Earth rotation and closure phase

Put each station on a spherical Earth

Use Earth-centered Cartesian axes: $X$ points toward longitude zero on the equator, $Y$ toward ninety degrees east, and $Z$ toward the north pole. A station has latitude $\varphi$ and east-positive longitude $L$ . The teaching Earth has radius $R=6.371\times10^6\,\mathrm m$ ; station heights and Earth’s flattening are omitted.

The source declination is $\delta=12.391123^{\circ}$ . Let $H$ be its west-positive Greenwich hour angle. Rotation increases $H$ by approximately $15^{\circ}$ per sidereal hour. Longitude enters the station position; the local hour angle is $H+L$ . Keeping those two angles distinct prevents a longitude from being counted twice.

\mathbf r=R(\cos\varphi\cos L,\;\cos\varphi\sin L,\;\sin\varphi),\qquad \mathbf b_{ij}=\mathbf r_j-\mathbf r_i

Subtract station positions first. Use the straight chord between stations for the electromagnetic baseline. A surface route curves away from that straight separation.

Reading: NRAO Essential Radio Astronomy, Chapter 3, aperture synthesis

Project onto east and north on the sky

The source direction in these Earth-fixed coordinates is $(\cos\delta\cos H,-\cos\delta\sin H,\sin\delta)$ . A unit east vector is $(\sin H,\cos H,0)$ . A unit north vector is $(-\sin\delta\cos H,\sin\delta\sin H,\cos\delta)$ . Each is perpendicular to the source direction, and they are perpendicular to one another. Taking scalar products with the baseline produces the projected components.

For a baseline with components $(X,Y,Z)$ , divide those two lengths by $\lambda$ to obtain the equations below. As $H$ changes, the measured point usually traces an elliptical arc. A constant physical baseline can therefore test many stripe orientations during one track.

u=\frac{X\sin H+Y\cos H}{\lambda},\qquad v=\frac{-X\sin\delta\cos H+Y\sin\delta\sin H+Z\cos\delta}{\lambda}

The output is in wavelengths. The browser retains a pair only when both stations see the source at least ten degrees above their local horizon.

Reading: NRAO Essential Radio Astronomy, Chapter 3, aperture synthesis

Count pairs, tracks and copied conjugates separately

With $n$ stations, choose the first station in $n$ ways and the second in $n-1$ ways, then divide by two because a reversed pair is its conjugate. That gives $n(n-1)/2$ distinct pairs. Four stations supply six pairs. The default samples nine hour angles across three hours, so all six visible pairs yield fifty-four complex measurements.

This count describes how many samples were recorded, not how many unrelated sky parameters are well determined. Nearby points in the spatial-frequency plane can carry nearly redundant information. The faint opposite dots in the coverage map are conjugates implied by a real sky. Counting those dots again would exaggerate the measurements.

An equally important omission is the zero baseline. No pair of distinct stations directly supplies $V(0,0)$ . The synthetic reconstruction receives a separate ideal total-power measurement of $0.600\,\mathrm{Jy}$ . For a real source with extended emission, a short-baseline amplitude and the compact-source total flux can differ.

Reading: NRAO Essential Radio Astronomy, Chapter 3, aperture synthesis · EHT Collaboration, First M87 Results IV, arXiv:1906.11241v1

Follow the phase once around a triangle

Represent station $i$ by a complex gain $g_i=a_i e^{i\alpha_i}$ , where $a_i$ is positive and $\alpha_i$ is its phase error in the chosen correlation convention. The observed visibility on $i\to j$ is multiplied by $g_i g_j^*$ . Form the product on $i\to j\to k\to i$ . Every station gain appears once conjugated and once unconjugated.

The product retains positive gain amplitudes but loses every station phase. Taking its argument gives the closure phase. This cancellation assumes the three measurements are simultaneous enough to share the same station errors. Baseline-dependent errors, polarization leakage and changes within an integration can leave nonclosing effects.

V^{\mathrm{obs}}_{ij}=g_i g_j^*V_{ij},\qquad V^{\mathrm{obs}}_{ij}V^{\mathrm{obs}}_{jk}V^{\mathrm{obs}}_{ki}=a_i^2a_j^2a_k^2V_{ij}V_{jk}V_{ki}

The closure phase is the argument of the rightmost sky product. The cancellation follows by multiplication; it does not require guessing the station phases.

\Phi_{ijk}=\arg(V_{ij}V_{jk}V_{ki})=\operatorname{wrap}(\phi_{ij}+\phi_{jk}-\phi_{ik})

If the table stores the last edge as i to k, conjugate it to traverse k to i. Wrap the final angle into one full cycle, conventionally from minus to plus 180 degrees.

Reading: EHT Collaboration, First M87 Results IV, arXiv:1906.11241v1 · EHT 2019-D01-01 M87 data release, pinned commit 27ae76a

A numerical triangle, and what it can tell us

Take sky phases $20^{\circ}$ , $50^{\circ}$ and $10^{\circ}$ on stored edges AB, BC and AC. Their directed closure is $60^{\circ}$ . Now add station errors $30^{\circ}$ , $-40^{\circ}$ and $15^{\circ}$ . The observed edge phases become $90^{\circ}$ , $-5^{\circ}$ and $25^{\circ}$ . Adding the first two and subtracting the last still gives $60^{\circ}$ .

A translated sky contributes a phase proportional to the baseline. The three directed baseline vectors sum to zero, so a translation also disappears from the triangular product. Closure phase can reveal asymmetry but does not recover an absolute sky position. A point-symmetric source has real visibilities after choosing its center, which gives closure values of zero or a half-cycle away from nulls.

Four stations have four possible triangles, but only three independent closure phases. More generally, subtract the independent station phase differences from the baseline phase count. One overall station phase is arbitrary, leaving the independent closure count below. Triangles that share baselines also share noise.

N_{\mathrm{closure}}=\frac{n(n-1)}2-(n-1)=\frac{(n-1)(n-2)}2

This is a generic count for a connected complete array with nonzero signals. Shared baselines correlate the triangle errors.

Reading: EHT Collaboration, First M87 Results IV, arXiv:1906.11241v1

A separate calibrator restores more information

The browser supplies an ideal simultaneous centered point calibrator, with known unit flux and phase-only station errors constant across the track. Its true visibility is real and known on every baseline. Set station A’s phase to zero, infer the other phases from its calibrator edges and undo their differences in the target measurements. The reconstructed complex target signals then match the error-free synthetic signals.

That is a deliberately favorable calibration experiment. The real EHT triangle used later comes with residual station phases as nuisance quantities. Its tutorial uses amplitudes and closure phase directly. A single triangle can already reject many symmetric candidate shapes. A fully phase-calibrated two-dimensional image requires more measurements and calibration information.

Reading: EHT 2019-D01-01 M87 data release, pinned commit 27ae76a · EHT Collaboration, First M87 Results IV, arXiv:1906.11241v1

Saved synthetic default and method

Four spherical-Earth stations, nine hour angles over a three-hour track, 1.3 mm wavelength. An exact discrete Fourier sum generates a noiseless finite-pixel source. A separate ideal point calibrator and total-power measurement enter both ninety-step constrained inversions.

Image grid and field	$24\times24,\quad120\,\mu\mathrm{as}$
Central Greenwich hour angle	$H=105^{\circ}$
Total flux	$F=0.600\,\mathrm{Jy}$
Recorded complex samples	54
First directed closure phase	$0.287631^{circ}$
Smooth-prior signal RMS	$7.0408,mathrm{mJy}$
Compact-prior signal RMS	$5.9340,mathrm{mJy}$

The full station coordinates, sampled signals and reconstructions are saved in default-synthetic-record.json; the six-preset summary is model-benchmarks.json. Image pixels store jansky per pixel, dirty images use jansky per unit-peak beam.

Separate new geometry from phase calibration

Compare One narrow direction with Let Earth turn. Predict which extra track directions should appear before inspecting the coverage.
Select Atmospheric phase errors. Compare its closure readout with Crossed baselines, then open the raw and calibrated image comparison.
Drag one station far enough that a pair approaches the horizon. Explain why the number of saved samples changes.

Exercises with explanations

A fifth station joins a four-station array. How many new pairs and independent closure phases become available?
Read the explanation
It creates four new baselines. The independent closure count rises from three to six, so three new independent closure phases appear. Ten triangles can be drawn among five stations; several contain the same phase information.
Only the AB phase receives an extra ten-degree error from a correlator defect. Does closure cancel it?
Read the explanation
No. A change attached to AB alone is baseline-dependent. It does not appear with the opposite sign on another edge. The triangle closure changes by ten degrees when it traverses AB in the positive direction.
Transfer the method to a satellite array: station positions now vary during the observation. Which step changes?
Read the explanation
Evaluate each directed position difference at the measurement time, then project it onto the sky. The Fourier relation and simultaneous triangular cancellation keep the same form. Timing accuracy and any non-simultaneous links need their own error model.

Sources and calculation scope

NRAO Essential Radio Astronomy, Chapter 3, aperture synthesis
The narrow-field Fourier relation and Earth-rotation synthesis, especially equations 3.180–3.200. The browser uses a spherical Earth, a single wavelength and a finite pixel sky.
EHT 2019-D01-01 M87 data release, pinned commit 27ae76a
Actual EHT-HOPS SR1 network-calibrated Stokes I from 10 April 2017, low band, 227.0707 GHz. The tutorial preserves the publisher CSV and selects one simultaneous AA–AZ–LM triangle. The repository licenses data under PDDL 1.0.
EHT Collaboration, First M87 Results IV, arXiv:1906.11241v1
Fixed version of the 2019 imaging paper, DOI 10.3847/2041-8213/ab0e85. Read Figures 1, 3, 5 and 7, the station-gain equation 7, regularized imaging equation 14, and Appendix A. Figures remain with the publisher and authors; this guide links to them.