Every essay — page 41
Stars
The diagram that sorted them, and the one quantity that decides a star's whole life.
Galaxies
Where the unknown stops being a number and becomes a profile — and most of it is not light.
A spiral drawn by velocities alone
A map of the Milky Way's gas is built by turning each cloud's velocity into a distance with a circular rotation curve. Gas in a spiral arm does not move on circles: it streams by ten or fifteen kilometres a second in a pattern locked to the arms, so the distance errors are locked to the arms too. Read that way, a disc with no arms in it at all comes out of the calculation with arms.
A galaxy born without a halo
When two galaxies collide, some of the gas flung out along their tidal tails condenses into new dwarf galaxies. Debris launched from a disc cannot take the dark halo with it, so these dwarfs should rotate at the speed their own gas and stars allow and no faster. The relation that predicts every ordinary galaxy's rotation from its baryons says they should rotate twice as fast. One rotation curve decides between a halo and a law — except that the inclination of a lumpy young disc can move the answer by more than the difference.
Cosmology
One object, seen once, from inside — and every number in it the output of a model.
A dimming short by what young stars repay
In an expanding universe a galaxy's surface brightness should fall as the fourth power of one plus its redshift; if light merely tired on its way through static space, as the first power. Measured on elliptical galaxies at redshift 0.85, it falls as about the two-and-a-half power — neither. The verdict on expansion rests on what the galaxies did in the meantime, and the decisive fact is not the size of the shortfall but its sign: expansion needs the galaxies to have been brighter when their stars were younger, and tired light needs them to have been fainter.
A hole too deep for the sky to have made
If the Milky Way sat inside a vast underdense region, everything around it would be expanding faster than the universe as a whole, and the locally measured Hubble constant would come out high — which is what Cepheids and supernovae find. The idea needs no new physics. It needs a void about half empty across the six hundred megaparsecs the supernovae sample, and the same cosmology that predicts the microwave background prices such a void at dozens of standard deviations.
Exoplanets
Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.
A zodi measured in heat and needed in light
The dust around nearby stars that will limit the search for Earth-like planets in reflected light has been measured — but in its own glow at ten microns, by an interferometer whose fringes hide the dust nearest the star. Converting that measurement into the visible brightness at the habitable zone needs the dust's radial profile and its grains' albedo and temperature, none of which was measured. The same ten-micron zodi can mean a quarter or ten times the visible dust a coronagraph will see.
An envelope spent so an ocean could stay
A planet in the eventual habitable zone of a small red dwarf spends its first tens of millions of years too close to a star that is still contracting and still bright, with its ocean boiled into steam and the steam being split and lost. A wisp of hydrogen left over from the nebula — a few hundredths of a per cent of the planet's mass — would change that, because the star's ultraviolet would spend itself removing the hydrogen first. The envelope that shields the ocean and the envelope that never leaves bracket a range a factor of twenty wide, and the amounts a rocky planet is expected to capture fall inside it.
Gravitation
Two bodies pulling on each other, and everything that goes wrong at three.
A rare event reached by cloning the near misses
Counting how often an ensemble of integrations goes unstable works for an event of one in a hundred and becomes hopeless for one in a million, because a ten per cent answer needs a hundred events and so a hundred million futures. Splitting gets there by a different route: wait for the members that come closest, copy them, and run the copies on. Each stage measures a common event and the rare one is a product of common ones — provided the copies forget their parents and the stages know how much time is left.
A relaxation that stops before it is finished
A cold, collapsing system of stars reaches virial equilibrium in a few crossing times without any star meeting another, because the potential changing in time reshuffles their energies. Lynden-Bell's theory predicted the state it should reach — the most probable one. Computed exactly, in a slab of self-gravitating sheets where every force is known, the collapse reaches equilibrium and not that state: the fluctuations that drive the mixing are what the mixing destroys, and they die before the job is done, leaving a profile and an ordering of energies that remember how the collapse began.
Orbits
Kepler's three laws, and the family of curves a single force allows.
A correction that tilts the arc it meant to aim
A transfer designed with Lambert's two-body solution and flown through the Earth's flattened field misses its target, and the standard repair is to aim at the target minus the miss and solve again. That converges fast — until the transfer angle approaches 180°. There the two end points and the Earth's centre are nearly in line, the arc's plane is decided by a sideways shift of the aim, and correcting the bulge's sideways miss rotates the whole arc. The iteration slows by a factor of a hundred, and with a larger perturbation it runs away. Correcting only within the plane fixes it, at the cost of a plane change paid at arrival.
A check that cannot see the clock
The four Lagrange coefficients that carry an orbit's state forward satisfy one identity exactly, and a propagation that breaks it is wrong by about as much as it breaks it. But one equation on four numbers leaves three directions unchecked, and two of them are the inputs every propagation depends on: the time and the strength of gravity. A state carried forward for the wrong time, or under the wrong gravitational parameter, satisfies the identity perfectly — and a solver can be written so that its own errors take exactly that form.
The observed sky
Coordinates, seasons, phases and shadows — geometry seen from inside it.
A year that depends on the season counting it
The tropical year is quoted to eight decimal places, and it is not one number. Counted from the June solstice it is fifty seconds shorter than counted from the December solstice, because the Earth's perihelion moves through the seasons and the planet covers the last of each year's shortfall at whatever speed it happens to be moving there. The Gregorian calendar was fitted to the year counted from the March equinox, which is why its average year is the length it is — and that year is itself drifting. On Mars the spread is fourteen minutes, and a calendar's leap rule can be more precise than the year it is meant to hold.
A polarisation capped by the light the planet catches
Light scattered by a planet's atmosphere is polarised, most strongly when the planet is half lit, and the polarisation angle turns as the planet orbits — so a planet's polarised light would weigh its atmosphere and draw its orbit on the sky without a transit. In 2008 it was reported for a hot Jupiter at a part in ten thousand of its star's light. That amplitude is more than the planet can send: no atmosphere reflects more than three-quarters of the light that falls on it or polarises more than all of it, and the light that falls on it is set by its size and its distance. The photons for a real detection are cheap; what limits it is a polarimeter stable to a part in a million.
Spaceflight
Celestial mechanics used forwards — where to burn, and what it costs.
A turn spread where the climb leaves room
An electric spacecraft spiralling from low orbit to geostationary has to remove its launch inclination on the way, and the classic steering law tilts its thrust out of the plane by the same angle all the way round each orbit. But out-of-plane thrust turns the plane only near the nodes, while in-plane thrust raises the orbit equally well everywhere. Letting the tilt follow the nodes — leaning hard where turning works and lying flat where it does not — saves 150 metres a second on a transfer from Cape Canaveral, with no coasting, because the climb fills exactly the parts of each orbit where the turn would be wasted.
A height the map already knows
A navigation service for the Moon's south pole needs only a handful of satellites, if they are put in elliptical orbits that hang for hours over the pole at the high point of each lap. But satellites hanging over a pole all stand at much the same height in the sky, on a cone, and four ranges from a cone cannot tell whether the receiver is lower or its clock is late. On the surface the receiver knows its height already, from a map of the terrain — and that one number, used as a measurement, does more for the service than two more satellites.
Starlight
Distance, brightness and colour: what can be measured when nothing can be visited.
A temperature read against the light's long tail
A star's total light depends on its temperature to the fourth power; its light in the infrared depends on it barely more than linearly, because the infrared lies on the long, flat tail of the Planck curve. The ratio of the two cancels the star's size and distance and leaves a steep function of temperature — and the only model it needs is the infrared surface flux, whose errors are divided by two and a half on the way to the answer. That is the infrared flux method, one of the two foundations of the stellar temperature scale, and the other foundation disagrees with it by up to a hundred and fifty kelvin.
Where a shrinking star stops boiling at its centre
A young star descends the Hertzsprung–Russell diagram at almost fixed temperature, convective from centre to surface, until something at its centre changes and it turns sharply left toward the main sequence. The something is the opacity. As the star shrinks its centre grows hotter and denser, the Kramers opacity there falls, and radiation becomes able to carry the luminosity without the steep gradient that drives convection. A Sun-like star crosses that line at about twice its present radius, two million years old; a star of a third of the Sun's mass never crosses it at all.
Stars
The diagram that sorted them, and the one quantity that decides a star's whole life.
A turnover whose position is an age
The mass function of a young cluster is measured by counting its objects by brightness and converting each brightness into a mass. Below the hydrogen-burning limit nothing holds a brown dwarf's brightness steady: it fades as it contracts, so the conversion needs the cluster's age, and an age wrong by a factor of two moves every mass by about forty per cent. The shape of the substellar mass function survives the error; its position on the mass axis and the ratio of brown dwarfs to stars do not — and comparing those between clusters is how the universality of star formation is tested.
Four clocks and the stars none of them reads
An isolated dwarf star can be dated four ways — by its position on the Hertzsprung–Russell diagram, by how fast it spins, by its oscillations, and by the ratio of two elements its galaxy made at different rates. Each clock has an age range where it works and masses it cannot read. Laid side by side for a Sun-like star they cover most of its life to ten per cent or so. For a star of eight-tenths of the Sun's mass only one clock is left, it works only while the star is young, and the commonest stars in the Galaxy's disc are dated one at a time no better than fifteen per cent at any age.
Gravitation
Two bodies pulling on each other, and everything that goes wrong at three.
