(chap:18)= # Observing design, error budgets, and feasibility calculations :::{admonition} Chapter opening :class: chapter-opening Observing design starts with the physical quantity to be measured and works backward to the telescope and instrument settings. A stellar angular diameter requires knowing where $|V|^2(B,\lambda)$ is most sensitive to size. An intensity-correlation measurement requires knowing the height of the zero-delay correlation peak and how much the detector response will dilute it. A polarization-rotation or time-delay measurement requires enough precision in the event-table time, frequency, and polarization tags. From the Hanbury Brown--Twiss Sirius experiment and the 32 Narrabri stellar-diameter measurements to VERITAS, MAGIC, and the photon-counting Vega experiment, intensity interferometry has used the same feasibility ledger: photon rate sets the statistical ceiling, correlation contrast sets the target signal, and baseline, bandwidth, time resolution, and calibration errors determine whether that signal survives in the data {cite:p}`1956Natur.178.1046H,1957RSPSA.242..300B,1958RSPSA.248..222B,1974MNRAS.167..121H,2006ApJ...649..399L,2020NatAs...4.1164A,2020MNRAS.491.1540A,2021MNRAS.506.1585Z`. ::: ## From science parameters to event tables Every science case corresponds to a set of physical parameters $\boldsymbol\theta_{\rm phys}$: stellar angular diameter, binary separation, oblateness and position angle of a rotating star, radius of a line-emitting region, polarization-angle rotation, time delay, or the contrast of a second-order correlation peak. The instrument does not measure those quantities directly. It produces the event table described in Eq. {eq}`eq:event-row` of Chapter {ref}`chap:01` and Eq. {eq}`eq:ch06-raw-event` of Chapter {ref}`chap:06`; the physical parameters have to be inferred from those event fields. The first design question is whether the recorded fields still contain the information needed for the target parameter. Are the time stamps good enough for nanosecond correlations? Do the position labels define the required baselines? Are wavelength and polarization labels preserved? Do the quality flags trace sky background, dead time, clouds, and clock state? If the data are collapsed immediately into an integrated image, nanosecond timing and polarization correlations cannot be recovered later. If the event table is kept, the same data can later be projected into light curves, $g^{(2)}(\tau)$, $|V|^2(B)$, polarization angles, or multiwavelength $u,v$ samples. The relation from physical parameters to data can be written as ```{math} :label: eq:ch18-likelihood-chain \mathcal L(\boldsymbol\theta_{\rm phys}) = P\!\left(d\,\middle|\, q_{\rm inst}(\boldsymbol\theta_{\rm phys}), {\bf c}_{\rm cal}, {\bf b}_{\rm sky} \right). ``` $\mathcal L$ is the likelihood. $q_{\rm inst}$ denotes the directly measurable instrument-space quantities, such as $|V|^2$, $g^{(2)}-1$, Stokes parameters, or arrival-time residuals. ${\bf c}_{\rm cal}$ contains calibration parameters: time zero point, filter bandpass, detector efficiency, and background correction. ${\bf b}_{\rm sky}$ contains sky glow, moonlight, dark counts, and host or field background. In a real observing proposal, it is not enough to say that a star will be observed. The useful statement is that the star's $m_B$, angular diameter $\theta$, visibility window, and model priors produce a particular set of $|V|^2$ points, and that those points reduce the uncertainty in $\theta$ to a specified level. Angular scale fixes the first baseline estimate. The Fourier relation between spatial frequency, visibility, and sky brightness was introduced in Eq. {eq}`eq:visibility` and in Eq. {eq}`eq:ch05-vcz` of Chapter {ref}`chap:05`. For observing design, its meaning is concrete: amplitude interferometry measures the amplitude and phase of $V$, while intensity interferometry measures $|V|^2$ in the thermal-light approximation. The proposal therefore has to say which visibility lobe the target angular scale falls in, which projected baselines and position angles are needed, and whether the missing phase information leaves model degeneracies {cite:p}`2003RPPh...66..789M,2009A&A...507.1719F,2012NewAR..56..143D,2014MNRAS.437..798M,2022MNRAS.512.1722K`. A useful resolution-scale estimate is ```{math} :label: eq:ch18-baseline-scale B \sim \frac{\lambda}{\theta} = 86\,{\rm m} \left(\frac{\lambda}{416\,{\rm nm}}\right) \left(\frac{1\,{\rm mas}}{\theta}\right). ``` A hot star with $\theta=2\,{\rm mas}$ begins to be resolved at only $\sim43\,{\rm m}$ at 416 nm. A $\theta=0.5\,{\rm mas}$ target needs $\sim170\,{\rm m}$, and a $\theta=0.1\,{\rm mas}$ target moves into the $\sim860\,{\rm m}$ regime. Narrabri's maximum baseline was about 188 m, so it was naturally strongest for bright, hot stars with angular diameters from a few tenths of a milliarcsecond to several milliarcseconds. CTA-scale kilometer arrays push the spatial frequency toward tens of microarcseconds, but the target must still provide enough photons {cite:p}`1974MNRAS.167..121H,2006ApJ...649..399L,2010SPIE.7734E..1TJ,2013APh....43..331D`. ## Photon rate, bandwidth, and coherence dilution Photon rate begins with magnitude. The AB-flux and photon-rate conversions were given in Eqs. {eq}`eq:ab-rate` and {eq}`eq:photon-rate`. At 550 nm, with a 10 nm bandpass and $\eta=0.25$, a 12 m telescope receives $R_\gamma\simeq2.8\times10^8\,{\rm s^{-1}}$ from an $m_{\rm AB}=5$ source and $1.8\times10^7\,{\rm s^{-1}}$ from an $m_{\rm AB}=8$ source. A 17 m MAGIC-scale aperture gives about $5.7\times10^8\,{\rm s^{-1}}$ and $3.6\times10^7\,{\rm s^{-1}}$ under the same assumptions. These are not yet the useful correlated-event rates. Spatial mode acceptance, polarization splitting, filter incidence angle, dead time, data-quality cuts, and background corrections all reduce the usable rate {cite:p}`2006ApJ...649..399L,2020MNRAS.491.1540A,2021MNRAS.506.1585Z,2024MNRAS.529.4387A`. ```{figure} ../_static/figures/generated/chapter_18/ch18_photon_rate_magnitude.png :name: fig:ch18-photon-rate :width: 86.0% Conversion from AB magnitude to narrowband photon rate. The curves use a 550 nm central wavelength, a 10 nm bandpass, and total efficiency η = 0.25. Aperture changes the collecting area linearly, while an increase of 5 magnitudes lowers the photon rate by a factor of 100. The gray region marks rates above $10^8\,{\rm s^{-1}}$, where high-speed detectors and data throughput become part of the feasibility calculation. ``` The height of an intensity-correlation signal is set jointly by the optical coherence time and the electronic response. The bandwidth conversion for coherence time is Eq. {eq}`eq:coherence-time`; the time dilution of a thermal-light peak is described by Eq. {eq}`eq:g2-dilution` and Eq. {eq}`eq:ch04-contrast-dilution` of Chapter {ref}`chap:04`. For the 416 nm, 13 nm bandpass commonly used by VERITAS, one obtains $\tau_c\simeq4.4\times10^{-14}\,{\rm s}$. If the effective width of the detector and electronics response is $\Delta t=4\,{\rm ns}$, the zero-baseline second-order correlation peak of unresolved thermal light is naturally diluted to a few $10^{-6}$. The zero-baseline normalized correlation $N_0\sim10^{-6}$ reported in the VERITAS papers lies at the same order of magnitude; the precise number also contains filter shape, detector response, polarization, background, and electronics-chain corrections {cite:p}`2020NatAs...4.1164A,2024ApJ...966...28A`. ```{figure} ../_static/figures/generated/chapter_18/ch18_coherence_dilution.png :name: fig:ch18-coherence-dilution :width: 86.0% A wider optical bandpass shortens the optical coherence time, while a slower electronic response dilutes the zero-delay correlation peak. The combination of 416 nm, 13 nm, and 4 ns produces contrast of order a few 10−6, so the correlation peak emerges only through long averaging and careful calibration. ``` The optical bandpass does not improve the SNR monotonically. For ideal thermal-light intensity interferometry, when the electronic bandwidth is much smaller than the optical bandwidth, the leading dependence of SNR on optical bandwidth approximately cancels: a narrower band gives fewer photons but a longer coherence time, while a wider band gives more photons but lower correlation contrast per time bin. Real instruments still care deeply about filters. PMT or SPAD current, sky background, angle-dependent filter response, and line-emission science all change the usable signal. In the low $f/D\simeq1$ MAGIC beam, an interference filter sees a broad range of incidence angles; the effective passband becomes wider and shifts blueward relative to a collimated-beam measurement. That changes the correlation-peak shape and SNR and has to be handled with both optical modeling and laboratory calibration {cite:p}`2020MNRAS.491.1540A,2024MNRAS.529.4387A,2021MNRAS.506.1585Z`. ## Signal-to-noise ratio for correlation measurements The standard two-telescope intensity-interferometry SNR scaling was given in Eq. {eq}`eq:ch05-snr`. At the design stage, the target magnitude, telescope area, total efficiency, electronic bandwidth, integration time, and target $|V|^2$ on the chosen baseline can be inserted directly. LeBohec and Holder used $A=100\,{\rm m^2}$, $\alpha=0.3$, $\Delta f=1\,{\rm GHz}$, $T=5\,{\rm h}$, and $|V|^2=0.5$ to obtain a $5\sigma$-level estimate at $m_V\simeq6.7$. Under the same assumptions, an $m_V=5$ source reaches an SNR above twenty, while an $m_V=9$ source falls below one sigma {cite:p}`2006ApJ...649..399L,2008AIPC..984..205L,2012NewAR..56..143D`. ```{figure} ../_static/figures/generated/chapter_18/ch18_snr_heatmap.png :name: fig:ch18-snr :width: 90.0% Dependence of intensity-interferometry SNR on magnitude and integration time. The calculation uses $A=100\,{\rm m^2}$, α = 0.3, $\Delta f=1\,{\rm GHz}$, and |V|2 = 0.5. The orange marker shows the LeBohec–Holder estimate: mV ≃ 6.7, 5 h, and roughly 5σ. ``` Equation {eq}`eq:ch05-snr` is a first-pass estimate, not a data-analysis pipeline. In real data, $T$ means the effective live time after quality cuts. $\Delta f$ means the effective bandwidth produced by the optical pulse, PMT or SPAD response, cables, amplifiers, digitizers, and software correlator. $|V|^2$ also changes with elevation and hour angle. VERITAS has four 12 m IACTs, giving six simultaneous baselines, a 416 nm, $\sim13$ nm filter, 4 ns sampling, and about 3.5 TB of raw data per hour. MAGIC uses two 17 m telescopes with central PMTs and narrowband filters; early demonstrations reported roughly an order-of-magnitude sensitivity gain over Narrabri. The photon-counting Vega experiment showed a different route: SPADs and software correlation gave $\langle g^{(2)}\rangle=1.0034\pm0.0008$ at zero baseline, while the $\sim2$ km projected baseline showed no correlation, consistent with Vega's $\sim3.3$ mas angular diameter being fully resolved {cite:p}`2020NatAs...4.1164A,2020MNRAS.491.1540A,2021MNRAS.506.1585Z,2024MNRAS.529.4387A`. Many baselines increase both data volume and geometric information. $N$ telescopes provide $N(N-1)/2$ simultaneous baselines: four telescopes give only six, while 60 telescopes give 1770. These baselines sample different $u,v$ points of the source structure; they are not repeated measurements of one number. If the target is a uniform-disk angular diameter, a few well-placed baselines may be enough. If the target is a rapidly rotating star, a binary, a disk wind, or an asymmetric line-emitting region, position-angle coverage is essential. CTA-layout studies repeatedly show the same tradeoff: long baselines resolve small structure but lose the overall scale; short baselines measure the global size but miss high spatial frequencies. For hot-star structures between $0.1$ and $3\,{\rm mas}$, a 30--2000 m baseline range matters more than the single longest baseline {cite:p}`2010SPIE.7734E..1CN,2010SPIE.7734E..1TJ,2012MNRAS.419..172N,2012MNRAS.424.1006N,2013MNRAS.430.3187R`. ## Baseline choice and parameter sensitivity The uniform disk is the usual first model. Its visibility and first null were given in Eqs. {eq}`eq:uniform-disk`, {eq}`eq:ch05-uniform-disk`, and {eq}`eq:ch05-first-null`. Baseline choice directly sets the angular-diameter information. If all baselines sit on the $|V|^2\simeq1$ plateau, curves for different $\theta$ are almost indistinguishable. If only very low visibility beyond the first null is observed, the correlation peak approaches the noise floor. The falling part of the first lobe and the region near the first null usually carry the strongest diameter information. Limb darkening, rapid rotation, and companions change the curve shape, so feasibility plots should show both the target model $|V|^2$ and $\partial |V|^2/\partial\theta$ {cite:p}`2003RPPh...66..789M,2020NatAs...4.1164A,2024ApJ...966...28A,2025ApJ...995..191A`. ```{figure} ../_static/figures/generated/chapter_18/ch18_visibility_fisher.png :name: fig:ch18-visibility-fisher :width: 94.0% Squared visibility and diameter sensitivity for a uniform disk. At 416 nm, the falling parts of the curves for 2 mas, 1 mas, and 0.5 mas targets fall at tens, hundreds, and several hundreds of meters, respectively. The right panel shows that most diameter information comes from baselines where the curve has a large slope; the short-baseline plateau |V|2 ≃ 1 is weakly sensitive to diameter. ``` The more general Fisher-information form was given in Eqs. {eq}`eq:fisher`, {eq}`eq:ch07-fisher`, and {eq}`eq:ch08-sii-fisher`. In observing schedules, its meaning is practical. $\mu_i$ may be $|V|^2$, the area of a $g^{(2)}$ peak, a polarization angle, or a delay. $\sigma_i$ should include both statistical noise and systematic terms. A baseline with a tiny error bar but nearly zero derivative contributes little to the target parameter. A baseline with a large derivative but $|V|^2$ too low for a detectable peak cannot carry the result by itself. The VERITAS-SII observation of the rapidly rotating star $\gamma$ Cas is a clear example of position-angle coverage: different projected baselines sample $|V|^2$ in different directions, which constrains oblateness and major-axis position angle. A single equivalent circular diameter is not enough for that target {cite:p}`2025ApJ...995..191A,2024ApJ...966...28A`. ```{figure} ../_static/figures/generated/chapter_18/ch18_uv_coverage.png :name: fig:ch18-uv-coverage :width: 94.0% How array scale changes u, v sampling. The left panel sketches a four-telescope coverage pattern: six baselines remain sparse even after Earth rotation. The right panel sketches a CTA-like many-telescope array, with short, intermediate, and long baselines present at the same time. Dense coverage does not automatically mean high SNR, but it determines whether asymmetric targets can be fit or reconstructed stably. ``` Wavelength is also part of baseline choice. The same physical baseline has a spatial frequency at 416 nm that is about 1.9 times higher than at 800 nm. The blue end gives finer resolution, but atmospheric transmission, mirror reflectivity, detector quantum efficiency, and filter angle of incidence are harder. Spectral-line observations must use line flux rather than broadband magnitude. In hot stars and Be stars, H$\alpha$, He, or metal lines may arise from regions larger than the continuum photosphere. A narrow band can separate geometry and velocity channel, but the event rate, filter leakage, and background must be recalculated for each channel {cite:p}`2010SPIE.7734E..0AD,2012NewAR..56..143D,2021MNRAS.506.1585Z,2024MNRAS.529.4387A`. ## Background, calibration, and quality cuts When background is uncorrelated, it mainly dilutes the correlation peak. The two-channel background-dilution factor and its assumptions were defined in Eqs. {eq}`eq:background-dilution` and {eq}`eq:ch06-background-dilution`. The observing design must turn that correction into background measurements made in the same channel, at the same gain state, and close in time. In the VERITAS analysis of $\beta$ UMa, off runs $0.5^\circ$ from the target were used to estimate nonstellar current. The typical off-source intensity was only about $3\%$ of the on-source intensity, so the effect on the radius fit was small. Moonlight, clouds, haze, city light, or detector afterpulsing can make this approximation fail and require separate quality cuts {cite:p}`2024ApJ...966...28A,2025ApJ...995..191A`. ```{figure} ../_static/figures/generated/chapter_18/ch18_background_dilution.png :name: fig:ch18-background-dilution :width: 94.0% Dilution of a correlation peak by uncorrelated background. If both ends have the same background-to-starlight ratio, the observed contrast falls as (1 + β)−2. β = 0.03 produces only a few-percent correction, while β = 0.5 suppresses the peak to about 44% of its original value. The right panel gives the inverse correction factor; larger corrections are more sensitive to uncertainty in the background estimate. ``` Timing calibration is a shared hardware boundary for intensity interferometry and photon counting. Correlation peaks are usually only a few nanoseconds wide. If the relative-delay model between telescopes is wrong by 1 ns, the peak is broadened or shifted out of the fitting window. In the Vega photon-counting experiment, the relative timing precision between zero-baseline subapertures was about $100\,{\rm ps}$. Between two telescopes, the analysis also had to handle light-travel time, instrumental delay, GPS or rubidium-clock drift, and fiber dispersion; the final correlations used $\sim400\,{\rm ps}$ time bins. The authors noted that a multi-telescope implementation needs roughly nanosecond-level synchronization to keep observing times on the scale of hours. VERITAS-SII uses White Rabbit 10 MHz clock distribution and 4 ns sampling, but still has to track the changing optical-path delay with hour angle in every run, together with run-to-run shifts in peak position {cite:p}`2021MNRAS.506.1585Z,2024ApJ...966...28A`. Calibrator stars deserve the same care as science targets. An ideal calibrator has a known angular diameter, stable brightness, a sky position near the target, a similar color, and enough photon rate. If the science target is a 0.5 mas hot star and the calibrator diameter is uncertain by $5\%$, that calibration term can easily dominate the final diameter uncertainty. Modern intensity-interferometry observations often use hot stars with known diameters as references: first check zero-baseline normalization, filter response, and the shape of $|V|^2(B)$, then observe the unknown target. The 2024 MAGIC performance paper used several reference stars to validate the system before reporting new stellar angular diameters. The VERITAS observations of $\beta$ UMa and $\gamma$ Cas show the same system moving from circular-disk diameters to oblateness and position-angle measurements {cite:p}`2024MNRAS.529.4387A,2024ApJ...966...28A,2025ApJ...995..191A`. ## Error budgets and observing proposals The final uncertainty should be separated into statistical and systematic terms. For a fitted parameter $\theta$, a useful budget is ```{math} :label: eq:ch18-error-budget \sigma_\theta^2 = \sigma_{\rm stat}^2 + \sigma_{\rm bg}^2 + \sigma_{\rm time}^2 + \sigma_{\rm cal}^2 + \sigma_{\rm model}^2 . ``` $\sigma_{\rm stat}$ comes from finite coincidence counts or correlation-peak fitting. $\sigma_{\rm bg}$ comes from off-source background interpolation and dark current. $\sigma_{\rm time}$ comes from clocks, geometric delay, sampling, and peak shape. $\sigma_{\rm cal}$ comes from calibrator stars, throughput, filters, and polarization. $\sigma_{\rm model}$ comes from the source model: treating a rapidly rotating star as a circular disk, representing a line-emitting disk as a Gaussian, or ignoring a companion. Statistical error usually falls as $T^{-1/2}$; the other terms often form a floor. Once that floor is above the target precision, adding more integration time does not fix the measurement. ```{figure} ../_static/figures/generated/chapter_18/ch18_error_budget.png :name: fig:ch18-error-budget :width: 94.0% Statistical and systematic terms in an error budget. The left panel shows statistical error falling with integration time while background, timing, calibration, and model terms form a floor. The right panel shows the contribution of each term to the total variance at 10 h. An observing proposal should connect every term to calibration data and state the error floor, not only the total SNR. ``` | Quantity | Common range | Use in planning | |:-------------|:---------------------|:-----------------------------| | $m_V,m_B$ | $-1$ to $9$ mag is the main current SII range | Convert to $n$ or $R_\gamma$; first decide whether the target is too faint. | | $A_{\rm eff}$ | $1-250\,{\rm m^2}$ per telescope | Enters photon rate and SNR; mirror aging, obscuration, and fiber coupling belong in the efficiency. | | $\Delta\lambda$ | $0.1-30\,{\rm nm}$ | Sets coherence time, PMT current, sky background, and spectral-line selection. | | $\Delta t$ | $0.1-5\,{\rm ns}$ | Sets the dilution of the correlation peak and the usable electronic bandwidth. | | $B_p$ | $10-2000\,{\rm m}$ | Sets angular scale and $u,v$ coverage; short and long baselines constrain different structures. | | $\beta$ | $0.01-1$ | Background-to-starlight ratio; sets contrast dilution and background systematics. | | $\sigma_{\rm cal}$ | $1-10\%$ in visibility or diameter scale | Often becomes the uncertainty floor after long integrations. | An executable observing proposal begins with the target parameter: for example, "measure the blue photospheric axis ratio of $\gamma$ Cas to $5\%$", or "measure the limb-darkened diameter of a 1 mas A-type star to $3\%$". The target description then gives magnitude, color, estimated angular diameter, visibility window, and existing CHARA, VLTI, Keck, or Narrabri constraints. The instrument section must state the number of telescopes, baseline range, filter central wavelength and bandwidth, sampling time, synchronization method, data rate, and quality cuts. SNR should be reported separately for the most informative baselines. The statistical precision comes from Eq. {eq}`eq:ch05-snr`, the background correction from Eq. {eq}`eq:background-dilution` or Eq. {eq}`eq:ch06-background-dilution`, and the systematic terms from Eq. {eq}`eq:ch18-error-budget`. The success criteria should also be quantitative: at least three baselines in the falling part of the first lobe reach ${\rm SNR}>5$; repeated calibrator observations show normalization drift below $3\%$; off-source background correction is less than $10\%$ of the total signal. If a long-baseline detection fails, the result can still set an angular-diameter upper limit or exclude an overly large emitting region. If the calibration floor is too high, the analysis still identifies which hardware upgrade is needed: timing synchronization, filters, or background control. The same chain of estimates applies beyond intensity interferometry. Amplitude interferometry, quantum-network telescopes, CMB polarization, FRB photon statistics, and laboratory teaching benches all need the same ingredients: event rate, mode number, response function, calibration terms, model derivatives, and error budget. What changes is the observable $\mu_i$. Science-case ranking should use the same structure, scoring scientific value and observability separately so that attractive targets are not confused with executable targets {cite:p}`2010SPIE.7734E..0AD,2012MNRAS.419..172N,2012MNRAS.424.1006N,2013APh....43..331D,2014SPIE.9146E..0ZD,2017MNRAS.472.4126G,2022SPIE12183E..0DK,2024SPIE13095E..0IT,2026arXiv260212717K`.