An interferometer observes a spectroscopic binary whose two stars sit too close…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Coherence and statistical optics · Coherence and statistical optics

What an Instrument Can See, Ch Spatial coherence and the van Cittert-Zernike theorem

Problem

An interferometer observes a spectroscopic binary whose two stars sit too close together for any single telescope to resolve, at wavelength

\lambda = 550\ \mathrm{nm}

. Each star is unresolved on its own, so it may be treated as a point (delta-function) incoherent source: a primary of flux

I_1

and a secondary of flux

I_2 = rI_1

, with brightness ratio

r \le 1

, separated on the sky by a small angle

\alpha

. The interferometer's baseline

\mathbf{B}

is aligned with the line joining the two stars, so the problem is one-dimensional, and the pair sits at angular positions

\theta = \mp\alpha/2

(primary, secondary) along that line. The van Cittert-Zernike theorem of the propagation-of-coherence section gives the complex coherence factor as the normalised Fourier transform of the source's angular brightness,

\mu_{12}(B) = \frac{\displaystyle\int I_0(\theta)\,e^{-i2\pi\theta B/\lambda}\,d\theta}{\displaystyle\int I_0(\theta)\,d\theta},

evaluated at the baseline

B

(the phase factor the theorem divides out is dropped throughout). ((a)) Write

I_0(\theta)

as two delta functions of weight

I_1

and

I_2

\theta=\mp\alpha/2

, and show that

\mu_{12}(B) = \frac{I_1\,e^{i\pi\alpha B/\lambda} + I_2\,e^{-i\pi\alpha B/\lambda}}{I_1+I_2}.

((b)) Show that the squared fringe visibility is

|\mu_{12}(B)|^2 = 1 - \frac{4r}{(1+r)^2}\sin^2\!\left(\frac{\pi\alpha B}{\lambda}\right),

and check that setting

r=1

recovers the equal-brightness result

|\mu_{12}(B)| = |\cos(\pi\alpha B/\lambda)|

, which does reach zero. ((c)) The measured visibility first dips to a minimum

|\mu_{12}| = 0.60

at a baseline

B_1 = 4.0\ \mathrm{m}

, and no shorter baseline shows a smaller value. Find the angular separation

\alpha

of the two stars, in milliarcseconds. ((d)) Find the brightness ratio

r

from that minimum, and explain why this binary's visibility curve never falls to zero on any baseline.

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