Atomic Data for Non-Equilibrium Modeling of Kilonovae: The Ionization Properties of Te I - III

This paper presents new level-resolved ionization cross-section calculations for Te I–III using the Flexible Atomic Code to address the lack of non-thermal electron interaction data, demonstrating that configuration average approximations yield accurate results that significantly improve ionization balance models for kilonova ejecta.

The Gist

The Cosmic Crash and the Atomic Puzzle

Imagine two neutron stars—city-sized spheres of matter so dense that a teaspoon weighs a billion tons—smashing into each other. This cosmic collision, which happened in 2017, created a spectacular explosion called a kilonova. It's like a firework display made of heavy elements, scattering the building blocks of the universe (like gold and platinum) into space.

Astronomers are trying to understand exactly what this explosion looks like by studying the light it emits. However, there's a problem: the standard rules of physics we usually use to predict how atoms behave (called "equilibrium") break down in this chaotic, expanding cloud of debris.

This is where Tellurium (a heavy element, atomic number 52) comes in. Astronomers think they see Tellurium in the light from these explosions, but to be sure, they need to know exactly how Tellurium atoms interact with electrons in this weird, non-equilibrium environment.

The Problem: Missing Instructions

Think of an atom like a complex machine. To predict how it works, you need a manual (a database) that tells you:

1. How much energy it takes to knock an electron off the machine (ionization).
2. How the machine reacts when hit by a fast-moving electron.

For most common elements, we have these manuals. But for heavy, rare elements like Tellurium, the manuals are missing pages. Scientists have been using "best guess" formulas (like the Lotz formula) to fill in the blanks. It's like trying to fix a Ferrari using a manual for a bicycle. It might work okay in a pinch, but it's not accurate.

The Solution: Building a Better Manual

The authors of this paper decided to build a new, high-precision manual for Tellurium (specifically for its neutral state and its first two ionized states). They used a powerful computer code called FAC (Flexible Atomic Code) to simulate billions of atomic interactions.

Here is the tricky part they had to solve:

• The "Ghost" Electrons: When an electron hits a Tellurium atom, it doesn't just knock an electron off directly. Sometimes, it bumps an electron into a high-energy "waiting room" (an excited state), and then that electron falls out. This is called Excitation Autoionization.
• The Analogy: Imagine a pinball machine.
  • Direct Ionization: You hit the ball, and it flies straight out of the machine.
  • Excitation Autoionization: You hit the ball, it bounces off a bumper (excites), hits a second bumper, and then flies out.
  • The authors found that their computer simulations were sometimes counting "ghost" bounces—predicting that the ball would fly out when, in reality, the physics of the machine (the atom) meant it shouldn't have. They had to tweak their simulation settings to make sure they weren't counting impossible bounces.

They discovered that a method called Configuration Average (which looks at the atom as a whole "cloud" rather than counting every single electron path) was actually a very good shortcut. It gave them results almost as accurate as the super-detailed method, but without the headache of the "ghost" errors.

The "Non-Thermal" Electron Storm

In a kilonova, the debris is radioactive. As it decays, it shoots out high-speed electrons (non-thermal electrons). These are like tiny, super-fast bullets flying through the gas cloud.

The authors used their new manual to simulate how these bullets slow down as they hit Tellurium atoms. They found that the old "bicycle manual" (the Lotz formula) underestimated how many bullets would keep going fast. This matters because:

• If the bullets slow down too fast, they can't knock off enough electrons to create the light we see.
• If they stay fast too long, they create too much ionization.

By using their new, accurate data, they found that the "bullets" behave differently than previously thought, which changes the predicted mix of Tellurium ions in the explosion.

Why This Matters

This paper is essentially a quality control check for the tools astronomers use to study the universe's most violent events.

1. Better Maps: By fixing the "manual" for Tellurium, astronomers can now look at the light from kilonovae and say, "Yes, that is definitely Tellurium," with much higher confidence.
2. The "Good Enough" Shortcut: They proved that for these heavy atoms, the "Configuration Average" method is a great middle ground. It's fast, accurate enough for big simulations, and avoids the pitfalls of trying to track every single electron.
3. Future Proofing: They are calling on the scientific community to build similar manuals for other heavy elements. Until we have these, our understanding of how the universe creates heavy elements (like the gold in your ring) will remain a bit fuzzy.

In a nutshell: The authors fixed a broken instruction manual for a specific heavy atom, showed that a "good enough" shortcut works well, and proved that using this new data changes how we understand the glowing debris of exploding stars.

Technical Summary

1. Problem Statement

Kilonovae, electromagnetic transients resulting from binary neutron star mergers, exhibit spectral signatures that evolve over time. While early phases can be modeled using Local Thermodynamic Equilibrium (LTE), the ejecta transitions to a non-equilibrium regime approximately one week post-merger. In this regime, the ionization balance is driven not by thermal electrons, but by non-thermal electrons generated by the beta-decay of radioactive r-process elements.

A critical bottleneck in modeling these systems is the lack of accurate electron-impact ionization cross-sections for high-Z (high atomic number) r-process elements like Tellurium (Te, Z=52).

• Existing databases often store only Maxwellian (thermal) rate coefficients, not the necessary energy-dependent cross-sections.
• When cross-sections are missing, researchers rely on semi-empirical formulas (e.g., the Lotz formula), the accuracy of which for high-Z elements is unknown.
• Inaccurate cross-sections lead to uncertainties in the Spencer-Fano equation (which calculates the non-thermal electron energy distribution) and subsequent ionization balance models, hindering the identification of spectral features (e.g., the 2.1 µm feature in AT2017gfo attributed to Te III).

2. Methodology

The authors performed new theoretical calculations for the ionization cross-sections of Te I, Te II, and Te III using the Flexible Atomic Code (FAC).

• Theoretical Framework:
  • Used a relativistic Distorted Wave (DW) formalism and the Binary Encounter Dipole (BED) approximation.
  • Calculated both Direct Ionization (DI) and Excitation Autoionization (EAI). EAI is critical for high-Z ions, where excited states above the ionization potential autoionize.
  • Generated extensive configuration lists involving single and double promotions from 5s, 5p, and 4d sub-shells (up to $n=20$).
• Optimization Schemes:
  • To address inaccuracies in the single central-field potential assumption, the authors tested two optimization strategies:
    1. Optimizing the potential on the ground state of the target ion (e.g., Te I).
    2. Optimizing the potential on the ground state of the ionized system (e.g., Te II).
  • They also investigated the impact of Configuration Interaction (CI), finding that unlimited CI led to poor agreement with measurements, while limiting mixing to non-relativistic configurations yielded better results.
• Validation & Comparison:
  • Compared results against experimental data (Freund et al. 1990 for Te I; Djuric et al. 1994 for Te II).
  • Compared against the Lotz (1967) empirical formula and the Configuration Average (CA) approximation.
• Non-Equilibrium Modeling:
  • Integrated the new cross-sections into a Spencer-Fano solver (pynonthermal) to generate non-thermal electron energy distributions ($y(E)$).
  • Convolved these distributions with ionization cross-sections to derive non-thermal ionization rates.
  • Coupled these rates with recombination coefficients (from AUTOSTRUCTURE) to solve for ionization balance under kilonova-like conditions (thermal electron densities $\sim 10^6$ cm$^{-3}$, temperatures 2200–4500 K).

3. Key Contributions

• New Atomic Data: Provided the first comprehensive, level-resolved ionization cross-sections for Te I–III, including metastable-resolved rates.
• Methodological Insight: Demonstrated that optimizing the central potential on the initial target ion (rather than the ionized product) yields the best agreement with experimental data for near-neutral p-shell ions.
• Evaluation of Approximations: Validated the Configuration Average (CA) approximation as a robust alternative to computationally expensive level-resolved calculations, particularly for mitigating errors associated with near-threshold resonances in EAI channels.
• Impact Assessment: Quantified how different cross-section datasets (Level-Resolved, CA, and Empirical/Lotz) alter the predicted ion fractions in kilonova ejecta.

4. Key Results

• Agreement with Experiment:
  • For Te I, the Level-Resolved (LR) calculation optimized on the Te I ground state, combined with the BED approximation, agreed well with measurements. However, the raw DW calculation overestimated the peak cross-section by >20%.
  • A significant finding was that Excitation Autoionization (EAI) contributions were highly sensitive to the energy of levels just above the ionization potential. In LR calculations, some computed levels were erroneously placed above the threshold, inflating the EAI cross-section. A scaling factor of ~0.15 was required to match Te I data, suggesting DW overestimates near-threshold resonances.
  • The Configuration Average (CA) approach naturally avoided these near-threshold resonance issues and agreed well with both experiment and scaled LR results without arbitrary scaling factors.
• Lotz Formula Limitations:
  • The Lotz formula consistently underestimated ionization cross-sections by nearly a factor of 2 for Te I–III compared to FAC calculations and measurements.
  • Empirical models are highly sensitive to the choice of the constant $a_i$ and shell occupation numbers ($n_i$), leading to order-of-magnitude variations in predicted ionization rates.
• Impact on Ionization Balance:
  • Non-Thermal Electron Distribution: The shape and magnitude of the non-thermal electron energy distribution ($y(E)$) were similar for CA and LR datasets but differed significantly from empirical (Lotz) models.
  • Ion Fractions:
    • Models using FAC data (CA or LR) predicted similar ion fractions (Te II–IV).
    • Models using empirical Lotz cross-sections (with standard parameters) predicted significantly lower ionization (higher Te II, lower Te III/IV) compared to FAC models.
    • Metastable Resolution: Explicitly tracking metastable levels (which are efficiently ionized) altered ion fractions by 5–30% compared to ground-state-only models, highlighting the need for metastable-resolved data in complex systems.

5. Significance

This work addresses a critical gap in laboratory astrophysics required for interpreting kilonova spectra.

• Diagnostic Capability: Accurate ionization balance is essential for identifying r-process elements in kilonova ejecta. The study shows that relying on empirical Lotz formulas introduces large systematic errors in predicted ion populations.
• Modeling Strategy: The authors propose a hybrid approach: using Configuration Average data to efficiently solve the Spencer-Fano equation (determining the non-thermal electron spectrum) and Level-Resolved (or metastable-resolved) data for the final ionization balance. This balances computational cost with physical accuracy.
• Future Directions: The paper emphasizes the need for coordinated efforts to generate collision data for the ~50 elements involved in r-process nucleosynthesis. The provided Te I–III data serves as a benchmark for testing the validity of the CA approximation in open d- and f-shell systems.

In summary, the paper demonstrates that Configuration Average calculations offer a practical and accurate path forward for non-equilibrium kilonova modeling, significantly improving upon current empirical estimates while avoiding the computational and resonance-related pitfalls of fully level-resolved calculations for high-Z ions.