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. By numerically inverting the Tolman-Oppenheimer-Volkov (TOV) equation using an explicitly isospin-dependent parametric Equation of State (EOS) of dense neutron-rich nucleonic matter, a restricted EOS parameter space is established using observational constraints on the radius, maximum mass, tidal deformability and causality condition of neutron stars (NSs). The constraining band obtained for the pressure as a function of energy (baryon) density is in good agreement with that extracted recently by the LIGO+Virgo Collaborations from their improved analyses of the NS tidal deformability in GW170817. Rather robust upper and lower boundaries on nuclear symmetry energies are extracted from the observational constraints up to about twice the saturation density ρ 0 of nuclear matter. More quantitatively, the symmetry energy at 2 ρ 0 is constrained to E sym ( 2 ρ 0 ) = 46 . 9 ± 10 . 1 MeV excluding many existing theoretical predictions scattered between E sym ( 2 ρ 0 ) = 15 and 100 MeV. Moreover, by studying variations of the causality surface where the speed of sound equals that of light at central densities of the most massive neutron stars within the restricted EOS parameter space, the absolutely maximum mass of neutron stars is found to be 2.40 M ⊙ approximately independent of the EOSs used. This limiting mass is consistent with findings of several recent analyses and numerical general relativity simulations about the maximum mass of the possible super-massive remanent produced in the immediate aftermath of GW170817. deformability

The density dependence of nuclear symmetry energy is among the most uncertain parts of the Equation of State (EOS) of dense neutron-rich nuclear matter. It is currently poorly known especially at suprasaturation densities partially because of our poor knowledge about isovector nuclear interactions at short distances. Because of its broad impacts on many interesting issues, pinning down the density dependence of nuclear symmetry energy has been a longstanding and shared goal of both astrophysics and nuclear physics. New observational data of neutron stars including their masses, radii, and tidal deformations since GW170817 have helped improve our knowledge about nuclear symmetry energy, especially at high densities. Based on various model analyses of these new data by many people in the nuclear astrophysics community, while our brief review might be incomplete and biased unintentionally, we learned in particular the following: (1) The slope parameter L of nuclear symmetry energy at saturation density ρ0 of nuclear matter from 24 new analyses of neutron star observables was about L≈57.7±19 MeV at a 68% confidence level, consistent with its fiducial value from surveys of over 50 earlier analyses of both terrestrial and astrophysical data within error bars. (2) The curvature Ksym of nuclear symmetry energy at ρ0 from 16 new analyses of neutron star observables was about Ksym≈−107±88 MeV at a 68% confidence level, in very good agreement with the systematics of earlier analyses. (3) The magnitude of nuclear symmetry energy at 2ρ0, i.e., Esym(2ρ0)≈51±13 MeV at a 68% confidence level, was extracted from nine new analyses of neutron star observables, consistent with the results from earlier analyses of heavy-ion reactions and the latest predictions of the state-of-the-art nuclear many-body theories. (4) While the available data from canonical neutron stars did not provide tight constraints on nuclear symmetry energy at densities above about 2ρ0, the lower radius boundary R2.01=12.2 km from NICER’s very recent observation of PSR J0740+6620 of mass 2.08±0.07M⊙ and radius R=12.2–16.3 km at a 68% confidence level set a tight lower limit for nuclear symmetry energy at densities above 2ρ0. (5) Bayesian inferences of nuclear symmetry energy using models encapsulating a first-order hadron–quark phase transition from observables of canonical neutron stars indicated that the phase transition shifted appreciably both L and Ksym to higher values, but with larger uncertainties compared to analyses assuming no such phase transition. (6) The high-density behavior of nuclear symmetry energy significantly affected the minimum frequency necessary to rotationally support GW190814’s secondary component of mass (2.50–2.67) M⊙ as the fastest and most massive pulsar discovered so far. Overall, thanks to the hard work of many people in the astrophysics and nuclear physics community, new data of neutron star observations since the discovery of GW170817 have significantly enriched our knowledge about the symmetry energy of dense neutron-rich nuclear matter.

Given an Equation of State (EOS) for neutron star (NS) matter, there is a unique mass–radius sequence characterized by a maximum mass MNSmax at radius R max. We first show analytically that the MNSmax and R max scale linearly with two different combinations of the NS central pressure P c and energy density ε c, by dissecting perturbatively the dimensionless Tolman–Oppenheimer–Volkoff (TOV) equations governing NS internal variables. The scaling relations are then verified via 87 widely used and rather diverse phenomenological as well as 17 microscopic NS EOSs with/without considering hadron–quark phase transitions and hyperons, by solving numerically the original TOV equations. The EOS of the densest NS matter allowed before it collapses into a black hole is then obtained. Using the universal MNSmax and R max scalings and Neutron Star Interior Composition Explorer and XMM-Newton mass–radius observational data for PSR J0740+6620, a very narrow constraining band on the NS central EOS is extracted directly from the data for the first time, without using any specific input EOS model.

HighlightsFeedback-regulation system is established between single Pt sites and MoS2 supportsThe Pt1-MoS2-def can expand the electron transfer path from single Pt sites to whole Pt-MoS2 supports in SO2 gas atmosphere.The Pt1-MoS2-def sensors exhibit high SO2 responses and extremely low limit of detection (3.14% to 500 ppb SO2) at room temperature.The Pt1-MoS2-def sensors array can realize real-time monitoring of SO2 for plant growth.Reasonably constructing an atomic interface is pronouncedly essential for surface-related gas-sensing reaction. Herein, we present an ingenious feedback-regulation system by changing the interactional mode between single Pt atoms and adjacent S species for high-efficiency SO2 sensing. We found that the single Pt sites on the MoS2 surface can induce easier volatilization of adjacent S species to activate the whole inert S plane. Reversely, the activated S species can provide a feedback role in tailoring the antibonding-orbital electronic occupancy state of Pt atoms, thus creating a combined system involving S vacancy-assisted single Pt sites (Pt-Vs) to synergistically improve the adsorption ability of SO2 gas molecules. Furthermore, in situ Raman, ex situ X-ray photoelectron spectroscopy testing and density functional theory analysis demonstrate the intact feedback-regulation system can expand the electron transfer path from single Pt sites to whole Pt-MoS2 supports in SO2 gas atmosphere. Equipped with wireless-sensing modules, the final Pt1-MoS2-def sensors array can further realize real-time monitoring of SO2 levels and cloud-data storage for plant growth. Such a fundamental understanding of the intrinsic link between atomic interface and sensing mechanism is thus expected to broaden the rational design of highly effective gas sensors.

We investigate how simultaneous mass and radius measurements of massive neutron stars can help constrain the properties of dark matter possibly admixed in them. Within a fermionic dark matter model that interacts only through gravitation, along with a well-constrained nuclear matter equation of state, we show that the simultaneous mass and radius measurement of PSRJ0740+6620 reduces the uncertainty of dark matter central energy density by more than 50% compared to the results obtained from using the two observables independently, while other dark matter parameters remain unconstrained. Additionally, we find that the dark matter fraction fD should be smaller than 2% when constrained by the observed neutron star maximum mass alone, and it could be even smaller than 0.3% with the simultaneous measurement of mass and radius, supporting the conclusion that only a small amount of dark matter exists in dark matter admixed neutron stars (DANSs).

The density dependence of nuclear symmetry energy Esym(ρ) remains the most uncertain aspect of the equation of state (EOS) of supradense neutron-rich nucleonic matter. Utilizing an isospin-dependent parameterization of the nuclear EOS, we investigate the implications of the observational crustal fraction of the neutron star (NS) moment of inertia ΔI/I for the Esym(ρ). We find that symmetry energy parameters significantly influence the ΔI/I, while the EOS of symmetric nuclear matter has a negligible effect. In particular, an increase in the slope L and skewness Jsym of symmetry energy results in a larger ΔI/I, whereas an increase in the curvature Ksym leads to a reduction in ΔI/I. Moreover, the ΔI/I is shown to have the potential for setting a lower limit of symmetry energy at densities exceeding 3ρ0, particularly when L is constrained to values less than 60 MeV, thereby enhancing our understanding of supradense NS matter.

. Determining the Equation of State (EOS) of dense neutron-rich nuclear matter is a shared goal of both nuclear physics and astrophysics. Except possible phase transitions, the density dependence of nuclear symmetry E sym ( ρ ) is the most uncertain part of the EOS of neutron-rich nucleonic matter especially at supra-saturation densities. Much progresses have been made in recent years in predicting the symmetry energy and understanding why it is still very uncertain using various microscopic nuclear many-body theories and phenomenological models. Simultaneously, significant progresses have also been made in probing the symmetry energy in both terrestrial nuclear laboratories and astrophysical observatories. In light of the GW170817 event as well as ongoing or planned nuclear experiments and astrophysical observations probing the EOS of dense neutron-rich matter, we review recent progresses and identify new challenges to the best knowledge we have on several selected topics critical for understanding astrophysical effects of the nuclear symmetry energy.

The Tolman–Oppenheimer–Volkoff (TOV) equations govern the radial evolution of pressure and energy density in static neutron stars (NSs) in hydrodynamical equilibrium. Using the reduced pressure and energy density with respect to the NS central energy density, the original TOV equations can be recast into dimensionless forms. While the traditionally used integral approach for solving the original TOV equations require an input nuclear Equation of State (EOS), the dimensionless TOV equations can be anatomized by using the reduced pressure and energy density as polynomials of the reduced radial coordinate without using any input nuclear EOS. It has been shown in several of our recent works that interesting and novel perspectives about NS core EOS can be extracted directly from NS observables by using the latter approach. Our approach is based on i ntrinsic and p erturbative a nalyses of the d imensionless (IPAD) TOV equations (IPAD-TOV). In this review article, we first discuss the length and energy density scales of NSs as well as the dimensionless TOV equations for scaled variables and their perturbative solutions near NS cores. We then review several new insights into NS physics gained from solving perturbatively the scaled TOV equations. Whenever appropriate, comparisons with the traditional approach from solving the original TOV equations will be made. In particular, we first show that the nonlinearity of the TOV equations basically excludes a linear EOS for dense matter in NS cores. We then show that perturbative analyses of the scaled TOV equations enable us to reveal novel scalings of the NS mass, radius and the compactness with certain combinations of the NS central pressure and energy density. Thus, observational data on either mass, radius or compactness can be used to constrain directly the core EOS of NS matter independent of the still very uncertain nuclear EOS models. As examples, the EOS of the densest visible matter in our Universe before the most massive neutron stars collapse into black holes (BHs) as well as the central EOS of a canonical or a 2.1 solar mass NS are extracted without using any nuclear EOS model. In addition, we show that causality in NSs sets an upper bound of about 0.374 for the ratio of pressure over energy density and correspondingly a lower limit for trace anomaly in supra-dense matter. We also demonstrate that the strong-field gravity plays a fundamental role in extruding a peak in the density/radius profile of the speed of sound squared (SSS) in massive NS cores independent of the nuclear EOS. Finally, some future perspectives of NS research using the new approach reviewed here by solving perturbatively the dimensionless TOV equations are outlined.

Using a meta model for nuclear Equation of State (EOS) with its parameters constrained by astrophysical observations and terrestrial nuclear experiments, we examine effects of nuclear EOS especially its symmetry energy E sym ( ρ )  term on the speed of sound squared C s 2 ( ρ ) and the critical density ρ t where C s 2 ( ρ t ) vanishes (indicating the onset of spinodal decomposition) in both dense neutron-rich nucleonic matter relevant for relativistic heavy-ion collisions and the cold n + p + e + μ matter in neutron stars at β -equilibrium. Unlike in nucleonic matter with fixed values of the isospin asymmetry δ , in neutron stars with a density dependent isospin profile δ ( ρ ) determined consistently by the β -equilibrium and charge neutrality conditions, the C s 2 ( ρ ) almost always show a peak and then vanishes at ρ t . The latter strongly depends on the high-density behavior of E sym ( ρ )  if the skewness parameter J 0 characterizing the stiffness of high-density symmetric nuclear matter (SNM) EOS is not too far above its currently known most probable value of about - 190 MeV inferred from recent Bayesian analyses of neutron star observables. Moreover, in the case of having a super-soft E sym ( ρ )  that is decreasing with increasing density above about twice the saturation density of nuclear matter, the ρ t is significantly lower than the density where the E sym ( ρ )  vanishes (indicating the onset of isospin-separation instability and pure neutron matter formation) in neutron star cores.

In this talk, we report results of our recent studies to delineate effects of the tensor force on the density dependence of nuclear symmetry energy within phenomenological models. The tensor force active in the isosinglet neutron roton interaction channel leads to appreciable depletion/population of nucleons below/above the Fermi surface in the single-nucleon momentum distribution in cold symmetric nuclear matter (SNM). We found that as a consequence of the high momentum tail in SNM the kinetic part of the symmetry energy Ekinsym(ρ) is significantly below the well-known Fermi gas model prediction of approximately 125(ρ/ρ0)2/3. With about 15% nucleons in the high momentum tail as indicated by the recent experiments at J-Lab by the CLAS Collaboration, the Ekinsym(ρ) is negligibly small. It even becomes negative when more nucleons are in the high momentum tail in SNM. These features have recently been confirmed by three independent studies based on the state-of-the-art microscopic nuclear many-body theories. In addition, we also estimate the second-order tensor force contribution to the potential part of the symmetry energy. Implications of these findings in extracting information about nuclear symmetry energy from nuclear reactions are discussed briefly.