The Bulk Properties of Isolated Neutron Stars Inferred from the Gravitational Redshift Measurements

The measurements of the bulk properties of most isolated neutron stars (INSs) are challenging tasks. Tang et al. have developed a new method, based on the equation of state (EoS) of neutron star (NS) material constrained by the observational data, to infer the gravitational masses of a few INSs whose gravitational redshifts are available. However, in that work, the authors only considered the constraints on the EoS from nuclear experiments/theories and the gravitational wave data of GW170817; the possible phase transition has not been taken into account. In this work, we adopt three EoS models (including the one incorporates a first-order strong phase transition) that are constrained by the latest multimessenger NS data, including in particular the recent mass–radius measurements of two NSs by Neutron Star Interior Composition Explorer, to update the estimation of the gravitational masses of RBS 1223, RX J0720.4-3125, and RX J1856.5-3754. In comparison to our previous approach, the new constraints are tighter, and the gravitational masses are larger by about 0.1M ⊙. All the inferred gravitational masses are within the range of the NS masses measured in other ways. We have also calculated the radius, tidal-deformability, and moment of inertia of these sources. The inclusion of the first-order strong phase transition has little influence on modifying the results.


INTRODUCTION
The mass determination of compact stars, the leftover products of stellar death, is fundamental in the researches of astronomy.As one of the basic quantities one can measure, the mass of neutron star (NS) may enable us to reveal the formation and evolution mechanism of stars, since different progenitors and evolution channels can leave different imprints on their final masses.For example, Kiziltan et al. (2013) found that the mass distribution of NSs in NS-white dwarf (NSWD) systems is peaked at heavier mass than that of NSs in binary NS systems, indicating significant mass accretion of NSs in NSWD via the so-called recycling process.Meanwhile, mass distribution of NSs can also be used to probe the currently unknown maximum mass of NS, which is important for determining the equation of state (EoS) of ultra-dense matter and whether stars collapse into NSs or black holes (Alsing et al. 2018;Shao et al. 2020).On the other hand, the birth mass of NS is also fascinating and can be used to check the theoretical expectations for remnants mass-produced by electron-capture versus Fe-core collapse SNe (Podsiadlowski et al. 2004;Kiziltan et al. 2013).
Isolated NSs (INSs) are believed to trace initial masses when they were born.However, the mass measurement for INSs is much more challenging than NSs in binary systems ( Özel & Freire 2016).There are some promising methods for such mass measurement: (1) the pulse profile modeling of the emission from the hotspots of NS, but the targets of the Neutron Star Interior Composition Explorer (NICER) mission currently have masses that are likely larger than their initial birth masses; (2) the glitches (sudden and temporary change in the NS spin) of some young (isolated) pulsars can be used to put constraints on the mass of NS (Pizzochero et al. 2017), but such a method may provide rather loose constraints; (3) the mass of INS with gravitational redshift measurement can be extracted by taking the advantage of EoS constraining results (Tang et al. 2020), which is benefited from the fact that most of the EoSs tend to give a unique map between the mass and the gravitational redshift of the NS.
In our previous work of Tang et al. (2020), the possibility of phase transition (PT) was not considered, and the constraints on EoSs mainly come from the nuclear constraints and gravitational wave (GW) data of GW170817.Since then, some important progresses have been made.Firstly, mass-radius measurements of two NSs have been successfully carried out by the NICER mission, i.e., PSR J0030+0451 (Miller et al. 2019;Riley et al. 2019) in the low-mass region and PSR J0740-6620 (Miller et al. 2021;Riley et al. 2021) in the high-mass region.This means more stringent constraint can be made on the EoS, and thus the masses of the other INSs that only have redshift measurement.Secondly, a new approach embedding both the PT and no phase transition (NPT) model as one has been proposed by Tang et al. (2021b), whose method can be used to take into analysis when estimating the mass from the gravitational redshift.
In view of these new progresses, we carry out this work, focusing on the mass estimation method proposed by Tang et al. (2020), and update the previous analysis by taking into account the possibility of PT in our EoS models and incorporating the constraints on EoS from the NICER's observations (i.e., the mass-radius measurement of PSR J0030+0451 and PSR J0740-6620) additional to the nuclear constraints and GW data of GW170817.In this work, we estimate the mass, radius, tidal-deformability, and the moment of inertia of the sources RBS 1223, RX J0720.4-3125, and RX J1856.5-3754 with different models.Each of these sources has its gravitational redshift been measured.
Our work is organized as follows.In Section 2 we introduce the methods.The results of the calculation are presented in Section 3. Section 4 is the conclusion and discussion.

The neutron star EoSs constrained with latest multimessenger data
Three EoS models, namely the four-pressure (4P) model, the phase transition (PT) model, and the NPT model, have been used in this work.The first so-called four-pressure model (Jiang et al. 2020;Tang et al. 2020) adopts four pressures {P 1 , P 2 , P 3 , P 4 } at the corresponding rest-mass densities of {1, 1.85, 3.7, 7.4}ρ sat to parameterize the EoS; the second/third are the hybrid parameterization method proposed in Tang et al. (2021b), which is capable to describe generic phenomenological EoS models both with and without a PT.This method constructs the adiabatic indices based on four widely used parameterization models, and the specific expression reads where ε, p, h, and ρ denote the energy density, the total pressure, the pseudo enthalpy, and the rest-mass density, respectively.Γ crust is determined by the tabulated low-density EoS.Γ nuc (ρ, x) is calculated under the parabolic expansion-based nuclear empirical parameterization (Steiner et al. 2010;Biswas et al. 2021).Υ(h, v k ) is the expansion function used in the causal spectral representation (Lindblom 2018).Γ m is the adiabatic index for the piece modeled by a single polytrope ( Özel & Psaltis 2009;Read et al. 2009).c q is the sound velocity that describes the high-density part of EoS with constant-speed-of-sound (CSS) parameterization (Alford et al. 2013).For the dividing densities, ρ 0 corresponds to the crust-core transition density, and ρ 1 is fixed to 1.85 ρ sat .The PT and NPT models mainly depend on the parameter Γ m , and we treat Γ m < 1.4 as the PT model (NPT model otherwise).For PT model, ρ 2 means the PT density, and ∆ρ measures the density jump.While for NPT model, ρ 2 and ρ 2 +∆ρ are just simply dividing densities.
With the EoS models, we can map EoS parameters to a series of mass-radius (or mass-tidal-deformability) relations (Lindblom & Indik 2014); on the contrary, with the observation data of NSs, we can constrain the EoS by Bayesian analysis.The observation data used in the Bayesian inference include the following: the tidal-deformability measurements from GW170817 (Abbott et al. 2017(Abbott et al. , 2018)), the mass-radius measurements of PSR J0030+0451 (Miller et al. 2019;Riley et al. 2019) and PSR J0740-6620 (Miller et al. 2021;Riley et al. 2021 (Raaijmakers et al. 2019;Jiang et al. 2020).Thus we only use the data of Riley et al. (2019) for PSR J0030+0451 1 and Riley et al. (2021) for PSR J0740-6620. 2 For GW data, we use the interpolated, marginalized likelihood of Hernandez Vivanco et al. (2020), which shows good consistency with the original GW data analysis.While for data of NICER, we use the Gaussian kernel density estimation (KDE) of the publicly distributed posterior samples of mass (GW) and radius to build the likelihood.The final likelihood is a production of these two parts and is sampled using the PyMultinest (Buchner 2016a)
Given an EoS, we can map the gravitational redshift z g with the compactness C through relation Here compactness is defined as C = GM/Rc 2 .c, G, M and R are the speed of light, Newton's gravitational constant, the gravitational mass, and circumferential radius, respectively.Since each posterior EoS sample in our model gives a monotonous relation between the mass and compactness, we can also uniquely find mass and radius by redshift.The tidal-deformability can also be determined if the mass is known (Jiang et al. 2019).Further, the moment of inertia I can be evaluated using an EoS-insensitive relation between tidal-deformability Λ and the dimensionless moment of inertia Ī = c 4 I/G 2 M 3 .The relation is called I-Love relation (Yagi & Yunes 2013;Landry & Kumar 2018) and reads as follows: where a n are fitting coefficients and are taken from Landry & Kumar (2018).
1 The data of ST+PST case is considered, see http://doi.org/10.5281/zenodo.3386449 2 The data file "STU/NICERxXMM/FI H/run10" from https://zenodo.org/record/4697625#.YKMcuy0tZQJ is taken into analysis.In each panel, the contour plots show the 68% confidence region of the results obtained from different models, while the upper and right sub-graphs show the marginalized probability density function (PDF) of the mass and radius, respectively.

RESULTS
We compare the results of 4P model in this work and those obtained in Tang et al. (2020).In the left panel of Fig. 1, we notice that the slope of the new result in mass-redshift relationship is smaller than that of previous work.The main reason is that we have included the measurements from NICER that favor stiffer EoS than the sole GW data.Another reason is that we do not put constraint on the upper limit of M TOV as Tang et al. (2020), which consequently reduces the slope in the high-mass region.As shown in right panel of Fig. 1, the results of mass-redshift relation constrained with the three EoS models are consistent with each other, especially at the low-redshift region because the EoS has already been constrained well for lower densities.These relations also well cover the ranges of redshift measurement of these sources.Therefore, it is straightforward to simultaneously estimate the mass and radius given the sample of  6620 had been successfully measured, and the radii seem to be larger than those suggested by GW170817.Moreover, the first-order strong PT has been properly incorporated in the parameterizing approach.These progresses motivate us to reestimate the bulk properties (including the masses, the radii, the tidal-deformability and moment of inertia) of a group of INSs.
In comparison to our initial estimates presented in Tang et al. (2020), the currently inferred gravitational masses of INSs are larger with smaller uncertainties.This is anticipated, because in the EOS constraints, we have added the mass-radius data of two NSs observed by NICER, and both NSs prefer a radius higher than that favored by GW170817 (for a given z g , the larger the radius, the higher the gravitational mass).We also find that there is little difference in all the results between PT model and NPT model.This may reflect the fact that most part of these compact objects are not dense enough for the presence of a first-order strong PT.
NICER is continuing to collect the data to reliably measure the mass-radius of a few more NSs.The LIGO/Virgo/KAGRA network is expected to run in the end of 2022, and many more NS mergers will be detected in the near future.With these data, the constraints on the EoS of NS material will be tightened, and the presence of a first-order strong PT or not will be further probed.These progresses will in turn yield more reliable measurements of the bulk properties of the INSs with known gravitational redshifts.

Figure 1 .
Figure 1.Panel (a) and (b) both represent the relationship between the gravitational redshift and the mass of NS.Panel (a) shows the comparison between the results of 4P model in this work (green lines) and that obtained in Tang et al. (2020) (red lines); where MTOV ≤ 2.3M⊙ was assumed).The shadow areas represent 68% confidence region.Panel (b) shows the comparison of results obtained from different models in this work.Red-, green-, and blue-band plots represent the 68% confidence regions of the results obtained from 4P, PT, and NPT model, respectively.Grey-, purple-, and yellow-band plots show the 95% highestposterior-density interval of gravitational redshift measurements for sources RBS 1223, RX J0720.4-3125, and RX J1856.5-3754,respectively.

Figure 2 .
Figure 2. Interpolated mass-radius distribution for three sources.Panel (a), (b), and (c) represent the results of RBS 1223, RX J0720.4-3125, and RX J1856.5-3754,respectively.Red, green, and blue lines denote the PT, NPT, and 4P model, respectively.In each panel, the contour plots show the 68% confidence region of the results obtained from different models, while the upper and right sub-graphs show the marginalized probability density function (PDF) of the mass and radius, respectively.

Figure 3 .Figure 4 .
Figure 3. Distribution of tidal-deformability.Red dashed-dotted line, green solid line, and blue dashed line represent the PT, NPT and 4P model, respectively.Left, middle, and right panels show the tidal-deformability of RBS 1223 and RX J0720.4-3125,respectively. package.

Table 1 .
Results of the mass, radius, tidal deformability, and moment of inertia for the three sources obtained with three EoS models (all in 68% confidence interval).