Fringe contrast against baseline, for four stellar diameters

At its defaults it draws fringe contrast against baseline, for four stellar diameters. The visibility of the fringes an interferometer would obtain at 550 nm, against the separation of its two apertures, for uniform discs of angular diameter 10, 20, 47, 100 milliarcseconds. Each curve is 2J₁(πθB/λ)/(πθB/λ), the transform of a uniform disc, and each first reaches zero at 13.84 m for 10 mas, 6.92 m for 20 mas, 2.94 m for 47 mas, 1.38 m for 100 mas. Dividing each of those by λ/θ returns the same number, 1.2197, which is the 1.22 in every textbook and is the first zero of J₁ divided by π — recovered here from the four curves rather than written into them. The practical content is that a smaller star needs a longer baseline, in exact inverse proportion, and that the measurement is of a contrast rather than of a picture. Michelson and Pease found the fringes from Betelgeuse vanishing at a 3.07 m separation in 1920 at a wavelength of 575 nm, which by the same arithmetic is a disc 47.1 milliarcseconds across — and no telescope resolved that star for another seventy years.

stellar-visibility is one function in lib/figures/optics.js — rays, lenses, mirrors and what light does to a surface. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The visibility of the fringes an interferometer would obtain at 550 nm, against the separation of its two apertures, for uniform discs of angular diameter 10, 20, 47, 100 milliarcseconds. Each curve is 2J₁(πθB/λ)/(πθB/λ), the transform of a uniform disc, and each first reaches zero at 13.84 m for 10 mas, 6.92 m for 20 mas, 2.94 m for 47 mas, 1.38 m for 100 mas. Dividing each of those by λ/θ returns the same number, 1.2197, which is the 1.22 in every textbook and is the first zero of J₁ divided by π — recovered here from the four curves rather than written into them. The practical content is that a smaller star needs a longer baseline, in exact inverse proportion, and that the measurement is of a contrast rather than of a picture. Michelson and Pease found the fringes from Betelgeuse vanishing at a 3.07 m separation in 1920 at a wavelength of 575 nm, which by the same arithmetic is a disc 47.1 milliarcseconds across — and no telescope resolved that star for another seventy years.

The degree of coherence of a 47 milliarcsecond disc at 550 nm, and its square, against the separation of two apertures. A Michelson interferometer measures the upper curve, because fringe contrast is |γ|. Correlating the intensities at the two apertures instead measures the lower one, because the excess correlation of two thermal beams is |γ|² — the same information about the source, since one curve determines the other, and reaching zero at the same baseline of 2.94 m. What is lost is the phase of γ, which the intensity correlation never sees; what is bought is that a path error of many wavelengths does not matter, because the quantity being correlated is a slow fluctuation of brightness rather than a wave. The half-coherence baselines differ — 1.71 m against 1.25 m — which is the practical statement that the squared curve is the steeper one to measure against.

The degree of coherence of a 47 milliarcsecond disc at 550 nm, and its square, against the separation of two apertures. A Michelson interferometer measures the upper curve, because fringe contrast is |γ|. Correlating the intensities at the two apertures instead measures the lower one, because the excess correlation of two thermal beams is |γ|² — the same information about the source, since one curve determines the other, and reaching zero at the same baseline of 2.94 m. What is lost is the phase of γ, which the intensity correlation never sees; what is bought is that a path error of many wavelengths does not matter, because the quantity being correlated is a slow fluctuation of brightness rather than a wave. The half-coherence baselines differ — 1.71 m against 1.25 m — which is the practical statement that the squared curve is the steeper one to measure against.

The visibility of the fringes an interferometer would obtain at 575 nm, against the separation of its two apertures, for uniform discs of angular diameter 10, 20, 47, 100 milliarcseconds. Each curve is 2J₁(πθB/λ)/(πθB/λ), the transform of a uniform disc, and each first reaches zero at 14.47 m for 10 mas, 7.23 m for 20 mas, 3.08 m for 47 mas, 1.45 m for 100 mas. Dividing each of those by λ/θ returns the same number, 1.2197, which is the 1.22 in every textbook and is the first zero of J₁ divided by π — recovered here from the four curves rather than written into them. The practical content is that a smaller star needs a longer baseline, in exact inverse proportion, and that the measurement is of a contrast rather than of a picture. Michelson and Pease found the fringes from Betelgeuse vanishing at a 3.07 m separation in 1920 at a wavelength of 575 nm, which by the same arithmetic is a disc 47.1 milliarcseconds across — and no telescope resolved that star for another seventy years.

Two-slit fringes formed by light from an incoherent disc 47 milliarcseconds across at 575 nm, computed by adding the intensities — never the amplitudes — of the patterns made by 601 independent points spread across it, for baselines of 0.5 m, 2 m, 3.2 m, 5 m. The wider the slits are set, the more the patterns from opposite edges of the source slide out of step with each other, and the shallower the sum becomes. The visibility measured off each drawn curve is 0.952 at 0.5 m, 0.401 at 2 m, 0.030 at 3.2 m, 0.073 at 5 m, against 0.952, 0.401, 0.030, 0.073 from van Cittert and Zernike's theorem. Nothing in the summation knows about that theorem: it is four hundred cosines added up. What the agreement means is that the contrast of a fringe pattern is a Fourier component of the source's shape, so an instrument that measures contrast is measuring the source without ever forming an image of it.

Fringe visibility against baseline for three different sources, chosen so that two of them lose their fringes at the same separation — 3.08 m at 575 nm. A uniform disc 47 milliarcseconds across falls to zero there and comes back in a small sidelobe. An equal double star, whose visibility is a cosine, falls to zero there as well and then returns all the way to one, over and over. A Gaussian source of comparable width never reaches zero at all. An observer who measured only the first null would report the same angular size for all three, and would be wrong about two of them. This is the honest statement of what an interferometer measures: not a diameter, but samples of the source's Fourier transform, one spatial frequency per baseline. Recovering the shape needs many baselines, which is what aperture synthesis is; recovering it uniquely also needs the phase, which a single pair of apertures through a turbulent atmosphere does not deliver.

physicscheck asserts something about stellar-visibility that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.