Modeling the refractive index profile n(z) of polar ice for ultra-high energy neutrino experiments

1. Problem Statement

Ultra-High Energy Neutrino (UHEN) experiments, such as the Askaryan Radio Array (ARA) and the proposed IceCube-Gen2, rely on detecting radio signals produced by neutrino interactions in polar ice. The sensitivity and effective volume of these detectors are heavily dependent on the refractive index profile, $n(z)$, of the ice as a function of depth.

• The Challenge: As radio signals propagate through ice, variations in density cause the refractive index to change, bending the ray paths (refraction). This creates a "shadowed zone" where neutrino interactions cannot be detected.
• Current Limitation: Most simulations assume a single exponential model for the refractive index (based on a self-gravitating fluid model). However, glaciological studies suggest that ice densification occurs in distinct stages (snow $\to$ firn $\to$ bubbly ice $\to$ deep ice), implying a more complex, piecewise density profile.
• Goal: To determine whether a multi-stage densification model provides a more accurate description of the ice refractive index than the standard single exponential model, using in-situ radio transmission data.

2. Methodology

The authors utilized in-situ radio transmission data from the ARA experiment at the South Pole to constrain the $n(z)$ profile.

• Data Source:
  • Transmitters: Two deep pulser sources were used:
    1. IDP (IceCube Deep Pulser): A stationary source at 1400m depth.
    2. SDP (SPICE Deep Pulser): A source lowered into the SPICE borehole, ranging from surface to ~1700m depth.
  • Receivers: ARA stations (A2–A5) equipped with antennas at depths of 170–200m, located 2–5 km horizontally from the transmitters.
• Measurement Technique:
  • The experiment measures the time difference ($dt(D,R)$) between the Direct (D) ray path and the Refracted/Reflected (R) ray path.
  • The D path travels through deeper ice, while the R path bends upward, traversing shallower layers (firn and snow).
  • $dt(D,R)$ is calculated using cross-correlation and leading-edge thresholding of the received waveforms.
• Simulation & Modeling:
  • Ray Tracing: The RadioPropa code (integrated into NuRadioMC) was used to simulate ray paths for different $n(z)$ models.
  • Models Tested:
    1. 1-Stage Model: Single exponential decay ($n(z) = 1.778 - B_0 e^{Cz}$).
    2. 2-Stage Model: Two exponential phases.
    3. 3-Stage Model: A piecewise function based on glaciological boundaries (Snow/Firn at ~550 kg/m³ and Firn/Bubbly Ice at ~758 kg/m³).
  • Density Conversion: Ice density data from the SPICE core was converted to refractive index using a linear relationship ($n = 1 + A\rho$) with a conversion factor $A = 0.845$.
• Snow Accumulation Correction: The authors accounted for snow accumulation over time (2011–2022) by analyzing the drift in $dt(D,R)$ over years, determining an effective accumulation rate of ~0.24 m/year for subsurface antennas, which was used to correct antenna depths for accurate simulation.

3. Key Contributions

• In-Situ Validation: The first direct comparison of multi-stage glaciological densification models against in-situ radio timing data over a large depth range (up to 1.7 km).
• Refinement of $n(z)$ Models: Demonstration that a 3-stage exponential model significantly outperforms the traditional single-exponential model in fitting the observed $dt(D,R)$ data.
• Shadow Zone Characterization: Detailed analysis of how different $n(z)$ models alter the size and location of the "shadowed zone" (the volume invisible to the detector) for different antenna depths.
• Impact on Sensitivity: Quantification of how the choice of ice model affects the calculated effective volume and neutrino sensitivity for future experiments.

4. Results

• Model Fit Quality:
  • The 3-stage model provided the best fit to the A2 station timing data, with a $\chi^2$ per degree of freedom of 9.3.
  • In contrast, the 1-stage and 2-stage models yielded poor fits ($\chi^2 \approx 406.5$ and $295.8$, respectively), failing to capture the timing differences across various transmitter depths.
• Comparison with Density Data:
  • The 3-stage model closely matched the converted SPICE core density data in the firn region (20.5m – 96.6m depth).
  • In the bubbly ice region (>96.6m), the 3-stage model slightly exceeded the converted density data, suggesting the linear density-to-refractive-index relationship might break down at greater depths or that further density data is needed.
• Shadow Zone Effects:
  • For 170m deep antennas (typical of ARA), the 3-stage model predicts a larger effective volume (smaller shadow zone) compared to the 1-stage model.
  • For 100m deep antennas (typical of RNO-G), the 3-stage model predicts a smaller effective volume (larger shadow zone) for sources deeper than 200m.
• Sensitivity Impact:
  • Simulations for ARA station depths (170m) using the 3-stage model show a 14% increase in neutrino sensitivity over the energy range $10^{18} - 10^{21}$ eV compared to the single exponential model.

5. Significance

• Improved Detector Calibration: Accurate $n(z)$ modeling is critical for reconstructing neutrino interaction points and energies. Using the wrong model introduces systematic errors in event reconstruction.
• Optimized Experiment Design: The findings suggest that detector sensitivity is highly dependent on antenna depth relative to the specific ice stratification. This informs the optimal deployment depth for future experiments (e.g., IceCube-Gen2 Radio, RNO-G).
• Glaciological Insight: The study validates the physical reality of multi-stage ice densification in polar ice sheets using radio propagation, bridging the gap between glaciology and high-energy astrophysics.
• Future Work: The authors note that while the 3-stage model fits timing data well, further density measurements in the deep bubbly ice region and analysis of signal amplitude/frequency content are needed to fully constrain the model and understand potential non-linearities in the density-refractive index relationship.