= Solution
\b[Delay-compensated geometry.] Choose an east-west baseline vector $\mathbf b=b\hat{\mathbf x}$ and let the upward unit vector be $\hat{\mathbf z}$. A ray direction at zenith angle $\beta$ in their plane has $\mathbf s(\beta)=(\sin\beta,0,\cos\beta)$. Its signed geometric path difference is $\mathbf b\cdot\mathbf s=b\sin\beta$. Removing the phase-centre delay $b\sin\theta/c$ leaves the residual path
$$
\delta\ell=b[\sin(\theta+\alpha)-\sin\theta]
\simeq b\alpha\cos\theta.
$$
With the receiver ordering chosen to give positive phase for this path, a complex correlator measures
$$
\boxed{V(\alpha)\propto e^{ik\delta\ell}
\simeq\exp(ikb\alpha\cos\theta).}
$$
The reverse ordering complex-conjugates the expression without changing its information. A single real multiplier supplies its cosine component; the stated exponential represents an in-phase/quadrature correlator. Equal cable lengths to the central correlator remove a fixed instrumental difference but do not remove the changing <geometric delay>. Pointing the <antennas> and electronically compensating the path are separate operations.
\Image[/past-exam-of-the-frontiers-of-observational-astrophysics-course-of-the-university-of-cambridge/2015/paper-1-interferometer-geometry.png]
{title=Original east-west interferometer geometry and normalized visibility of a centred one-dimensional uniform source}
\b[Resolved uniform source.] Contributions from mutually incoherent sky elements add as brightness-weighted <complex visibilities>. Put $I(\alpha)=I_0$ over $-\Delta\alpha/2<\alpha<\Delta\alpha/2$ and zero elsewhere, and define $q=kb\cos\theta$. Integrating gives
$$
V=I_0\int_{-\Delta\alpha/2}^{\Delta\alpha/2}e^{iq\alpha}d\alpha
=I_0\Delta\alpha\,\frac{\sin(q\Delta\alpha/2)}{q\Delta\alpha/2}.
$$
Therefore
$$
\boxed{\frac{V(b)}{V(0)}=\operatorname{sinc}\left(\frac{kb\Delta\alpha\cos\theta}{2}\right),
\qquad\operatorname{sinc}x=\frac{\sin x}{x}.}
$$
The zero-argument value is defined by continuity as one. Normalization by $V(0)=I_0\Delta\alpha$ isolates the visibility shape from total <flux>. The first null occurs when the source spans one fringe period, $\Delta\alpha=\lambda/(|b\cos\theta|)$. A negative real visibility beyond a null is a phase reversal, not negative <surface brightness>.
\b[Path-compensation precision.] A finite observing band sets a coherence envelope. For a rectangular band of width $\Delta\nu$, an uncorrected residual delay $\delta\tau=\delta\ell/c$ multiplies the correlation by
$$
e^{2\pi i\nu_0\delta\tau}\operatorname{sinc}(\pi\Delta\nu\delta\tau).
$$
Coarse delay steps must leave $\Delta\nu|\delta\tau|\ll1$ if decorrelation is to be small. Thus a coarse path-step scale must be well below the <coherence length> $c/\Delta\nu$, with the precise bound determined by the tolerated amplitude loss and maximum quantization residual. A digital fine-delay or phase-rotation system can separately correct the central-frequency phase.
If there is no such carrier-phase correction, the steps must also be a small fraction of $\lambda$ to avoid large untracked phase jumps or time-averaging loss. For a step $d_\ell$ rounded to the nearest setting, $|\delta\ell|\le d_\ell/2$ and the maximum carrier-phase error is $\pi d_\ell/\lambda$. \b[<Bandwidth> controls coarse group-delay decorrelation; <wavelength> controls the precision of untracked carrier phase.] No numerical delay step can be inferred from the central <frequency> alone without a <bandwidth>, loss tolerance and compensation architecture. Earth rotation additionally sets how rapidly the delay setting must be updated.
\b[Largest sampled angular scale.] For a near-zenith view take the shortest projected baseline as $b_{\min}\simeq0.5\ {\rm m}$. At $\nu=30\ {\rm GHz}$, $\lambda=c/\nu=9.9931\times10^{-3}\ {\rm m}$. The inverse shortest <spatial frequency> gives
$$
\boxed{\Theta_{\max}\sim\frac{\lambda}{b_{\min}}
=0.0199862\ {\rm rad}=68.7\ {\rm arcmin}\approx1.15\text{ degrees}.}
$$
This is an order-of-magnitude largest fringe scale. A convention defining maximum recoverable source size as $0.6\lambda/b_{\min}$ would instead give about $41$ arcminutes; source profile, weighting and projected baseline coverage determine the operational value. The <antenna> diameter and <primary beam> can further restrict the usable field but were explicitly neglected in the source's initial idealization. Ten <antennas> supply $10\times9/2=45$ pairwise baselines; their number improves sampling and sensitivity, not the inverse-shortest-baseline scale by a factor of ten. The longest baseline is not supplied, so the finest synthesized resolution is not numerically fixed.
\b[A suitable observing programme.] A compact centimetre-wave array is well suited to <CMB> temperature anisotropy on degree to sub-degree scales sampled by its baselines, and, with sufficiently long baselines and suitable foreground control, thermal Sunyaev-Zeldovich structure in <galaxy clusters>. At $30\ {\rm GHz}$ the thermal Sunyaev-Zeldovich signal is a decrement relative to the <CMB>. The compact baselines retain sensitivity to extended brightness features; the <spatial frequencies> measured by each baseline have a direct statistical interpretation in <CMB> analysis.
Compared with lower radio <frequencies>, synchrotron foregrounds are generally reduced, while thermal dust rises toward higher <frequencies>; observing near this centimetre-wave atmospheric window is therefore useful, although foregrounds are not absent. Correlation and phase switching reject mean uncorrelated receiver offsets and some instrumental drifts. Receiver <thermal noise> still contributes <variance>, and atmosphere, ground pickup or cross-talk can be correlated and require control. Additional long baselines can identify compact radio-source contaminants for subtraction, but the question does not specify enough array geometry to guarantee that capability. There is no zero baseline in ordinary pairwise cross-correlation, so the array does not directly measure the sky monopole or arbitrary larger-scale smooth emission.
Back to article page
