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I know the exact time a radio telescope detected a transient event. I also know the exact location of the telescope, and the galactic coordinates (galactic longitude, latitude) and right ascension and declination of the beam pointing center.

I conjecture that the event was also observed by another receiver at another location. How can I correct for the time difference between the two locations (i.e. the event reached one observer earlier than the other due to that they are in different locations)? Currently, I’m taking the dot product of a vector between the known observer and the source, and a vector between the known observer and the potential observer, and dividing that by the speed of light. To obtain the vector between the observers, I calculate their ‘n-vectors’ ( https://www.movable-type.co.uk/scripts/latlong-vectors.html ) and take the difference. However, I’m getting nonsensical delay values.

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  • $\begingroup$ It's hard to tell right now exactly what level of detail you need, but I've added an answer. Please feel free to add more details or explain what else you'll be needing! $\endgroup$ – uhoh Jul 2 at 7:14
  • $\begingroup$ Rather than asking essentially the same question twice or even thrice!, it's better to explain more clearly the first time exactly what you need. If you needed more detail you should have mentioned it right here rather than accepting my self-proclaimed "partial answer" and then re-asking. In Stack Exchange we try to avoid answer fragmentation by not spreading answers out over several posts or different sites. $\endgroup$ – uhoh Jul 14 at 9:55
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This may be a partial answer depending on what you do or don't know how to do at the moment, and the level of accuracy you require. Feel free to add some feedback.

If you'd like to use (or learn to use) Python then you can solve this problem trivially using Skyfield!

It depends on the level of accuracy that you need. The time difference will be of the order of 20 milliseconds for a 6000 km difference in light-path distance for example, but in that time the Earth moves (in some direction) about 0.02 * $\sqrt{GM/a}$ where GM is 1.327E+20 m^3/s^2 and $a$ is about 1.5E+11 meters, or about 600 meters.

If you don't want to worry about small corrections right now, just calculate the dot product between the normal vector pointing from the direction of the event and the vector drawn from the known observation site to the proposed site, that's the path length difference. Divide that by the speed of light to get the time difference.

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    $\begingroup$ Thanks for your help! In what time units would this be? I’m trying to use the formula, however I’m getting nonsensical values (like 4.31+E12). To determine the vector between the sites, I’ve computed the “n-vector” (bit.ly/3fJndSO) at both sites, and subtracted them. Then, (starting by assuming the object is directly above the first site), I multiplied the first vector by the distance to the object (in meters). Then, I took the dot product of the vectors and divided by the speed of light (in m/s). But I get huge values? Am I doing something wrong/is there a better way to do this? $\endgroup$ – PerplexedDimension Jul 19 at 9:01
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    $\begingroup$ Unfortunately, I can’t use skyfield. $\endgroup$ – PerplexedDimension Jul 19 at 9:02
  • $\begingroup$ @PerplexedDimension I'm sorry but I don't click on blind links. Can you add that to your question so that I or anyone else can answer directly? Also it's probably best if you un-accept this answer until you've really had your question completely answered. Thanks! $\endgroup$ – uhoh Jul 19 at 9:32
  • $\begingroup$ ..........done! $\endgroup$ – PerplexedDimension Jul 19 at 17:23

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