Stellar Intensity Interferometer Lab: Hanbury Brown-Twiss Correlation, Baseline and Star Diameter

Setting up the lab bench...

Presets

Live readouts

Graphs

Excess correlation V^2 against baseline: theory and one simulated run

Running correlation estimate converging on V^2 (grey: 1 sigma band)

Signal to noise against baseline (red line: SNR = 5)

Numerical checks

running...

How to use

Choose a preset or drag the sliders: the blue curve is the exact visibility squared, the orange points are one simulated observing run, and the green line is your current projected baseline.
Move the telescope separation toward the first null and watch the correlation vanish; past it, it rises again as a weak side lobe.
Increase integration time or mirror size to shrink the error bars, then check that the fitted diameter matches the star.
Tilt the star toward the horizon: the projected baseline shortens, so the correlation rises.
Keys: Space pauses, R resets the running correlator.

The 3D page shows two telescopes on a rail (drawn up to 2 times larger than real, one scene unit is 10 m). The coloured pattern on the ground is the squared coherence |gamma|^2 around telescope 1: telescope 2 sees a strong correlation where the ground is bright and none in the dark rings. The dishes and beams follow the star position sliders; the screen above the hut is the live correlator.

The science

Two telescopes look at the same star. Starlight is chaotic (thermal), so the intensity at each telescope fluctuates in a noisy way. When the two mirrors are closer than the coherence length of the light on the ground, those fluctuations are partly the same: Hanbury Brown and Twiss (1956) found that the correlation of the two photocurrents measures the square of the fringe visibility of the star, with no phase information and no path-length control.

g2(b, 0) = 1 + |gamma12(b)|^2, gamma12 = (Fourier transform of the sky brightness) [Siegert relation, Van Cittert-Zernike theorem]

For a uniform disk of angular diameter θ the transform is an Airy function, and for linear limb darkening I(μ) = 1 - u(1 - μ) there is a closed form:

The first zero is at x = 3.8317, i.e. b = 1.2197 λ/θ. Measuring several baselines and fitting the curve gives the stellar diameter. With θ = 5.9 mas at 440 nm the first null is near 19 m, which is why the 1960s Narrabri instrument (up to 188 m) could resolve stars.

Signal to noise in the original analysis (photon noise limited, narrow electronic band Δf inside a wide optical band):

SNR = A α n |gamma|^2 √(Δf T / 2)

A is the mirror area, α the quantum efficiency, n the photon flux per unit optical frequency bandwidth and T the integration time. SNR does not depend on the optical bandwidth, which is why intensity interferometry tolerates poor optics and turbulence. It is a weak signal: the per-sample correlation is about 10^-5, so hours of data are needed.

Honest simplifications. The star is a uniform or linearly limb-darkened disk (no spots or oblateness). The photon flux uses a rough zero-point (about 1000 photons s^-1 cm^-2 A^-1 for magnitude 0 near 440 nm) and the diameters of Rigel and Betelgeuse are approximate. The projected baseline uses a flat site. The Monte Carlo is rescaled: it draws N = (SNR0/0.1)^2 samples with a strong per-sample correlation 0.1 V^2, which has the same statistical SNR as the real 10^10-sample measurement. Earth rotation, atmospheric transparency, filter bandwidth and electronic delays are not modelled. This is a teaching model, not a data-reduction tool.