---
title: "Atmospheric Neutrino Zenith Experiment Interactive Lab | Sandbox Physics"
description: "How can the direction atmospheric neutrinos come from reveal that some change flavor?"
canonical_url: "https://sandboxphysics.com/labs/atmospheric-neutrino-zenith/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/atmospheric-neutrino-zenith/"
last_modified: "2026-10-27"
---

Neutrinos · D04 · A planet-sized baseline

# Atmospheric Neutrino Zenith Experiment

**How can the direction atmospheric neutrinos come from reveal that some change flavor?** Direction tells you how far the neutrino has traveled: a short way through the air from straight overhead, the whole Earth from straight below. Compare each direction with what you would expect there without oscillation. A shortfall that depends on both distance and energy points to flavor change; an error in the flux or the detector's acceptance would not produce that pattern.

Follow a neutrino through Earth, collect a sample of events from the whole sky, and compare the counts in each direction with two predictions. Switch energy bands to see which travel distances matter.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This is a two-flavor teaching model in vacuum. Neutrinos are produced at a fixed height, and their direction and energy are reconstructed perfectly. All counts come from one generated record with a stated source and acceptance. The events are simulated; Super-Kamiokande data and matter effects inside Earth are not included. Paths are drawn to scale, but the markers are enlarged. The full assumptions and primary sources are below.

A fixed sample of simulated sky events

## Where do the events go missing?

Preparing the sky sample…

Upward through Earth

$\text{—}$ sampled events

No-oscillation prediction $\text{—}$

A longer journey

Downward from the nearby atmosphere

$\text{—}$ sampled events

No-oscillation prediction $\text{—}$

A shorter journey

Both directions use four bins, excluding the two nearest the horizon. The two cards share one bar scale.

Select a direction bin to inspect its ray. Use the arrow keys when the chart is focused. The sample stays frozen.

Sampled counts Oscillation prediction No-oscillation prediction

Error bars show the square root of the observed count. They give the size of counting fluctuations; they are not confidence intervals. The predictions use the same source and acceptance that generated the events. The up/down ratio leaves out the two central bins next to the horizon.

Read the ten direction bins

Energy bands use perfect true energy and direction. These simplified muon-flavor event counts omit backgrounds, lepton scattering angles and resolution; they cannot reproduce the experiment’s fitted significance.

[How the atmosphere makes neutrinos →](https://sandboxphysics.com/stories/neutrinos/atmospheric-neutrino-production/) [Super-K and the missing upward events →](https://sandboxphysics.com/stories/neutrinos/super-k-zenith-angle/) [Super-K · 1998 original paper ↗](https://arxiv.org/abs/hep-ex/9807003)

Physics tutorial

## The source is all around us. The baseline is not.

**Background** Cosmic-ray cascades in the atmosphere produce neutrinos. A detector at the surface receives some that travel a short way down from above and others that travel a long way up through Earth. In 1998, Super-Kamiokande used the shortfall of events in some directions as evidence for oscillation.

**Why it matters** A flavor probability is not yet a prediction of how many events you will count. This Lab combines propagation with a stated teaching source spectrum, an exposure and a direction-dependent acceptance, then compares the generated counts with the integrated expectations.

Start with the essentials

- **Focus question**: Does the upward deficit remain when source and acceptance are symmetric between the two hemispheres?
- **One-sentence intuition**: Compare each direction with its own no-oscillation expectation: flux or acceptance alone can make the histogram uneven, but a flavor deficit over long baselines adds its own dependence on energy and direction.

### Core mathematical model

#### Path length from the production shell to the detector

$$
\begin{aligned}c&=\cos\theta_z\\L(c,h)&=\sqrt{R^2c^2+2Rh+h^2}-Rc\end{aligned}
$$

The detector sits on the surface of a sphere of radius 6371 km. A positive cosine means the neutrino is going down; a negative one means it is coming up. All neutrinos are produced at one fixed height above the sphere, and the formula comes from intersecting a ray with that shell.

#### Muon-flavor survival

$$
\begin{aligned}P_{\mu\mu}&=1-\sin^2(2\theta)\sin^2\phi\\\phi&=1.266933\,\frac{\Delta m^2}{\mathrm{eV}^2}\frac{L}{\mathrm{km}}\frac{\mathrm{GeV}}{E}\end{aligned}
$$

This is two-flavor, ultrarelativistic propagation in vacuum. A lower survival probability removes events from this simplified muon-flavor sample; it does not mean that Earth absorbed the neutrinos.

#### Assumed source and acceptance shapes

$$
\begin{aligned}f(c)&=1+a(1-c^2)\\\epsilon(c)&=1-b(1-c^2)\end{aligned}
$$

The first factor raises the flux in horizontal directions; the second lowers the acceptance there. Both are the same for upward and downward directions. They are illustrative functions, not a fitted atmospheric flux or a detector calibration.

#### From spectrum to expected counts

$$
\begin{aligned}\lambda_B&=\frac{N_0}{2I}\int_B q\,dE\,dc\\q&=E^{-1.7}f(c)\epsilon(c)P_{\mu\mu}\end{aligned}
$$

Energy enters as its numerical value in GeV and is limited to 1–20 GeV; the normalizing integral in the denominator is the power-law weight integrated over that range. The event weight multiplies a teaching flux with power −2.7 by an interaction weight proportional to energy. The reference exposure is the expected total with no oscillation and both angular adjustments at zero.

#### Normalization check for the full no-oscillation sample

$$
\frac{N_{\mathrm{null}}}{N_0}=1+\frac{2}{3}(a-b)-\frac{8}{15}ab
$$

Integrating the even angular polynomial gives this independent check on the normalization. A larger acceptance loss lowers the expected total. With both angular adjustments off, the no-oscillation histogram is flat.

### Common difficulties

#### Upward does not mean produced underground

**Typical misconception** The upward sample comes from a source inside Earth.

**Better mental model** These neutrinos are produced in the atmosphere on the far side of Earth. The arrow points from where they are produced to the same surface detector. Earth only supplies a longer path, not a neutrino source.

#### A detector measures outgoing particles

**Typical misconception** The plotted neutrino energy and direction are directly measured without uncertainty.

**Better mental model** This Lab assumes perfect reconstruction. In reality the muon direction differs from that of its parent neutrino, and a real analysis must include energy, efficiency, backgrounds and nuclear interactions. The simplified bins here are not the collaboration’s published bins.

### Run the experiment

1. 01

   #### Turn a direction into a distance

   Pick Through Earth, Horizon and Overhead in turn. Use Path detail to see the short downward ray.

   **What to observe:** From overhead, the distance equals the production height. Straight up from below, it adds Earth’s diameter. Even a horizontal path is longer than the height, because Earth is curved.

2. 02

   #### Find the upward deficit

   Acquire Oscillating sky. Compare the sampled points with the green prediction and the dashed no-oscillation line.

   **What to observe:** Both predictions use the same flux, acceptance and exposure, so the gap between them comes from the survival probability. The points scatter because they are a finite sample of events.

3. 03

   #### Use energy as a second check

   Select each recorded energy band, then change only the preview energy.

   **What to observe:** Each band button selects from the same recorded events and integrates both predictions over the same energy range. The preview energy changes only one hypothetical ray. At a given distance, a higher energy changes the oscillation phase.

4. 04

   #### Run a control without oscillation

   Try No oscillation, then Uniform control. Change the horizontal flux and acceptance in the advanced controls and acquire again.

   **What to observe:** The counts can vary with direction even without oscillation. Because the source and acceptance functions are even, the expected up-to-down ratio stays at one when there is no oscillation, but a finite sample need not have equal counts.

5. 05

   #### Collect more events and recount them

   Try Long exposure and vary the random seed. Export the record and sum the individual event rows into the ten bins.

   **What to observe:** Summing the CSV event rows gives back the displayed counts, and each bin also lists its two integrated predictions. A longer exposure shrinks the relative counting fluctuations. This teaching model does not report a discovery significance.

Physical conclusion

The direction sets the baseline and the energy sets the scale of the oscillation phase. A convincing count comparison also needs a source model and an acceptance, and real ones carry uncertainties beyond this controlled teaching example.

## Related experiments and articles

Continue exploring

### How flavor change is predicted and tested

Propagation through vacuum and matter predicts a flavor probability. How do solar, atmospheric, reactor and beam experiments test it?

1. [Lab **Neutrino Flavor Propagation** Start with coherent vacuum propagation and ideal flavor trials.](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)
2. [Lab **Neutrino Matter Resonance** Follow a rotating matter Hamiltonian and compare slow with sudden conversion.](https://sandboxphysics.com/labs/neutrino-matter-resonance/)
3. [Lab **Solar Neutrino Flavor Observatory** Separate electron flavor from total active solar flux with three reaction responses.](https://sandboxphysics.com/labs/solar-neutrino-flux/)
4. [You are here **Atmospheric Neutrino Zenith Experiment** Turn Earth-crossing baselines into direction-dependent event counts.](https://sandboxphysics.com/labs/atmospheric-neutrino-zenith/)
5. [Lab **Reactor Near–Far Spectra** Fit the same near and far counts while testing calibration and normalization.](https://sandboxphysics.com/labs/reactor-near-far-spectra/)
6. [Lab **Long-Baseline CP Observatory** Compare both accelerator beam polarities on frozen spectra and separate CP phase, matter and response assumptions.](https://sandboxphysics.com/labs/long-baseline-cp/)

Neutrinos · Reading path

### Understand flavor change and mass

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How does the atmosphere make neutrinos all around us?**](https://sandboxphysics.com/stories/neutrinos/atmospheric-neutrino-production/)
2. [Lab **Cherenkov Light & Detector Rings**](https://sandboxphysics.com/labs/cherenkov-rings/)
3. [Article **Super-Kamiokande: the neutrinos that took the long way**](https://sandboxphysics.com/stories/neutrinos/super-k-zenith-angle/)
4. [Article **2015: how did two very different experiments establish flavor change?**](https://sandboxphysics.com/stories/neutrinos/nobel-2015-two-neutrino-evidence-paths/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)