Astronomy & Celestial Mechanics
the sky, computed
Generator

The uv-synthesis generator

The transform plane an array of 9 actually samples
The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them.

Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is *Hermitian*: a real sky forces V(−u,−v) = V*(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly *sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them.

5 essays call uv-synthesis. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them. Starlight

Resolution without a mirror

Two telescopes a kilometre apart do not make a kilometre-wide telescope. They measure one number — the Fourier component of the sky at the spatial frequency their separation sets — and an image is what you get by collecting enough of those.

 20 closure phases, unmoved by an atmosphere that ruins every baseline. Closure phase measured against closure phase true, for all 20 triangles of a 6-antenna array observing a binary 3.2 mas apart with a flux ratio of 0.35. Each antenna has been given an independent atmospheric phase of 65° rms, which corrupts the individual baseline phases by 102° rms — several times the 31.2° the source itself produces, so no single visibility phase in this simulation carries usable information. Every point here nonetheless sits exactly on the diagonal: the per-antenna terms cancel identically round any triangle, and the largest departure over all 20 is 2.5e-14 degrees, which is round-off. The price is in the counting. 6 antennas give 15 baseline phases of which 5 are consumed by the unknowns, so of the 20 triangles only 10 closures are independent — a fraction (N−2)/N = 0.667 of the phase information. For two antennas that fraction is zero and there is no closure at all; the Event Horizon Telescope's image rests on quantities of this kind and on no absolute phase whatever. Starlight

A phase that survives what corrupts it

An atmosphere over each antenna adds an unknown to the phase of every baseline that antenna takes part in. Sum the phases round a triangle and every one of those unknowns cancels identically — which is the reason an image can be made across ten thousand kilometres, and the reason it has no position on the sky.

 A hole's shadow at 17° to its spin axis. The critical curve — the outline of a black hole's shadow on a distant observer's sky — for spins from 0 to 0.998, seen 17 degrees from the spin axis, in gravitational radii. Every curve is the projection of the unstable spherical photon orbits, and nothing about an accretion flow enters it. The sizes are 5.20, 5.12, 4.92, 4.84: a spread of 6.9 per cent across the entire range of spin, which is what makes an image of one of these a mass measurement with almost no model in it. What spin does instead is push the curve sideways and flatten one edge — the fast hole's outline sits 0.70 gravitational radii off centre, because photons co-rotating with the hole are dragged round and escape from closer in than counter-rotating ones can. That displacement is the spin signal, and it is measured against a centre nothing else marks. Gravitation

A ring whose size is almost only a mass

The dark patch at the centre of an image of a black hole is the projection of the orbits light cannot escape from, and its size changes by seven per cent across the whole range of spin and inclination. That insensitivity is what makes an image a mass measurement with almost no model in it — and what leaves the spin in the part of the picture that barely moves.

 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. Starlight

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.

 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. Starlight

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.

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