Astronomy & Celestial Mechanics
the sky, computed
Generator

The visibility-curve generator

Fringe visibility for a 47 mas disc at 575 nm
Fringe visibility for a 47 mas disc at 575 nm. Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the same observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star.

Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the *same* observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star.

7 essays call visibility-curve. 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.

Fringe visibility for a 47 mas disc at 575 nm. Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the same observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star. Starlight

An angle of five hundredths of an arcsecond

No telescope has ever resolved a star other than the Sun, and stellar diameters are measured anyway — by finding the separation of two apertures at which the star's interference fringes vanish. What that returns is an angle; the radius arrives only when a distance is brought in, and the distance is the worse-known half.

 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 star 1.24 times wider than it is tall, and 19 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.9257 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.245 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.768, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.329 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.320 at the theoretical exponent 0.25 and 1.232 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.19 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo. Starlight

A temperature that depends on where the observer stands

A star turning near its break-up rate is half again as wide as it is tall, and its equator is thousands of degrees cooler than its poles. Neither of those is a small correction to a spectrum — the effective temperature and the luminosity such a star appears to have are partly statements about which way its axis happens to point.

 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. Starlight

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.

 A colour that predicts an angle. The surface-brightness parameter Fᵥ — a combination of the V magnitude and the angular diameter that depends only on how bright each unit of the star's surface is — against the colour V − K, the ratio of the star's light at 0.55 and 2.2 microns. The line is the relation log θ = 0.5175 + 0.2753 (V − K) − 0.2 V, fitted to interferometric diameters of dwarfs and subgiants: Fᵥ = 3.962 − 0.1376 (V − K). The dots are 40 calibrating stars drawn about it with a 1 per cent scatter in angular diameter, the precision of a good interferometric measurement; refitting them returns a slope of −0.1374 and a scatter of 0.0023 in Fᵥ, which is 1.1 per cent in angle. The dashed curve is what blackbodies of 3,500 to 9,000 K would give, pinned to the Sun: it departs from the stars' line by up to 0.035 over the calibrated range, because real atmospheres have molecular bands and limb darkening a blackbody lacks — which is why the relation is calibrated on measured stars rather than derived. Given a star's V and V − K, the line returns its angular diameter without resolving it. Starlight

An angle too small to measure, read from a colour

A giant star in the Large Magellanic Cloud subtends two microarcseconds, a hundred times below anything an interferometer resolves. Its angle is read instead from its colour, through a relation calibrated on nearby stars that can be resolved — and because dust dims a star and reddens it by amounts that nearly cancel in that relation, the distance it gives hardly depends on the dust at all.

 The cancellation that protects the method, and where it fails. The error in distance from 0.1 magnitudes of extinction left uncorrected, against temperature, for the angle read from V and V − K through the blackbody relation's local slope, and for a standard candle in V (dashed, −4.7 per cent). For late-type stars the dimming and the reddening nearly cancel — the empirical relation gives +1.0 per cent. For hotter stars the reddening term grows and nothing offsets it: the error matches the standard candle's near 13,500 K and reaches +7.5 per cent at 20,000 K and +11.9 at 30,000 K, worse than the candle the method was meant to improve on. Starlight

The cancellation that leaves with the hot stars

The angular size of a cool giant can be read from its colour, and dust barely spoils the answer: the dimming and the reddening dust causes push in opposite directions and nearly cancel. Carried to hot stars the method gets weaker on both counts. Above about 10,000 K the colour V − K stops tracking temperature, a colour error costs more angle, and the reddening outweighs the dimming. Near 13,500 K leaving the dust uncorrected costs as much as it would a standard candle.

 The visibility precision a one-per-cent diameter needs. The precision in fringe visibility needed to measure a uniform disc's angular diameter to 1%, on the best baseline up to 330 m, against the diameter, at three wavelengths. Below the resolution limit the required precision falls as the square of the diameter. Even at its best, a visibility measures a diameter only about as precisely as it is itself measured, so a one-per-cent diameter needs visibilities better than a per cent. The shaded band, one to three per cent, is what a single calibrated visibility typically achieves; averaging ten or more measurements brings the effective precision to a few tenths of a per cent (dotted). At 0.3% the limit is about 0.13 mas in the visible, 0.19 mas in the red and 0.38 mas in the near-infrared H band, where most stellar diameters have been measured. Starlight

A diameter smaller than the fringes can see

The surface-brightness method gives distances good to a per cent for binaries of cool giants, and loses that precision for the hot stars that dominate other galaxies' bright binaries. The repair is an empirical calibration from the angular diameters of nearby hot stars — and those diameters are a few tenths of a milliarcsecond, below what the longest optical baselines comfortably resolve. A diameter measured at the edge of resolution is a small departure from perfect fringe contrast, and it costs visibility precision as the square of how small the star is.

The whole library · All essays