Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 29

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 561–580 of 909.

Galaxies

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

The distance a proton keeps its energy, against the energy. The energy-loss length of a cosmic-ray proton — the distance over which it would lose all its energy at its current rate — against its energy, computed from the photon density of the microwave background at 2.725 K and the cross-section for photoproducing a pion, which peaks at the Δ resonance. The dashed line is the loss to cosmic expansion alone, about 4,283 Mpc. Only above about 6.1·10¹⁹ eV does the combined loss length fall below a gigaparsec, because below that few background photons are energetic enough in the proton's frame to cross the 145 MeV threshold. Above it the loss length collapses: 144,327 Mpc at 3.16·10¹⁹ eV, 155 Mpc at 10²⁰ and 19 Mpc at 10²¹, where every proton is above threshold and the loss is set by the photon density alone. A proton seen above 10²⁰ eV has come from within a hundred megaparsecs or so — a horizon drawn by light at three kelvin. Pair production on the same photons, which costs about a gigaparsec near 10¹⁹ eV, is not drawn.

A horizon drawn by light at three kelvin

A proton above about fifty exaelectronvolts cannot cross the universe, because the microwave background, seen from the proton, is a bath of gamma rays that it loses energy to by making pions. The cut-off in the cosmic-ray spectrum sits where that horizon predicts. But a source that simply cannot accelerate particles any harder would put a cut-off in the same place, and what the particles turn out to be made of says the second explanation may be the right one.

5 figures · Cosmic rays A disc galaxy walked down the plane by three dry mergers. Specific angular momentum against stellar mass, both logarithmic. The solid line is the relation disc galaxies follow, j proportional to M to the two-thirds, normalised to 1000 kpc km/s at 10¹¹ solar masses; the dashed line is the spheroids', a fifth as high. The points follow the median galaxy built from discs that start on the line at 2.51·10¹⁰ solar masses by successive equal-mass mergers, each between two galaxies with the same history as each other, adding spin vectors in random directions and keeping none of the orbital angular momentum. Each merger doubles the mass and moves the point right by 0.3 of a decade; the vector sum keeps on average only two thirds of the specific angular momentum while the line rises by the two-thirds power of two, so the point falls below it: to 0.45 of the line after one merger, 0.18 after two and 0.08 after three. Two dry major mergers carry a disc to where spheroids are measured.

Two spins that cancel, and an orbit that leaves

Disc galaxies and ellipticals of the same stellar mass differ by a factor of five in how much angular momentum each gram of their stars carries. Adding spin vectors in random directions explains most of it — two mergers of equals, with the orbit's angular momentum handed to the dark halo, carry a disc down to where ellipticals sit — and whether a merger makes a spheroid or rebuilds a disc turns on one number the vectors do not supply.

5 figures · Galaxy spin

The observed sky

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

The rate at which the solar wind opens the Earth's field. The rate at which reconnection at the dayside magnetopause turns closed field lines into open ones, as a voltage in kilovolts — webers of flux per second — against the clock angle of the interplanetary field in the plane perpendicular to the Sun–Earth line, 0° pointing north and 180° south, for a wind at 450 km/s carrying transverse fields of 5, 10, 20 nT. The law is empirical: the rate rises as the speed to the four-thirds, in proportion to the field, and as the sine of half the clock angle to the power 4.5. Due south, a 5 nT field opens flux at 57 kV; turned to within 20° of north, at 22 volts. The steepness in angle is the whole of the solar wind's leverage on the magnetosphere: the wind's pressure decides how large the cavity is, and one component of its field decides whether anything gets in.

A substorm timed by webers and volts

The solar wind opens the Earth's magnetic field at a rate that can be quoted as a voltage — tens of kilovolts, which is webers of flux per second — and the tail closes it again. Half a gigaweber loaded at fifty kilovolts takes three hours, and three hours is how often substorms recur. The aurora's position is the open flux, its surges are the unloading, and above a certain driving the ionosphere starts to limit the circuit that drives it.

5 figures · Magnetosphere When a target can be observed from a telescope that cannot look near the Sun. The days of the year (across) on which a target at ecliptic longitude 0° and a given ecliptic latitude (up) can be observed by a telescope allowed to point only between 85° and 135° from the Sun, the constraint of a large infrared telescope at the Sun–Earth L2 point. The Sun moves along the ecliptic once a year, and the allowed band sweeps round with it. A target on the ecliptic is inside the band twice a year, for about 51 days each time, a little before and a little after its opposition — never at it, since opposition is 180° from the Sun. The two windows widen with latitude and merge into one above 45°, where the elongation at opposition no longer exceeds 135°; within 5° of either ecliptic pole, where the elongation at conjunction no longer falls below 85°, the target is never outside the band at all, which is the continuous viewing zone. The pattern depends on ecliptic, not equatorial, coordinates: the season of a space telescope is set by the Sun's path, and the Earth's axis has no part in it.

A season without a night

A telescope in space has no horizon and no night, and it can still only look at most of the sky for part of the year. Its season is a band of solar elongation swept round the ecliptic by the Sun, so an object on the ecliptic can be seen for two stretches of about seven weeks and never at opposition, while the few degrees round the ecliptic poles can be watched all year. From low orbit the Earth takes the night's place, and the zones it never hides are carried round the sky every eight weeks by the planet's equatorial bulge.

5 figures · Sidereal time

Orbits

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

One surface, two ages, and a kink between them. The cumulative number of craters per square kilometre larger than a given diameter, both on logarithmic axes, for a surface that formed 3.9 billion years ago and was flooded 3.3 billion years ago by a layer 200 m thick. The dashed lines are the isochrons of the two ages under the lunar chronology. Craters with rims taller than the flood — larger than 5.6 km, since a simple crater's rim stands about 0.036 of its diameter high — survived it, so above the kink the count follows the 3.9-billion-year isochron. Everything smaller was buried, and the craters counted there formed after the flood, so below the kink the count follows the 3.3-billion-year isochron. Fitting the branches separately returns 3.90 and 3.30 billion years. The kink is not noise and not a change in the impactors: its diameter is a measurement of how thick the flood was.

A kink that measures how deep the lava was

A surface that was flooded partway through its life carries two crater populations — the old craters too big to bury, and the young ones made since — and its crater count follows one age above a certain diameter and another below it. The diameter of the kink is the thickness of the flood, read through the height of a crater's rim, and a single age fitted across it describes nothing that ever happened.

5 figures · Surface chronology The spike, and where one passage through the planets sends it. The distribution of the reciprocal semi-major axis 1/a of long-period comets, in units of 10⁻⁴ AU⁻¹ and each curve scaled to its own peak. The narrow shaded peak is the Oort spike: comets arriving from the Oort cloud for the first time, with 1/a of about 4·10⁻⁵ AU⁻¹ and a spread of a few times 10⁻⁵. The broad curve is where those same comets go after one passage through the planetary region, whose perturbations change 1/a by a typical 5·10⁻⁴ AU⁻¹ — ten times the spike's width. 47 per cent end with negative 1/a, on hyperbolic orbits that leave the Solar System; only 4 per cent remain inside the spike's width. A comet in the spike is therefore on its first passage, almost by definition: the planets cannot put a comet back into the spike once it has been through them.

A spike only first-time visitors can be in

Long-period comets arriving from the Oort cloud have orbital energies packed into a spike a few hundred-thousandths of an inverse astronomical unit wide. One passage through the planets changes that energy by ten times as much, so no comet can be in the spike twice — and the spike's position, near thirty thousand astronomical units, is where the Galaxy's own tide is just strong enough to lift a comet's perihelion over Jupiter and Saturn in a single orbit.

5 figures · Hyperbolic orbits

Spaceflight

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

The cheapest circular orbit to stop in at Mars is not the lowest. The speed change needed to capture into a circular orbit round Mars, in km/s, against the orbit's radius in Mars radii, for arrival excesses of 2.65, 4, 6 km/s — the first being a minimum-energy transfer from the Earth. One burn at periapsis, where the spacecraft is fastest, turns the hyperbola into the circle. The cost has a minimum, at a radius of 2μ/v∞² from the planet's centre, where it is exactly v∞/√2: 3.6 radii and 1.87 km/s for 2.65 km/s; 1.6 radii and 2.83 km/s for 4 km/s; 0.7 radii and 4.24 km/s for 6 km/s. Lower orbits cost more because the circular speed there is higher than the gain from arriving deeper in the well; higher orbits cost more because the burn is made where the spacecraft is moving slowly and the Oberth effect is lost. A faster arrival moves the optimum inward, and at a large enough excess it moves below the planet's surface, where every circular orbit is on the expensive side of the minimum.

The cheapest circle to stop in

A spacecraft arriving at a planet has to lose speed to stay, and the cheapest circular orbit to lose it into is not the lowest. It sits at a radius set only by the arrival speed — twice the planet's gravitational parameter divided by the square of the excess — where the braking costs exactly the excess divided by the square root of two. Most orbiters want something else, and the arithmetic of what they do instead decides which planets are cheap to stop at and which are not.

5 figures · Escape The same porkchop with Mars's orbit tilted, and the ridge that appears. Contours of departure energy C₃ for transfers from the Earth to Mars, over departure date and flight time, with Mars's orbit inclined by its real 1.85° to the ecliptic and every point a solved three-dimensional Lambert problem. The dashed line is where the transfer angle is exactly 180°. Along it the contours pile up into a ridge reaching C₃ of 346 km²/s² and more, against 7.9 at the cheapest point: a transfer half-way round the Sun has to lie in a plane through the Sun and two nearly opposite points, and reaching a target a fraction of a degree out of the ecliptic tilts that plane towards the pole. The ridge splits the plot into two basins — Type I transfers, less than half-way round, below and to the left, and Type II, more than half-way, above — and the cheapest transfers sit in one basin or the other, never on the line between them.

A wall across the porkchop

Draw the cost of going to Mars with Mars's orbit in the Earth's plane and the porkchop plot is a smooth pair of basins. Tilt Mars's orbit by its real 1.85 degrees and a wall appears between them, hundreds of times higher than the valley floor, along the line where the transfer goes exactly half-way round the Sun. The wall is a plane change forced by geometry, its height changes from one launch window to the next, and a single burn half-way to Mars removes it.

6 figures · Launch windows

Cosmology

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

The sky-averaged 21-centimetre signal, and the trough that was claimed. The brightness temperature of neutral hydrogen's 21-centimetre line relative to the microwave background, averaged over the sky, in millikelvin, against observed frequency (the redshift is 1420.4 MHz over the frequency, minus one). In this model the first stars' ultraviolet couples the hydrogen's spin to the gas near redshift 17, X-rays heat the gas near 11, and reionisation is half complete at 7.7. Because the gas has cooled faster than the background since it decoupled, coupling makes it absorb: the trough reaches −171 mK at 71 MHz (redshift 19.1). Heating turns the signal into a weak emission, and reionisation removes it above about 197 MHz. The dashed curve is the deepest absorption standard physics allows at each frequency — gas that has only cooled, fully coupled — and the point is the one claimed detection, −500 mK at 78 MHz: about 1.9 times deeper than that floor.

A shadow cast on the microwave background

Before the first stars lit it up, the neutral hydrogen between the galaxies was colder than the microwave background behind it, and once starlight tied its 21-centimetre line to that cold, it absorbed. The absorption, redshifted to radio frequencies near 70 megahertz, is a thermometer for gas nobody has seen — and the one claimed detection was twice as deep as cooling alone allows, which is either a discovery about dark matter or a lesson about antennas.

5 figures · Reionisation A void: a deficit of matter, and the outflow it drives. The density contrast of a typical cosmic void against distance from its centre in units of its radius (solid), from the profile fitted to stacked voids in simulations, with a central density 15 per cent of the mean and a compensating overdense ridge just outside the edge. The dashed curve is the excess expansion rate the deficit drives, v/(Hr) = −(f/3)Δ(r), for a growth rate f = 0.75: the centre expands 21 per cent faster than the universe around it, and the outflow speed, which is this rate times the radius, peaks near 0.7 void radii. Beyond the ridge the flow reverses: matter at the ridge is pulled back towards the denser wall. A void is a small open universe expanding inside a flat one, and every galaxy in it is moving outward.

An emptiness that expands faster

A cosmic void is not a hole in the universe but a region with a fifth of the average density, and gravity makes it behave like a small open universe: it expands faster than its surroundings, every galaxy in it streams outward, and in a redshift survey it is stretched along the line of sight — the opposite of what the same gravity does around a cluster. The stretch measures how fast structure grows, and the average void's roundness measures the geometry of the universe.

5 figures · Large-scale structure A clumped plasma recombines earlier. The fraction of hydrogen still ionised against redshift, decreasing to the right, on a logarithmic axis, from Peebles's three-level atom: for a smooth plasma (solid), and for one whose baryons are clumped into two equal volumes at 1.71 and 0.29 times the mean density, a clumping factor of 0.5 (thick dashed, averaged by mass; thin dashed, each zone alone). Recombination goes as the density squared, so the dense zone recombines first and carries most of the mass; the average ionisation falls through one half at redshift 1292 instead of 1273, and at redshift 1100 it is 0.123 against 0.145. The photons cannot see the individual patches, which are smaller than their mean free path; they see only the average, and the average recombines earlier.

A field that would have moved the ruler

A magnetic field left over from the early universe, too weak to matter to any galaxy, could still have pushed the baryons into small clumps before the universe became transparent. Clumped hydrogen recombines earlier, the sound waves in the plasma stop sooner, and the ruler that the microwave background is read with comes out shorter — which raises the Hubble constant the background implies, in the direction of the disagreement that has resisted every other explanation.

5 figures · Primordial fields

Gravitation

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

A bead on a rotating wire: when gas is flung off a disc. The effective potential — gravity plus the centrifugal potential in the frame rotating with the footpoint — for gas sliding along a rigid magnetic field line that leaves a Keplerian disc at 20°, 30°, 40° from the vertical, in units of GM over the footpoint radius, against distance along the line in units of that radius. The field line corotates with the disc where it is anchored, like a wire whipped round, and gas on it behaves like a bead. Close to vertical the potential rises away from the disc and the gas is bound to its footpoint. Inclined by more than 30° it falls, and gas is flung outward along the line with nothing but the rotation to drive it. The 30° threshold is exact for a Keplerian disc — the second derivative of the potential at the footpoint changes sign there — and it was the result that made magnetised disc winds a mechanism rather than a speculation.

A disc that throws its spin upward

A disc does not need turbulence to accrete. A magnetic field threading it can fling gas off its surface along field lines tilted more than thirty degrees from the vertical, and each gram flung out leaves with several times the angular momentum it had — so a wind carrying a few per cent of the inflow can drive the rest onto the star. Because it pulls on the disc's faces rather than across its edge, a wind outpulls the turbulence as soon as the field's pressure is a few ten-thousandths of the gas's.

5 figures · Magnetorotational instability Supply and demand in a cluster's core. The rate at which hard binaries heat a star cluster's core, against the core's central density, for binary fractions of 1%, 5%, 20%, both on logarithmic axes and in units that put the balance for five per cent at one. The heating rate per binary does not depend on how hard the binary is — the energy released per encounter is proportional to its binding, and the chance of an encounter to its size, so the two cancel — and summed over a core whose radius shrinks as its density rises, the total grows as the square root of the density. The horizontal line is the demand: the energy the rest of the cluster carries away by relaxation through its half-mass radius, which the core has no say in. Each supply curve meets it at one density: 25 for 1%, 1.0 for 5%, 0.063 for 20%. The core contracts or expands until it produces exactly what is demanded, and a core rich in binaries does so at a lower density.

A core that makes what the cluster asks for

After core collapse, a star cluster's binaries heat it — but how much they heat it is not theirs to decide. The rest of the cluster carries energy outward at a rate fixed by its own size and mass, and the core contracts or swells until its binaries supply exactly that. So the core's structure follows from the halo's demand, and the cluster expands at a rate that forgets how it was born.

6 figures · Binary heating A chaotic orbit that waits, and then moves. The momentum of one trajectory of the standard map at K = 1.2, unwrapped and counted in whole turns of 2π, against the iteration number, over 39,892 iterations. The motion inside each band is chaotic from the first step — the trajectory's Lyapunov time is 3.7 iterations — but it lingers near one integer resonance for thousands of iterations before reaching the next: the waits were 3,098, 1,464, 9,461, 15,957, 6,058, 3,854 iterations. Each arrival at the next integer resonance, drawn dashed, is marked. Between the integer resonances lie the remnants of the broken invariant curves, which no longer block the trajectory but still hold it, and each passage is quick compared with the wait before it.

A barrier that leaks at a rate that can be computed

When the last invariant curve between two resonances breaks, it does not vanish. It becomes a cantorus — a curve full of gaps — and a chaotic orbit has to find the gaps to get through. The time that takes grows as the cube of how close the system is to the threshold, and it has nothing to do with the Lyapunov time, which is why an orbit can be hopelessly unpredictable and still stay put for the age of the solar system.

6 figures · Chaos

Cosmology

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

Three fits of the dark energy's equation of state, and the line they all cross. The equation of state w of the dark energy against redshift, for the three combined fits published with the second year of acoustic-scale data from the largest spectroscopic galaxy survey — the same galaxy and microwave-background data with three different supernova compilations — each written as w = w₀ + wₐ z/(1 + z). A cosmological constant is the horizontal line at −1. All three fits have w above −1 today (between −0.84 and −0.67) and below it in the past, and they cross it at redshift 0.35, 0.44, 0.41. Below −1 lies the phantom region, where a density grows as the universe expands; a single field with ordinary kinetic energy can be on either side of the line but cannot cross it. The preference for these fits over a constant is quoted at 2.8, 3.8 and 4.2 standard deviations, and the only thing that differs between the three is the supernovae.

A crossing that may belong to the fitting form

Allow the dark energy's equation of state to change and three independent fits say it has: above −1 today, below it before redshift half. The data hold w well at one redshift only, a line crossed from below is one no single field can cross, and a straight line in the scale factor fitted to a curved history crosses it anyway.

6 figures · Dark energy The helium the first three minutes make, against the neutron's lifetime. The primordial helium mass fraction against the free-neutron lifetime, integrated through the first three minutes with the weak conversion rates normalised on that lifetime and free decay at the same rate, at the baryon density the microwave background gives. The line rises by 1.96 parts in ten thousand per second: a longer-lived neutron means slower conversions, an earlier freeze-out and less decay before the neutrons are locked into helium. The two laboratory lifetimes — 877.75 s from storing ultracold neutrons in a bottle, 887.7 s from counting protons in a beam — give 0.2468 and 0.2487, a difference of 19 in the fourth decimal place. The band is helium in metal-poor galaxies (Aver et al. 2015), 0.2449 ± 0.004; read back through the line it says the neutron lives 869 s, anywhere from 848 to 889 — a measurement of a laboratory constant from galaxies, good to about 20 seconds, and twice too coarse to referee a ten-second disagreement. The absolute level of the line is set to a full network's value for the bottle lifetime; the slope is computed.

The first three minutes use two lifetimes

A free neutron lives 877.75 seconds in a bottle and 887.7 in a beam. Nucleosynthesis uses the lifetime twice — once to set how fast neutrons and protons interconvert, once to set how fast the leftovers decay — and if a neutron can decay into something invisible, those are the two different numbers the laboratories measure.

5 figures · Nucleosynthesis

Exoplanets

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

How planet occurrence rises with the host star's iron, by kind of planet. Planet occurrence relative to its value at solar metallicity, on a logarithmic axis, against the host star's iron abundance [Fe/H] (zero is the Sun; one unit is a factor of ten). Each class is a power law, occurrence ∝ 10^(β[Fe/H]). Giant planets within a few astronomical units follow β = 2, the relation found in the first large radial-velocity survey to measure it: 12 per cent of stars at [Fe/H] = +0.3 have one, against 3 per cent at solar and under 0.3 per cent at −0.5 — a factor of 158 across the range drawn. The small planets found by transits rise far more gently; the slopes drawn, β = 0.5 for sub-Neptunes and 0.2 for super-Earths, are representative of published fits, which range from nearly flat to about a doubling across this interval depending on orbital period.

A threshold that makes iron matter only to giants

A star with twice the Sun's iron is four times as likely to have a giant planet, and barely more likely to have a small one. The steep law and the flat one look like two mechanisms. A core racing its disc, run over a spread of discs, produces both — steep where winning is rare, flat where it is common.

5 figures · Occurrence rates The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone.

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

5 figures · Spin–orbit alignment

Galaxies

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

One stream after three different populations of dark subhaloes. The density of stars along the middle sixteen kiloparsecs of a thin stream, relative to its mean, after four billion years of encounters with dark subhaloes drawn from a mass function falling as the −1.9 power of the mass between 10⁵ and 10⁸ solar masses. Top: the abundance cold dark matter predicts — in this realisation 26 encounters, 4 of them above a million solar masses — which leave 32 per cent of the stream's length more than 30 per cent below the mean density. Middle: 0.3 of that abundance, 8 encounters and 7 per cent of the length in gaps. Bottom: none at all, where the only variation is the counting noise of the 60,000 model stars. No single dip in the top trace is a clean gap: they overlap, and a large one and several small ones are superposed at every point. What differs between the three is the statistics of the whole trace, not any feature in it.

A spectrum that counts what no single gap can

One gap in a stellar stream measures a subhalo's mass multiplied by an unknown time. A stream crossed by dozens of them measures something better — how many subhaloes of each size there are — but only as a statistic of the whole stream's density, where no feature can be pointed to and one stream is one throw of the dice.

5 figures · Stellar streams Three arrangements of gas, three velocity–delay maps. The response of a broad emission line to a flash of the continuum, as the density of emitting gas in delay (up, days) and line-of-sight velocity (across, km/s; positive is receding), for gas between 10 and 30 light-days from a hundred-million-solar-mass black hole in three kinematic states, 6,000 clouds each. Left: a rotating disc inclined at 30° with orbits tilted up to 15° from its plane — an ellipse symmetric in velocity, with the fastest gas, nearest the hole, responding soonest. Middle: gas falling in at the free-fall speed — the near side is moving away from the observer and responds first, so the red side leads and the blue side lags, by 27 days on average beyond 1,500 km/s. Right: the same gas flowing out — the mirror image, the blue side leading and the red lagging by 27 days. The three maps are as different as maps can be; their projections onto either axis alone are not.

A delay that tells which way the gas is moving

A broad emission line echoes the continuum after a delay, and the delay is a radius. Measure it separately in the line's blue wing and its red wing and it becomes a map — symmetric for gas in orbit, lagging on the blue side for gas falling in, on the red side for gas blown out — and the one route to the geometry factor for an individual black hole.

5 figures · Quasars

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