The collection

Every essay — page 39

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 761–780 of 909.

Spaceflight

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

The cost of turning an orbit, against how much of each revolution the engine runs. The velocity spent per unit of inclination change, relative to an instantaneous turn at a node, when a small out-of-plane thrust is applied only in arcs centred on the two nodes, against the half-width of each arc (blue). Thrusting all the time costs π/2, 1.571; arcs of ±60° cost 1.209; ±30°, 1.047; ±10°, 1.0051. The price is time: the engine is off for the rest of each orbit, and the turn takes longer by the inverse of the duty times the efficiency (orange, right-hand scale) — relative to a push that was always fully effective, 1.57 times as long for continuous thrust, 3.14 for ±30°, 9.0 for ±10°. Most of Edelbaum's penalty is spent in the quarter-orbits near the points where a push turns the node instead of the plane.

A plane turned only where turning is cheap

A low-thrust engine that turns an orbit's plane continuously pays π/2 times what an impulse at the node would cost, because for half of every revolution it is pushing where a push mostly swings the node instead of tilting the plane. Thrusting only in arcs about the nodes recovers almost all of that: arcs of ±30 degrees cost five per cent more than an impulse, at the price of taking twice as long. It is how geostationary satellites hold their planes, and it stops working for large turns, where a chord is shorter than any arc.

6 figures · Plane change

Starlight

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

A spot a slow star hides inside its velocity

A dark spot on a rotating star removes light from one part of each spectral line, and the line's centre seems to move: a velocity of metres per second that no planet caused. In principle the spot betrays itself, because it changes the line's shape and a planet does not. In practice the betrayal is a few per cent of the signal on a star that rotates as slowly as the Sun, and seeing it takes a signal-to-noise of hundreds per pixel across thousands of lines.

6 figures · The Doppler effect

Spectra that age, and spectra that stop ageing. Synchrotron spectra in a 3-microgauss field, divided by the straight power law the electrons were injected with so that a spectrum with no break would be a flat line at zero, at ages of 10, 30, 100 million years. Dashed: electrons left alone, whose break falls steadily to lower frequency. Solid: the same electrons stirred throughout, gaining energy at a rate of one e-fold every 30 million years against their losses. The stirred spectra's break moves down at first and then stops, at the frequency radiated by electrons of Lorentz factor 41,763, where gain and loss balance. Read as passive spectra, the stirred ones give ages of 8.5, 18.9, 28.8 million years: the last is 3.5 times younger than the electrons are.

A lobe that reads the time since it was stirred

If a radio lobe's electrons are re-energised in place by the turbulence around them, the break in their spectrum stops falling and its spectral age stops growing, settling at the time the stirring takes to double their energy a few times over. The spectrum's shape does not betray it: a stirred lobe fits an unstirred one exactly. What betrays it is a map that stops ageing with distance, and a bill for energy that grows exponentially with the time the stirring lasts.

6 figures · Synchrotron radiation

Stars

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

The rise in survival, and what two detectors see of it. The survival probability of ⁸B electron neutrinos against neutrino energy (grey), and what two detectors measure of it against the energy of the electron they record: the ratio of the counted spectrum to the one expected without conversion, for electron scattering in water (blue) and for charged-current capture on deuterium (orange), both smeared by a water detector's energy resolution. Between 3 and 10 MeV the survival probability rises by 0.122. Scattering shows a rise of 0.044 over the same recoil energies, 36 per cent of it; capture on deuterium shows 0.063, 52 per cent.

A rise a detector shrinks to a tilt

Between 10 and 3 MeV the chance that a solar neutrino arrives as it left should rise by a third of its value, and no detector has clearly seen it. Counting is not the obstacle: a large water detector has many times the exposure the rise needs. What hides it is that scattering shows only a third of the rise, spread over the spectrum as a tilt of a few per cent, and that half of the evidence for the tilt sits where the radioactive background is and the other half where an error in the energy scale looks the same.

6 figures · Solar neutrinos

A lone star is dated only as it leaves

A star in a cluster is dated by the bend its neighbours trace. A star alone has one position on the Hertzsprung–Russell diagram, and an age for it means finding which isochrone passes through that point. On the main sequence they almost all do: a Sun-like star measured as well as a parallax and a spectrum allow is anywhere from three and a half to five and a half billion years old. A star the same mass on its way off the main sequence is dated to two per cent.

6 figures · The HR diagram

Cosmology

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

The echo of a one-day flare, for four void fields. The 1–10 GeV flux of the cascade echo against the time since the start of a one-day teraelectronvolt flare of 10 Crab-days, from a blazar 500 Mpc away, both on logarithmic scales, for void fields of 10⁻²¹ G, 10⁻²⁰ G, 10⁻¹⁹ G, 3 × 10⁻¹⁹ G. The echo carries 1.0 × 10⁻⁴ erg cm⁻² in the band whatever the field — the field decides only when it arrives, and so how thinly it is spread. Half of it has arrived 14 hours after the flare began in the weakest field and 4.1 years in the strongest, and the flux at that moment falls from 1.1 × 10⁻⁹ to 1.3 × 10⁻¹³ erg cm⁻² s⁻¹. The dashed curve is the flux a space telescope needs to detect a source averaged over that long, falling as the square root of the time to a floor of 10⁻¹² erg cm⁻² s⁻¹. Counting only what arrives after the flare has ended, three of the four echoes clear it.

An echo heard only from the weakest fields

A missing gamma-ray halo gives the field in the voids a lower limit and nothing more. A flare's echo could give it a value — the gigaelectronvolt light that arrives late because the void field bent its path. But the echo carries a fixed amount of energy and the field decides only how long it is spread over, so it can be heard only from fields weak enough to deliver it quickly, and only after flares brighter than almost any yet seen from a blazar far enough away to make one.

7 figures · Primordial fields

A conversion read galaxy by galaxy, with or without an alignment. The mean ellipticity of disc galaxies along a tidal template, in bins of each galaxy's own spin estimator — the quantity whose population average is the Lee–Pen alignment — for 60,000 discs a fifth as thick as they are wide, in three samples: spins set by the torque (net alignment 0.36), 85% of them randomised (0.054), and all of them random (0.002). The points trace the same curve through the origin in every sample — not quite straight, flattening at both ends, because a disc's shape saturates when its spin lies in the sky's plane — and a straight line fitted through each sample (dashed) has slope 0.279, 0.290, 0.292. That slope is the projection factor, what a disc's shape shows of its spin, and it does not care whether the population is aligned. A sample with no alignment at all measures the conversion as well as one with a strong one; only the population's mean, where all its galaxies pile up, moves.

A conversion measured without an alignment

The factor that turns a disc galaxy's spin alignment into the shape alignment a lensing survey fits looked like something only a sample with a detected alignment could test. It is not. It is a relation each galaxy obeys on its own, so it can be read off a few hundred discs with velocity maps whether or not their spins point anywhere in particular — and what is left for the hard measurement to test is only whether a disc's shape is anything more than its projected spin.

7 figures · Tidal torque theory

Exoplanets

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

A fraction read in each channel's own units

Hot Jupiters arrive by two routes, one that ends at twice the planet's Roche distance and one that ends where the disc stopped. Fitting the mixture in Roche distances alone, as the edge invites, gives a fraction that drifts by almost half if the planets weigh more than assumed. Each route has to be read in the units in which it does not move — and then the answer is limited not by how well the planets are measured but by how their radii have changed since they arrived.

6 figures · Planet migration

A helium line that weighs the star first

The metastable helium line at 1083 nanometres made the escape of a planet's atmosphere measurable from the ground. But the atoms it sees exist only in a balance between the star's extreme-ultraviolet light, which fills the level, and its mid-ultraviolet, which empties it — so the same outflow is six times deeper around an active K dwarf than around an F star, one depth allows a fifty-fold range of mass-loss rates across the wind's possible temperatures, and a rate inferred from it inherits a share of every error in a spectrum that cannot be observed.

6 figures · Atmospheric escape

Galaxies

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

$The angular momentum a halo has and the angular momentum a disc needs. The fraction of the mass with specific angular momentum below j, against j in units of the mean, for a halo's material — the universal profile fitted to simulated haloes, with shape parameter μ of 1.1, 1.25, 1.5 — and for an exponential disc with a flat rotation curve (black). Both have the same mean, as a disc that kept its halo's angular momentum would. They do not have the same shape. Below a tenth of the mean a disc holds 2% of its mass; a halo with μ = 1.25 holds 11%, and one with μ = 1.1, 15%. The halo also stops at a largest j, 4.0 times the mean for μ = 1.25, beyond which the disc still has a thin tail. A disc built from a halo's gas without any exchange of angular momentum would be too concentrated at its centre and cut off at its edge.$

A halo with too much slow gas for a disc

A disc's size follows from its halo's average spin, and the average is all the simplest account of disc formation uses. The whole distribution does not match. A halo holds five times as much of its material at low angular momentum as an exponential disc can accommodate, so gas that kept its angular momentum exactly would build a dense centre, a thin body and a hard edge — and the bulgeless discs that are common in the nearby universe say that something removed a fifth to a quarter of the slowest gas before it settled.

6 figures · Galaxy spin

A stirring supernovae cannot pay for

The gas in a disc forming stars at redshift two moves at fifty kilometres a second in random directions, five times faster than in a spiral today, and turbulence that fast decays in a few tens of millions of years. Supernovae are the obvious source of the energy, and they run out at about thirteen kilometres a second however fast the disc forms stars. What holds the discs up is gravity — gas stirring itself by falling inward — and a disc stirred that way is also a disc pouring its gas into its own centre.

6 figures · Tully–Fisher

Gravitation

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

Sixty futures that began a trillionth apart. The momentum of the standard map at K = 3, a kicked rotor whose phase space is almost entirely chaotic, for 60 members started 10⁻¹² apart in angle, against iteration. For 41 iterations they are one trajectory; then the separation, growing by a factor of about 2.0 each step, reaches the size of the system and they become independent random walks. Three of the 60 cross the threshold at |p| = 90 within 400 iterations (dark), the stand-in for an orbit that goes unstable. Which members cross is decided by differences far below anything that could be measured; how many cross is a property of the dynamics, and it is the only thing the ensemble can be asked for.

A probability for a system with one future

The Solar System will do one thing, and an integration of it is wrong about which thing within a hundred million years. What long integrations report instead is a probability — one per cent that Mercury's orbit goes unstable before the Sun dies — and a probability of a single event is a strange object. It is a count over an ensemble of futures that differ by less than anything measurable, which makes sense only because chaos makes those futures independent; and how many of them are needed is a binomial question with an unforgiving answer for rare events.

6 figures · Numerical integration

The flattening a halo has, and the flattening an orbit feels. Three model haloes with flat rotation curves, each built from thin similar-shell layers whose axis ratio changes with radius (dashed), and the axis ratio a nearly circular orbit at each radius responds to (solid) — the a²-weighted mean over the shells inside the orbit, because a shell exerts no force on anything within it. A halo flattened to 0.85 everywhere is felt as 0.85 everywhere. A halo round inside and flattened to 0.8 beyond about 30 kpc is felt as round by every orbit within 20 kpc — 0.988 at 20 — and at 60 kpc as 0.83, where its own shells are 0.80. A halo flattened inside and round outside is felt as flattened well beyond where its shells have become round. An orbit reports a running average of everything inside it, weighted towards its own radius, and nothing at all about what lies outside.

A shell outside the stream turns nothing

A flattened halo turns the plane of every orbit inside it, and a stream of stars several orbits long records how far. But the turning is set only by the halo inside the orbit — a flattened shell enclosing it exerts no force on it at all — and within that, mostly by the shells just inside. So a stream measures a running average of the halo's shape out to its own radius, a halo round inside and flattened outside looks round to every stream in its core, and reading the shape of each layer back out of the streams costs more precision the further out the layer lies.

6 figures · Shell theorem

Orbits

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

Which pairs of planets the angular momentum deficit can bring together. For each adjacent pair of planets, the system's angular momentum deficit divided by the critical deficit at which that pair's orbits could touch, on a logarithmic scale; below one the pair cannot collide under secular evolution, whatever happens. Blue: using only the deficit of the pair's own subsystem — the four terrestrial planets (7.8 × 10⁻⁹ in units of a solar mass times √AU) or the four giants (4.3 × 10⁻⁶). Orange: using the whole Solar System's deficit, which the giants hold almost entirely. On their own budget the inner pairs sit at 0.21, 0.12, 0.34 — safe. On the whole budget they sit at 112, 65, 187. The giants are safe either way, at 0.05, 0.18, 0.36. The terrestrial planets are stable because almost none of the giants' deficit reaches them, not because there is too little of it.

A deficit the giants hold and Mercury could borrow

The angular momentum the planets' orbits lack of circular motion is conserved by every secular interaction, to every order, so it is a budget no planet can overdraw. Compared with what it would take to bring a pair of orbits into contact, it proves without an integration that the terrestrial planets cannot collide among themselves — and shows that they are safe only because the giants, who hold five hundred times as much, lend them almost none. A loan of half a per cent of Jupiter's would be enough to unsettle Mercury.

6 figures · Secular theory

Where a spinning sphere of rubble is steepest. The slope of the surface of a spinning sphere of loose material against latitude — the angle between the local vertical and the combined pull of gravity and rotation — for spins of 30%, 60%, 80%, 90%, 97% of the critical rate, measured in its square, at which the equator would lift off: for a density of 1.19 g cm⁻³ the critical period is 3.03 hours and these are periods of 5.5, 3.9, 3.4, 3.2, 3.1 hours. The slope is zero at the pole and at the equator, where the pull and the vertical are aligned by symmetry, and steepest in between: at 30% it peaks at 10° at latitude 40°, at 97% at 70° at 10°. The band marks the angle of repose of loose rock, 30° to 40°. From 80% of the critical rate the steepest band is past it: the surface begins to slide well before the equator has lost its weight.

A surface that slides before the body sheds

A rubble pile spun towards its limit does not wait for its equator to lose its weight. Long before that, its mid-latitudes grow steeper than loose rock can stand, and the surface flows — downhill, which on a spinning body means towards the equator. The ridge that so many small asteroids wear is built in that interval, at spins a sixth slower than the one that would fling material off, and whether it is built at all depends on a cohesion of less than a pascal.

6 figures · Rubble piles

The observed sky

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

$The Moon's Cassini states as it recedes. The angle between the Moon's spin axis and its orbit normal in each Cassini state, signed so that the present state is positive, against the Earth–Moon distance in Earth radii, for the Moon's present shape (β = (C − A)/B fixed) and orbital inclination. Today the Moon sits in Cassini state 2 at 6.68°. Closer in, two more states exist, a low-obliquity state 1 and an unstable state 4; they meet and vanish at 33.8 Earth radii, where the spin-axis precession, which weakens as the cube of the distance, falls to the critical fraction of the orbit's regression. A Moon that formed close to the Earth sat in state 1 at small obliquity, growing to 21° as it receded; at 33.8 Earth radii that state disappeared from under it, and it had to move to state 2 — which at that distance is at 46°. Since then, the obliquity has relaxed along state 2 to its present value.$

A state the Moon had to jump to

The Moon's spin axis sits a degree and a half from the ecliptic pole, locked in a geometry Cassini described in 1693. The lock is an equilibrium between the Earth's torque on the Moon's bulge and the slow regression of its orbit, and both depend on the Moon's distance. When the Moon was closer than about 34 Earth radii there were two more equilibria — and the one it almost certainly occupied vanished as it receded, leaving it to jump to the present one at an obliquity near 45°.

6 figures · Libration

Whether a field helped, depending on what limits the escape. The ratio of the atmospheric loss of a magnetised Earth to that of the same Earth without a field, against the Sun's age, both on logarithmic scales, set at 2 today — the magnetised planet losing more — for two ways an unmagnetised planet's loss can scale. If both planets lose atmosphere in proportion to the solar-wind power they intercept (blue), the magnetosphere's cross-section shrinks as the young Sun's denser wind compresses it while the bare planet's does not, and the ratio falls to 0.46 at 0.7 billion years: the field protected the young Earth, on balance from before 1.9 billion years. If instead the bare planet's loss is set by the ions the Sun's ultraviolet makes above it (orange), the ultraviolet rises more slowly into the past than the wind's pressure does, and the ratio grows to 3.7: the field cost the young Earth more than it saves today. The answer to whether a magnetic field protects an atmosphere is not a property of fields; it is a statement about which process limits the escape, and the young Sun amplifies the difference.

A shield that depends on what limits the leak

Today the Earth, with its magnetic field, loses ions to space about as fast as Venus and Mars without one — a little faster, by most measures. Whether that was also true under the young Sun, whose wind pressed eighty times harder, depends on something no measurement of the present can settle. If escape is limited by the wind's power, the field protected the young Earth; if it is limited by the ions the Sun's ultraviolet makes, the field cost it more than it does now. The young Sun does not decide the question. It amplifies whichever answer is right.

6 figures · Magnetosphere

Spaceflight

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

Two arrivals with the same weak thruster. Two spacecraft entering Mars's sphere of influence (outer circle, 577 thousand km) from the right, each thrusting continuously against its velocity at 0.3 mm s⁻² — about what an electric thruster gives a spacecraft of a few tonnes — and aimed to pass 6 planetary radii from the centre. Arriving at 0.40 km/s (blue), the thrust removes the excess while the craft crosses the sphere: it is captured 10 days after entry, having spent 268 m/s, on an orbit whose apoapsis lies inside the sphere, where the planet rather than the Sun keeps it, and the drawn path follows the same thrust winding it inward until it is within one and a half planetary radii. Arriving at 0.80 km/s (orange), it passes through on a bent hyperbola and leaves again: the thrust had too little time, and too little leverage, to remove an excess twice as large.

A thruster that must arrive slow

A rocket arriving at a planet stops by burning hard at the bottom of the gravity well, where every metre a second it spends removes the most energy. An electric thruster cannot burn hard anywhere. It can capture only arrivals slower than a few hundred metres a second, so it has to arrive with almost no excess at all — and then it pays the full circular speed of its final orbit to spiral down, two and a half times what a burn from the same start would cost, because a spiral spends its thrust at every speed on the way rather than at the highest one.

6 figures · Escape

The last correction: accurate if late, cheap if early. For a spacecraft approaching a moon at 6 km/s with a 30 km aim error left by navigation from the Earth, which locates the moon by camera to 20 microradians and acts an hour after its last image: the error remaining at the flyby after one final correction (blue, km) and that correction's cost (orange, m/s), against how long before the flyby it is made, all on logarithmic scales. The error falls in proportion to the time-to-go, because the moon is seen from closer, until it meets a floor: 3.1 km from how well the limb of a Europa-sized moon can be fitted once it fills the camera's view, with the burn's own two-per-cent error, 0.6 km, beneath it. The cost rises as one over the time-to-go, because the same error has to be removed in less time. The floor is reached 6.2 hours out, and a correction later than that costs more and buys nothing. A correction a day out costs 0.35 m/s and delivers 10.8 km; one six hours out, 1.4 m/s for 3.2 km. With 1 m/s to spend, the best is 4.2 km, at 8.7 hours out.

An aim bought with the last day of the approach

A spacecraft that photographs the moon it is approaching can aim its flyby far more precisely than the ground can tell it to, because it sees the target from closer every hour. But each hour closer leaves less time for a correction to act, so every kilometre of aim is bought with propellant — and below a few kilometres the camera stops helping at all, because a moon that fills the view can be located only as well as its limb can be fitted. For a small craft with a metre a second to spend, the aim comes out at four kilometres, set by geometry and not by the antenna.

6 figures · Gravity assist

Starlight

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

A lobe, the shell it pushed out, and two sight lines through it. A radio lobe of radius 20 kpc drawn in the plane containing the line of sight, the observer to the right, 60 kpc from the host in the same halo as before. Growing at Mach 2, it swept the gas that filled its volume into a shell compressed 2.29 times and 2.9 kpc thick, and the field in that shell was amplified along the boundary. A sight line through the lobe's centre crosses the shell head-on, where the amplified field lies across the line of sight and contributes nothing: the shell there turns the light by about 6 rad m⁻². A sight line at 85 per cent of the radius crosses it obliquely, along 4.8 kpc of shell with the amplified field partly along the path, and the shell turns it by about 18 rad m⁻². The rotation the halo model attributes to the gas in front of the lobe is partly written by the lobe itself, and it is written most strongly around its edge.

A rotation measure the lobe writes on its own boundary

The magnetic fields of hot group and cluster gas are mostly measured through the Faraday rotation of the radio galaxies that live in it, on the assumption that a lobe is a passive lamp behind a screen. But a lobe made its own room. The gas that filled it is now a shell around it, compressed if the lobe grew faster than sound, and the shell rotates the light too — so the inferred field comes out too low for a lobe that grew gently and up to three times too high for one that drove a shock. Only sources behind the halo, which carry no shell, can say which.

7 figures · Faraday rotation

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