself — so the calibrators are the stars whose errors happened to push them nearer. Correcting each star's distance afterwards does not repair it, because the correction needs a picture of where the sample's stars really lie, and that picture depends on the luminosity being calibrated. What works is to choose nothing, keep every star with the negative parallaxes included, and fit the candle by predicting the parallaxes rather than inverting them.

6 figures · Parallax

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

$One production rate, many Suns. Global isotope production, as a fraction of today's average, over the modulation potential and the Earth's dipole moment. Each contour is one production rate — 0.85, 1, 1.2, 1.4, 1.7 times today's — and every point on it is a combination of Sun and Earth an isotope record cannot distinguish. Production at today's dipole of 7.8 × 10²² A m² and a modulation potential of 650 MV is reproduced exactly by a dipole of 10 with a potential of 449 MV. Over the last ten thousand years the dipole is thought to have ranged between about 6 and 11 in these units, the band across the middle of the plane. So a record of production becomes a record of the Sun only after a record of the dipole has been divided out, and the solar history inherits every error in the geomagnetic one.$

A Sun read through a shield that also changed

Carbon-14 in tree rings and beryllium-10 in ice extend the record of solar activity ten thousand years past the first telescope, because the Sun's field keeps cosmic rays out of the atmosphere that makes them. But the Earth's own field keeps them out too, and over those ten thousand years it changed by as much as the Sun does over a cycle. An isotope record measures the two shields together, so the Sun comes out of it only after the Earth has been divided out — and every slow trend in the reconstructed Sun is, until shown otherwise, a trend in that division.

6 figures · Solar cycle $X-ray activity against Rossby number, with and without a tachocline. The X-ray luminosity of a star as a fraction of its total, against its Rossby number — the rotation period divided by the convective turnover time — for a synthetic sample of 600 stars of 0.4 to 1.2 solar masses, which have a radiative core and a tachocline below their convection zones, and 220 of 0.1 to 0.35 solar masses, which are convective to the centre and have neither. Both are drawn from one relation: saturated at a ratio of 10⁻³·¹³ below a Rossby number of 0.13 and falling as its −2.7th power above, with 0.25 dex of scatter. The saturated stars of the two kinds average −3.09 and −3.16 in the logarithm; 56 of the fully convective stars rotate slowly enough to lie on the unsaturated branch. This is the shape the measured relation has: the stars with no tachocline lie on the same line as those with one, saturated at the same level and falling along the same slope.$

A dynamo that did not need the shear layer

The Sun's magnetic cycle has long been placed in the thin layer of shear beneath its convection zone, where a rigidly rotating interior meets a differentially rotating envelope. Stars below about a third of a solar mass have no such layer — they are convective to the centre — and they are as magnetically active as the Sun, saturate at the same level, and fade with slowing rotation along the same line. The saturation is the cleaner half of that evidence, because the slope depends on a turnover time that was itself calibrated from activity.

6 figures · Internal rotation

Cosmology

One object, seen once, from inside — and every number in it the output of a model.

The same ruler, and a wave that arrives early. The acoustic oscillation in the galaxy power spectrum — the ratio to a smooth spectrum, minus one — against wavenumber, for no free-streaming species and for three neutrinos, with the sound horizon held at 99.0 h⁻¹ Mpc in both so that only the phase differs. The two agree on the largest scales and separate as the wavenumber grows: by 0.3 h/Mpc the wave with neutrinos runs 0.21 radians ahead, so its crests sit at smaller wavenumber — near 0.2 h/Mpc by 2.2 thousandths of an h/Mpc. The amplitude 0.05 and the damping of the feature by bulk flows over 7 h⁻¹ Mpc are illustrative; the phase is computed.

A wave pulled ahead by particles that outran it

The neutrinos of the early universe moved at the speed of light while the sound wave that became the acoustic feature moved at little more than half of it, and their gravity dragged the wave forward. The drag survives as a phase in the galaxy power spectrum, a shift that no change in the ruler's length can copy, so galaxy surveys can count the species that streamed freely without asking the microwave background.

5 figures · Baryon acoustic oscillations

The trough a smooth fit leaves behind. A trough 150 mK deep, centred at 75 MHz and 15 MHz wide at half depth (dashed), and what remains of it after the same foreground fit that removes the sky, with 3, 5, 7 terms. The fit cannot tell which part of the spectrum is sky and which is signal, so it absorbs whatever part of the trough looks smooth: what remains is 197, 62, 71 mK deep at its lowest, narrower than the injected trough and flanked by bumps above zero that were never in the sky. Measured jointly with the foreground, the trough is worth, in signal-to-noise, a trough 48, 29, 15 per cent as deep with no foreground to remove. What a global experiment measures is not the trough but its projection onto whatever the foreground fit leaves free.

A depth that belongs to the fit as much as the sky

The trough the first stars should have cast is found by subtracting a smooth function from a sky ten thousand times brighter, and a smooth function flexible enough to remove that sky also removes most of any trough as wide as the models predict. What a global 21-centimetre experiment reports is not the trough but the part of it the fit could not imitate, and that part depends on how many terms the fit was given.

5 figures · Reionisation

Exoplanets

Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.

A shoreline measured in light already spent

In the solar system one line of slope four, in the plane of escape velocity against sunlight, separates every world with air from every world without. Drawn against the light a planet receives today it puts the temperate planets of a small red dwarf safely on the side with air. Drawn against the stripping light their star has already spent, it moves them an order of magnitude toward the airless side — and leaves most of them inside a band that the line's unknown height cannot resolve.

5 figures · Exoplanet atmospheres $A second peak, seen through three precisions. The distribution of measured core fractions for a population in which 90 per cent of planets sit at their stars' prediction (0.32, with an intrinsic scatter of 0.03) and 10 per cent at 0.65 (scatter 0.08), each planet compared with its own star, observed with measurement errors of 0.04, 0.08, 0.15 in core fraction. At 0.04 the dense group is a separate hump; at 0.08 it is a shoulder; at 0.15 it has become a long upper tail on a single peak, indistinguishable by eye from a population that is simply broad. The current measurements sit between the second and third curves.$

A second peak the errors have to let through

Whether the densest small planets are a population of their own or the far edge of the ordinary one is not a question about any single planet. It is a question about the shape of a histogram seen through measurement errors as wide as the gap it is looking for, and the number of planets it needs rises fifteen-fold for every factor of three in lost precision — which puts the answer a few dozen planets beyond the samples that first suggested it.

6 figures · Planet composition

Galaxies

Where the unknown stops being a number and becomes a profile — and most of it is not light.

$A polarisation fraction that should change with frequency. The polarisation fraction of dust emission relative to its value at 353 GHz, against frequency on a logarithmic axis. A single population at a single temperature gives a fraction that does not change at all. If the carbon grains are unaligned and warmer than the silicates, they take a growing share of the emission toward high frequency and dilute the polarisation: at 1200 GHz the fraction falls to 0.90 of its 353 GHz value for a 3 K difference and 0.85 for 5 K. Half-aligned carbon halves the effect. The shaded band is flat to within ten per cent between 100 and 1200 GHz, the precision with which the satellite and balloon measurements across this range agree with no change.$

The grains that glow and the grains that align

Interstellar dust polarises starlight and polarises its own glow, and for decades the standard account gave the two jobs to different grains — aligned silicates, unaligned carbon. Two populations at two temperatures predict a polarisation fraction that falls toward high frequency; what satellites and balloons measured was a fraction that does not change. That flatness, which no grain shape can fake, says the grains that glow are the grains that align.

5 figures · Dust polarisation

A disc weighed by how far its stars climb

The rotation curve cannot say how much of a galaxy's speed belongs to its disc, because any split between disc and halo fits it equally well. The stars' motions perpendicular to the disc can, with no halo in the argument at all — and when that was done for thirty face-on spirals the discs came out light. But the vertical weighing needs the disc's thickness, which a face-on galaxy hides, and it needs the dispersion of the stars that hold the disc up, which the light measured may not belong to.

6 figures · Rotation curves

Gravitation

Two bodies pulling on each other, and everything that goes wrong at three.

A Love number measured at nuclear density

The number that told Cassini there was water under Titan's ice is defined the same way for a neutron star, and in the last tenth of a second before two of them merge it changes the shape of the gravitational wave they emit. What the wave measures is mostly not the Love number itself but the star's size, raised to the sixth power — which makes a faint tidal signal into one of the few direct measurements of how large matter at nuclear density lets a star be.

5 figures · Love numbers

Six moons, and how long each takes to move. The time each of Saturn's six largest moons would take to move outward by its own orbital radius, a divided by its rate of change, on a logarithmic scale. Bars: a constant tidal quality factor tuned to the inner moons' measured motion, under which the time grows steeply with distance and falls with mass — 6 billion years for Enceladus, 210 billion for Titan. Dashed: Titan's measured migration, 11 centimetres a year, which is 11 billion years — 19 times faster than the constant-Q prediction and within a factor of two of the inner moons, where the constant-Q picture puts them a factor of 33 apart. Resonance locking predicts exactly that: one time scale for every locked moon, set by the planet rather than by the moon.

A migration timed by the planet and not the moon

Titan is Saturn's most distant large moon, and by the standard account of tides it should be the slowest to move — a two-hundred-billion-year journey across its own orbit. It is moving eleven centimetres a year, twenty times faster, on the same time scale as the small moons close to the planet. A quality factor cannot do that. An oscillation inside Saturn, tuned to each moon in turn and dragged along by the planet's own slow evolution, can.

5 figures · Tidal heating

Orbits

Kepler's three laws, and the family of curves a single force allows.

An orbit seen at an angle keeps everything but its focus

No orbit outside the solar system is seen face on. It is seen as a shadow on the sky, and the shadow of an ellipse is an ellipse, swept at equal areas in equal times, indistinguishable in shape and timing from a real orbit. Only one thing gives the tilt away. The star the companion circles projects to where the true focus projects, and that is not where the apparent ellipse has its focus.

6 figures · Conic sections

A two-body arc flown in the real field. How far from its target a spacecraft arrives when it flies the velocity Lambert's two-body problem gives, through the Earth's real field with its equatorial bulge, against the angle of the transfer, from a 400 km orbit to a point on a 420 km orbit, on a logarithmic scale. In a pure two-body field every solution arrives within a metre of the target; exactly 180° is left out, where the two points and the Earth's centre are in line and the transfer's plane is not determined. In the real field the miss grows with the transfer: at 51.6° inclination, 4.2 km after 60° and 43.0 km after 300°. An equatorial orbit misses most, because the bulge's extra pull is strongest over the equator and acts on it all the way round; an inclined orbit feels it only part of the time but also has its plane turned.

An answer from two bodies, flown through a flattened Earth

Lambert's problem gives the exact velocity that carries a spacecraft between two points in a given time — exactly, for a spherical Earth. The real Earth bulges at the equator, by a thousandth of its central pull, and a spacecraft flying the exact two-body answer between two points in low orbit arrives tens of kilometres from where it was sent. The repair is to keep the two-body solver and lie to it about the target, and the lie converges by a factor of a hundred each time it is told.

5 figures · Lambert's problem

The observed sky

Coordinates, seasons, phases and shadows — geometry seen from inside it.

Where in the sky satellites still shine. The whole sky above the horizon seen from below, zenith at the centre, north at the top and the Sun 20° below the western horizon. Shaded: directions in which a satellite at 550 km is sunlit — 71 per cent of the sky by solid angle. The lit region is a cap pressed against the western horizon and reaching up past the zenith, because satellites seen low in the west are far away and high above the shadow; the eastern sky is dark. Ignoring the penumbra moves its edge by under a degree.

The shadow that also switches off the satellites

After sunset the sky overhead darkens as the Earth's shadow climbs through the air, and the same shadow, climbing on past the top of the atmosphere, decides which satellites still catch sunlight. For a satellite five hundred and fifty kilometres up the zenith stays lit until the Sun is twenty-three degrees down — five degrees past the end of astronomical twilight — and at the latitudes of most large observatories' summers there is no hour of the night when it is not.

5 figures · Twilight

A year that asks for more than a day on some days. The observing time requested per day through the year, with each target's request spread evenly over the days it can be seen, for the same programme filling 90 per cent of the year. The telescope supplies one day per day (dashed). On 104 days the visible targets ask for more than that, peaking at 1.6 days per day when the Galactic centre is in its window; on the rest there is slack. The annual total fits; the seasons do not.

A schedule written by where the Sun is

A space telescope that must keep the Sun at a fixed range of angles sees each part of the sky in windows set by the date, and the sky its users want is not spread evenly. A schedule can fill around a crowded window as long as the crowd fits inside it, by sending everything else to other dates. Once the demand for one region exceeds the share of the year that region is visible, no choice of proposals elsewhere can recover the excess — the limit is each window, not the total.

5 figures · Sidereal time

Spaceflight

Celestial mechanics used forwards — where to burn, and what it costs.

Four satellites, and which four. Two sets of four navigation satellites seen on the sky from below, zenith at the centre and the horizon at the rim, with circles at 30° and 60° elevation. Left: one overhead and three low and spread around the horizon. Right: four bunched in one part of the sky. Each set fixes a position, and the error it leaves is the range error multiplied by the dilution of precision — 2.0 for the spread set, 62.0 for the bunched one. The same four range measurements, made with the same precision, give positions 31 times more uncertain when the satellites are bunched.

Four satellites in view, and which four

A satellite navigation receiver needs four satellites to fix its position, and a constellation is designed to keep at least four above every place. Four is the count, not the answer. The same four ranges, measured equally well, give a position thirty times worse when the satellites are bunched in one part of the sky than when they are spread around it, and the constellations carry their extra satellites to pay for the moments when the geometry, not the count, fails.

5 figures · Ground tracks

Too slow to shatter. The kinetic energy a projectile brings per gram of a 2,000 kg satellite, against the impact speed on logarithmic axes, for projectiles of 1, 10, 100 kg. Above 40 joules per gram (dashed) the standard breakup model counts the collision as catastrophic — the whole target is shattered; below it, only a crater is made. The speeds needed are 12.6, 4.0, 1.3 km/s respectively. In the geostationary ring, where nothing crosses faster than about 0.8 km/s (shaded), only a projectile of hundreds of kilograms — another satellite — can shatter a satellite; in low orbit, at ten kilometres a second, a kilogram can.

A ring where nothing leaves and little shatters

Every argument about orbital debris in low orbit ends in the atmosphere, which pulls fragments down in years or centuries. The geostationary ring has no atmosphere, so its removal term is exactly zero and anything left there stays for millions of years. It is saved by a second fact of the same geometry. Everything in the ring moves at nearly the same speed in nearly the same direction, so collisions there are slow, and a slow collision usually leaves a crater where a fast one leaves a cloud.

5 figures · Orbital debris

Starlight

Distance, brightness and colour: what can be measured when nothing can be visited.

How fast to switch. The rms false polarisation left by a 1 per cent rms brightness fluctuation, against the modulation frequency, both on logarithmic scales, for fluctuations whose spectrum turns down at 5, 20, 100 Hz. Below the turnover the false signal is the whole fluctuation, about 0.7 per cent; above it, it falls steeply as the pairs are taken faster than the air changes. To reach a part in ten thousand (dashed) — the precision at which stellar and planetary polarisation becomes interesting — the switching must run at 3,311 Hz, 13,183 Hz, 66,069 Hz respectively: switching at hertz is useless, and a single beam needs kilohertz or more.

A polarisation measured faster than the air changes

Polarisation is a difference between two brightnesses, and the brightness of a star seen through the air changes by a per cent from one millisecond to the next. Measure the two one after the other and the air's flicker becomes a false polarisation a hundred times larger than the signals worth having. The cure is to switch between them faster than the air can change, or to measure both at once and then swap them — turning a calibration problem into a differential one.

5 figures · Polarimetry

A profile that averages a decade of depth

Each point on a star's disc reports the temperature at a different depth, so a limb-darkening profile looks like a depth sounding of the atmosphere waiting to be read. It is one, but a blurred one. Every angle averages a whole decade of optical depth, the averages overlap, and recovering the temperature structure from them is the inversion of a Laplace transform — a problem whose resolution grows only as the logarithm of the data's precision.

5 figures · Limb darkening

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

A limit written for free electrons

The Eddington limit says a star cannot shine brighter than the point where its light pushes its own gas away, and the push is calculated for free electrons scattering light. In a magnetic field of a hundred trillion gauss an electron is not free. It is bound to the field, it can barely respond to light polarised across it, and the limit calculated for it is the wrong number by a factor of hundreds — which is one of two ways a neutron star can shine five hundred times brighter than it is supposed to.

5 figures · Eddington limit

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