Neutron stars host the strongest magnetic fields that we know of in the Universe. Their magnetic fields are the main means of generating their radiation, either magnetospheric or through the crust. Moreover, the evolution of the magnetic field has been intimately related to explosive events of magnetars, which host strong magnetic fields, and their persistent thermal emission. The evolution of the magnetic field in the crusts of neutron stars has been described within the framework of the Hall effect and Ohmic dissipation. Yet, this description is limited by the fact that the Maxwell stresses exerted on the crusts of strongly magnetised neutron stars may lead to failure and temperature variations. In the former case, a failed crust does not completely fulfil the necessary conditions for the Hall effect. In the latter, the variations of temperature are strongly related to the magnetic field evolution. Finally, sharp gradients of the star's temperature may activate battery terms and alter the magnetic field structure, especially in weakly magnetised neutron stars. In this review, we discuss the recent progress made on these effects. We argue that these phenomena are likely to provide novel insight into our understanding of neutron stars and their observable properties.
An accretion outburst onto a neutron star transient heats the neutron star's crust out of thermal equilibrium with the core. After the outburst the crust thermally relaxes toward equilibrium with the neutron star core and the surface thermal emission powers the quiescent X-ray light curve. Crust cooling models predict that thermal equilibrium of the crust will be established $\approx 1000 \, \mathrm{d}$ into quiescence. Recent observations of the cooling neutron star transient MXB 1659-29, however, suggest that the crust did not reach thermal equilibrium with the core on the predicted timescal…
Neutron stars cooling after sustained accretion outbursts provide unique information about the neutron star crust and underlying dense matter. Comparisons between astronomical observations of these cooling transients and model calculations of neutron star crust cooling have frequently been used to constrain neutron star properties such as the mass, radius, crust composition, and presence of nuclear pasta. These calculations often use a fixed pressure at which the crust-core transition happens, though this quantity depends on the dense matter equation of state. We demonstrate that varying the c…
We study thermal relaxation in a neutron star after internal heating events (outbursts) in the crust. We consider thin and thick spherically symmetric heaters, superfluid and non-superfluid crusts, stars with open and forbidden direct Urca processes in their cores. In particular, we analyze long-term thermal relaxation after deep crustal heating produced by nuclear transformations in fully or partly accreted crusts of transiently accreting neutron stars. This long-term relaxation has a typical relaxation time and an overall finite duration time for the crust to thermally equilibrate with the c…
Pulsars undergoing crustquake release strain energy, which can be absorbed in a small region inside the inner crust of the star and excite the free superfluid neutrons therein. The scattering of these neutrons with the surrounding pinned vortices may unpin a large number of vortices and effectively reduce the pinning force on vortex lines. Such unpinning by neutron scattering can produce glitches for Crab like pulsars and Vela pulsar of size in the range $\sim 10^{-8} - 10^{-7}$, and $\sim 10^{-9} - 10^{-8}$, respectively. Although we discuss here the crustquake initiated excitation, the propo…
We study breaking stress of deformed Coulomb crystals in a neutron star crust, taking into account electron plasma screening of ion-ion interaction; calculated breaking stress is fitted as a function of electron screening parameter. We apply the results for analyzing torsional oscillation modes in the crust of a non-magnetic star. We present exact analytic expression for the fundamental frequencies of such oscillations and show that the frequencies of all torsional oscillations are insensitive to the presence of the outer neutron star crust. The results can be useful in theoretical modeling of…
We study the effect of physics input on thermal evolution of neutron stars in soft X-ray transients (SXTs). In particular, we consider different modern models of the sources of deep crustal heating during accretion episodes and the effects brought about by impurities embedded in the crust during its formation. We simulate thermal structure and evolution of episodically accreting neutron stars under different assumptions on the crust composition and on the distribution of heat sources and impurities. For the nonaccreted crust, we consider the nuclear charge fluctuations that arise at crust form…
The evolution of the magnetic field in neutron star crusts because of the Hall effect has received significant attention over the last two decades, which is strongly justified because of the dominance of this effect in highly magnetised neutron stars. However, the applicability of the Hall effect is based on the assumption that the crust does not fail and sustains its rigidity. This assumption can be violated for substantially strong magnetic fields. If this is the case, the evolution of the magnetic field is described by a different set of equations, which include the effects of a non-rigid c…
It is generally accepted that the Coulomb crystal model can be used to describe matter in the neutron star crust. In [1] we study the properties of deformed Coulomb crystals and how their stability depends on the polarization of the electron background; the breaking stress in the crust $σ_{\max}$ at zero temperature was calculated based on the analysis of the electrostatic energy and the phonon spectrum of the Coulomb crystal. In this paper, I briefly discuss the influence of zero-point and thermal contributions on $σ_{\max}$.
The crust of accreting neutron stars plays a central role in many different observational phenomena. In these stars, heavy elements produced by H-He burning in the rapid proton capture (rp-) process continually freeze to form new crust. In this paper, we explore the expected composition of the solid phase. We first demonstrate using molecular dynamics that two distinct types of chemical separation occur, depending on the composition of the rp-process ashes. We then calculate phase diagrams for three-component mixtures and use them to determine the allowed crust compositions. We show that, for …
Using a theoretical model, we track the thermal evolution of a cooling neutron star crust after an accretion induced heating period with the goal of constraining the crustal parameters. We present for the first time a crust cooling model $-\text{ } NSCool\text{ } -$ that takes into account detailed variability during the full outburst based on the observed light curve. We apply our model to KS 1731-260. The source was in outburst for $\sim$12 years during which it was observed to undergo variations on both long (years) and short (days-weeks) timescales. Our results show that KS 1731-260 does n…
In accreting neutron star X-ray transients, the neutron star crust can be substantially heated out of thermal equilibrium with the core during an accretion outburst. The observed subsequent cooling in quiescence (when accretion has halted) offers a unique opportunity to study the structure and thermal properties of the crust. Initially crust cooling modelling studies focussed on transient X-ray binaries with prolonged accretion outbursts (> 1 year) such that the crust would be significantly heated for the cooling to be detectable. Here we present the results of applying a theoretical model to …
Treatment of the vortex motion in the superfluids of the inner crust and the outer core of neutron stars is a key ingredient in modeling a number of pulsar phenomena, including glitches and magnetic field evolution. After recalculating the microscopic vortex velocity in the inner crust, we evaluate the velocity for the vortices in the outer core for the first time. The vortex motion between pinning sites is found to be substantially faster in the inner crust than in the outer core, $v_0^{\rm crust} \sim 10^{7}\mbox{\cms} \gg v_0^{\rm core} \sim 1\mbox{\cms}$. One immediate result is that vorte…
It is assumed that accreting neutron stars (NSs) in LMXBs are heated due to the compression of the existing crust by the accreted matter which gives rise to nuclear reactions in the crust. It has been shown that most of the energy is released deep in the crust by pycnonuclear reactions involving low-Z elements. We discuss if NSs in very-faint X-ray transients (VFXTs; those which have peak X-ray luminosities < 1E36 erg/s) can be used to test this model. Unfortunately we cannot conclusively answer this because of the large uncertainties in our estimates of the accretion rate history of those VFX…
We consider the dynamics of a mechanical failure induced by a shear stress in a strongly magnetized neutron-star crust. We show that even if the elastic properties of the crust allow the creation of a shear crack, the strongly sheared magnetic field around the crack leads to a back-reaction from the Lorentz force which does not allow large relative displacement of the crack surfaces. Instead, the global evolution of the crack proceeds on a slow resistive time scale, and is unable to release any substantial mechanical energy. Our calculations demostrate that for {\it some} magnetic-field config…
We revisit the mechanism of vortex unpinning caused by the neutron-vortex scattering \cite{prad1} in the inner crust of a pulsar. The strain energy released by the crustquake is assumed to be absorbed in some part of the inner crust and causes pair-breaking quasi-neutron excitations from the existing free neutron superfluid in the bulk of the inner crust. The scattering of these quasi-neutrons with the vortex core normal neutrons unpins a large number of vortices from the thermally affected regions and results in pulsar glitches. We consider the geometry of a cylindrical shell of the affected …
A quasi-statistical equilibrium model is constructed to simulate the multicomponent composition of the crust of an accreting neutron star. The ashes of rp-process nucleosynthesis are driven by accretion through a series of electron captures, neutron emissions, and pycnonuclear fusions up to densities near the transition between the neutron star crust and core. A liquid droplet model which includes nuclear shell effects is used to provide nuclear masses far from stability. Reaction pathways are determined consistently with the nuclear mass model. The nuclear symmetry energy is an important unce…
Despite the absence of viscous drag, the neutron superfluid permeating the inner crust of a neutron star can still be strongly coupled to nuclei due to non-dissipative entrainment effects. Neutron superfluidity and entrainment have been systematically studied in all regions of the inner crust of a cold non-accreting neutron star in the framework of the band theory of solids. It is shown that in the intermediate layers of the inner crust a large fraction of "free" neutrons are actually entrained by the crust. The results suggest that a revision of the interpretation of many observable astrophys…
The neutron superfluid permeating the inner crust of mature neutron stars is expected to play a key role in various astrophysical phenomena like pulsar glitches. Despite the absence of viscous drag, the neutron superfluid can still be coupled to the solid crust due to non-dissipative entrainment effects. Entrainment challenges the interpretation of pulsar glitches and suggests that a revision of the interpretation of other observed neutron-star phenomena might be necessary.
Neutron stars in low-mass binary systems are subject to accretion. The common assumption in studying the properties of the neutron star crust is the fully accreted crust approximation. However, observations of some X-ray transient sources indicate that the original crust has not been completely replaced by accreted material, but is partly composed of the compressed original crust. A two-part (or hybrid) crust made of the original crust that is compressed and of the accreted material crashing onto it was reconstructed as a function of the accretion stage. The differences in the composition and …
Nuclear reactions heat and cool the crust of accreting neutron stars and need to be understood to interpret observations of X-ray bursts and of long-term cooling in transiently accreting systems. It was recently suggested that previously neglected neutron transfer reactions may play a significant role in the nuclear processes. We present results from full nuclear network calculations that now include these reactions and determine their impact on crust composition, crust impurity, heating, and cooling. We find that a large number of neutron transfer reactions indeed occur and impact crust model…
The Vlasov equation is used to determine the dispersion relation for the eigenmodes of magnetized nuclear and neutral stellar matter, taking into account the anomalous magnetic moment of nucleons. The formalism is applied to the determination of the dynamical spinodal section, a quantity that gives a good estimation of the crust-core transition in neutron stars. We study the effect of strong magnetic fields, of the order of $10^{15}-10^{17}$ G, on the extension of the crust of magnetized neutron stars. The dynamical instability region of neutron-proton-electron ($npe$) matter at subsaturation …
We investigate the effects of the symmetry energy on nuclear pasta phases and the crust-core transition in neutron stars. We employ the relativistic mean-field approach and the coexisting phases method to study the properties of pasta phases presented in the inner crust of neutron stars. It is found that the slope of the nuclear symmetry energy at saturation density plays an important role in the crust-core transition and pasta phase properties. The correlation between the symmetry energy slope and the crust-core transition density obtained in this study is consistent with those obtained by ot…
We study equation of state (EOS) of an accreting neutron star crust. Usually, such EOS is obtained assuming (implicitly) that the free (unbound) neutrons and nuclei in the inner crust move together. We argue, that this assumption violates the condition $μ_n^\infty={\rm const}$, required for hydrostatic (and diffusion) equilibrium of unbound neutrons ($μ^\infty_n$ is the redshifted neutron chemical potential). We construct a new EOS respecting this condition, working in the compressible liquid-drop approximation. We demonstrate that it is close to the catalyzed EOS in most part of the inner cru…
We study the effects of high magnetic fields on the structure and on the geometry of the crust in neutron stars. We find that the crust geometry is substantially modified by the magnetic field inside the star. We build stationary and axis-symmetric magnetized stellar models by using well-known equations of state to describe the neutron star crust, namely the Skyrme model (Sky) for the inner crust and the Baym, Pethick, and Sutherland (BPS) equation of state for the outer crust. We show that the magnetic field has a dual role, contributing to the crust deformation via the electromagnetic intera…
Nonequilibrium conditions imposed by neutrino cooling through the liquid-solid transition lead to disorder in the solid crust of neutron stars. Disorder reduces the superfluid fraction, $ρ_s/ρ$, at densities above that of neutron drip, $ρ_d \approx 4\times 10^{11}\,g/cm^3$. For an amorphous solid crust the suppression of $ρ_s$ is small, except in the highest density regions of the crust. In contrast to the strong reduction in neutron conduction predicted for coherent Bragg scattering in a crystalline crust, the disordered solid crust supports sufficient neutron superfluid density to account fo…
The observed thermal relaxation of transiently accreting neutron stars during quiescence periods in low-mass X-ray binaries suggests the existence of unknown heat sources in the shallow layers of neutron-star crust. Making use of existing experimental nuclear data, we estimate the maximum possible amount of heat that can be deposited in the outer crust of an accreting neutron star due to electron captures and pycnonuclear fusion reactions triggered by the burial of X-ray burst ashes.
Even though the "free" neutrons in the inner crust of a neutron star are superfluid, they are still strongly coupled to nuclei due to non-dissipative entrainment effects. These effects have been systematically studied in all regions of the inner crust in the framework of the band theory of solids. Using concepts from solid-state physics, it is shown that the density of conduction neutrons, i.e. neutrons that are effectively "free", can be much smaller than the density of unbound neutrons (by an order of magnitude in some layers) due to Bragg scattering. These results suggest that a revision of…
Some neutron star low-mass X-ray binaries have very long outbursts (lasting several years) which can generate a significant amount of heat in the neutron star crust. After the system has returned to quiescence, the crust then thermally relaxes. This provides a rare opportunity to study the thermal properties of neutron star crusts, putting constraints on the thermal conductivity and hence the structure and composition of the crust. KS 1731-260 is one of only four systems where this crustal cooling has been observed. Here, we present a new Chandra observation of this source approximately 8 year…
We explore the spectrum of low-energy collective excitations in the crust of a neutron star, especially in the inner region where neutron-proton clusters are immersed in a sea of superfluid neutrons. The speeds of the different modes are calculated systematically from the nuclear energy density functional theory using a Skyrme functional fitted to essentially all experimental atomic mass data.
We study the effects of the symmetry energy on the neutron drip density and properties of nuclei in neutron star crusts. The nonuniform matter around the neutron drip point is calculated by using the Thomas--Fermi approximation with the relativistic mean-field model. The neutron drip density and the composition of the crust are found to be correlated with the symmetry energy and its slope. We compare the self-consistent Thomas--Fermi approximation with other treatments of surface and Coulomb energies and find that these finite-size effects play an essential role in determining the equilibrium …
Based on the formalism by Kaminker et al. (Astron. Astrophys. 343 (1999) 1009) we derive an analytic approximation for neutrino-pair bremsstrahlung emissivity due to scattering of electrons by atomic nuclei in the neutron star crust of any realistic composition. The emissivity is expressed through generalized Coulomb logarithm which we fit by introducing an effective potential of electron-nucleus scattering. In addition, we study the conditions at which the neutrino bremsstrahlung in the crust is affected by strong magnetic fields. The results can be applied for modelling of many phenomena in …
Observed temperatures of transiently accreting neutron stars in the quiescent state are generally believed to be supported by deep crustal heating, associated with non-equilibrium exothermic reactions in the crust. Traditionally, these reactions are studied by considering nuclear evolution governed by compression of the accreted matter. Here we show that this approach has a basic weakness, that is that in some regions of the inner crust the conservative forces, applied for matter components (nuclei and neutrons), are not in mechanical equilibrium. In principle the force balance can be restored…
Nuclear pairing is studied both in atomic nuclei and in neutron-star crusts in the unified framework of the energy-density functional theory using generalized Skyrme functionals complemented with a local pairing functional obtained from many-body calculations in homogeneous nuclear matter using realistic forces.
Crustquake events may be connected with both rapid spin-up `glitches' within the regular slowdown of neutron stars, and high-energy magnetar flares. We argue that magnetic field decay builds up stresses in a neutron star's crust, as the elastic shear force resists the Lorentz force's desire to rearrange the global magnetic-field equilibrium. We derive a criterion for crust-breaking induced by a changing magnetic-field configuration, and use this to investigate strain patterns in a neutron star's crust for a variety of different magnetic-field models. Universally, we find that the crust is most…
The limiting rotational frequency of neutron stars may be determined by the strength of their crusts. As a star spins up from accretion, centrifugal forces will cause the crust to fail. If the crust breaks unevenly, a rotating mass quadrupole moment will radiate gravitational waves (GW). This radiation can prevent further spin up and may be a promising source for continuous GW searches. We calculate that for a breaking strain (strength) of neutron star crust that is consistent with molecular dynamics simulations, the crust may fail at rotational frequencies in agreement with observations.
A simple description of superfluid hydrodynamics in the inner crust of a neutron star is given. Particular attention is paid to the effect of the lattice of nuclei on the properties of the superfluid neutrons, and the effects of entrainment, the fact that some fraction of the neutrons are locked to the motion of the protons in nuclei.
We use the effective relativistic mean-field (E-RMF) model to study the crustal properties of the neutron star. The unified equations of state (EoS) are constructed using recently developed E-RMF parameter sets, such as FSUGarnet, IOPB-I, and G3. The outer crust composition is determined using the atomic mass evaluation 2020 data [Chinese Physics C {\bf 45}, 030002 (2021)] along with the available Hartree-Fock-Bogoliubov mass models [Phys. Rev. C {\bf 88}, 024308 (2013)] for neutron-rich nuclei. The structure of the inner crust is estimated by performing the compressible liquid drop model calc…
By means of bootstrap technique, we perform a full error analysis on the Duflo-Zucker mass model. We illustrate the impact of such study on the predicted chemical composition of the outer crust of a non-accreting neutron star. We define an existence probability for each nuclear species as a function of the depth of the crust. We observe that, due to statistical uncertainties, instead of having a well defined transition between two successive layers, we have a mixture of two species.
I construct a model of the inner crust of neutron stars using interactions from chiral effective field theory (EFT) in order to calculate its equation of state (EOS), shear properties, and the spectrum of crustal shear modes. I systematically study uncertainties associated with the nuclear physics input, the crust composition, and neutron entrainment, and estimate their impact on crustal shear properties and the shear-mode spectrum. I find that the uncertainties originate mainly in two sources: The neutron-matter EOS and neutron entrainment. I compare the spectrum of crustal shear modes to obs…
Thanks to recent measurements of tidal deformability and radius, the nuclear equation of state and structure of neutron stars are now better understood. Here, we show that through resonant tidal excitations in a binary inspiral, the neutron crust generically undergoes elastic-to-plastic transition, which leads to crust heating and eventually meltdown. This process could induce $\sim \mathcal{O}(0.1)$ phase shift in the gravitational waveform. Detecting the timing and induced phase shift of this crust meltdown will shed light on the crust structure, such as the core-crust transition density, wh…
To make best use of multi-faceted astronomical and nuclear data-sets, probability distributions of neutron star models that can be used to propagate errors consistently from one domain to another are required. We take steps toward a consistent model for this purpose, highlight where model inconsistencies occur and assess the resulting model uncertainty. Using two distributions of nuclear symmetry energy parameters - one uniform, the other based on pure neutron matter theory, we prepare two ensembles of neutron star inner crust models. We use an extended Skyrme energy-density functional within …
Giant pulsar frequency glitches as detected in the emblematic Vela pulsar have long been thought to be the manifestation of a neutron superfluid permeating the inner crust of a neutron star. However, this superfluid has been recently found to be entrained by the crust, and as a consequence it does not carry enough angular momentum to explain giant glitches. The extent to which pulsar-timing observations can be reconciled with the standard vortex-mediated glitch theory is studied considering the current uncertainties on dense-matter properties. To this end, the crustal moment of inertia of glit…
Calculations of the effects of band structure on the neutron superfluid density in the crust of neutron stars made under the assumption that the effects of pairing are small [N. Chamel, Phys. Rev. C 85, 035801 (2012)] lead to moments of inertia of superfluid neutrons so small that the crust alone is insufficient to account for the magnitude of neutron star glitches. Inspired by earlier work on ultracold atomic gases in an optical lattice, we investigate fermions with attractive interactions in a periodic lattice in the mean-field approximation. The effects of band structure are suppressed when…
The equilibrium properties of the outer crust of cold nonaccreting magnetars (i.e. neutron stars endowed with very strong magnetic fields) are studied using the latest experimental atomic mass data complemented with a microscopic atomic mass model based on the Hartree-Fock-Bogoliubov method. The Landau quantization of electron motion caused by the strong magnetic field is found to have a significant impact on the composition and the equation of state of crustal matter. It is also shown that the outer crust of magnetars could be much more massive than that of ordinary neutron stars.
The temperature in the crust of an accreting neutron star, which comprises its outermost kilometer, is set by heating from nuclear reactions at large densities, neutrino cooling, and heat transport from the interior. The heated crust has been thought to affect observable phenomena at shallower depths, such as thermonuclear bursts in the accreted envelope. Here we report that cycles of electron capture and its inverse, $β^-$ decay, involving neutron-rich nuclei at a typical depth of about 150 m, cool the outer neutron star crust by emitting neutrinos while also thermally decoupling the surface …
Magnetic field evolution in neutron-star crusts is driven by the Hall effect and Ohmic dissipation, for as long as the crust is sufficiently strong to absorb Maxwell stresses exerted by the field and thus make the momentum equation redundant. For the strongest neutron-star fields, however, stresses build to the point of crustal failure, at which point the standard evolution equations are no longer valid. Here, we study the evolution of the magnetic field of the crust up to and beyond crustal failure, whence the crust begins to flow plastically. We perform global axisymmetric evolutions, explor…
We present a new mechanism for deep crustal heating in accreting neutron stars. Charged pions ($π^+$) are produced in nuclear collisions on the neutron star surface during active accretion and upon decay they provide a flux of neutrinos into the neutron star crust. For massive and/or compact neutron stars, neutrinos deposit $\approx 1\textrm{--} 2 \, \mathrm{MeV}$ of heat per accreted nucleon into the inner crust. The strength of neutrino heating is comparable to the previously known sources of deep crustal heating, such as from pycnonuclear fusion reactions, and is relevant for studies of coo…
With the improved accuracy of neutron star observational data, it is necessary to derive new equation of state where the crust and the core are consistently calculated within a unified approach. For this purpose we describe non-uniform matter in the crust of neutron stars employing a compressible liquid-drop model, where the bulk and the neutron fluid terms are given from the same model as the one describing uniform matter present in the core. We then generate a set of fifteen unified equations of state for cold catalyzed neutron stars built on realistic modelings of the nuclear interaction, w…
The neutron star crustal EoS and transition point properties are computed within a unified meta-modeling approach. A Bayesian approach is employed including two types of filters: bulk nuclear properties are controlled from low density effective field theory (EFT) predictions as well as the present knowledge from nuclear experiments, while the surface energy is adjusted on experimental nuclear masses. Considering these constraints, a quantitative prediction of crustal properties can be reached with controlled confidence intervals and increased precision with respect to previous calculations: {$…
We analyze how the crust equation of state affects several neutron star properties and how it impacts on possible constraints inferred from astrophysical observations. Using three distinct crusts, we generate three sets of model-independent equations of state describing stellar matter from a Taylor expansion around saturation density. The equations of state are thermodynamically consistent, causal, and compatible with astrophysical observations. The relations between the tidal deformability $Λ$ and compactness $C$, Love number $k_2$ and radius of neutron star with mass $M$ are studied, and the…
We review the nuclear and condensed matter physics underlying the thermal and transport properties of the neutron star inner crust. These properties play a key role in interpreting transient phenomena such as thermal relaxation in accreting neutron stars, superbursts, and magnetar flares. We emphasize simplifications that occur at low temperature where the inner crust can be described in terms of electrons and collective excitations. The heat conductivity and heat capacity of the solid and superfluid phase of matter is discussed in detail and we emphasize its role in interpreting observations …
A compressible liquid-drop approach adjusted to uniform matter many-body calculations based on chiral EFT interactions and to the experimental nuclear masses is used to investigate the neutron star crust properties. Eight chiral EFT hamiltonians and a representative phenomenological force (SLy4) are confronted. We show that some properties of the crust, e.g. clusters mass, charge, and asymmetry, are mostly determined by symmetric matter properties close to saturation density and are therefore mainly constrained by experimental nuclear masses, while other properties, e.g., energy per particle, …
We consider how superfluidity of dripped neutrons in the crust of a neutron star affects the frequencies of the crust's fundamental torsional oscillations. A nonnegligible superfluid part of dripped neutrons, which do not comove with nuclei, act to reduce the enthalpy density and thus enhance the oscillation frequencies. By assuming that the quasi-periodic oscillations observed in giant flares of soft gamma repeaters arise from the fundamental torsional oscillations and that the mass and radius of the neutron star is in the range of 1.4 < M/M_SUN < 1.8 and 10 km < R < 14 km, we constrain the d…
