Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 27

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 521–534 of 534.

Spaceflight

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

The relative orbit when the target's own is not circular. Free relative motion of a chaser about a target, over one revolution, for four eccentricities of the target's orbit — drawn by propagating two exact Keplerian orbits and differencing them in the target's own rotating frame rather than by any linearised solution. The chaser differs from the target by 0.0035 in eccentricity, so the two semi-major axes are identical. Along-track distance runs across the page. At e = 0 the path is the familiar closed ellipse, 2.00 times as long along the track as across it — recovered here from an exact propagation, which is the check that the calculation is right. Raise the target's eccentricity and the path stops being an ellipse and starts breathing: the separation varies by a factor of 2.00 over one revolution at e = 0 and 2.26 at e = 0.7, because the frame itself is now turning at a rate that changes through the orbit and the chaser is swept around it fastest at periapsis. Every one of them still closes exactly, and that is the one statement of the circular theory that survives — two orbits of equal semi-major axis have equal periods and cannot drift apart, whatever else differs between them.

The relative orbit that breathes

Every result in the close-approach frame — the two-to-one ellipse, the burn that moves the wrong way, the eight kilometres behind — assumes the target's own orbit is circular. Around an eccentric target none of the geometry survives. One statement does, and it is about energy rather than shape.

5 figures · Rendezvous Two ranges that look identical until something is spent. The bearing to a target, as a camera measures it, for two relative orbits that differ only in size — one 2.5 times further away than the other in every coordinate at every instant. Before the manoeuvre the two histories are identical to machine precision, and that is not a numerical accident but a property of the equations: the linearised relative dynamics are homogeneous, so scaling a whole trajectory gives another trajectory, and a direction is unchanged by scaling. An angles-only tracker is therefore exactly blind to range, and no amount of observing improves it. At 42 per cent of the way through, both vehicles fire the same 0.05 m/s impulse — and the histories separate, by up to 2.3 degrees, because an impulse measured in metres per second is the one quantity in the problem that does not scale with the trajectory. The manoeuvre is what makes range observable, which is why an autonomous vehicle navigating on a camera alone flies a deliberate zig-zag rather than a straight approach, and why the manoeuvre is chosen for what it reveals as much as for where it goes.

A camera that cannot see how far

The equations that make a close approach tractable are linear and homogeneous, so scaling an entire relative trajectory gives another valid one with every bearing identical. A vehicle navigating on angles alone is therefore not merely imprecise about range — it is exactly blind to it, and the only thing that restores the scale is spending propellant.

5 figures · Rendezvous

Starlight

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

Three pictures, one set of measurements. A one-dimensional sky, the transform of it sampled at 10 spatial frequencies out of 128, and three images reconstructed from those samples. Every one of the three reproduces every measured visibility exactly — checked to a part in a million — so the data cannot choose between them. The dirty image is the samples transformed back and nothing else, complete with the negative sidelobes that incomplete sampling produces. The other two are the members of the same family that minimise a penalty: one the total curvature, one the total absolute brightness. They differ from each other by 7 per cent of the peak, and where they differ is exactly where the array did not look. The smooth reconstruction blurs the two compact sources into one another and keeps the broad component; the sparse one splits them cleanly and loses the broad one. Neither is a lie and neither is the sky. A published image is the data plus a criterion, and the criterion is a choice made by whoever ran the reduction.

How much of the picture is the prior

An array measures the sky's transform at a few dozen spatial frequencies and at no others, so the data do not determine an image. They determine a family of them, and the reconstruction picks one member by a criterion that is not in the data — smoothness, or sparsity, or entropy. Three reconstructions of one measurement fit it exactly and disagree by a tenth of the peak.

5 figures · Interferometry The position closure discards, and what it costs to buy back. The astrometric error left after phase referencing, against how far away the calibrator is, for three switching cadences, on a 8000-kilometre baseline at 1.3 mm. Referencing gives up the immunity closure quantities have: instead of forming a combination the atmosphere cannot enter, it measures the atmosphere on a nearby source and subtracts it, which recovers the absolute position and leaves behind whatever differs between the two lines of sight. That difference is governed by the Kolmogorov structure function, which rises as the five-thirds power of a separation, so the phase residual and therefore the position error rise as the five-sixths power of the separation, which the drawn spatial term reproduces exactly. Below about 1.7 degrees at the fastest cadence the curves flatten, because there the atmosphere's change between one visit to the calibrator and the next is the larger of the two differences and the separation has stopped mattering. The consequence is a premium on finding a close calibrator: at one degree the error is 2 microarcseconds and at eight it is 5, and the sky is not dense in compact bright sources. The three curves are the other half of the trade — a faster cycle freezes the atmosphere better and spends more of the observation looking at the calibrator, so the optimum is where the two losses meet rather than as fast as the hardware allows.

The position closure throws away

A closure phase is immune to the atmosphere because it cannot see anything that looks like a per-antenna error — and a displacement of the source on the sky looks exactly like one. Recovering where a source is means giving that immunity up on purpose, measuring the atmosphere on a neighbour instead of cancelling it, and paying for the difference between two directions.

5 figures · Interferometry A sight line stops where its own optical depth is one. The source function of a grey atmosphere in radiative equilibrium — a straight line, S = (3/4)(τ + 2/3) in units of the flux — with the depth each sight line reads off it. The emergent intensity at an angle is the source function integrated along the ray with everything in front of it attenuating, and for a source function linear in optical depth that integral is exactly the source function evaluated at τ = μ. Not approximately: the Eddington–Barbier relation is an identity for a linear source function, and the numerical integral agrees with it here to a part in ten thousand, which is the quadrature's error and not the relation's. A vertical ray reads the source function at τ = 1 and a ray at 78 degrees reads it at τ = 0.2, which is higher in the atmosphere and therefore cooler — so the limb is dimmer than the centre by the amount the source function has risen between those two depths. Everything about limb darkening is the slope of this one line, and the famous 2/3 is the intercept: the average ray, over the whole disc, reads the source function at two thirds of a unit of optical depth, which is the depth a stellar "surface" actually means.

A sight line stops at two thirds

The whole of limb darkening is one sentence: what leaves a star at an angle is the source function evaluated where the slant optical depth is one. For a source function linear in depth that statement is exact rather than approximate, and it produces the grey atmosphere's (2+3μ)/5 with no fitting anywhere in it.

5 figures · Limb darkening The sign of one gradient decides which way the disc is shaded. The emergent intensity across a stellar disc, against the fractional radius, for four gradients of the source function with depth. Every curve is the Eddington–Barbier relation evaluated at the appropriate μ and checked against a numerical integration of the transfer equation. The contrast between centre and limb is the gradient, and its sign is the gradient's sign. A source function rising inward — the ordinary case, since temperature rises inward and the source function follows it — darkens the limb: a sight line at the edge leaves from higher up, where the gas is cooler. A flat source function produces a uniformly bright disc whatever the geometry. And a source function that falls inward brightens the limb, because now the shallower ray is reading a hotter layer. That last case is not hypothetical: above a star's photosphere the temperature stops falling and begins to rise, so at wavelengths that see those layers — millimetre continuum, the cores of strong lines, the ultraviolet — the Sun's limb is brighter than its centre. Limb darkening is therefore not a property of stars but of a gradient, and the same atmosphere shows both signs at different wavelengths.

A limb brighter than the middle

Limb darkening is not a property of stars. It is a property of a gradient, and the same relation that produces it produces the opposite whenever the source function falls inward — which it does above every stellar photosphere. The Sun's limb is darker in the visible and brighter at a millimetre, and both are the same equation.

5 figures · Limb darkening

Galaxies

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

Two ways to stop, and the same curve shifted. The fraction of galaxies that have stopped forming stars, against their stellar mass, at four local densities. Two things are happening at once. Mass quenching turns the curve upward above about 10^10.6 solar masses, and it does so at the same mass whatever the environment — the curves rise in step. Environment quenching lifts the whole curve, and it does so by the same fraction of what is left at every mass, which is why the four curves look like one curve with its floor raised rather than like four different shapes. At 10⁹·⁵ solar masses the red fraction runs from 13 per cent in the field to 97 in a cluster core, and almost all of that difference is environmental; at 10¹¹·⁵ almost everything is red everywhere and the environment has nothing left to do. The two mechanisms are separable, which is the census's central quantitative result and the thing any account of either one has to explain.

Two ways to stop that do not interact

A galaxy's chance of having stopped forming stars depends on its mass and on where it is, and the census says the two dependences multiply. The efficiency with which environment quenches a galaxy is the same number at every mass, across two and a half decades of it — which is surprising, because both mechanisms work on the same gas.

5 figures · Galaxy populations Three colours watching one galaxy stop. One star-formation history — steady, then shut off with an e-folding time of 0.4 billion years — followed in three colour indices, each plotted as a fraction of the gap its own sequences actually span. The three curves are the same galaxy. Each band's luminosity is the history convolved with that band's own decline after a burst: NUV at 0.1 Gyr, u at 0.35 Gyr, g at 0.9 Gyr, r at 3 Gyr, because the ultraviolet comes from stars that live a hundred million years and the red from stars that live ten billion. In the first three hundred million years the ultraviolet index reddens at 1.56 magnitudes per billion years and the optical one at 0.10 — a factor of 15 — so a galaxy that has just stopped is already conspicuous in one and indistinguishable from a star-forming one in the other. That is what an ultraviolet band buys: not a better measurement of the same thing, but sensitivity to a phase the optical colours arrive too late to see.

A crossing time that is the filter's

The emptiness of the gap between the two galaxy populations was read as a timescale, and the reading used one colour. Each passband remembers star formation for a different length of time, so the same galaxies give a crossing time of one billion years in the ultraviolet and three in the optical — and the disagreement between two filters is the measurement the single filter could not make.

5 figures · Galaxy populations

Orbits

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

A cycle and a radiation, racing for the same pericentre. Two integrations of the same compact triple, in units of the cycle's own timescale at the starting separation. The dashed curves are the semi-major axis and the solid ones the eccentricity. Radiation alone shrinks the orbit slowly and smoothly, reaching a tenth of its starting size after 1136 cycle times — which the closed form for a circular inspiral predicts to within a few per cent, and which is the check that the integrator is right. With the cycle running, the distant companion drives the eccentricity to 0.9104, the pericentre falls by two orders of magnitude, and the radiation — whose rate goes as the inverse seventh power of one minus the square of the eccentricity — empties the orbit in 18. That is a factor of 62, obtained without changing a single parameter of either mechanism. The two do not merely add: the cycle manufactures, at its own peak, exactly the configuration in which the radiation is most efficient, and the radiation then shrinks the orbit until the relativistic precession switches the cycle off.

A cycle that builds what the wave needs

Gravitational radiation empties an orbit at a rate that goes as the inverse seventh power of one minus the square of the eccentricity. A distant companion drives exactly that eccentricity, over and over. The two are not independent mechanisms that happen to act on the same binary — the first manufactures, at its own peak, the configuration the second is most efficient in.

5 figures · Kozai–Lidov A ceiling the outer orbit imposes. The greatest eccentricity a cycle can drive an inner orbit to, against the mutual inclination it started at, for four ratios of the inner orbit's angular momentum to the outer one's. The bottom curve is the test-particle case and it is the familiar closed form, √(1 − (5/3)cos²i₀), reaching one at ninety degrees — recovered here by a solver that knows nothing about it, which is the check. Every other curve is lower. The reason is conservation: the cycle works by moving angular momentum out of the inner orbit, and it has nowhere to put it but the outer one. A test particle can give up all of its angular momentum because the outer orbit is infinite by comparison; an inner binary carrying 1 times as much as its companion cannot, because tilting the outer orbit by the required amount changes the mutual inclination and shuts the cycle down. At ninety degrees the ceiling falls from 1.000 to 0.866. The pericentre a mechanism can deliver is bounded by what the perturber can absorb, and the test-particle formula is an upper bound rather than a prediction.

A ceiling the outer orbit imposes

The closed form for how far a cycle can drive an eccentricity assumes the inner orbit is a test particle, so the angular momentum it gives up goes nowhere. A real inner binary of comparable mass has to be paid for: the outer orbit absorbs what the inner one loses, tilts, and changes the very inclination the cycle depends on.

5 figures · Kozai–Lidov

Cosmology

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

A blur undone with the field that made it. The acoustic feature in the correlation function, as it was laid down and as bulk flows leave it, with three reconstructions. The broad band is left out entirely — an analysis divides it away and fits the position of what remains — so what is drawn is the feature itself. Every galaxy has moved since recombination, by several megaparsecs, and the two ends of a pair separated by 150 megaparsecs move almost independently: the pairwise dispersion of those displacements, Σ = 8.5 Mpc here, convolves the feature and broadens it from 35 to 43 megaparsecs. Reconstruction estimates the displacement from the observed density and moves every galaxy back along it. The estimate has to be smoothed, because the relation between density and displacement is only linear on large scales, and whatever the smoothing removes survives as a residual — Σ falls to 1.1 at a 10-megaparsec kernel and would fall further at a finer one, until the noise the kernel is there to suppress took over. The feature is sharpened by the same field that blurred it, and what limits the sharpening is not the idea but how well the field can be measured.

A blur undone with the field that made it

The acoustic feature is smeared by the same bulk flows whose displacements can be estimated from the density they produced. Moving every galaxy back along that estimate sharpens the peak and removes most of its second-order shift — and what limits the procedure is not the idea but how well a field can be measured from the objects it moved.

5 figures · Baryon acoustic oscillations Fine along one axis, coarse across the other two. The smallest scales a forest survey can measure, against how many quasars a square degree it has sightlines to, at redshift 2.33. Along a sightline the sampling is set by the spectrograph: a 69 km/s pixel is 0.29 comoving megaparsecs at this redshift, so the radial wavenumber reaches 10.8 Mpc⁻¹ and it does not depend on the survey's size at all. Across the sky it is set by how densely the quasars sit: at 30 a square degree the mean separation is 18 megaparsecs and the transverse wavenumber reaches 0.17. The ratio is 63, and improving it means finding more quasars, which are a finite population on the sky. That anisotropy is the inverse of a galaxy survey's, where the transverse direction is sampled by a wide sky and the radial one is limited by a redshift shell and contaminated by peculiar velocities — so the forest measures the expansion rate at its redshift better than it measures the distance to it, which is the reverse of every galaxy survey.

A ruler read in absorption

Beyond the reach of any galaxy survey the only tracer left is the hydrogen between here and a quasar, sampled finely along each sightline and coarsely across the sky. That inverts the balance every galaxy measurement is built on — the forest measures an expansion rate better than a distance, which is the harder of the two to get any other way.

5 figures · Baryon acoustic oscillations

Exoplanets

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

Two references, failing in opposite places. The faintest companion each technique can detect, against separation in resolution elements, in units of the speckle noise a perfectly matched reference would leave. Angular differential imaging builds its reference from the target's own exposures, so the reference matches the instrument's state exactly and the residual speckle is small — and the companion is in every frame of it, so close in the companion subtracts most of itself and the achievable contrast collapses. Reference-star differential imaging builds the reference from archival exposures of other stars. The companion is in none of them, so the throughput is one at every separation and the curve is flat; what is lost is the match, because two stars observed at different times through different air with a differently flexed telescope have different speckles. The two cross, and where they cross depends on how large the library is: 1.1 resolution elements at 20 frames, 1.4 resolution elements at 200 frames, 1.8 resolution elements at 2000 frames. A library of thousands of archival exposures is therefore worth building, and the thing it buys is the inner working angle rather than the depth.

A reference built from other stars

Subtracting a star's halo using the star's own exposures removes part of the planet with it. Building the reference from a library of archival exposures of other stars removes none of the planet and matches the halo worse — so the two techniques fail in opposite places, and which one wins at a given separation depends on how many exposures somebody has kept.

5 figures · Direct imaging

The observed sky

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

A free libration does not last. How long a free libration survives before tidal dissipation removes it, against the tidal quality factor, for four locked bodies — with the age of the solar system marked. A lightly damped oscillator loses its amplitude in about Q of its own periods, and a free libration's period is years, so the decay time is thousands to hundreds of thousands of years for any plausible Q. Against 4.5 billion years that is instantaneous. A free libration that is observed today was excited within the last geological instant, and one that is not observed bounds how much has happened. The Moon's is detected, at a few arcseconds, and requires something recent — an impact, a core–mantle interaction, or a resonance passage. Mercury's has been searched for and not found, and the upper limit is the constraint.

A rocking nobody started

A locked body is a pendulum, and a pendulum has a period of its own. That free libration is damped by tides in thousands of years, so any body still oscillating at its own frequency was disturbed in the last geological instant — and the Moon is, while Mercury is not.

5 figures · Libration

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