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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 62 , Issue 2 : 029503(2019) https://doi.org/10.1007/s11433-017-9188-4

Dense matter with eXTP

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  • ReceivedDec 7, 2017
  • AcceptedFeb 11, 2018
  • PublishedAug 30, 2018
PACS numbers

Abstract

In this White Paper we present the potential of the Enhanced X-ray Timing and Polarimetry (eXTP) mission for determining the nature of dense matter; neutron star cores host an extreme density regime which cannot be replicated in a terrestrial laboratory. The tightest statistical constraints on the dense matter equation of state will come from pulse profile modelling of accretion-powered pulsars, burst oscillation sources, and rotation-powered pulsars. Additional constraints will derive from spin measurements, burst spectra, and properties of the accretion flows in the vicinity of the neutron star. Under development by an international Consortium led by the Institute of High Energy Physics of theChinese Academy of Sciences, the eXTP mission is expected to be launched in the mid 2020s.


Acknowledgment

ALW and TER acknowledge support from ERC Starting (Grant No. 639217 CSINEUTRONSTAR). AP acknowledges support from a Netherlands Organization for Scientific Research (NWO) Vidi Fellowship. YC is suported by the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Global Fellowship (Grant No. 703916). SKG, KH and AS are supported in part by the DFG through Grant SFB 1245 and the ERC (Grant No. 307986 STRONGINT).The Chinese team acknowledges the support of the Chinese Academy of Sciences through the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA15020100).


Contributions statement

This paper is an initiative of eXTP's Science Working Group 1 on Dense Matter, whose members are representatives of the astronomical community at large with a scientific interest in pursuing the successful implementation of eXTP. The paper was primarily written by Anna Watts, WenFei Yu, Juri Poutanen, and Shu Zhang with major contributions by Sudip Bhattacharyya (Fe lines), Slavko Bogdanov (rotation powered pulsars), Long Ji (spin measurements), Alessandro Patruno (spin measurements) and Thomas Riley (pulse profile modelling technique). Contributions were edited by Anna Watts. Other co-authors provided input to refine the paper.


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  • Figure 1

    Schematic structure of a NS (not to scale). The outer layer is a solid ionic crust supported by electron degeneracy pressure. Neutrons begin to leak out of ions (nuclei) at densities $\sim4\times~10^{11}$ g cm$^{-3}$ (the neutron drip density, which separates inner from outer crust), where neutron degeneracy also starts to play a role. At the very base of the crust, nuclei may become very deformed (the pasta phase). At densities $\sim2~\times~10^{14}$ g cm$^{-3}$, the nuclei dissolve completely and this marks the crust-core boundary. In the core, densities could reach up to ten times $\rho_\mathrm{sat}$ $\sim$ $2.8\times~10^{14}$ g cm$^{-3}$ ($\rho_\mathrm{sat}$ being the density in normal atomic nuclei). The matter in the core is highly neutron-rich, and the inner core may contain stable states of strange matter in deconfined quark or baryonic form. See also Figure 1 of ref. [5].

  • Figure 2

    Hypothetical states of matter accessed by NSs and current or planned laboratory experiments, in the parameter space of temperature, baryon number density and nuclear asymmetry $\alpha~=~1-2Y$ where $Y$ is the hadronic charge fraction ($\alpha~=~0$ for matter with equal numbers of neutrons and protons, and $\alpha~=~1$ for pure neutron matter). NSs access unique states of matter that are either difficult to create in the laboratory—such as nuclear superfluids and perhaps strange matter states like hyperons—or cannot be created, such as deconfined quarks and the color superconductor phase. For simplicity, the transition region is shown only in projection on the density-temperature axis. Figure adapted from Figure 2 of ref. [12]; for an alternative visualisation in the parameter space of temperature and baryon chemical potential, see Figure 2 of ref. [5].

  • Figure 3

    The pressure-density relation (EOS) (a) and the corresponding $M$-$R$ relation (b) for some example models with different microphysics. Nucleonic (neutrons, protons): models AP3 and AP4 from ref. [27], also used in ref. [28]. Quark (u, d, s quarks): models from refs. [29,30]. Hybrid (inner core of uds quarks, outer core of nucleonic matter): models from ref. [31]. Hyperon (inner core of hyperons, outer core of nucleonic matter): Model from ref. [32]. CEFT: range of nucleonic EOS based on Chiral Effective Field Theory (CEFT) from ref. [33]. pQCD: range of nucleonic EOS from ref. [34]that interpolate from CEFT at low densities and match to perturbative QCD (pQCD) calculations at higher densities than shown in this figure. All of the EOS shown are compatible with the existence of $\sim~2$ $M_\odot$ NSs.

  • Figure 4

    Stellar rotation modulates emission from a hot region (hotspot), generating an X-ray pulsation. Relativistic effects encode information about $M$ and $R$ in the normalisation and harmonic content of the pulse profile. These effects are key observables exploited by the pulse profile modelling technique, and include Doppler boosting and gravitational redshifting, time-delays, and light bending (which renders the far side of the star partially visible). The Figure illustrates these effects for a rapidly spinning, oblate star. We compare pulse profiles generated by a photosphere embedded in a Minkowski exterior spacetime to those generated by a photosphere embedded in a Schwarzschild exterior spacetime (see text). For the purpose of illustration, we use: a gravitational mass of 1.8 $M_\odot$; an equatorial coordinate radius of 14 km; a coordinate spin frequency of 600 Hz; and a distance of 1 kpc. The observer is in the equatorial plane. The local photospheric radiation field is completely specified by the local comoving blackbody temperature. The temperature field is non-evolving in the corotating reference frame, and is constituted by a hotspot of angular radius of 60$^\circ$, centred at a colatitude of 60$^\circ$. Its temperature falls smoothly from 2.5 keV at the centre to 0.5 keV at the boundary, where the latter is the temperature everywhere outside the spot boundary. (a) Monochromatic profiles at two energies (2- 10 keV). (b) The resolved stellar photosphere at two rotational phases. The colour corresponds to the redshifted temperature on a distant image-plane.

  • Figure 5

    We illustrate the response of monochromatic pulse profiles to variation of the (circumferential) equatorial radius, whilst all other parameters are fixed at the values implemented in Figure 4. We use the realistic exterior spacetime described in Figure 4. The two synthetic stars shown are of equal gravitational mass and spin frequency, but have equatorial radii of 10 and 12 km; these stars require distinct EOS models to exist. The pulse profiles are clearly sensitive to the equatorial radius, and it is the need to detect such differences that drives the design requirements for eXTP.

  • Figure 6

    The pulse profile as well as the phase-dependence of the PD and PA. The black solid line gives the contribution of two antipodal spots, while the blue dashed and red dotted lines correspond to the contribution of the primary and secondary spot separately. The pulse profile and PD are degenerate to exchanging $i$ and $\theta$, while the PA shows dramatically different behaviour allowing both angles to be obtained (adapted from ref. [78]).

  • Figure 7

    Constraints from pulse profile modelling (PPM) and burst spectral fitting expected for eXTP. EOS models as in Figure 3. The orange dashed contours show the 1$\sigma$ constraints obtained from the pulse profiles of the AMP SAX J1808.4–3658 using RXTE data [63,90]. The constraints expected from pulse profile modelling and polarization data using the LAD and PFA on eXTP, for the same source, from an observation of 100 ks, are shown by the solid orange contour. Spectral evolution during PRE bursts (observed from this source) as determined using direct fits with atmosphere spectral models [25]produces more perpendicular constraints on $M$ and $R$, given by dotted orange curves. Combining these methods constrains the mass and radius of the neutron star to lie within the overlapping region (filled orange), with errors of a few % on both parameters. Pulse profile modelling of burst oscillations from this source will provide an entirely independent constraint on $M$ and $R$ with a similar level of accuracy. These techniques can then be applied to other known sources (including several AMPs that also have burst oscillations) to deliver the multiple measurements necessary to map the EOS.

  • Figure 8

    Constraints from pulse profile modelling of rotation-powered pulsars with eXTP. The orange error contours are for four MSPs for which masses are known precisely: PSR J1614$-$2230, PSR J2222$-$0137, PSR J0751$+$1807 and PSR J1909$-$3744. The 1$\sigma$ contours shown correspond to $\sim$ 5%-10% constraints on $R$ with the extent in $M$ corresponding to the 1$\sigma$ bounds from the radio measurements. This should be achievable using eXTP exposures of $\sim$ 1 Ms for each source. The most stringent constraints on the EOS are likely to come from the highest mass target, and here more exposure time would be merited. EOS models as in Figure 3—the underlying model assumed in the simulations is the AP3 nucleonic EOS (red).

  • Figure 9

    Spin limits on the EOS. The mass-shedding limit can be recast as an upper limit on radius for a star of a given spin rate [102]. This means that NSs of a given spin rate must extend to the left of the relevant limit in the $M$-$R$ plane (shown as thick violet lines, for various spins). EOS models as in Figure 3. The current record holder, which spins at 716 Hz [103]is not constraining. However, given a high enough spin individual EOS can be ruled out. Above 1000 Hz, for example, some individual EOS in the pQCD band, and one of the quark star models, would be excluded.

  • Figure 10

    EOS constraints from relativistic Fe line modelling. (a) (Adapted from ref. [146]): $M$ versus $r_{\rm~in}c^2$/GM curves for two reasonable values of spin, and for two currently viable EOS models: one very stiff (A) and one of intermediate stiffness (B). On the straight (near horizontal) portions, $r_{\rm~in}$=$R$; elsewhere $r_{\rm~in}=r_{\rm~ISCO}$. Since the geometrically thin accretion disk, which gives rise to the broad relativistic line, may be truncated by radiation pressure or a stellar magnetic field at a radius larger than $r_{\rm~ISCO}$ and $R$, the observationally inferred $r_{\rm~in}c^2$/GM is an upper limit. If $M$ can be measured independently (e.g. by one of the other methods described in this white paper), this upper limit on $r_{\rm~in}c^2$/GM provides an extra constraint on the EOS. In addition, if the upper limit of $r_{\rm~in}c^2$/GM is sufficiently small, softer EOS models can be ruled out without a mass measurement. (b) Simulated relativistic Fe line spectrum for a 30 ks observation with eXTP (residual plot). We assume a wabs(bbody+diskbb+powerlaw+diskline) XSPEC model with reasonable values of 2-20 keV flux ($6.4\times10^{-9}$ erg cm$^{-2}$ s$^{-1}$) and line equivalent width (124 eV).

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