Astronomy & Celestial Mechanics
the sky, computed
Concept

Baseline — where it appears

The vector between two antennas or two telescopes observing one source together. It sets the angular resolution of an interferometric measurement, which improves in inverse proportion to its length, and it is why the deep-space complexes sit a third of the world apart.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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.

starlight · Interferometry A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky.

An angle measured against a quasar

A tracking station measures how fast a spacecraft is receding, which is one number where three are wanted. The two missing angles come from the Earth's rotation, slowly, and near a planetary encounter there is no time for slowly — so the position is instead measured directly, as a difference of arrival times between two antennas, referred to a quasar a few degrees away.

spaceflight · Radiometric navigation A few per cent of diameter, hidden in the second lobe. Visibility against baseline in the natural variable πθB/λ, for stars with linear limb-darkening coefficients of 0, 0.3, 0.6, 0.9, each rescaled to the uniform disc that best fits its own first lobe. In the first lobe the four curves are within 0.72 per cent of one another; past the first null they differ by up to 4.2 per cent. That is the whole difficulty of measuring a stellar diameter. A limb-darkened star has a faint edge, so a uniform-disc fit returns a diameter 9.7 per cent too small at u = 0.9 and 2.5 per cent too small at u = 0.3 — and the information needed to tell which is in a region where the visibility is under five per cent and the calibration errors of a real interferometer are comparable to the signal. The correction from what is measured to what is wanted is taken from a model atmosphere, because the observation that would supply it is the hardest one there is. The fit here is done by least squares on the drawn curves rather than read from a conversion table, and the zero-coefficient case returns the uniform disc to 0.00 per cent, which is the fit checking itself.

A diameter that depends on a model atmosphere

An interferometer measures fringe visibilities and somebody fits a disc. A uniform disc and a limb-darkened one agree to within a per cent across the whole of the first lobe and differ by five in the second — where the visibility is under five per cent and the calibration errors are the same size.

starlight · Angular diameter

Named alongside it

The objects these essays reach for when they reach for this one.

Angular accuracyAngular diameterAngular resolutionAperture synthesisBolometric fluxClock offsetDeconvolutionDeep-space networkDelta-DORDifferential measurementEffective temperatureEncounter navigation

All concepts