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NASA NTRS2014

Path Toward a Unifid Geometry for Radiation Transport

The Direct Accelerated Geometry for Radiation Analysis and Design (DAGRAD) element of the RadWorks Project under Advanced Exploration Systems (AES) within the Space Technology Mission Directorate (STMD) of NASA will enable new designs and concepts of operation for radiation risk assessment, mitigation and protection. This element is designed to produce a solution that will allow NASA to calculate the transport of space radiation through complex computer-aided design (CAD) models using the state-of-the-art analytic and Monte Carlo radiation transport codes. Due to the inherent hazard of astronaut and spacecraft exposure to ionizing radiation in low-Earth orbit (LEO) or in deep space, risk analyses must be performed for all crew vehicles and habitats. Incorporating these analyses into the design process can minimize the mass needed solely for radiation protection. Transport of the radiation fields as they pass through shielding and body materials can be simulated using Monte Carlo techniques or described by the Boltzmann equation, which is obtained by balancing changes in particle fluxes as they traverse a small volume of material with the gains and losses caused by atomic and nuclear collisions. Deterministic codes that solve the Boltzmann transport equation, such as HZETRN [high charge and energy transport code developed by NASA Langley Research Center (LaRC)], are generally computationally faster than Monte Carlo codes such as FLUKA, GEANT4, MCNP(X) or PHITS; however, they are currently limited to transport in one dimension, which poorly represents the secondary light ion and neutron radiation fields. NASA currently uses HZETRN space radiation transport software, both because it is computationally efficient and because proven methods have been developed for using this software to analyze complex geometries. Although Monte Carlo codes describe the relevant physics in a fully three-dimensional manner, their computational costs have thus far prevented their widespread use for analysis of complex CAD models, leading to the creation and maintenance of toolkit-specific simplistic geometry models. The work presented here builds on the Direct Accelerated Geometry Monte Carlo (DAGMC) toolkit developed for use with the Monte Carlo N-Particle (MCNP) transport code. The workflow for achieving radiation transport on CAD models using MCNP and FLUKA has been demonstrated and the results of analyses on realistic spacecraft/habitats will be presented. Future work is planned that will further automate this process and enable the use of multiple radiation transport codes on identical geometry models imported from CAD. This effort will enhance the modeling tools used by NASA to accurately evaluate the astronaut space radiation risk and accurately determine the protection provided by as-designed exploration mission vehicles and habitats

Lee, Kerry↗

DOE OSTI2020

Analytic Insights into the Neutronic Characteristics of Neutron Moderators from MCNP Calculations

In the course of re-processing the ENDF/B VIII.0 thermal scattering law data, formulas and data checking insights were developed from MCNP calculations into the neutronic characteristics of moderators. Infinite medium test calculations were carried out in MCNP as part of the verification effort for the ACE formatted thermal scattering files. Edit results from MCNP were combined with derived results to formulate several equations which give analytic insights into the neutronic behavior of moderators.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

On the Origin of Pulsar and Magnetar Magnetic Fields

Abstract In order to address the generation of neutron star magnetic fields, with particular focus on the dichotomy between magnetars and radio pulsars, we consider the properties of dynamos as inferred from other astrophysical systems. With sufficiently low (modified) Rossby number, convective dynamos are known to produce dipole-dominated fields whose strength scales with convective flux, and we argue that these expectations should apply to the convective protoneutron stars (PNSs) at the centers of core-collapse supernovae. We analyze a suite of three-dimensional simulations of core collapse, featuring a realistic equation of state and full neutrino transport, in this context. All our progenitor models, ranging from 9 M ⊙ to 25 M ⊙ , including one with initial rotation, have sufficiently vigorous PNS convection to generate dipole fields of order ∼10 15 Gauss, if the modified Rossby number resides in the critical range. Thus, the magnetar/radio pulsar dichotomy may arise naturally in part from the distribution of core rotation rates in massive stars.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Proceedings for the Workshop on Applied Nuclear Data Activities 2024

The Workshop for Applied Nuclear Data Activities (WANDA) is designed to increase communication among nuclear data (ND) users in multidisciplinary federal programs, ND producers, ND funders, and other ND experts. It also presents an opportunity to cross-pollinate ideas as well as introduce ND gaps identified by federal programs to ND experts and ND capabilities to the various federal ND users. WANDA 2024 included five technical sessions, three of which focused on Fusion Energy Sciences (FES)—FES Fusion Neutronics, FES Tritium Production, and FES Material Damage—and two stand-alone sessions—Isotopes and Targetry for Nuclear Data and Uncertainty Quantification. The FES sessions successfully brought new voices to the WANDA discussions, expanding the application space in which nuclear data are critical. FES programs need accurate nuclear data with realistic uncertainty quantification to properly estimate, for example, shielding, activation, tritium production, helium production, structural material integrity, and superconducting magnet operation. This includes a variety of projectile (neutrons, photons, charged particles) and target atoms. One of the action items common to all the FES sessions was a need to perform sensitivity studies to identify the prioritization of nuclear data needs. The Isotopes and Targetry session highlighted the many capabilities available to produce high-quality targets for nuclear data measurements, including 3D printing with spherical powders, combustion synthesis coupled with spin coating & electrospraying, inkjet printing, and isotopic doping. These new methods open doors for more accurate measurement, but it was also stressed that sample characterization following any method of fabrication is of the highest importance to accurately interpret nuclear data measurement results that used that sample. The Uncertainty Quantification (UQ) session was broken into two categories: nuclear data uncertainty quantification and the use of that uncertainty quantification. Thematic to the UQ session was the loss of information when going from nuclear data measurement, to evaluation, to evaluated file, and finally to neutron transport calculations. Current evaluated ND libraries typically only contain covariances, which assume that the probability distributions are Gaussian. Beyond being a simplified assumption for many evaluations, this can lead to negative values on many observables when attempting to sample the covariance. The covariance format, however, is very efficient in that a simple set of linear equations can transform uncertainty from parameters or cross sections to the application of interest. Focused collaboration is needed between nuclear data evaluators and nuclear data users to ensure that needs are being met.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS2015

Space Radiation Transport Code Development: 3DHZETRN

The space radiation transport code, HZETRN, has been used extensively for research, vehicle design optimization, risk analysis, and related applications. One of the simplifying features of the HZETRN transport formalism is the straight-ahead approximation, wherein all particles are assumed to travel along a common axis. This reduces the governing equation to one spatial dimension allowing enormous simplification and highly efficient computational procedures to be implemented. Despite the physical simplifications, the HZETRN code is widely used for space applications and has been found to agree well with fully 3D Monte Carlo simulations in many circumstances. Recent work has focused on the development of 3D transport corrections for neutrons and light ions (Z < 2) for which the straight-ahead approximation is known to be less accurate. Within the development of 3D corrections, well-defined convergence criteria have been considered, allowing approximation errors at each stage in model development to be quantified. The present level of development assumes the neutron cross sections have an isotropic component treated within N explicit angular directions and a forward component represented by the straight-ahead approximation. The N = 1 solution refers to the straight-ahead treatment, while N = 2 represents the bi-directional model in current use for engineering design. The figure below shows neutrons, protons, and alphas for various values of N at locations in an aluminum sphere exposed to a solar particle event (SPE) spectrum. The neutron fluence converges quickly in simple geometry with N > 14 directions. The improved code, 3DHZETRN, transports neutrons, light ions, and heavy ions under space-like boundary conditions through general geometry while maintaining a high degree of computational efficiency. A brief overview of the 3D transport formalism for neutrons and light ions is given, and extensive benchmarking results with the Monte Carlo codes Geant4, FLUKA, and PHITS are provided for a variety of boundary conditions and geometries. Improvements provided by the 3D corrections are made clear in the comparisons. Developments needed to connect 3DHZETRN to vehicle design and optimization studies will be discussed. Future theoretical development will relax the forward plus isotropic interaction assumption to more general angular dependence.

Wilson, John W.↗

NASA NTRS1972

The ORNL-SNAP shielding program

The effort in the ORNL-SNAP shielding program is directed toward the development and verification of computer codes using numerical solutions to the transport equation for the design of optimized radiation shields for SNAP power systems. A brief discussion is given for the major areas of the SNAP shielding program, which are cross-section development, transport code development, and integral experiments. Detailed results are presented for the integral experiments utilizing the TSF-SNAP reactor. Calculated results are compared with experiments for neutron and gamma-ray spectra from the bare reactor and as transmitted through slab shields.

Mynatt, F. R.↗

DOE OSTI2025

Monte Carlo Simulations of Polarized Radiative Transfer in Neutron Star Atmospheres

Soft X-ray emission from neutron stars affords powerful diagnostic tools for uncovering their surface and interior properties, as well as their geometric configurations. In the atmospheres of neutron stars, the presence of magnetic fields alters the photon-electron scattering cross sections, resulting in nontrivial angular dependence of intensity and polarization of the emergent signals. This paper presents recent developments of our Monte Carlo simulation, MAGTHOMSCATT, which tracks the complex electric field vector for each photon during its transport. Our analysis encompasses the anisotropy and polarization characteristics of X-ray emission for field strengths ranging from nonmagnetic to extremely magnetized regimes that are germane to magnetars. In the very low field domain, we reproduced the numerical solution to the radiative transfer equation for nonmagnetic Thomson scattering, and provided analytical fits for the angular dependence of the intensity and the polarization degree. These fits can be useful for studies of millisecond pulsars and magnetic white dwarfs. By implementing a refined injection protocol, we show that, in the magnetar regime, the simulated intensity and polarization pulse profiles of emission from extended surface regions becomes invariant with respect to the ratio of photon (ω) and electron cyclotron (ω B ) frequencies once ω/ω B ≲ 0.01. This circumvents the need for simulations pertinent to really high magnetic field strengths, which are inherently slower. Our approach will be employed elsewhere to model observational data to constrain neutron star geometric parameters and properties of emitting hot spots on their surfaces.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Estimating List-Mode Data Sensitivities to Nuclear Data with MCNP6

Nuclear data are a vital component of predictive simulations used in applications like experiment design, stockpile stewardship, nuclear nonproliferation/safeguards, health physics, and criticality safety. A singular simulation requires the coalescence of different areas of nuclear data such as cross sections, angular distributions, and energy distributions of emitted neutrons for different materials and energy ranges. Improving nuclear data and thus reducing the uncertainty in simulated parameters could enable smaller, better-informed safety factors and ultimately reduce operational and procedural costs. There is a constant effort to garner a better understanding of the physical quantities represented by nuclear data through experiments. Integral experiment benchmarks use simulated and measured results to validate current nuclear data values. In the past, benchmarks primarily focused on the effective multiplication factor (k eff ); however, this limited scope has caused compensating errors and areas of nuclear data that lack validation. Compensating errors are inaccuracies in nuclear data that are obfuscated by cancellation when observing integrated values such as k eff . Diverse integral benchmark experiments that look for quantities of interest other than k eff and include multiple responses minimize the possibility of compensating errors and provides validation to areas of nuclear data previously lacking experimental validation. Benchmark experiments can be optimized during the design process to be highly dependent on specific areas of nuclear data. The dependence of a response in an experiment to a specific area/type of nuclear data is defined as sensitivity. A larger sensitivity means that nuclear data uncertainties will play a larger role in the response(s) resulting in larger bias. Currently, the sensitivity capabilities of the Monte Carlo N-Particle (MCNP ®1 ) transport code are limited to responses of k eff and tallied values (e.g., flux, surface current). As a part of the EUCLID project, this work explores estimating list-mode nuclear data sensitivities that can be used to design experiments aimed to constrain and reduce compensating errors in nuclear data by focusing on responses other than k eff . Tallied values are ideal quantities that are estimated with detectors during experiments. List-mode data (a list of neutron collection times) are the direct output of detector systems in subcritical neutron noise experiments. Expanding MCNP sensitivity capabilities to include the sensitivity of responses estimated from list-mode data, such as the prompt neutron decay constant (α) and multiplicity estimates (S and D), enables more direct comparison of simulated and measured experimental quantities. Additionally, deterministic tools such as SENSMG are capable of obtaining sensitivities to a wide variety of responses; however, these tools cannot handle complex geometries due to the assumptions made in discretizing the phase-space variables of the Boltzman transport equation.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2023

Secular outflows from 3D MHD hypermassive neutron star accretion disc systems

Magnetized hypermassive neutron stars (HMNSs) have been proposed as a way for neutron star mergers to produce high electron fraction, high-velocity ejecta, as required by kilonova models to explain the observed light curve of GW170817. The HMNS drives outflows through neutrino energy deposition and mechanical oscillations, and raises the electron fraction of outflows through neutrino interactions before collapsing to a black hole (BH). Here, in this study, we perform 3D numerical simulations of HMNS–torus systems in ideal magnetohydrodynamics, using a leakage/absorption scheme for neutrino transport, the nuclear APR equation of state, and Newtonian self-gravity, with a pseudo-Newtonian potential added after BH formation. Due to the uncertainty in the HMNS collapse time, we choose two different parametrized times to induce collapse. We also explore two initial magnetic field geometries in the torus, and evolve the systems until the outflows diminish significantly ($\sim\!\! 1\!\! - \!\!2\ \mathrm{s}$). We find bluer, faster outflows as compared to equivalent BH–torus systems, producing M ∼ 10 −3 M ⊙ of ejecta with Y e ≥ 0.25 and v ≥ 0.25c by the simulation end. Approximately half the outflows are launched in disc winds at times $t\lesssim 500 \ \mathrm{ms}$, with a broad distribution of electron fractions and velocities, depending on the initial condition. The remaining outflows are thermally driven, characterized by lower velocities and electron fractions. Nucleosynthesis with tracer particles shows patterns resembling solar abundances in all models. Although outflows from our simulations do not match those inferred from two-component modelling of the GW170817 kilonova, self-consistent multidimensional detailed kilonova models are required to determine whether our outflows can power the blue kilonova.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Implementation of the Windowed Multipole Method in Shift

The windowed multipole (WMP) method has been implemented in the Shift Monte Carlo (MC) radiation transport code with support for both CPU and GPU execution. With this method, small WMP data libraries (~100 MB) can be used to accurately Doppler broaden cross sections to arbitrary temperatures “on the fly” during an MC simulation. This approach yields significant memory savings relative to traditional methods, making it ideal for high-fidelity analysis such as coupled multiphysics simulations. This document provides the exact forms of the WMP equations used by Shift, as well as a detailed description of the structure of WMP HDF5 data files provided by the Massachusetts Institute of Technology (MIT). The Shift implementation has been validated against the OpenMC radiation transport code, with excellent agreement demonstrated for 70 nuclides across an operative range of temperatures. CPU and GPU performance testing using a small module reactor (SMR) problem demonstrated that this method decreases the neutron tracking rate by a factor of ~2 on the Summitdev machine. A new set of WMP data being developed in-house will employ novel methods to improve tracking rates.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Coupling coarse-mesh CFD with fine-mesh CFD for modeling molten-salt reactors in the Virtual Test Bed (VTB)

The Nuclear Energy Advanced Modeling and Simulation (NEAMS) program aims at developing a simulation tool kit to accelerate the development and deployment of nuclear power technologies. NEAMS multiphysics tools have been designed to provide numerical simulation support for the design and licensing of GEN IV reactors. Pronghorn is NEAMS's coarse-mesh computational fluid dynamics (CFD) tool, which is designed to run 3D core transients in GEN IV reactors at a reduced computational cost. To increase their accuracy, coarse-mesh CFD simulations require calibrated closure coefficients. One way of computing these coefficients is via the Nek5000, NEAMS's high-fidelity CFD tool. This article discusses our current research lines in informing Pronghorn closure coefficients via Nek5000 to enable multiphysics simulations of the core cavity of the molten-salt fast reactor. We present an application in which Pronghorn mixing length turbulent viscosity has been calibrated from Nek5000 simulations. The resulting Pronghorn thermal-hydraulics model is then coupled to Griffin, the NEAMS neutron transport solver, to solve for the steady-state configuration of the molten-salt fast reactor. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2022

Setup and verification of a SCALE/KENO platform for generic FHR benchmark calculations

The work presented in this article is preliminary to downstream analysis of a generic fluoride salt-cooled high-temperature reactor (gFHR) core performed by the University of Tennessee in collaboration with Kairos Power (KP). A Monte Carlo transport model of the publicly available gFHR equilibrium core is developed in SCALE/KENO with multigroup energy treatment. Several output quantities of interest are used to verify the simulations against a benchmark model developed by KP using the continuous energy Monte Carlo code Serpent 2. Good agreement is seen in flux and fission rate profiles with a maximum relative difference of 1.4% and 2.8% respectively. Furthermore, an effective multiplication factor bias of 44 pcm was observed between the two simulations. The fuel temperature reactivity coefficient calculated with SCALE is within uncertainty to the reference model. This verification acts as a publicly reproducible benchmark for the gFHR in SCALE/KENO. A simplified depletion model is also presented where a single fuel pebble is depleted to discharge burnup through the equilibrium core while the equilibrium core is assumed to be invariant. This method produces results that intercept the equilibrium core concentrations in every case, however, an interesting artifact of this particular depletion model is uncovered. The phenomenon is shown to be a fundamental feature of the differential rate equations and inspires questions about how this system behaves when the time evolution of the equilibrium core is considered. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2025

Advection algorithms for quantum neutrino moment transport

Neutrino transport in compact objects is an inherently challenging multidimensional problem. Here, this difficulty is compounded if one includes flavor transformation—an intrinsically quantum phenomenon requiring one to follow the coherence between flavors and thus necessitating the introduction of complex numbers. To reduce the computational burden, simulations of compact objects that include neutrino transport often make use of momentum-angle-integrated moments (the lowest order ones being commonly referred to as the energy density and flux) and these quantities can be generalized to include neutrino flavor, i.e., they become quantum moments. Numerous finite-volume approaches to solving the moment evolution equations for classical neutrino transport have been developed based on solving a Riemann problem at cell interfaces. In this paper we describe our generalization of a Riemann solver for quantum moments, specifically decomposing complex numbers in terms of a (signed) magnitude and phase instead of real and imaginary parts. We then test our new algorithm in numerous cases showing a neutrino fast flavor instability, varying from toy models with analytic solutions to snapshots from neutron star merger simulations. Compared to previous algorithms for neutrino transport with flavor mixing, we find uniformly smaller growth rates of the flavor transformation along with concomitantly larger length-scales, and that the results are a better match with the growth rates seen from multiangle codes.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1993

Calculations of cosmic-ray helium transport in shielding materials

The transport of galactic cosmic-ray helium nuclei and their secondaries through bulk shielding is considered using the straight-ahead approximation to the Boltzmann equation. A data base for nuclear interaction cross sections and secondary particle energy spectra for high-energy light-ion breakup is presented. The importance of the light ions H-2, H-3, and He-3 for cosmic-ray risk estimation is discussed, and the estimates of the fractional contribution to the neutron flux from helium interactions compared with other particle interactions are presented using a 1977 solar minimum cosmic-ray spectrum.

Cucinotta, Francis A.↗

DOE OSTI2023

(U) Average Scattering Angle Cosine Sensitivities

The Evaluated Nuclear Data File (ENDF) provides covariances of the average scattering angle cosine $\overline{μ}$ for 117 nuclides, and the NJOY nuclear data processing code can process these covariances. To propagate them to the uncertainty of a response due to uncertainties in $\overline{μ}$ requires sensitivities of the response with respect to $\overline{μ}$. This report uses NJOY’s definition of $\overline{μ}$ and derives a relationship between the sensitivity of a response to $\overline{μ}$ and the sensitivity of the response to the anisotropic P 1 elastic scattering cross section. This relationship has been implemented in the multigroup neutron sensitivity code SENSMG, which relies on the discrete ordinates code PARTISN for the neutron transport calculations. Kodeli has also calculated sensitivities with respect to $\overline{μ}$ and associated uncertainties. However, the formal derivation of this paper shows that the definition of the sensitivity with respect to $\overline{μ}$ is ambiguous because there are multiple ways to change $\overline{μ}$ by changing scattering cross sections. Here we derive two such ways. Both are implemented in SENSMG. This report is organized as follows. Section II discusses $\overline{μ}$ and its significance. Section III presents the equations used for the sensitivity of a response with respect to $\overline{μ}$, and Section IV discusses how to compute these sensitivities with SENSMG and MCNP6.3. Section V is a fast-spectrum test problem, and Sec. VI is a thermal-spectrum problem. Section VII is a summary. The example problem inputs are listed in the appendix.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1986

Calculations of cosmogenic nuclide production rates in the Earth's atmosphere and their inventories

The production rates of cosmogenic isotopes in the Earth's atmosphere and their resulting terrestrial abundances have been calculated, taking into account both geomagnetic and solar-modulatory effects. The local interstellar flux was assumed to be that of Garcia-Munoz, et al. Solar modulation was accounted for using the heliocentric potential model and expressed in terms of the Deep River neutron monitor count rates. The geomagnetic field was presented by vertical cutoffs calculated by Shea and Smart and the non-vertical cutoffs calculated using ANGRI. The local interstellar particle flux was first modulated using the heliocentric potential field. The modulated cosmic-ray fluxes reaching the earth's orbit then interacted with the geomagnetic field as though it were a high-pass filter. The interaction of the cosmic radiation with the Earth's atmosphere was calculated utilizing the Bolztmann transport equation. Spallation cross sections for isotope production were calculated using the formalism of Silberberg and Tsao and other cross sections were taken from standard sources. Inventories were calculated by accounting from the variation in solar modulation and geomagnetic field strength with time. Results for many isotope, including C-14, Be-7 and Be-10 are in generally good agreement with existing data. The C-14 inventory, for instance, amounts to 1.75/sq cm(e)/s, in excellent agreement with direct estimates.

Obrien, K.↗

DOE OSTI2022

A new moment-based general-relativistic neutrino-radiation transport code: Methods and first applications to neutron star mergers

ABSTRACT We present a new moment-based energy-integrated neutrino transport code for neutron star merger simulations in general relativity. In the merger context, ours is the first code to include Doppler effects at all orders in υ/c, retaining all non-linear neutrino–matter coupling terms. The code is validated with a stringent series of tests. We show that the inclusion of full neutrino–matter coupling terms is necessary to correctly capture the trapping of neutrinos in relativistically moving media, such as in differentially rotating merger remnants. We perform preliminary simulations proving the robustness of the scheme in simulating ab-initio mergers to black hole collapse and long-term neutron star remnants up to ${\sim }70\,$ ms. The latter is the longest dynamical space-time, 3D, general relativistic simulations with full neutrino transport to date. We compare results obtained at different resolutions and using two different closures for the moment scheme. We do not find evidences of significant out-of-thermodynamic equilibrium effects, such as bulk viscosity, on the post-merger dynamics or gravitational wave emission. Neutrino luminosities and average energies are in good agreement with theory expectations and previous simulations by other groups using similar schemes. We compare dynamical and early wind ejecta properties obtained with M1 and with our older neutrino treatment. We find that the M1 results have systematically larger proton fractions. However, the differences in the nucleosynthesis yields are modest. This work sets the basis for future detailed studies spanning a wider set of neutrino reactions, binaries, and equations of state.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1982

Neutron star evolutionary sequences

Detailed numerical calculations which are solutions of the full set of general relativistic equations describing the evolution of a spherical star are presented, for the case of the evolution of neutron stars that are cooling over the central temperatures range of 10 to the 10th to 10 to the 7th K. The effects of nucleon superfluidity in the inner crust and core are included, and models are constructed with and without a pion condensate at high densities. It is found that the localized neutrino cooling which dominates the early evolution of neutron stars is so rapid that heat transport within the star cannot keep pace, and temperature distribution is not isothermal. The residual contraction of the neutron star during the early cooling phase contributes little to the heat budget of the star, and most of the gravitational energy released raises the Fermi energy of the degenerate nucleons. It is concluded that since calculations with and without pion condensate are consistent with the upper limits of current observations, these are not sufficient in distinguishing between the various models of neutron star cooling.

Richardson, M. B.↗