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In the popular press, in recent months, we have heard a lot about high-energy neutrinos from far outside our solar system reaching our detectors....

But I wonder...

If a single neutrino from a great distance randomly, rarely, smacks a xenon atom in a detector, how the heck can they 'triangulate' it's direction?

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You correctly state that neutrinos do not interact too often. The physical parameter describing that is the effective cross-section. So what you observe in a detector is not the neutrino itself, but secondary particles, e.g. muons. Colloquially put, you may regard anything with high mass (density) in between the neutrino source and your instrument (to detect mainly the secondary particles) as detector for the neutrinos. This might be a large chunk of ice (like for the icecube experiment), a mountain range (as for the Gran Sasso lab), or even a complete planet, if the incident neutrinos originate from a star located on the other side of Earth at the moment of observation. The latter is possible since neutrinos e.g. from the Sun may pass through the whole planet without interacting with its matter (at night-time for the scientists working for the detector).

Let me try to explain with a drawing I took from a 2018 blog entry of the Antarctic Muon And Neutrino Detector Array (AMANDA). The red dots are on the ice surface, the vertical lines are the boreholes into the Antarctic ice shield and the little black dots are the PMTs, the Photomultiplier Tubes which detect the Cherenkov light generated by some fast muons, in the picture on the top of the blue cone. The readings of the PMTs (detecting the Cherenkov light) are displayed as color-code, I assume the color scale somehow relates to the time passed since the trigger event.

AMANDA sketch

So how do we now know where the neutrino came from? We can use particle physics to derive the secondary particle's velocity (vector) and from that where the neutrino actually came from. This is not trivial and requires quite a bit of computer power, but it is possible.

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    $\begingroup$ Consequence: We can locate any nuclear reactor operating anywhere in the world by maxing the Antarctic data with the Japan data. $\endgroup$ – Joshua Mar 23 at 18:52
  • $\begingroup$ > I assume the color scale somehow refers to the time passed since the trigger event. You assume correctly. The color in each dot shows the time when the Cherenkov light has hit that particular Photomultiplier. $\endgroup$ – macKaiver Mar 25 at 5:59
  • $\begingroup$ @macKaiver That's exactly what I mean. Photomultipliers show signals at different times. The trigger event would be when the first PMT received a signal, the color scale is the time passed since then. $\endgroup$ – B--rian Mar 25 at 7:54
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High energy muon neutrinos occasionally interact and produce a muon. Energy and momentum must be conserved in the process and the muon heads off in the same direction as the neutrino.

The relativistic muon can then be tracked by a network of detectors which are sensitive to the Cerenkov radiation produced when muons travel faster than the speed of light in ice.

The muon track allow its kinematics to be measured and hence the reconstruction of where the originating neutrino came from. This can potentially be done to within a few degrees.

There is less angular resolution in the case of high energy electron neutrinos. Any muons are produced in secondary interactions and there is a "cascade" of charged leptons from which a rough direction can be reconstructed.

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    $\begingroup$ To be a little clearer, the secondary muon follows almost the same track the neutrino would have if it hadn't interacted. Thus, by tracking the muon, you may determine the direction the neutrino came from. $\endgroup$ – John Doty Mar 23 at 15:07

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