🔭 astrophysics

Modeling the Effect of the Heliospheric Magnetic Field on Cosmic Ray Muon Shadows

Using 13 years of MINOS detector data, this study analyzes cosmic ray muon shadows cast by the Sun to evaluate the Parker spiral model of the Heliospheric Magnetic Field, finding it most accurate during solar minimum but suggesting that more detailed models are needed to account for spectral hardness and solar maximum conditions.

Original authors: MINOS+ Collaboration, :, P. Adamson, I. Anghel, A. Aurisano, G. Barr, A. Blake, S. V. Cao, T. J. Carroll, C. M. Castromonte, R. Chen, S. Childress, J. A. B. Coelho, S. De Rijck, J. J. Evans, G. J. Fe

Published 2026-05-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: MINOS+ Collaboration, :, P. Adamson, I. Anghel, A. Aurisano, G. Barr, A. Blake, S. V. Cao, T. J. Carroll, C. M. Castromonte, R. Chen, S. Childress, J. A. B. Coelho, S. De Rijck, J. J. Evans, G. J. Feldman, W. Flanagan, S. Fogarty, M. Gabrielyan, H. R. Gallagher, S. Germani, R. A. Gomes, M. C. Goodman, P. Gouffon, N. Graf, N. Grossman, K. Grzelak, A. Habig, S. R. Hahn, J. Hartnell, R. Hatcher, A. Holin, J. Huang, L. W. Koerner, M. Kordosky, A. Kreymer, J. Krueger, K. Lang, P. Lucas, W. A. Mann, M. L. Marshak, N. Mayer, R. Mehdiyev, J. Meier, W. H. Miller, G. Mills, D. Naples, J. K. Nelson, R. Nichol, J. OConnor, R. B. Pahlka, Ž. Pavlović, G. Pawloski, A. Perch, M. M. Pfützner, D. D. Phan, R. K. Plunkett, N. Poonthottathil, X. Qiu, A. Radovic, P. Sail, M. C. Sanchez, A. Schreckenberger, R. Sharma, N. Skuza, A. Sousa, N. Tagg, J. Thomas, M. A. Thomson, A. Timmons, J. Todd, S. C. Tognini, R. Toner, D. Torretta, P. Vahle, A. Weber, L. H. Whitehead, S. G. Wojcicki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Earth is constantly being pelted by a steady rain of invisible, high-speed bullets called Cosmic Rays. Most of these bullets are protons (tiny bits of atoms) zooming in from deep space. When they hit our atmosphere, they smash into air molecules and create a secondary spray of particles, including muons. These muons are like the "ghosts" of the original cosmic rays; they are tough enough to punch through miles of rock and reach detectors buried deep underground, like the one in the Soudan Mine in Minnesota used in this study.

Here is the simple story of what the scientists did, using some everyday analogies:

The Cosmic "Shadow" Game

Usually, this rain of muons falls evenly on the detector, like snow falling on a flat roof. However, the Sun and the Moon are big, solid objects floating in space. When a cosmic ray hits them, it gets absorbed. This means that if you look at the sky from Earth, there should be a tiny, dark "shadow" where no muons are coming from, right behind the Sun and Moon.

The scientists found these shadows. But here's the twist: the shadows aren't where they should be.

The Invisible Wind (The Magnetic Field)

Think of the space between the Sun and Earth not as empty vacuum, but as a river of invisible wind. This is the Heliospheric Magnetic Field (HMF). It's like a giant, swirling magnetic hose that the Sun is spraying out.

Because cosmic rays are electrically charged, they don't travel in straight lines through this magnetic wind. Instead, they get pushed and pulled, curving their paths like leaves caught in a strong, twisting stream. By the time they reach Earth, the "shadow" cast by the Sun has been shifted or distorted by this magnetic wind.

The Experiment: Backtracking the Rain

The MINOS+ team wanted to figure out exactly how this magnetic wind works. They used a clever trick: Time Reversal.

The Setup: They took the muons that hit their detector and traced their paths backward in time, all the way from the mine, through the atmosphere, and out into space toward the Sun.
The Simulation: They built a computer model of the magnetic wind (using a famous model called the Parker Spiral, which imagines the magnetic field as a spiral shape, like a garden hose spinning while you spray water).
The Test: They simulated millions of particles moving backward through this magnetic wind to see where they would have come from if the model was perfect.

What They Found

The scientists looked at three different time periods:

Solar Minimum: When the Sun is quiet and the magnetic wind is calm.
Solar Maximum: When the Sun is active, stormy, and the magnetic wind is chaotic.
The Whole 13 Years: A mix of both.

The Results:

The Quiet Sun: When the Sun was quiet (Solar Minimum), the computer model worked surprisingly well. The simulated shadows matched the real shadows almost perfectly. It was like the magnetic wind was a gentle, predictable breeze.
The Stormy Sun: When the Sun was active (Solar Maximum), the model struggled. The real shadows were shifted in a way the simple model couldn't explain. It was as if the magnetic wind had suddenly become a violent, unpredictable hurricane that the simple garden-hose model couldn't predict.

The "Energy" Puzzle

The scientists also realized that the "speed" (energy) of the cosmic rays matters a lot.

The Analogy: Imagine throwing a ping-pong ball and a bowling ball into the same strong wind. The light ping-pong ball gets blown off course easily. The heavy bowling ball barely moves.
The Finding: The simple model only worked if they assumed the cosmic rays were much heavier (higher energy) than they actually are. To make the math work, they had to pretend the particles were "super-bowling balls" (about 23 TeV of energy) instead of the "bowling balls" (about 10 TeV) we know they are.

The Conclusion

The paper concludes that the simple "garden hose" model (Parker Spiral) is a good starting point, especially when the Sun is calm. However, it's not the whole story.

When the Sun is active, the magnetic field is likely more complex than a simple spiral. The scientists suggest there might be "waves" or "tilts" in the magnetic field (like a ballerina's spinning skirt) that they didn't include in their model. These complex structures are what are causing the shadows to shift in ways the simple model can't predict.

In short: The Sun casts a shadow in the cosmic ray rain. By studying where that shadow lands, scientists can map the invisible magnetic wind of the solar system. They found that their map is accurate when the Sun is quiet, but needs to be redrawn with more detail when the Sun is having a stormy day.

Technical Summary: Modeling the Effect of the Heliospheric Magnetic Field on Cosmic Ray Muon Shadows

Problem Statement
Cosmic rays (CRs), primarily protons and alpha particles, are deflected by electromagnetic fields as they traverse the solar system. While the Earth's geomagnetic field (GMF) is well-mapped, the Heliospheric Magnetic Field (HMF) is complex and difficult to model. The Sun and Moon absorb incident CRs, creating "shadows" in the cosmic ray flux observable from Earth. However, these shadows are not static; they are shifted and distorted by the HMF and GMF. The MINOS+ Collaboration utilized muon data from the MINOS far detector to locate these solar shadows during different phases of the solar cycle (solar minimum, solar maximum, and the full 13-year span). The central problem addressed is whether the observed shifts and distortions of the solar muon shadow are consistent with the standard Parker spiral model of the HMF combined with a dipole model of the GMF, and if not, what modifications to the models or parameters are required to reconcile theory with observation.

Methodology
The study utilized 13 years of muon data (July 2003–July 2016) from the MINOS far detector, located 2070 meters water equivalent underground in the Soudan Mine. The analysis focused on single muon events with incident zenith angles less than 80° and sufficient track length to reconstruct trajectories.

Shadow Localization: The team employed a $\Lambda$ probability test (a log-likelihood ratio) to locate the shadow. This involved comparing observed particle counts ( $N_{obs}$ ) near the Sun against a theoretical flat background ( $N_{back}$ ) and a theoretical shadow template ( $N_{th}$ ). The template accounted for the Sun's angular size (0.52°) and scattering effects in the rock overburden.
Statistical Validation: Monte Carlo simulations were used to generate random background histograms to determine the significance of the observed shadow locations. The probability of random fluctuations mimicking the shadow was calculated, yielding high confidence levels (up to 6 $\sigma$ for the full dataset).
Computational Modeling: To interpret the shadow shifts, the authors simulated the time-reversed trajectories of CRs from the detector back to the Sun.
- Field Models: The simulation superimposed the Parker spiral model for the HMF and a dipole model for the GMF. The Parker spiral parameters included a constant solar wind velocity ( $u_{sowi} = 400$ km/s) and solar rotation frequency. The GMF was modeled using a dipole moment and converted into a Sun-centered coordinate system.
- Particle Dynamics: The Lorentz force equation was integrated using a 4th-order Runge-Kutta scheme. Particles were treated as relativistic, with energies following a power-law distribution ( $E^{-2.75}$ ).
- Scenarios: Simulations tested various magnetic field polarity ratios (50/50, 33/67, 67/33) and two shadow template types: a simple circular shadow and a "distorted" shadow derived from the actual $\Lambda$ probability histograms.
- Parameter Variation: The primary cosmic ray energy cutoff ( $E_0$ ) was treated as a free parameter to determine if a different energy spectrum could reconcile the model with the observed shadow positions.

Key Contributions

Shadow Characterization: The paper successfully located solar muon shadows with high statistical significance across three distinct timeframes: the full 13-year dataset, the solar minimum (2007–2009), and the solar maximum (2012–2014).
Template Development: The authors introduced a method for creating "distorted" shadow sampling templates based on the observed $\Lambda$ probability distributions, moving beyond the assumption of a perfectly circular shadow.
Model Testing: The study rigorously tested the Parker spiral model against observational data, quantifying the agreement between simulated and observed shadow shifts in Right Ascension ( $\Delta RA$ ) and Declination ( $\Delta DEC$ ).
Energy Spectrum Analysis: The paper investigated the sensitivity of shadow positions to the primary CR energy spectrum, specifically testing the impact of the low-energy cutoff ( $E_0$ ).

Results

Shadow Shifts: The observed shadows showed distinct shifts depending on the solar cycle phase. The solar minimum shadow was shifted differently than the solar maximum shadow, and the full dataset represented a combination of these shifts.
Model Consistency:
- The Parker spiral model showed the best agreement with the solar minimum data.
- The model showed the least agreement with the solar maximum data, particularly regarding the $\Delta DEC$ shift.
- Distorted shadow templates performed better for solar maximum and full data periods, while simple round templates were more effective for solar minimum data.
Polarity Effects: The simulations confirmed that the magnetic field polarity ratio directly affects the $\Delta DEC$ position of the shadow. The data appeared to favor a positive polarity-dominated HMF, especially during solar maximum.
Energy Cutoff Discrepancy: To align the simulated shadow means with the theoretical center (0,0) using the Parker spiral model, the primary CR energy cutoff ( $E_0$ ) had to be adjusted to 22.8 $\pm$ 1.2 TeV. This is significantly higher (2–3 times) than the experimentally determined value of $\approx$ 10 TeV.
Zenith Correction: A correction for the zenith angle dependence of the energy cutoff was applied. While this adjusted the spectrum to better match the observed muon energy distribution, it was insufficient to fully explain the observed $\Delta RA$ shift.

Significance and Conclusions
The paper concludes that the cosmic ray solar muon shadows are observable and can be used to probe the structure of the HMF. The results indicate that the solar minimum shadow is most consistent with the standard Parker spiral model combined with a dipole GMF. However, the model fails to fully reproduce the solar maximum shadow and the full dataset without invoking an unrealistically high energy cutoff.

The authors assert that the discrepancy suggests the Parker spiral model is an oversimplification for the conditions present during solar maximum. They propose that unmodeled factors, such as the neutral current sheet tilt (the "Ballerina Model"), shock fronts, and complex polarity variations, likely contribute to the observed shifts. The study establishes that while the general direction of deflection is consistent with the Parker spiral, the detailed structure of the HMF during solar maximum requires more complex modeling than the simple spiral model provides. The work demonstrates the utility of muon shadow observations for constraining HMF models but highlights the need for more sophisticated simulations to account for solar cycle variations.