Neutrinos · D12 · The inverse problem

IceCube: From Sensor Hits to a Track

Can the light recorded by IceCube’s sensors reveal which way a particle was going? Yes: fit the times at which light reaches many sensors spread through the ice. Scattering and absorption in the ice smear those times. Collecting more light does not, on its own, make the direction more accurate or free of bias.

Generate a simulated track in an ice array, hide its true direction and fit it from the times at which the sensors see light. Change scattering and absorption to see when even a bright event becomes harder to reconstruct.

Hit sensors $\text{—}$

Recorded photoelectrons $\text{—}$

Fitted polar angle $\text{—}$

Fitted azimuth $\text{—}$

Timing residual RMS $\text{—}$

Angular error · truth comparison $\text{—}$

Reconstruct a track direction from hit times

Infer a direction

Reveal generated directionGold is truth; teal is the fit. Hidden by default.Generated polar angle from upward vertical

\theta

65\,\mathrm{^\circ}

Generated azimuth in the horizontal plane

\phi

35\,\mathrm{^\circ}

Known reference-point horizontal position

x_0

0\,\mathrm{m}

Effective scattering length

\ell_s

120\,\mathrm{m}

Absorption length

\ell_a

160\,\mathrm{m}

Slider changes apply to the next generated event. The fit uses only the recorded hits and the known reference point. Large residuals signal that the model does not match the data; they are not a calibrated confidence interval.

This teaching array is inspired by IceCube: 81 sensors on nine strings, not the real detector geometry or data. A relativistic charged particle travels on an infinite straight line through a known reference point. The ice is uniform, with a refractive index of 1.32 at every wavelength. Light collection follows a simple empirical rule, scattering adds exponential delays and each clock has 2 ns of Gaussian noise. The fit searches over direction only, with a free start time. It does not measure neutrino energy, fit a vertex, classify cascades or give a calibrated uncertainty.

Preparing a reproducible teaching record…

Read a sample of the record

Scientific reference ↗Read the worldwide history →2026 Nobel Prize in Physics →

Start with the essentials

Focus question: How much of the direction can still be recovered after absorption and scattering?
One-sentence intuition: The fitted direction is inferred from the recorded hits. Collecting more light does not by itself make the fit better or unbiased.

Core mathematical model

Arrival time of the earliest direct light in uniform ice

t_i=t_0+\frac{l_i+d_i\sqrt{n^2-1}}{c}

The track is infinite and moves at the speed of light in vacuum. l is the sensor’s position along the track and d its perpendicular distance from it. The ice has index 1.32 at every wavelength, so the phase and group speeds of light are equal.

Fitting the direction with an unknown start time

\widehat{\mathbf u}=\arg\min_{\mathbf u}\sum_i(t_i-t_{\mathrm{geo},i}(\mathbf u)-\widehat t_0)^2

For each trial direction, the start time is set to the mean difference between measured and predicted times. A coarse search over all directions is followed by a local refinement. This deliberately simple fit assumes direct light, so late scattered photons can bias it. The RMS residual shows how well the times fit; it is not the angular uncertainty.

Common difficulties

Fit the recorded times, then inspect the residuals

Typical misconceptionThe fitted charged-particle track points exactly along the neutrino’s direction.

Better mental modelThe model leaves out the kinematics of the interaction. It is a timing fit for a charged track through a known point. It does not classify events, measure neutrino energy or perform a full IceCube reconstruction.

Run the experiment

01
Reconstruct blind
Generate an event with the true direction hidden. Fit the recorded hits, then reveal the generated direction.
What to observe: The fitted arrow and the angular error appear only after the fit, and the fit never reads the generated direction.
02
Break the propagation model
Choose Strong scattering. Fit the new event and compare timing residuals.
What to observe: The first unscattered photons can still point the way, but sensors whose first hit arrives late can pull the fit off.
03
Lose photons
Choose Strong absorption and compare hit count and residuals across several generated events.
What to observe: Fewer sensors see light, so the fit has less to go on. The display gives no calibrated confidence interval, and the error need not grow in every single random event.

IceCube: From Sensor Hits to a Track

Inspect the measurement

Fit the recorded times, then inspect the residuals

Core mathematical model

Arrival time of the earliest direct light in uniform ice

Fitting the direction with an unknown start time

Common difficulties

Fit the recorded times, then inspect the residuals

Run the experiment

Reconstruct blind

Break the propagation model

Lose photons

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