Part 3
Reconstructing the Sky
Detecting a neutrino is not astronomy. Astronomy needs a direction, and ideally an energy. IceCube gets them from two different kinds of event, and the frustrating part โ the thing that shapes every analysis the collaboration does โ is that no single event gives you both well.
Why a muon is the best messenger
A muon loses energy in two ways. At low energy it ionises atoms at a roughly constant rate; at high energy it radiates โ bremsstrahlung, pair production, photonuclear losses โ at a rate proportional to its energy. Together:
\[ -\frac{dE}{dx} = a + bE \]
with a โ 0.26 GeV/m and b โ 3.6 ร 10โปโด mโปยน in ice (approximate values). Separating and integrating from E down to zero gives the range:
\[ R = \int_0^{E}\frac{dE'}{a + bE'} = \frac{1}{b}\ln\!\left(1 + \frac{bE}{a}\right) \]
The logarithm is the whole story. Energy buys range, but only logarithmically:
| Muon energy | Range in ice |
|---|---|
| 100 GeV | 0.36 km |
| 1 TeV | 2.4 km |
| 100 TeV | 13.7 km |
| 1 PeV | 20.1 km |
A 100 TeV muon travels fourteen kilometres through ice โ fourteen times the width of the detector. This is the quiet reason IceCube works far better than its instrumented volume suggests: a neutrino can interact a long way outside the array and still send a muon through it. The detector's effective volume for muon events is much larger than the cubic kilometre Part 2 paid for.
Two event shapes, two trade-offs
Which shape you get depends on the interaction the primer described, and on the flavour of the neutrino that arrived.
Tracks
A muon neutrino interacts by charged current and makes a muon, which crosses the detector in a straight line. Timing the Cherenkov light along a kilometre-long lever arm fixes the direction to roughly half a degree.
But the muon was made outside and leaves still carrying energy, so the deposited light is only a lower bound on the neutrino's energy.
Cascades
A neutral-current interaction, or an electron neutrino, dumps its energy into a shower only a few metres long โ effectively a point source of light inside the array. Contained, so nearly all the energy is measured: roughly 15% accuracy.
But a glowing sphere has almost no directional information. Angular resolution is ten to fifteen degrees โ a patch of sky containing many candidate sources.
What actually limits the pointing
Naively the angular resolution should follow from timing alone. With sensors timed to a few nanoseconds and a lever arm L of a kilometre, the angular error is roughly
\[ \Delta\theta \sim \frac{c\,\Delta t}{n\,L} \]
which for ฮt = 3 ns gives under a tenth of a degree. The real figure is several times worse, and the reason is the ice itself: photons scatter many times on their way to a sensor, so the arrival time carries a long random tail. The limit is not the clock, it is the medium.
This is why mapping the optical properties of the ice โ layer by layer, through dust bands laid down over a hundred thousand years โ turned out to be as much a part of the experiment as the electronics. The detector had to learn the glacier it was frozen into.
Check your understanding
- Why does a search for a point source use tracks, while a measurement of the diffuse spectrum prefers cascades?
Answer: a point source needs direction, which only tracks supply; a spectrum needs energy, which only contained cascades supply. - At what energy does the muon range first exceed the detector size, and what changes there?
Answer: around 500 GeV. Above it the muon no longer stops inside, so events become through-going: direction improves, contained energy is lost. - If you could halve the sensor timing jitter, would the pointing improve by a factor of two?
Answer: no. Photon scattering in the ice dominates, so the clock is not the binding constraint.
What this leaves out
The energy-loss coefficients a and b themselves depend weakly on energy, and radiative losses are stochastic rather than smooth โ a muon can lose a large fraction of its energy in a single burst, which both brightens the track and spoils the simple picture of steady decline. Resolution figures quoted here are representative rather than exact; they depend on energy and on the reconstruction used.