Where Does Tracing of Cosmic Ray in Real Atmosphere Terminate?

This paper investigates realistic physical termination criteria for cosmic-ray backtracing in the atmosphere, demonstrating that the simplified sharp-boundary approximation should be replaced by altitude-dependent thresholds (at least 50 km for protons and higher for heavy nuclei) determined by the combined effects of Bethe-Bloch energy loss and hard scattering interactions.

The Gist

Imagine you are trying to figure out where a specific raindrop came from. You see it hit your window, and you want to trace its path backward through the storm to see if it fell from a cloud high above or if it was just splashed up from a puddle on the ground.

In the world of space physics, scientists do something similar with cosmic rays—tiny, high-speed particles zipping through space. They use computer simulations to "backtrace" these particles from where they are detected (like on a satellite) all the way back to see if they originated from deep space (primary cosmic rays) or if they were just local noise.

For a long time, scientists used a very simple, "one-size-fits-all" rule to decide when to stop this backward tracing. They essentially drew an invisible, sharp line in the sky at a specific altitude (like 40 km or 100 km) and said, "If the particle goes below this line, we stop looking. We assume it hit the air and stopped."

This paper argues that drawing a sharp line is like guessing where a car stops by looking at a map, rather than checking if the car actually ran out of gas or hit a wall. The authors, Du-Xin Zheng, Long Chen, and Ran Huo, say we need to look at the actual physics of what happens when a cosmic ray hits our atmosphere.

The Two "Brakes" on the Cosmic Ray

The paper identifies two specific physical "brakes" that stop a cosmic ray from traveling backward through the atmosphere. Think of these as the reasons a car would stop moving:

1. The "Friction" Brake (Bethe-Bloch Energy Loss):
   Imagine a runner sprinting through a thick crowd. Every time they bump into someone, they lose a tiny bit of speed. In the atmosphere, as a cosmic ray particle moves through air molecules, it constantly bumps into electrons. This is a slow, continuous drag.

   • When it matters: This is the main reason particles stop when they are moving relatively slowly (low energy). It's like the runner getting tired and slowing down gradually until they can't go on.

2. The "Crash" Brake (Hard Scattering):
   Now imagine that same runner suddenly smacks into a solid brick wall. They don't just slow down; they bounce off or shatter instantly. In the atmosphere, this happens when a cosmic ray smashes directly into an atomic nucleus.

   • When it matters: This is the main reason particles stop when they are moving very fast (high energy). It's a sudden, violent collision that ends the journey immediately.

The New "Stop" Sign

The authors ran detailed simulations using a realistic model of Earth's atmosphere (updated with current carbon dioxide levels) to see exactly where these "brakes" become strong enough to stop the particle.

They found that the old "sharp line" rules were often too low.

• For light particles (like protons): The particle can actually travel deeper into the atmosphere before these brakes become effective. The authors suggest the "stop line" should be raised to at least 50 km.
• For heavy particles (like iron nuclei): These are like heavy trucks; they are harder to stop. The "stop line" needs to be raised even higher, by about 15 km more than the proton line.

Why Does This Matter?

The paper uses a few helpful analogies to explain the impact:

• The "Penumbra" (The Fuzzy Edge):
  Imagine a shadow cast by a tree. The very edge of the shadow isn't a sharp black line; it's a fuzzy gray area where some light gets through and some doesn't.
  The authors explain that because cosmic rays stop due to random collisions (the "Crash" brake), there is no perfect, sharp line between "allowed" and "forbidden" particles. It's a fuzzy zone. By using a sharp line at the wrong altitude, scientists were either throwing away valid data (thinking a particle stopped when it didn't) or keeping bad data.

• The "Allowed Cone":
  Imagine looking up at the sky through a telescope. You can only see a certain cone of the sky. If you move your "stop line" up from 40 km to 50 km, you slightly widen that cone.
  The authors calculate that this small change allows scientists to see about 1% to 1.7% more valid cosmic ray events. For an experiment like AMS-02, which has been collecting data for 15 years, this tiny percentage translates to billions of extra data points that were previously being ignored or misclassified.

The Bottom Line

The paper doesn't propose a new machine or a new drug. It proposes a better math rule.

Instead of saying, "Stop tracing when you hit 40 km," the authors suggest a smarter rule: "Stop tracing when the particle has lost enough energy to friction or has had a high chance of crashing into an atom."

This makes the "map" of where cosmic rays come from more accurate, ensuring that scientists don't accidentally throw away the most interesting particles from deep space just because they were tracing them to the wrong altitude.

Technical Summary

Technical Summary: Where Does Tracing of Cosmic Ray in Real Atmosphere Terminate?

Problem Statement
In backtracing simulations used to determine cosmic-ray trajectories within the geomagnetic field, the atmosphere is conventionally approximated as an artificial sharp boundary at a fixed low altitude (e.g., 20 km, 40 km, or 100 km). At this boundary, the backward propagation of a charged particle is abruptly terminated. The authors argue that these altitudes are ad hoc and lack a foundation in the actual microscopic physical processes that halt particle propagation. Specifically, the cessation of a trajectory should be dictated by interactions between the cosmic-ray particle and atmospheric nuclei, rather than an arbitrary geometric cutoff.

Methodology
The paper investigates two specific microscopic mechanisms that terminate backward-propagating cosmic rays:

1. Bethe-Bloch Energy Loss: The continuous deceleration of the particle due to cumulative Coulomb scatterings, which violates the energy-conservation premise required for standard backtracing.
2. Hard Scattering: Discrete elastic or inelastic scattering events with atmospheric atoms that deflect the particle from its path.

To quantify these effects, the authors introduce two dimensionless variables:

• The relative rigidity shift due to energy loss ($\Delta R/R$).
• The expected number of hard scattering events ($\langle N \rangle$).

The study employs the following models and data:

• Atmospheric Model: The US Standard Atmosphere 1976, updated to reflect 2025 carbon dioxide levels, with partial densities of dominant species (H, He, C, N, O, Ar) modeled as exponential functions of altitude ($\rho_T \propto e^{-0.14h/\text{km}}$).
• Cross Sections: Hard scattering cross-sections are calculated using the Glauber-Gribov formalism, utilizing nucleon-nucleon cross-section data from the PDG 2022 compilation and Savitzky-Golay filtered fits.
• Trajectory Geometry: The authors first analyze a simplified straight-line trajectory tangent to the atmosphere at a perigee altitude $h$. They then extend this to curved trajectories characterized by a local radius of curvature $r = R/B$ near the perigee.

Key Contributions and Results

• Factorization of Dependencies: The study demonstrates that the altitude dependence of both $\Delta R/R$ and $\langle N \rangle$ can be factorized as approximately $\exp(-0.14h/\text{km})$. Furthermore, the effect of the local trajectory curvature can be factorized using a fitted function $g(r)$, which scales with the radius of curvature.
• Dominance Regimes: The analysis reveals a transition in the dominant termination mechanism based on particle rigidity ($R$). Bethe-Bloch energy loss dominates at low rigidities (specifically for protons below $\sim 0.57$ GV), while hard scattering becomes the dominant factor at higher rigidities.
• Revised Termination Criteria:
  • For protons, the authors propose that a simplified sharp-boundary altitude should be at least 50 km to satisfy the condition $\Delta R/R + \langle N \rangle \lesssim 1$.
  • For heavy nuclei, such as iron, the required altitude is significantly higher due to larger cross-sections and charge factors. The authors calculate that the boundary for iron must be raised by more than 15 km (to $\sim 66$ km) compared to the proton boundary to maintain the same physical termination criterion.
• Impact on Cutoff Rigidity: The paper illustrates that the traditional 40 km or 100 km boundaries introduce errors in determining geomagnetic cutoff rigidities. For instance, a 1 GV proton at a 40 km perigee yields a termination sum of 3.4 (indicating interaction), whereas at 50 km it is 0.93. Consequently, events previously discarded by a 40 km cutoff might be valid primaries under the new physics-based criterion.

Significance and Claims
The paper claims that replacing ad hoc altitude boundaries with a physics-based criterion ($\Delta R/R + \langle N \rangle$) improves the accuracy of cosmic-ray backtracing.

• Penumbra and Probabilistic Nature: The authors note that because hard scattering is probabilistic, the boundary between "allowed" and "forbidden" rigidities is inherently smeared, challenging the traditional concept of a sharp penumbra.
• Quantitative Impact: The shift in boundary altitude affects the calculated "allowed cone" for cosmic rays in Low-Earth Orbit (LEO). Using a 50 km boundary instead of 20 km reduces the allowed cone by approximately 1.0% for protons and 1.5% for iron, while a 100 km boundary reduces it by 1.7% for protons. This implies that space dosimetry estimates using lower boundaries may be overestimated by 1.0–1.5%, and conversely, experiments like AMS-02 could validly utilize an additional 1.2–1.7% of events if the boundary is optimized.
• Future Application: While the current work provides a parametric scaling law, the authors state that a fully consistent treatment integrating these processes along true curved trajectories for every event is reserved for future publication. The present study serves to establish the order-of-magnitude effects and the factorized scaling result necessary to replace arbitrary boundaries.