The collection

Every essay — page 37

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 721–740 of 909.

The observed sky

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

A shadow 77 mm across at 144.3 m: an occulter 1.42 m wide and a 50 mm aperture inside its umbra. Side view of an eclipse made by two spacecraft, distances along the axis in metres and widths across it in metres, the vertical scale magnified about fifty times. Sunlight arrives from the left. The occulting disc, 1.42 m across, casts an umbra — the cone from which none of the Sun's disc is visible — which narrows because the Sun is not a point: its edges converge at a rate set by the Sun's angular radius, 959.6 arcseconds, and would meet 153 m behind the disc. At 144.3 m, where the coronagraph sits, the umbra is 77 mm across and the aperture 50 mm; the penumbra, from which part of the Sun is visible, extends to 2.76 m. From the aperture the occulter's edge hides the Sun out to 1.058 solar radii, so the corona is visible from just beyond the limb.

An eclipse held in place to a millimetre

A total eclipse of the Sun works because the Moon's disc happens to match the Sun's almost exactly, and it lasts a few minutes. Two spacecraft flying 144 metres apart can make one that lasts six hours: a disc 1.4 metres across, with a coronagraph sitting in its shadow. The geometry that sets the shadow's size also sets the price. At that separation the umbra is 77 millimetres across around a 50-millimetre aperture, so the formation has to hold itself in line with the Sun to a centimetre — and to two millimetres in January, when the Sun is at its largest.

5 figures · Phases and eclipses

Exoplanets

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

$The radius a rocky planet would have, by core fraction: at 5 M⊕ an Earth-like and a Mercury-like planet differ by 8%. Radius against mass for rocky planets of one to eight Earth masses, from Zeng and collaborators' interior models, for core mass fractions of 0, 0.33, 0.5, 0.7, 1 — from a planet with no iron core to one that is all iron. The shaded band is the range of core fractions planets would have if they kept their host stars' iron, magnesium and silicon in the stars' own proportions, for stars whose iron-to-magnesium ratio is within 0.15 dex of the Sun's: 0.26 to 0.41. The Earth, at 0.325, sits in it. At five Earth masses an Earth-like planet and one with a core fraction of 0.7, like Mercury, differ in radius by 8 per cent — a difference that transit and radial-velocity measurements can now resolve.$

More iron than the star could give

A rocky planet is made of what its star was made of, minus the gases: iron, magnesium and silicon in the star's proportions, which for the Sun predict an iron core of a third of the planet's mass — and the Earth's core is 32.5 per cent. A handful of small exoplanets are so dense that their cores must hold seven-tenths of their mass, more iron than any star known to host planets could have supplied. Either something removed most of their rock after they formed, or the rock and the metal were separated before they did.

5 figures · Planet composition

Four rocky planets against the bare-rock limit: TRAPPIST-1b at 0.99, TRAPPIST-1c at 0.87, LHS 3844b at 1.01, GJ 367b at 0.99. Published day-side brightness temperatures of four small planets orbiting M dwarfs, each divided by the temperature its day side would have as dark bare rock re-radiating where it absorbs (dot, with its uncertainty), with the value for heat spread evenly over the whole planet marked for comparison (grey tick, at 0.71). TRAPPIST-1b: 503 ± 26 K against a bare-rock 508 K; TRAPPIST-1c: 380 ± 31 K against a bare-rock 434 K; LHS 3844b: 1040 ± 40 K against a bare-rock 1031 K; GJ 367b: 1728 ± 90 K against a bare-rock 1741 K. Three sit at the bare-rock limit within their errors, which leaves room for no thick atmosphere on dark rock. TRAPPIST-1c sits about one in eight below it, which a thin atmosphere could explain, and so could a moderately reflective surface. None comes near the even-redistribution value that a thick atmosphere would give.

A day side as hot as bare rock

A planet with no air can be no hotter on its day side than bare rock that re-radiates every photon where it lands. An atmosphere carries some of that heat to the night side, so the day side comes out cooler. For a rocky planet too small for its spectrum to be read, that difference is the test: three of the first four small planets around red dwarfs measured this way sit at the bare-rock limit, and the fourth sits below it by an amount that a thin atmosphere, or a brighter surface, could explain equally well.

5 figures · Exoplanet atmospheres

Spaceflight

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

Three sails against the two routes that have flown. Inclination reached against years since launch. The Venus staircase and the Jupiter step are the arithmetic of the launch-window figures — repeated Venus flybys at a v∞ of 15 km/s reaching 23°, and a single Jupiter flyby after 1.2 years reaching 90° on a heavy launch. The sails, of lightness number 0.020, 0.040, 0.080, spiral in to 0.48 AU — flat lines, 4.6, 2.3, 1.1 years — and then climb at a constant rate until they are polar. They pass the Venus route after 9.0, 4.5, 2.2 years; only the lightest catches Jupiter inside the decade, and none ever stops climbing.

A climb out of the ecliptic paid for in orbits

A solar sail can tilt its orbit without propellant by leaning against sunlight and changing which way it leans twice an orbit. Each orbit then adds the same few degrees whatever its size, so the climb runs faster the closer to the Sun the sail goes. Nothing in the orbit fixes how close that can be. The sail film's temperature does.

6 figures · Launch windows

The one axis the field will not take

A satellite that keeps one face towards the Earth can empty its reaction wheels into the Earth's magnetic field with coils, using no propellant. Its own rotation, once an orbit, cancels every steady torque except those about one axis, the pitch axis normal to the orbit. In an orbit near the equator the field lies along that same axis, and a coil cannot push about the direction of the field.

5 figures · Attitude control

Starlight

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

How many stars it takes to tell a shared axis from chance. The number of stars whose measured sin i, with an error of 0.1, is needed on average before an aligned cluster is preferred over random axes by a likelihood ratio of 370 to one — three standard deviations — against the inclination of the common axis, on a logarithmic scale, for perfectly aligned, κ = 20, κ = 5. Perfect alignment needs a handful of stars at any angle — 4 at 30°, 10 at 85°. A loose alignment, κ = 5, needs 29 about an axis at 30° and 38 about one at 85°, and 985 about one at 55°: a loose cone about an axis near 54.7°, where 3cos² i₀ = 1, has the same mean cos² i as random axes and is the hardest of all to tell from them.

A shared axis hides at the angle where it does no harm

If the stars of a cluster were born spinning about a common axis, their inclinations would show it. A tight alignment shows in a handful of stars at any orientation. A loose one about an axis tilted 54.7° from the line of sight has the same average inclination as random axes and takes a thousand stars to find. At that same angle, a shared axis also stops biasing the cluster's brightnesses.

5 figures · Gravity darkening

The cancellation that protects the method, and where it fails. The error in distance from 0.1 magnitudes of extinction left uncorrected, against temperature, for the angle read from V and V − K through the blackbody relation's local slope, and for a standard candle in V (dashed, −4.7 per cent). For late-type stars the dimming and the reddening nearly cancel — the empirical relation gives +1.0 per cent. For hotter stars the reddening term grows and nothing offsets it: the error matches the standard candle's near 13,500 K and reaches +7.5 per cent at 20,000 K and +11.9 at 30,000 K, worse than the candle the method was meant to improve on.

The cancellation that leaves with the hot stars

The angular size of a cool giant can be read from its colour, and dust barely spoils the answer: the dimming and the reddening dust causes push in opposite directions and nearly cancel. Carried to hot stars the method gets weaker on both counts. Above about 10,000 K the colour V − K stops tracking temperature, a colour error costs more angle, and the reddening outweighs the dimming. Near 13,500 K leaving the dust uncorrected costs as much as it would a standard candle.

5 figures · Angular diameter

Stars

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

$The share of white dwarfs whose centres have begun to freeze. The fraction of a population of carbon–oxygen white dwarfs, with masses spread as a Gaussian of mean 0.62 and width 0.12 M☉, whose centres have begun to crystallise, against cooling age, overall and in three mass ranges, in the toy model of the other figures. If a field appears when freezing begins, this is the shape its incidence must take: it switches on first among the heaviest white dwarfs, half of which have begun freezing by 0.5 Gyr, and last among the typical ones near 0.6 M☉, by 2.5 Gyr. A fossil field, present from the white dwarf's birth, would show no such delay and no such ordering by mass.$

A field that waits for the core to freeze

A field inherited from the parent star would be present from a white dwarf's birth. Yet magnetic white dwarfs of ordinary mass are rare among the young and common among the old, as if their fields were switched on later. The proposed switch is the freezing of the carbon–oxygen core, which stirs the liquid above it. If that is right, the delay is set by when each mass begins to freeze, and the heaviest should go first.

5 figures · Flux freezing $Larger and cooler, along a line of constant light. The fractional change in effective temperature against the fractional change in radius for a star whose luminosity is unchanged, and for luminosities −10% and +10%, from L = 4πR²σT⁴. Holding the luminosity, a radius 10 per cent larger needs a temperature 4.7 per cent lower: about half the radius change, with the sign reversed. The shaded box is the range reported for red dwarfs in eclipsing binaries against standard models — radii 5 to 15 per cent too large, temperatures 3 to 5 per cent too low — and it lies along the line of constant luminosity.$

Too large and too cool by just enough to keep the light

Red dwarfs in eclipsing binaries, whose radii are measured directly, come out several per cent larger than stellar models predict for their masses and a few per cent cooler. They are not brighter or fainter. The light is decided deep inside, by mass and opacity, and whatever disturbs the convecting outer layers can change only the radius and temperature through which that light leaves.

5 figures · Energy transport

Cosmology

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

Where solved fields land in the plane the surveys fit. The w₀–wₐ plane, with the cross at a cosmological constant. Each small point is the best two-parameter description, through the same mock survey as the earlier figures, of a scalar field solved from its own equation: four thawing potentials at seven values of w₀ from −0.97 to −0.62, and freezing trackers on inverse-power potentials with exponents from a quarter to two. The thawing fields all land on one line through the constant, wₐ ≈ −1.49(1 + w₀), with no point more than 0.08 from it; the freezing fields land on the other side, with wₐ positive. The three published fits (larger points, with their quoted errors) lie below the thawing line by −0.38, −0.59, −0.49 in wₐ: steeper than any thawing field with their present value would be. The diagonal is w₀ + wₐ = −1. The thawing line is steeper than it, so every thawing field is fitted as having crossed −1 — but early: these fits cross between redshift 1.7 and 3.3, where a line through the constant with this slope must, near z = 1/(1.49 − 1) = 2.0. The published fits cross near redshift 0.4.

Every thawing field falls on one line

A scalar field that is the dark energy sits still while the universe expands quickly, and begins to roll only when the expansion has slowed to the field's own mass. Solve the field instead of fitting a line to it, and four quite different potentials give one history and one line through the plane the surveys fit — a line the published fits do not sit on.

6 figures · Dark energy

How much each change moves the growth rate. The percentage difference in fσ₈ from growth in relativity inside a cosmological constant's expansion, against redshift, for three alternatives started from the same early amplitude. Changing the expansion to the evolving fit with DES-SN5YR and keeping relativity moves it by −1.2 per cent at redshift 0.5, and by at most 2.1 per cent anywhere below redshift 2. Keeping the constant's expansion and changing gravity to γ = 0.68 moves it by −8.6 per cent; doing both, −10.3. The shaded band is ±8 per cent, about the error of one of the better redshift-space measurements so far. The difference between a constant and the evolving fit is several times smaller than any single measurement; the difference between two laws of gravity is comparable to one. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is gravity.

The growth that tells a substance from a law

In relativity an expansion history leaves no freedom in how fast structure grows inside it, so every fitted history predicts its own growth. The evolving dark energy the surveys prefer changes that growth by one or two per cent; a change in the law of gravity, with the expansion held fixed, changes it by nine. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is there at all.

5 figures · Dark energy

Exoplanets

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

The Earth-sized bin is where diluted planets land

A star with an unresolved companion shares its aperture with that companion's light, and every planet found around either star is measured too small. The error runs one way only, so the smallest radius bin receives planets from every larger one and returns none. With a companion beside three stars in ten, more than a quarter of the planets called Earth-sized are something larger.

5 figures · Detection bias $The cadence that finds a planet and the cadence that weighs it. The fraction of caustic crossings — seven tracks across the planetary caustic at offsets spanning it, each sampled at 160 phases — in which a survey of a given cadence detects the anomaly (three points deviating from the single-lens curve by more than five times a 1 per cent photometric error, solid) and in which it also measures the source size (at least two of those points where the source's finite size changes the magnification by more than five per cent, which happens only during a fold crossing; dashed), for a Jupiter and an Earth around a third-of-a-solar-mass star. For the Jupiter, detection stays at 100 per cent at a four-hour cadence and 48 per cent at one day, while the source size is measured in 43 per cent of crossings at fifteen minutes, 15 at four hours and 0 at a day. For the Earth what limits the measurement is the track rather than the cadence: the source size is measured in 43 per cent of crossings at fifteen minutes and still 43 hourly — the tracks that cross a fold — while detection falls from 100 to 54 per cent, and at four hours both are 0. A cadence that finds giant planets reliably measures few of them; for small planets the cadence that finds them is already the one that measures them.$

A cadence that finds a planet is not one that weighs it

A planetary anomaly on a microlensing event is found by a handful of points that deviate, and it is turned into a mass only by points taken while the source crosses the caustic — a few hours for a Jupiter whose anomaly lasts days. Sampled every four hours, every Jupiter crossing in a simulated survey is found and fewer than one in six is weighed; the field layout that finds the most planets is not the one that weighs the most.

5 figures · Microlensing

Galaxies

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

A dispersion that weighs the sheet only if the orbits are known

A lens's images cannot tell how much of its mass is in the galaxy and how much is spread through a sheet around it, and the time delay — and so the Hubble constant — scales with the answer. The stars' velocity dispersion can tell, because orbits feel mass in three dimensions where light only integrates along a line. One dispersion through one aperture weighs the sheet only to sixteen per cent, because the shape of the orbits moves it as much; the same dispersion measured in rings weighs it to four.

5 figures · Strong lensing $How far a sample of lenses can take the Hubble constant. The fractional uncertainty in H₀ from a sample of N lenses, both axes logarithmic: the independent errors of each lens — 2 per cent from its delay, 3 from the rest of its lens model, 2.5 from its line of sight and its own mass sheet as well as its kinematics fix it — fall as 1/√N; the shared errors — 1 per cent from the calibration of the line of sight, and the orbital bias of the previous figure — do not. With one aperture dispersion per lens the independent error is 10.1 per cent per lens and the floor 1.6 per cent, reached at about 42 lenses; with six resolved rings per lens, 5.9 and 1.1 per cent, reached at about 29. Dotted: the two floors. A sample that adds lenses without changing how each is measured stops improving at its floor; the resolved kinematics lower both the slope's starting point and the floor.$

A hundred lenses and one assumption

A time-delay lens gives the Hubble constant to five or ten per cent, and the hope is that a hundred give it to one. The errors that differ from lens to lens — its delay, its own line of sight, its own internal sheet — do average away. The errors every lens shares because every lens was analysed the same way do not, and they set a floor that a sample reaches after a few dozen lenses and never goes beneath.

5 figures · Strong lensing

Gravitation

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

The binaries in a fed core are spread evenly in energy. The steady distribution of binding energies among the hard binaries of a core where new binaries enter at x = 5 and leave by ejection from a cluster whose escape speed is 4 times its dispersion: the number per unit binding energy, against binding energy in units of mσ², from 20,000 simulated binaries weighted by the time each spends at each energy. Between twice the entry energy and a third of the typical ejection energy (about 120) it is flat to within 5 per cent, and it falls away smoothly near ejection because a single encounter can throw a binary out from anywhere above about 24. A flat distribution is what a population with a constant flux through energy and a constant drift must have: binaries pass every energy at the same rate, and spend the same time at each.

The binaries a core keeps are spread evenly in energy

A hard binary in a star cluster gains binding energy at the same average rate however hard it already is. So in a core that is fed new binaries and loses hardened ones, every binding energy is passed through at the same speed, and the steady population is flat in energy — which, counted per factor of energy, means most binaries are within a factor of ten of being thrown out. Where they were born hardly matters; where they leave decides everything.

5 figures · Binary heating

Ejected from a globular, merged in a nucleus

A hard binary in a star cluster hardens until something ends it: an encounter that throws it out, or its own stars touching. Which comes first is decided by one comparison — the separation at which the cluster's escape speed is exceeded against the separation at which the stars meet — and the first shrinks as the square of the cluster's velocity dispersion. In a globular cluster binaries leave while their stars are still apart; in the nucleus of a galaxy they merge first, and the heating that holds the cluster up becomes a factory of stars more massive than they should be.

5 figures · Binary heating

Orbits

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

Jupiter keeps only the visitors already nearly bound

An interstellar object passing Jupiter can lose enough speed to stay, but one flyby can only turn the visitor's velocity relative to the planet, never shorten it, and the turn removes at most a few kilometres a second from its speed around the Sun. So capture is common for visitors that arrive with less than a kilometre a second to spare and almost impossible for the speeds the three known visitors had. A census of the slow tail of the interstellar population, and the orbits a capture leaves — more than a third of them backwards — follow from the same arithmetic.

5 figures · Hyperbolic orbits

Spun apart, or run out of ice: two clocks against size. Numbers of orbits against nucleus radius, both axes logarithmic, for a comet on 41P's orbit losing mass at 41P's rate per unit area. Solid: the orbits until the nucleus has lost all its mass to sublimation, growing in proportion to the radius. Dashed: the orbits until outgassing torques, changing sense at random from one orbit to the next, spin it past the rate at which a strengthless body of density 500 kg/m³ comes apart (a period of 4.7 hours) — growing as the fourth power of the radius, because each orbit's change of spin goes as the inverse square and a random walk needs the square of the number of steps. With a torque efficiency of 0.01, about what other comets' measured spin changes suggest, the two clocks cross at a radius of 1.2 km; with 41P's own 0.044, at 3.3 km. Smaller nuclei are spun apart before they run out of ice; larger ones run out first. The crossing moves as the two-thirds power of the efficiency.

The jet that pushes a comet also turns it

The gas a comet sheds near the Sun pushes its orbit, and because the vents are not arranged symmetrically about its spin axis, it also twists the nucleus. In 2017 one small comet's day more than doubled in two months. The twist per orbit grows as the inverse square of the nucleus's radius while its supply of ice grows only as the radius, so below about a kilometre a comet is spun apart before it runs out of ice — and the shortage of small comet nuclei may be the result.

5 figures · Non-gravitational forces

Stars

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

What DI Herculis's orbit should do, and what it does. The rate at which the periastron of the eclipsing binary DI Herculis advances, in degrees per century. Top: the prediction for stars whose spins are aligned with the orbit — 2.34 from general relativity, computed from the masses (5.17 and 4.52 solar masses), the 10.550164-day period and the eccentricity of 0.489; and 1.93 from the stars' tidal and rotational distortion, as published stellar models give it, of which the rotational flattening is 1.50 and the tides 0.43. Total 4.27. Bottom: the observed rate, between about 0.6 and 1.4 degrees per century in the published timing analyses — a quarter to a seventh of the prediction. For a quarter of a century after the discrepancy was established it was cited as possible evidence against general relativity, which contributes more than half the predicted total by itself.

A precession slowed by stars lying on their sides

The eclipsing binary DI Herculis turns the long axis of its orbit at a quarter to a seventh of the rate relativity and the stars' own distortion predict, and for a quarter of a century that gap was offered as evidence against general relativity. The answer was in the stars: both spin nearly on their sides, and a flattened star tipped past 54.7 degrees drags the orbit's periastron backwards instead of forwards. Explained, the slow precession becomes a measurement of something no spectrum shows — how far the spins lean towards the Earth.

5 figures · Binary stars

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