Using Neutron Star Observations to Determine Crust Thicknesses, Moments of Inertia, and Tidal Deformabilities

Andrew W. Steiner, Stefano Gandolfi, Farrukh J. Fattoyev, William G. Newton

I Introduction

Recent neutron star (NS) mass and radius observations have provided new constraints on the neutron star mass-radius curve and on the equation of state (EOS) of dense matter Lattimer and Prakash 2001. The EOS, in turn, is a fundamental property of quantum chromodynamics, which probes cold and dense matter which is otherwise difficult to access in experiment.

In the near future, mass and radius observations may be complemented by other constraints on NS structure. Although thousands of pulsars have been observed, there is only one binary system where both NSs are radio-active pulsars, PSR J0737-3039. The ability to observe pulsations from both NSs and the extreme nature of the system Burgay et al. 2003; Kramer et al. 2006, enables a potential measurement of the moment of inertia of one of the neutron stars Damour and Schäfer 1988. Also, the Laser Interferometer Gravitational-Wave Observatory is expected to measure the gravitational wave signal from a NS merger within the near future Abadie and et al. 2010, and a sufficiently large signal to noise observation will enable a measurement of the neutron star tidal deformability Flanagan and Hinderer 2008; Hinderer et al. 2010; Damour et al. 2012; Del Pozzo et al. 2013; Read et al. 2013; Lackey et al. 2014 (denoted by $\lambda$ and sometimes also called “tidal polarizability”). It turns out these two types of new observations are intimately related: the moment of inertia of a NS is strongly correlated with its tidal deformability Yagi and Yunes 2013a; Yagi and Yunes 2013b.

NSs can accrete matter from main-sequence companions which results in the emission of X-rays and the heating of the NS crust. If the accretion stops (referred to as “quiescence” since the X-rays from accretion subside), then the cooling NS crust can be directly observed Rutledge et al. 2002. The timescale for this cooling is proportional to the square of the NS crust thickness Lattimer et al. 1994, and thus the crust thickness is important for determining the properties of the crust from observations of crust cooling Lewin and van der Klis 2006; Shternin et al. 2007; Brown and Cumming 2009; Page and Reddy 2013.

Another potential constraint of NS structure comes from pulsar glitches. Previous papers Link et al. 1999; Andersson et al. 2012; Chamel 2013 have shown that, if NS crusts are believed to be the location of the angular momentum reservoir which contributes to the glitch spin up, then a significant fraction of the NS’s moment of inertia must lie in the superfluid component of the crust. Thus glitches are sensitive to the crustal fraction of the moment of inertia, denoted $\Delta I/I$ .

Finally, many of these quantities are (at least weakly) correlated with the nuclear symmetry energy Steiner et al. 2005; Tsang et al. 2012; Li et al. 2014. The nuclear symmetry energy is the difference between the energy per baryon of neutron matter and that of nuclear matter (we ignore quartic terms, see Ref. Steiner 2006). We denote the symmetry energy $S(n_{B})$ , where $n_{B}$ is the baryon number density, and $S\equiv S(n_{0})$ , where $n_{0}$ is the nuclear saturation density. The quantity $3n_{0}S^{\prime}(n_{0})$ is denoted as $L$ . The value of $L$ determines the pressure of neutron-rich matter at the saturation density. The pressure of neutron-rich matter, in turn, is related to all of the above NS structure quantities given above.

For the first time, we use existing NS mass and radius observations to predict the expected ranges of NS properties such as moments of inertia, tidal polarizabilities and crustal thicknesses which are measurable by a diverse range of ongoing observational programs. We generate these expected ranges based on Monte Carlo simulations using parameterizations which explore the full variation which is possible given current uncertainties in the nature of dense matter. Our EOS models are based on recent progress in the microscopic calculation of neutron-rich matter near the nuclear saturation density. At high densities, we assume no additional correlation with matter at lower densities, and use models which allow for strong phase transitions. Our method is in contrast to several previous papers which have computed theoretical predictions of moments of inertia, crust thicknesses, and tidal polarizabilities for smaller samples of representative EOSs Kalogera and Psaltis 1999; Lattimer and Prakash 2001; Morrison et al. 2004; Bejger et al. 2005; Steiner et al. 2005; Lattimer and Schutz 2005; Lattimer and Prakash 2007; Fattoyev and Piekarewicz 2010; Fattoyev et al. 2013; Lattimer and Lim 2013. Future observations will consitute direct tests of the theoretical framework we use and of the systematics of current mass-radius observations.

II Method

For the second model, we use the parametrization from Ref. Hebeler et al. 2013 [“Hebeler-Lattimer-Pethick-Schwenk (HLPS)”], and the results on neutron matter from Ref. Tews et al. 2013 from an interaction based on chiral effective theory. At each point, we fix $\alpha,\gamma,\eta$ , the three parameters which control nuclear saturation, to fix the saturation density, the binding energy, and the incompressbility. Note that this $\alpha$ is distinct from the parameter with the same name in the GCR model. The remaining two parameters $\alpha_{L}$ and $\eta_{L}$ , which control the properties of neutron matter, are again reparameterized in terms of $S$ and $L$ . The range of $S$ from Fig. 1 of Ref. Tews et al. 2013 is between 29.5 and 36.1 MeV and the range of $L$ is between $44.0$ and $65.0$ MeV.

For the NS crust, we use the tabulated crust EOSs based on the work in Ref. Newton et al. 2013. The advantage of this crust EOS, relative to the older work in Ref. Baym et al. 1971, is that we can employ the same values of $S$ and $L$ which we use in the EOS of neutron-rich matter at the saturation density. A two-dimensional grid of crust EOSs with varying values of $S$ and $L$ were computed and this grid was interpolated to generate the crust for general values of $S$ and $L$ in our simulations. For the transition between the crust and the core, we use the correlation between $n_{t}$ and $S$ and $L$ ,

where $L_{70}$ is $L$ in units of 70 MeV and $S_{30}$ is $S$ in units of 30 MeV. For our ranges of $S$ and $L$ , this correlation gives transition densities between 0.06 fm-3 and 0.1 fm-3, consistent with those obtained in Ref. Oyamatsu and Iida 2007. We use this transition density from Eq. 1 to compute the transition pressure, and find values between 0.30 and 0.82 MeV/fm3, slightly larger than the range 0.25 $-$ 0.65 MeV/fm3 found in Ref. Lattimer and Prakash 2001.

For the EOS above the saturation density, we either use a set of three piecewise polytropes (only five parameters since the transition to the first polytrope is already fixed by the EOS at the saturation density) referred to as “Model A” in Ref. Steiner et al. 2013. Alternatively, we use a set of four line segments in the $(\varepsilon,P)$ plane, “Model C” Steiner et al. 2013. This latter model is useful because it provides an alternative model which tends to favor stronger phase transitions in the core. We do not employ Model B or Model D from Ref. Steiner et al. 2013 because they do not typically provide significantly different results from Model A at the current level of accuracy. The choice of either GCR or HLPS near saturation density and either Model A or Model C at high densities gives a total of four EOS models to use with our three data sets.

For each of the above data sets and EOS models, we perform a Markov chain Monte Carlo simulation as first outlined in Ref. Steiner et al. 2010. To obtain our final results for a fixed data set we choose the smallest range which encloses all of the EOS models, as performed in Ref. Steiner et al. 2013. This procedure is a relatively simple version of a fully hierarchical Bayesian analysis which is currently too computationally expensive.

III Results

IV Discussion

If a measurement of the moment of inertia of PSR J0737-3039A was far outside our predicted range, then that implies a conflict with the mass and radius observations. This conflict could be resolved with modification of strong-field GR. However, this modification may have to be finely tuned in order to modify the NS structure without spoiling the agreement with GR found in the observations of the post-Keplerian parameters in the PSR J0737-3039 system Kramer and Wex 2009.

A large increase in the NS maximum mass, such as that implied by Refs. van Kerkwijk et al. 2011; Romani et al. 2012, would significantly change these results. Larger maximum masses imply larger radii (larger pressure is needed at smaller densities to compete with gravity as the mass becomes larger), and thus larger moments of inertia and tidal deformabilities.

V Acknowledgments

We thank N. Chamel, C. Fryer, W.C.G. Ho, J.M. Lattimer, S. Reddy and J.R. Stone for helpful discussions. A.W.S. is supported by DOE Grant No. DE-FG02-00ER41132. S.G. is supported by DOE Grants No. DE-AC02-05CH11231, by the NUCLEI SciDAC program, and by the LANL LDRD program. F.J.F. is supported by the NSF under Grant No. PHY-1068022. F.J.F. and W.G.N. are supported by NASA through the Science Mission Directorate under Grant No. NNX11AC41G. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.