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The difference in plane-wave propagation time between two antennas is the projected baseline along the source direction divided by the speed of light. Its sign depends on the ordering of the receivers. A monochromatic signal acquires relative phase $2\pi \nu {\tau }_g$; its change with Earth rotation is compensated by fringe stopping for the chosen reference direction.

The wavelength is $\lambda =c/\nu \simeq 1.00\mathrm{mm}$. Thirty dishes give $N(N-1)/2=435$ interferometric baselines. The longest projected baseline sets the finest angular scale, while the individual dish beam sets the instantaneous field of view. At the maximum $10\mathrm{km}$ separation,A $10\mathrm{m}$ aperture instead has primary-beam width of order $\lambda /D=10^{-4}\mathrm{rad}$, about $21$ arcseconds; the Airy first-null convention gives about $25$ arcseconds. Illumination taper changes the precise beam definition. The array therefore resolves many synthesized beams within one pointing, but it does not image an arbitrarily wide field at that resolution.

The shortest useful baseline determines missing large-scale structure. A common estimate is ${\theta }_{\max }\sim 0.6\lambda /B_{\min }$. If dishes can be packed with usable center separations near $10\mathrm{m}$, this is of order $12$ arcseconds. A more extended layout has a smaller recoverable scale. The site diameter does not specify $B_{\min }$ or the actual Fourier coverage, and low-elevation shadowing can remove short spacings. Mosaicking expands sky coverage but does not supply all zero-spacing information; suitable total-power or compact-array data help recover extended emission. Earth rotation and reconfiguration improve sampled baseline coverage, sidelobes and image reconstruction, with a tradeoff between resolution and surface-brightness sensitivity.

For dish aperture efficiency ${\eta }_A$, $A_{\mathrm{eff}}={\eta }_A\pi D^2/4$. The system-equivalent flux density of one dish isFor independent receivers, natural weighting, correlator efficiency ${\eta }_c$ and $n_{\mathrm{pol}}$ equally sensitive polarization products contributing to intensity, the unresolved-source thermal rms isHere $\Delta \nu$ is usable bandwidth and $t$ integration time. A numerical sensitivity is therefore conditional on these unspecified observing choices. For an illustrative $1\mathrm{GHz}$ bandwidth, one hour, two polarizations and ${\eta }_c=1$, it is about $0.089/{\eta }_A\mathrm{mJy}$, or $0.127\mathrm{mJy}$ at ${\eta }_A=0.7$. Resolved-source brightness sensitivity also depends on beam area and missing flux. The sensitivity expression assumes naturally weighted data and the stated polarization convention.^[ref]

Earth rotation changes geometric delay. At a baseline $B$, ${\tau }_g$ can be of order $B/c$, here $33\mu \mathrm{s}$, and a raw fringe-rate bound isThe actual rate depends on geometry and source direction. Correlator delay tracking and fringe stopping compensate the chosen phase center rather than averaging rapidly rotating uncorrected fringes away. Off-center directions retain residual delays and rates.

Finite bandwidth causes chromatic phase averaging and radial bandwidth smearing. A direction offset $\theta$ has residual delay of order $B\theta /c$; a channel should satisfy $\delta \nu B\theta /c\ll 1$ for small decorrelation. Across a field of order $10^{-4}$ radians on the longest baseline, the one-cycle frequency scale is about $300\mathrm{MHz}$, so substantially narrower channels are required for accurate wide-field imaging. Similarly, time averaging should keep $\Delta t{\Omega }_{E}B\theta /\lambda \ll 1$; the associated one-cycle scale is about $14$ seconds in this example. Frequency-resolved visibilities and shorter dumps control these field-dependent averaging losses in radio interferometry; they are averaging losses, not ordinary glass-lens chromatic focusing.

At $300\mathrm{GHz}$, atmospheric water vapor produces absorption, thermal emission and path-length fluctuations. A high dry site reduces these effects, while phase fluctuations can still destroy coherence on long baselines. A one-radian phase corresponds to path fluctuation $\lambda /(2\pi )\simeq 0.16\mathrm{mm}$, motivating frequent phase calibration or atmospheric path monitoring. Surface errors must also be small relative to the wavelength: the Ruze efficiency factor is $\exp [-(4\pi {\sigma }_h/\lambda {)}^2]$. Pointing errors matter relative to a primary beam only tens of arcseconds wide.

Important systematic errors include antenna gains and bandpasses, delay and baseline-coordinate errors, polarization leakage, pointing, atmospheric opacity, correlator response and flux-scale calibration. Nearby compact calibrators, bandpass standards, appropriate total-power information and repeated configurations address different parts of this problem. High nominal collecting area and longest-baseline resolution alone do not determine image dynamic range or reliable extended-source flux.

Choose the baseline vector from the first antenna to the second. Apart from a sign convention for receiver delay, the geometric delay is ${\tau }_g=\mathbf{b}\cdot \mathbf{s}/c$, and the phase difference is $\varphi =2\pi \nu {\tau }_g$. For equal received source powers $P_a$, adding the voltages and averaging over many wave periods givesUncorrelated receiver noise adds a mean level but not the source's coherent cross term. For an east-west baseline and one choice of hour-angle sign, ${\tau }_g=(b/c)\cos \delta \sin H(t)$. Earth rotation therefore sweeps the phase; locally the fringe frequency is $(b/\lambda )\cos \delta \cos H\dot{H}$. Tracking the antenna pointing keeps the source in the primary beam. Electronic fringe stopping is a separate operation: it can remove this phase evolution, making a source at the phase centre constant in the corrected complex visibility. The oscillating power above is the geometrical fringe before that correction.

For a small angular displacement $d\theta$ along the projected baseline, $d\varphi =2\pi b_{p}d\theta /\lambda$. One full fringe therefore spans $\Delta \theta \sim \lambda /b_p$. Here $b_p=|\mathbf{b}-(\mathbf{b}\cdot \mathbf{s})\mathbf{s}|$. This is the angular period of the measured Fourier component and the characteristic resolution scale when such long baselines are synthesized; precise beam width and image ambiguity also depend on baseline coverage and weighting. A single pair by itself does not give an unambiguous complete image.

Writing $x=h\nu /(k_{B}T)$, the expansion $e^x-1=x+O(x^2)$ immediately gives the Rayleigh-Jeans law,The factor two counts the two independent modes of polarization of waves. A small brightness-temperature perturbation thus has $\Delta I_{\nu }=2k_B{\nu }^2\Delta T_{\mathrm{RJ}}/c^2$ in this limit.

Hot intracluster electrons inverse-Compton scatter the microwave-background photons. The scattering optical depth is $d\tau =n_e{\sigma }_{T}dl$ and the characteristic thermal fractional energy transfer is of order $k_B{T}_e/(m_e{c}^2)$. The integrated distortion is therefore controlled byScattering conserves photon number while shifting photons toward higher frequencies, giving a low-frequency decrement and a high-frequency increment. Thus the printed estimate correctly gives a negative change of order $y{T}_{\mathrm{CMB}}$. Its numerical coefficient, however, is not the standard Rayleigh-Jeans coefficient. For nonrelativistic electrons and small $y$, the Kompaneets equation gives $\Delta n=y{x}^{-2}{\partial }_x(x^4{\partial }_x{n}_0)$. At small $x$, $n_0\simeq 1/x$, so $\Delta n\simeq -2y/x$ andEquivalently the thermodynamic-temperature spectral factor is $x\coth (x/2)-4\to -2$. The original PDF really prints the estimate with $-y$, not $-2y$; both numerical conventions are retained below. Relativistic corrections at the supplied $T_e$ are small compared with this factor-of-two difference.^[ref]

The uniform cube has a constant line-of-sight depth $L=6\times 10^{22}\mathrm{m}$. Using the supplied cross-section rather than replacing it with a more precise value givesAdopt $T_{\mathrm{CMB}}=2.72548\mathrm{K}$, explicitly an external measured constant, not a temperature supplied in the question. The values $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and $m_e\simeq 9.10938\times 10^{-31}\mathrm{kg}$ set the physical units. Hence the printed relation gives $\Delta T\simeq -22\mu \mathrm{K}$; the standard Rayleigh-Jeans result gives $\Delta T\simeq -44\mu \mathrm{K}$. Keeping the answer as $-8.14\times 10^{-6}{T}_{\mathrm{CMB}}$, or twice that, avoids needing a numerical background temperature.^[ref]

The face is a square, so its solid angle is $\Omega =(6\pi /10800{)}^2=3.046\times 10^{-6}\mathrm{sr}$, not the area of a circular six-arcminute aperture. At $15\mathrm{GHz}$,Thus $\Delta S_{\nu }\simeq -4.67\times 10^{-4}\mathrm{Jy}=-0.47\mathrm{mJy}$ using the printed estimate, or $-0.93\mathrm{mJy}$ using the standard factor two. The sign denotes missing brightness relative to the surrounding microwave background. At this frequency $h\nu /(k_B{T}_{\mathrm{CMB}})\simeq 0.264$; the exact nonrelativistic spectral intensity differs from the strict Rayleigh-Jeans approximation by about one percent. This precision does not affect the estimates.

For the array noise, the system temperature sets the receiver noise scale; let $A_e={\eta }_A\pi d^2/4$ be the effective area of each dish, and let ${\eta }_c$ describe correlation losses. No efficiencies or polarization of waves count are supplied, so first take ${\eta }_A={\eta }_c=1$ and a single independent polarization of waves. An unpolarized point source produces antenna temperature $T_a=S_{\nu }{A}_e/(2k_B)$, and the system-equivalent flux density is $\mathrm{SEFD}=2k_B{T}_{\mathrm{sys}}/A_e$. The interferometric radiometer equation gives one baseline an rms flux error $\mathrm{SEFD}/\sqrt{2\Delta \nu t}$. There are $N(N-1)/2=45$ cross-correlation baselines; averaging their independent thermal noise for a point source givesThe ideal single-dish effective area is $4\pi {\mathrm{m}}^2$, giving $\mathrm{SEFD}=6.59\times 10^3\mathrm{Jy}$ andCombining two independent parallel-hand polarization of waves measurements gives $6.3\mu \mathrm{Jy}$. Nonunit efficiencies increase these estimates by $1/({\eta }_A{\eta }_c)$. Here signal equal to noise means a weak-source signal-to-noise ratio of one after integration; it does not mean equating the source to the entire unaveraged receiver-noise power.^[ref]

At the band centre, $\lambda =0.01999\mathrm{m}$. Taking a representative maximum projected baseline of $20\mathrm{m}$ gives ${\theta }_{\mathrm{res}}\sim 3.4$ arcminutes. A corner-to-corner baseline of $20\sqrt{2}\mathrm{m}$ would give about $2.4$ arcminutes before projection and weighting. The exact antenna locations and source declination are not supplied, so neither a unique synthesized beam shape nor an exact FWHM is determined. The individual-dish primary-beam scale, roughly $\lambda /4\mathrm{m}\sim 17$ arcminutes, is the field of view, not the synthesized resolution.

To extract the cluster signal, retain frequency-resolved complex visibilities and fit an extended SZ template together with contaminating compact sources. For the ideal uniform square centred at the phase centre and with a nearly constant primary beam across it, the Fourier transform isThis is why the integrated decrement must not be treated as a point-source flux on every baseline. The first axial visibility null occurs at $b=\lambda /\Theta \simeq 11.5\mathrm{m}$. Short baselines carry most of the large-scale decrement; the longest baselines are useful for separating unresolved positive radio sources. natural visibility weighting or a suitable taper gives a clearer extended-source image than emphasizing the longest baselines, while a visibility-domain matched filter uses all samples with weights proportional to template amplitude divided by noise variance. Its amplitude estimate is $\hat{a}=\mathrm{Re}(T^{†}{C}^{-1}d)/(T^{†}{C}^{-1}T)$ for a real template amplitude. A joint fit propagates radio-source uncertainty instead of subtracting exact, noise-free source models.

Earth-rotation and multifrequency synthesis improve Fourier coverage, but each channel must use its own baseline in wavelengths, primary beam and SZ spectrum. Use the longer baselines and spectra to identify variable point sources, flag bad data, and test the decrement with time, antenna and frequency splits. Primary-beam and absolute-flux calibration must be included. Large-scale microwave-background fluctuations and residual ground or atmospheric signals can remain even below the thermal-noise estimate. The narrow low-frequency band alone gives limited separation of SZ and primary microwave-background temperature structure. There is no zero-spacing interferometric datum: integrated flux is model dependent unless sufficiently short baselines or additional total-power information are available. Exact cluster detection significance cannot be obtained from the point-source noise alone without the actual projected baseline distribution and weighting.