Sky localisation and arrival times

Compare the same physical feature on calibrated clocks

A disturbance arriving at separated receivers reaches them at different times. Those differences can constrain its direction even when the receivers do not form a conventional telescope image. The timing information becomes useful only after specifying which feature of the signal the clocks are comparing.

A trigger timestamp can include a threshold decision and processing latency. A reconstructed waveform reference time can depend on the waveform and response model. Different messenger emissions can also occur at different moments at the source. These are distinct timing questions, and treating all of them as the same clock reading can manufacture a false geometrical inconsistency.

Use a declared common time convention and account for clock offsets, receiver positions and motion, signal propagation and response phase where relevant. The LIGO calibration study illustrates why a strain time series is a calibrated reconstruction. Fairhurst’s triangulation analysis additionally examines how timing and waveform uncertainties affect localisation.

We first remove those complications by stipulation, not by claiming that real observations lack them. The following original geometry has stationary receivers in one flat frame, synchronized calibrated clocks, a far-source plane wave moving at c, and the same identifiable signal feature at each receiver. There is no atmosphere, dispersion or receiver motion in this toy.

A baseline measures one component of a direction

Let n be a dimensionless unit vector from the receiver origin toward the source. The wave propagates in direction −n. Put receiver A at the origin and another receiver j at baseline vector b_j, measured in metres from A. Let c = 299,792,458 m s⁻¹ be the vacuum light speed.

For a plane of constant phase moving in direction −n, the arrival time at position b_j is t_j = t_A − n·b_j/c. Thus the signed delay is Δt_j = t_j − t_A = −n·b_j/c. A receiver displaced toward the source, with positive n·b_j, encounters that same wavefront earlier and has a negative delay.

The units are metres divided by metres per second, giving seconds. Since n is a unit vector, |n·b_j| cannot exceed |b_j|. Therefore |Δt_j| ≤ |b_j|/c. This bound tests consistency with this specified wave, speed, geometry and clock model; it is not an assumption-free test of fundamental physics.

A single measured delay fixes one projection n·b_j. Directions with different perpendicular components can have that same projection. On the unit sky sphere, an interior projection generally gives a circle of possible directions rather than one point; uncertainty broadens the corresponding constraint.

The supplied first delay fixes n_x, not the whole sky position

Choose A = (0, 0, 0) m, B = (2,997,924.58, 0, 0) m and C = (0, 2,997,924.58, 0) m. Each of AB and AC has length c × 0.010 s, so its light-crossing time is exactly 10 ms in this model. These are invented receiver coordinates, not a real observatory network.

Supply t_B − t_A = +6 ms. The x-baseline relation gives +0.006 s = −n_x × 0.010 s, so n_x = −0.6. The positive delay means B receives the chosen feature later than A under the declared convention.

The unit-vector condition n_x² + n_y² + n_z² = 1 then leaves n_y² + n_z² = 0.64. Thus the allowed perpendicular components form a circle of radius 0.8 in the n_y–n_z component plane. The circle is a representation of allowed direction components, not a map of source distance.

No source distance has entered this far-plane-wave calculation. Changing the unknown emission time shifts all arrivals together and cancels from the differences. A direction constraint from relative timing should not be presented as a distance determination.

Both signs of the remaining component fit the same delays

Now supply t_C − t_A = −4 ms. The y-baseline relation is −0.004 s = −n_y × 0.010 s, giving n_y = +0.4. Substituting into the circle leaves n_z² = 0.64 − 0.16 = 0.48.

There are two directions: n = (−0.6, +0.4, +√0.48) and n = (−0.6, +0.4, −√0.48). The third components are approximately ±0.692820323. Each vector has squared norm 0.36 + 0.16 + 0.48 = 1, and each reproduces both supplied signed delays.

All receiver positions lie in the x–y plane. Reflecting n_z to −n_z changes neither dot product with an in-plane baseline, so timing alone cannot distinguish this pair. The figure plots n_y and n_z at the fixed n_x = −0.6 with one common linear component scale. Its dashed line n_y = +0.4 intersects the first circle at the two solutions.

Fairhurst’s discussion identifies this coplanar timing ambiguity in a more general localisation analysis. Additional amplitude, phase and antenna-response information, another independent geometry or an electromagnetic direction can supply further constraints under its own model. Adding information is different from silently choosing one of the two solutions.

A failed geometric check asks us to inspect the measurement model

Suppose an alternative supplied delay on AB were 11 ms. Dividing by its 10 ms light-crossing time would require |n_x| = 1.1, which no unit direction can have. The data and assumptions would be inconsistent. First check clock offsets, feature matching, receiver coordinates, propagation and response before assigning a physical speed anomaly.

Real measurements have timing uncertainty and may involve waveform parameters correlated with the reference time. The corresponding sky information can be extended, uneven and multimodal. A best-fit point discards that structure, so two nearby best fits need not convey the same association evidence as two full, well-modelled localisation maps.

A probability or uncertainty region has meaning within its stated statistical construction and assumptions. It is not necessarily a patch with uniform probability, nor does an overlapping catalogue object automatically become the source. The same background and selection questions that apply to other associations still matter.

For different messenger species, an observed relative arrival time can also include an intrinsic emission lag. The GW170817 gamma-ray analysis explicitly states emission-time and propagation assumptions when drawing physical constraints. Its methodological lesson here is to retain those assumptions; no numerical bound or new speed measurement is imported.

The next lesson asks how often unrelated events can agree in a chosen time and sky search. The timing ring is geometrical information. Turning a coincidence into evidence for a shared source requires an explicit association test and a suitable background.

Worked examples

Try the reasoning

1. With n toward the source, does a receiver displaced toward that source encounter the same wavefront earlier or later?

   Show the answer and reasoning

   Earlier, giving a negative t_j − t_A.

   Its baseline dot product n·b_j is positive, and the delay relation has a minus sign.

2. Does this single-baseline delay establish the far source’s distance?

   Show the answer and reasoning

   No.

   It constrains a direction projection in the plane-wave model; no distance parameter occurs in the supplied timing relation.

3. Why do the two reflected solutions have identical delays here?

   Show the answer and reasoning

   Every supplied baseline lies in the x–y plane.

   Changing n_z to −n_z leaves its dot product with each in-plane baseline unchanged.

4. Does a compatible timing direction by itself prove that a nearby catalogue object is the source?

   Show the answer and reasoning

   No.

   One must use the uncertainty, selection, backgrounds and the physical association model rather than only a matching point.

What to carry forward

• State the source-direction convention and compare the same calibrated signal feature before interpreting a delay.
• One baseline gives a direction projection; the supplied two in-plane baselines retain a reflected pair.
• Localisation consistency and common-source evidence require separate treatment of uncertainty, backgrounds and emission assumptions.