Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 42

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 821–840 of 909.

Gravitation

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

A cluster sinking by Chandrasekhar's formula alone. The orbital radius of a 2.0 × 10⁵ solar-mass globular cluster released on a circular orbit at 1 kpc in the two Fornax halo models, decaying under Chandrasekhar's dynamical friction with a velocity dispersion of 11.7 km/s and a Coulomb logarithm of 3. In the cusp it reaches the centre in 6.7 billion years. In the core the formula gives no arrival within 12 billion years, crossing into the core at 2.8. The formula already slows the cluster inside the core, because the circular speed there drops below the stars' velocity dispersion and few stars move slower than the cluster; but it still lets the cluster creep inward without limit. N-body experiments of clusters sinking into cored haloes find something stronger: the decay stops altogether near the core, which the local formula cannot produce at any setting of its logarithm.

A drag that stops where every orbit keeps time

Chandrasekhar's formula reads the density and the velocities where a sinking body is, and nothing about the orbits those stars are on. Inside a constant-density core every orbit has the same frequency, the differential motion that drags a body inward disappears, and the sinking stops — which is why the five globular clusters of the Fornax dwarf galaxy are still where they are.

6 figures · Dynamical friction

Orbits

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

A wall that leans the way the damping body says. The orbital period inside which a hot Jupiter's eccentricity is damped within 3 billion years, against the planet's mass, around a Sun-like star with planet Q′ = 10⁶, star Q′ = 10⁷ and a planetary radius of 1.1 Jupiter radii. If the planet's own tide did all the damping, the wall would fall with mass, from 5.5 days at one Jupiter mass to 3.2 at ten. If the star's tide did, it would rise, from 1.5 to 2.5 days. Together they give 5.5 and 3.4. Points are measured planets: filled where the orbit is eccentric, open where it is circular. The massive planets HAT-P-2 b, XO-3 b and WASP-14 b keep eccentric orbits at periods where lighter planets are all circular, which is the lean the planet's tide predicts and the opposite of the star's.

A wall that leans towards the body that damps

A hot Jupiter's eccentric orbit is rounded off by two tides at once, one dissipated in the planet and one in its star, and they depend on the planet's mass in opposite ways. So the period inside which planets are circular should fall with planet mass if the planet does the work and rise if the star does — and the heavy hot Jupiters that keep eccentric orbits where light ones never do say which.

6 figures · Circularisation What colour removes, and what it can only estimate. The share of a synthetic family's linked membership that is not part of the family, against the clustering cutoff, with the parts a colour test can and cannot see. The true interloper fraction rises with the cutoff, from 0 per cent at 20 m/s to 84 per cent at 110. A colour cut removes the interlopers of the other type — 12 per cent of the list at 60 m/s — and leaves those of the family's own type, 6 per cent, which can only be estimated by scaling the detected ones by the background's mix. That estimate (dashed) tracks the truth on average and scatters about it, because a few dozen interlopers are a few dozen random draws.

A family checked by a colour it was not chosen by

Asteroid families are found by clustering orbits, and the clustering absorbs background asteroids that happen to orbit nearby. Fragments of one body share its composition, and the survey that found them measured their colours for free — so colour is a second membership test the clustering never used. It removes the interlopers that look different, and it can only estimate the ones that look the same.

6 figures · Asteroid families

The observed sky

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

A reflector made worse so the light lands where it is wanted. The intensity returned by a 3.8 cm fused-silica corner cube against angle from the exact retro-direction, on a logarithmic scale relative to a perfect cube's peak, at 532 nm. A perfect cube returns an Airy pattern whose central lobe is 17 microradians in radius. Offsetting each of its three dihedral angles by 1.25 arcseconds splits the return into six beams, each from a sixth of the face, deflected by (4√6/3)·n·ε = 29 µrad. In the band of offsets a LAGEOS station sees, 32–38 µrad (shaded), the spoiled cube returns 10 times as much light as the perfect one, along the direction of each lobe. Each lobe is drawn here as the Airy pattern of a circular aperture of a sixth of the area, which is an approximation to the true sector shape.

A reflector built wrong so the light comes home

A corner-cube reflector sends light straight back the way it came — in its own frame. On a satellite crossing the sky the return is aberrated by twice the satellite's sideways speed over the speed of light, fifty microradians for a low orbit, which is further than a small perfect cube spreads its light. So the cubes on geodetic satellites are made imperfect on purpose, their angles offset by about an arcsecond to throw the returning beam out to where the telescope will be.

6 figures · Aberration The rhythms in the rock, set by the Moon's distance. The periods of the climatic precession cycles — the precession constant plus the perihelion frequency of each of the planets that dominate the Earth's eccentricity — and of the obliquity cycle, the precession constant plus the node frequency s₃, against the Earth–Moon distance. Today they are 23.8, 22.4, 19.0 thousand years for precession and 41 for obliquity. At 341,000 km they were 14.7, 14.2, 12.8 and 20.0. The planetary frequencies are held at today's values, which the chaos of the inner solar system makes uncertain beyond a few tens of millions of years; the precession constant is the term that moves.

A rhythm in the rock that measures the Moon

The Earth's axis precesses because the Sun and Moon pull on its equatorial bulge, and two-thirds of the pull is the Moon's. When the Moon was nearer and the day shorter, the axis turned faster, and the climate cycles that precession drives were shorter. Sediments laid down 1.4 billion years ago record those cycles — and with nothing in the rock but layer thicknesses, the ratio of two rhythms gives the Moon's distance at the time.

6 figures · Precession

Spaceflight

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

Two satellites moved apart by the attitude they hold. The along-track separation of two identical 5 kg satellites at 475 km, starting together, when one flies broadside to the airflow and the other edge-on for 15 days and then they swap, so that their relative drift ends at zero. The separation grows quadratically and then levels off. At solar minimum the manoeuvre moves the pair 2,797 km apart; mid-cycle 8,299; at solar maximum 24,620 — 206° of the orbit's 43,060 km circumference. No propellant is spent; the same command produces a tenfold different result depending on the state of the Sun.

A rendezvous paid for in air

Two satellites in the same low orbit can be moved along it relative to each other with no propellant at all, by turning one broadside to the thin air and the other edge-on. The draggier one sinks and speeds up, the gap grows as the square of the time, and swapping attitudes halfway brings the drift back to zero. The cost moves from fuel to time, to altitude, and to knowing an atmosphere that changes tenfold with the Sun.

6 figures · Rendezvous An arrival held without a burn, and released. A body approaching a planet from a heliocentric orbit just outside the planet's, drawn in the planet's rotating frame (the Sun far to the left, the planet's orbital motion upward), in units in which the Hill radius is 0.69. It drifts in along the shear of the differential rotation, passes through the neck at the Lagrange point L2 into the region the planet controls, loops round the planet 3.3 times and leaves through a neck again. No burn is drawn: the capture and the release are both the Sun's tide working against the planet's pull. For Mars one unit of distance is 1.56 million km and an orbit round the planet near the Hill radius takes about a year.

An arrival the Sun hands over

A spacecraft that reaches a planet with no speed to spare can be captured with no burn at all, passing through the neck between the planet's gravity and the Sun's into a loose, looping orbit. The capture is temporary and chaotic — a few revolutions, and arrivals a few hundred kilometres apart end differently — but while it lasts the spacecraft is already bound, and the burn that keeps it is taken from there.

6 figures · Escape

Starlight

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

The same two lines, seen as frequencies. The amplitude of the Fourier transform of the two line profiles of the previous comparison, on a logarithmic scale, against frequency in cycles per kilometre per second. At low frequency — the overall width — they agree, as the profiles did. At higher frequency they separate completely: the first star's transform falls to zero at 0.110 cycles per km/s and the second's at 0.132, because the rotation kernel's transform vanishes at 0.660/v sin i and no Gaussian-like broadening has a zero anywhere. The horizontal dashed line marks a noise level of 0.0005 in the transform, which a high signal-to-noise spectrum averaged over several lines reaches; the dip and the side lobe beyond it rise above it for both stars, so the rotation of each is measured by where its transform dips, independent of the macroturbulence. In a noisier spectrum the second star's weaker side lobe would be lost first.

A rotation found where the transform goes dark

A star's rotation and the turbulent flows on its surface both widen its spectral lines, and a profile fitted for one can be refitted for the other with almost no change. Convolution becomes multiplication in Fourier space, though, and the transform of a rotating disc has zeros — frequencies where it is exactly nothing — while the transforms of turbulence and of the spectrograph have none. Where the transform of a line goes dark is a measurement of rotation that turbulence cannot fake.

7 figures · Line formation The same data searched with a box and with a sine. The signal-to-noise ratio of the best fit at each trial period, for the light curve of the previous figure, searched two ways: with a box of three trial durations at every phase, and with a sinusoid. The box search peaks at 3.85 days with a signal-to-noise ratio of 7.3, against a true period of 3.85. The sinusoid's best peak anywhere is 3.3, at 1.59 days, and at the true period it reaches only 1.5 — no higher than the peaks the noise makes elsewhere in the search. The planet is a clear detection to one method and invisible to the other, in the same numbers.

A periodogram shaped like a shadow

A periodogram that fits sinusoids sees a transiting planet only through the fundamental harmonic of its box-shaped dip, which for a short transit is a few per cent of the signal. Folding the data at each trial period and fitting a box recovers all of it — five times the signal-to-noise for a transit two per cent of the orbit long, eighteen times for an Earth crossing the Sun. The box pays for its extra parameters in trials, and the price is about one unit of signal-to-noise.

6 figures · Periodograms

Stars

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

Three energies a freezing white dwarf radiates late. The age at which a white dwarf has finished crystallising, against its mass, built up from bare Mestel cooling and three energies released at the crystallisation luminosity: the latent heat of freezing (0.85 kT per ion), the gravitational energy of neon-22 settling (0.6), and the gravitational energy of carbon and oxygen separating (0.8). At 0.6 solar masses the three together add 3.20 billion years to 3.56 of cooling; at 1.1 they add 4.36 to 4.85. The separation term is, on these numbers, the second largest of the three. None of them is fuel; all are energy the star already had, released at a luminosity so low that a little of it takes a long time to radiate.

A star that distils itself as it freezes

A carbon–oxygen white dwarf does not freeze at one composition. The crystals that form at its centre take more oxygen than the liquid they form from, the carbon-rich liquid left behind is lighter and rises, and the star slowly sorts its own interior, oxygen down and carbon up. The gravitational energy that releases is radiated at a luminosity so faint that it holds the star up for about a billion years — a delay comparable to the latent heat of freezing, and one of the larger uncertainties in every white-dwarf age.

6 figures · White dwarf cooling One population, two period–luminosity relations. The period–luminosity relation of 400 synthetic Cepheids spread across the instability strip, each behind a random amount of dust up to E(B − V) = 0.6: absolute magnitude in V (upper scatter) and in the Wesenheit index W = V − 2.44(V − I) (lower). In V the scatter about the fitted line is 0.57 magnitudes — 0.20 from the strip's width alone, the rest from the dust. In W it is 0.07: the dust is gone by construction, and most of the strip's width has gone with it, because a star that is hotter at a given period is bluer by about as much as dust would make it redder. That leftover 0.07 magnitudes is 3 per cent in distance per star.

A scatter that points the way dust does

A Cepheid's period fixes its luminosity only to within the width of the instability strip, and at a given period the hotter stars are the brighter ones. Dust moves stars in the colour–magnitude diagram along a line too, and the two lines point nearly the same way. A magnitude built to cancel the dust — the Wesenheit index — therefore cancels most of the strip's width as well, which is why the tightest period–luminosity relations are the ones that ignore the extinction law's own advice.

6 figures · Variable stars

Exoplanets

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

One parabola, two causes. Transit times of a planet on a 1.0914-day orbit over 12 years, minus the best straight-line ephemeris, with timing errors of 30 seconds — the shape a period shrinking by 29 milliseconds a year produces, a parabola 116 seconds deep. The curve is drawn once because it is the same curve for both explanations: an orbit decaying at that rate, or a star accelerating towards the observer at 2.9e-6 m/s², which shortens the light-travel time to each successive transit by the same steadily growing amount. Nothing in the transit times can tell them apart.

A clock that slows, or a star that falls

Transit times that curve away from a straight line can mean a planet's orbit is shrinking — the tides draining it into its star — or that the star itself is accelerating along the line of sight under a distant companion's pull. Both draw the same parabola, and the transit times cannot choose. The star's velocities can, but only for the shortest orbits: the acceleration that would fake a given decay grows as one over the period, and beyond a day or two an unseen Saturn can mimic more decay than the tides produce.

6 figures · Transit timing

Cosmology

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

A family that runs the full height of the plane. The spectral tilt and the tensor-to-scalar ratio (logarithmic) predicted by inflationary potentials, against the measured tilt (0.9649 ± 0.0042, shaded, two standard deviations) and the upper bound on the ratio (0.036, solid line). The two curves are the T-model α-attractor, V ∝ tanh²(φ/√6α), for α from ten thousandths to a thousand, at 50 and 60 e-folds before the end of inflation: at large α the model is a quadratic potential and sits above the bound; as α falls the ratio drops without limit while the tilt settles at 1 − 2/N. The dots on each curve are α = 1, close to Starobinsky's model, at a ratio of 0.0046 and 0.0032. The short segments are power-law potentials, all excluded or nearly so. The dashed line at 0.001 is a target of the next generation of polarisation experiments: a null result there would exclude α = 1 and leave every smaller α standing.

A null that removes a model and not its family

The plateau potentials that survive every measurement of inflation predict a tensor-to-scalar ratio a few thousandths high, and the next polarisation experiments are designed to reach it. A null result there would exclude Starobinsky's model. It would not exclude the family it belongs to: the α-attractors stretch one shape by one parameter, their tilt stays where the measurement put it, and their ratio slides down as α does, without a floor.

6 figures · Inflation One likelihood, read with three priors. A Gaussian likelihood in the effective neutrino mass sum, peaking at −0.08 eV with a width of 0.05 eV — of the shape that combinations of microwave-background and baryon-acoustic data have been reported to give. Read with no prior it says the data prefer a negative sum. Read with the physical prior, sum ≥ 0 (shaded right of zero), its 95 per cent upper limit is 0.059 eV, at the normal-ordering floor. Read with the oscillation prior, sum ≥ 0.059 eV, its limit is 0.102 eV — but only 0.3 per cent of the likelihood lies above the floor at all, which is the more informative number: the data and the oscillation experiments disagree at about 2.8 standard deviations.

A bound below the floor is not a mass

Oscillation experiments prove that neutrinos weigh at least 0.059 eV between them. Cosmology's latest upper bound on their total is 0.064 eV, and a bound computed differently would sit below the floor. The usual reading is that cosmology has nearly weighed the neutrino. The better one is that the fit's likelihood peaks at a negative mass: the data see more clustering than any real neutrino allows, and the bound is a symptom of that tension rather than a measurement of anything.

6 figures · Density parameters

Exoplanets

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

More pairs on the target list than in the sky. The fraction of a magnitude-limited survey's targets that are unresolved binaries, against the fraction of stars in the volume that are, for companions with mass ratios spread evenly between a tenth and one. For solar-type stars, of which about 45 per cent have a companion, the target list holds 53 per cent binaries if the stars fill space uniformly, 50 if the survey reaches beyond the disc. The excess is modest on average because most companions are faint; it is concentrated in the near-equal pairs, which a magnitude limit over-represents by up to a factor of 2.8.

A target list that counts pairs twice

A planet survey chooses its stars by how bright they look, and an unresolved pair of stars looks brighter than either alone. A twin pair is seen from 2√2 times the volume a single star is, so a magnitude-limited target list holds more binaries than the Galaxy does, weighted towards exactly the near-equal pairs that dilute a transit most. The survey then counts its planets per target, and a target is not a star.

6 figures · Detection bias

Galaxies

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

A size–luminosity relation with a population below it. The lag of the broad Hβ line behind the continuum, in light-days, against the continuum luminosity at 5100 ångströms, for a synthetic population of active nuclei with accretion rates spread from slow to well above the Eddington rate. The line is the radius–luminosity relation calibrated on slowly accreting nuclei, log R = 1.527 + 0.533 log L₄₄. Nuclei accreting at dimensionless rates below 3 scatter about it by about 0.13 dex; those above (filled) lie systematically below it, by 0.41 dex on average in this model, in which the shortening grows as the accretion rate to the power −0.4. A mass computed from the line for one of them takes a radius that is too large.

A lag too short for the light

Most black-hole masses beyond the nearby universe come from one spectrum and one relation: the size of the broad-line region scales with the quasar's luminosity, calibrated on nuclei that accrete slowly. Nuclei accreting near or above the Eddington rate have lags shorter than the relation predicts, so their masses are overestimated — and since the accretion rate is computed from the mass, it is underestimated, which hides the very shortening that caused the error.

6 figures · Quasars Velocities that stop pointing along the stream. The angle between the stream's direction on the sky, as seen from the Galactic centre, and the mean motion of its stars, along the stream, for Large Magellanic Cloud masses of 0.0, 0.5, 1.5, 2.5 × 10¹¹ solar masses. In a static halo the misalignment stays at 1° or below — the small offset any stream has from its progenitor's orbit. With the Cloud it grows, peaking at 4°, 13°, 32° for the three masses, on the stretch of stream that passed nearest the Cloud. The misalignment is a measurement: with the halo's own potential known, its size along the stream gives the Cloud's mass.

A stream that points the wrong way

In a halo that holds still, the stars of a tidal stream move along it. The Large Magellanic Cloud, falling into the Milky Way for the first time, pulls on a passing stream directly and drags the Galaxy's own centre towards itself, and the stream is bent sideways faster than its stars' motions can follow. Their velocities stop pointing along the track — and how far they stop is a measurement of the Cloud's mass.

6 figures · Stellar streams

Cosmology

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

An electroweak field decays toward one line, and helicity decides where it meets it. The comoving strength of a magnetic field against its coherence length, both on logarithmic scales. A field made at the electroweak transition, at 100 GeV, starts on a hundredth of the Hubble length then (1 AU today) with a tenth of the radiation's energy density (1 × 10⁻⁶ G), and decays as its turbulence is processed until the largest eddy is the one an Alfvén wave can cross by recombination — the straight line, 10⁻⁸ G at 1 Mpc. With no net helicity it falls as λ^−5/2 and stops at 0.038 pc and 4 × 10⁻¹⁶ G. Maximally helical, it falls only as λ^−½ and stops at 4.7 kpc and 5 × 10⁻¹¹ G. The shaded floor is what the missing gamma-ray halos exclude: a field coherent over more than a megaparsec must exceed 9 × 10⁻¹⁷ G, and the bound rises as the inverse square root of the coherence length below the leptons' cooling length of 0.6 Mpc. The non-helical field ends 3.0 decades below it; the helical one 4.6 decades above it. Above 10⁻⁹ G at a megaparsec the microwave background excludes the field.

A twist that decides where the field ends

A magnetic field made in the first trillionth of a second starts strong and absurdly small, and its turbulence spends the next three hundred thousand years trading strength for size. How far it gets depends on one conserved quantity — how twisted its field lines are — and for a field made at the electroweak transition, that twist is the difference between failing the void bound by three decades and clearing it by five.

6 figures · Primordial fields A sightline through the web, and where its field would be. The density of matter along a 300-megaparsec sightline through a model cosmic web, relative to the mean, on a logarithmic scale: a lognormal field whose logarithm has a width of 1.2. Galaxies sit at the 8 density peaks above the mean, and each has blown a magnetised bubble 1.4 Mpc in radius (shaded columns). Together they cover 7% of the sightline. The dots along the bottom are where 60 teraelectronvolt photons, stopped at random along the path, made their pairs: a photon stops where it stops, regardless of what is there, so the pairs fall on the empty stretches as often as on the crowded ones. Four of the 60 are made inside a bubble and are turned by its field. The other 56 are made in unmagnetised space, and their light forms a halo at full strength whatever the bubbles hold.

A field the galaxies cannot spread far enough

The missing gamma-ray halos are read as a field older than the galaxies because the galaxies seem unable to have put one there. The reason is not strength. A galaxy's wind carries a field that, spread over a megaparsec, is still ten million times stronger than the halos require. What it cannot do is reach the empty volume where most of the pairs are made — and a halo is dimmed only by the share of space the field fills.

6 figures · Primordial fields

Exoplanets

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

An envelope that boils away in a million years, or keeps for longer than the universe. The time a 5-Earth-mass planet takes to lose an envelope of 5.0% of its mass through an isothermal wind driven by its own heat and the star's light, against the escape parameter Λ at the envelope's outer edge — the ratio of the gravitational binding of a molecule there to its thermal energy — for equilibrium temperatures of 500, 800, 1200 K, on a logarithmic time axis. The rate is exponential in Λ, so the time grows by a factor of e for every unit: each step of ten in Λ is more than four decades of time. The envelope goes in under a million years below Λ ≈ 19, 18, 18 and lasts longer than a billion years above Λ ≈ 26, 26, 26. The window between a million and a billion years is a band only about seven units of Λ wide, so a planet is either boiling or effectively sealed.

An envelope that boils down before the starlight strips it

The usual story of how small planets lose their hydrogen starts with the star's ultraviolet light and runs for a hundred million years. It skips the first million. When the gas disc that fed a young planet disappears, an envelope that was swollen and held in by the disc's pressure is suddenly exposed, and it pours off as a wind powered by ordinary heat — until it has shrunk to the size at which the same number near thirty that governs every escaping atmosphere shuts the flow off.

6 figures · Atmospheric escape

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