Neutron Star EoS, masses and radii Structure and layers Plus an atmosphere and an envelope ... 2007.04427 Neutron star interiors Radius: 10 km Mass: 1-2 solar Density: above the nuclear Strong magnetic fields Why important? 1903.04648, a short review on M-R measurements related to EoS Astrophysical point of view Astrophysical appearence of NSs is mainly determined by: • Spin • Magnetic field • Temperature • Velocity • Environment The first four are related to the NS structure! Equator and radius ds2=c2dt2e2Φ-e2λdr2-r2[dθ2+sin2θdφ2] In flat space Φ(r) and λ(r) are equal to zero. • t=const, r= const, θ=π/2, 0<Φ<2π l=2πr • t=const, θ=const, φ=const, 040 MO. The case of zero metallicity Woosley et al. 2002 No intermediate mass range for NS formation. DNS 1706.09438 DNS parameters 1902.03300 Individual masses of DNS 1902.03300 Individual masses of DNS 1902.03300 Binary pulsars See 1502.05474 for a recent detailed review Relativistic corrections and measurable parameters For details see Taylor, Weisberg 1989 ApJ 345, 434 Shapiro delay PSR 1855+09 (Taylor, Nobel lecture) Mass measurements PSR 1913+16 Taylor Uncertainties and inverse problems 1502.05474 PSR B1534+12. Pbdot depends on the Shklovskii effect. So, if distance is not certain, it is difficult to have a good measurement of this parameter. It is possible to invert the problem. Assuming that GR is correct, one can improve the distance estimate for the given source. Double pulsar J0737-3039 Lyne et al. astro-ph/0401086 Masses for PSR J0737-3039 Kramer et al. astro-ph/0609417 The most precise values. New mass estimates have uncertainties <0.001 DNS J1829+2456 mass measurements 2007.07565 Tests of theories of gravity 1802.09206 J1713+0747 Testing strong equivalence principle with triple pulsar PSR J0337+1715 1401.0535 NS+WD+WD NS+WD binaries Some examples PSR J0437-4715. WD companion [0801.2589, 0808.1594 ]. The closest millisecond PSR. MNS=1.76+/-0.2 solar. The case of PSR J0751+1807. Initially, it was announced that it has a mass ~2.1 solar [astro-ph/0508050]. However, then in 2007 at a conference the authors announced that the result was incorrect. Actually, the initial value was 2.1+/-0.2 (1 sigma error). New result: 1.26 +/- 0.14 solar [Nice et al. 2008, Proc. of the conf. “40 Years of pulsars”] It is expected that most massive NSs get their additional “kilos” due to accretion from WD companions [astro-ph/0412327 ]. Very massive neutron star arXiv: 1010.5788 Binary system: pulsar + white dwarf PSR 1614-2230 Mass ~ 2 solar About the WD see 1106.5497. The object was identified in optics. About formation of this objects see 1103.4996 Why is it so important? arXiv: 1010.5788 The maximum mass is a crucial property of a given EoS Collapse happens earlier for softer EoSs, see however, 1111.6929 about quark and hybrid stars to explain these data. Interestingly, it was suggested that just <0.1 solar masses was accreted (1210.8331) In the future specific X-ray sources (eclipsing msec PSR like SWIFT J1749.4−2807) can show Shapiro delay and help to obtain masses for a different kind of systems, see 1005.3527 , 1005.3479 . 2.01 solar masses NS 1304.6875 PSR J0348+0432 39 ms, 2.46 h orbit WD companion The NS mass is estimated to be: 1.97 – 2.05 solar mass at 68.27% 1.90 – 2.18 solar mass at 99.73% confidence level. System is perfect for probing theories of gravity as it is very compact. 1904.06759 J0740+6620 2.14 solar masses 2.14 solar mass NS New data confirm it: 2.01-2.17 Msun (1-sigma) 2104.00880 The most extreme (but unclear) example 1009.5427 BLACK WIDOW PULSAR PSR B1957+20 2.4+/-0.12 solar masses New estimates from gamma eclipses 2301.10995 PSR B1957+20 𝑀psr = 1.81 ± 0.07𝑀sun Fermi observation ~50 black widow and red backs For several gamma-ray eclipses are found. This allows to obtain good estimates of inclination. A massive NS in PSR J2215+5135 1805.08799 Different lines provide different velocity as they are emitted from different sides of the companion. Different sides of the companion move with different velocity. Thus, a correct model provides new mass determination. New calculations confirm high mass of the NS (2002.12483). High mass of PSR J1810+1744 2101.09822 Black widow – like system Detailed studies of companion are necessary to measure mass. Keck light curves PSR J0952-0607: the heaviest 2207.05124 Pspin=1.41 msec Porb=6.42 hour low magnetic field ~6 107 G PSR-WD masses 2009.12283 Light helium white dwarf companions are shown as purple circles, and the systems with massive white dwarf (CO WD) companions are shown as green squares. Triangles – non-recycled PSRs (WD formed first). How much do PSRs accrete? 1010.5429 M=1.4+0.43(P/ms)-2/3 Millisecond pulsars are ~0.2 solar masses more massive than the rest ones. DNS and NS+WD binaries 1011.4291 1.35+/-0.13 and 1.5+/-0.25 Cut-off at ~2.1 solar masses can be mainly due to evolution in a binary, not due to nuclear physics (see 1309.6635) Neutron stars in binaries Study of close binary systems gives an opportunity to obtain mass estimate for progenitors of NSs (see for example, Ergma, van den Heuvel 1998 A&A 331, L29). For example, an interesting estimate was obtained for GX 301-2. The progenitor mass is >50 solar masses. On the other hand, for several other systems with both NSs and BHs progenitor masses a smaller: from 20 up to 50. Finally, for the BH binary LMC X-3 the progenitor mass is estimated as >60 solar. So, the situation is tricky. Most probably, in some range of masses, at least in binary systems, stars can produce both types of compact objects: NSs and BHs. Mass determination in binaries: mass function mx, mv - masses of a compact object and of a normal star (in solar units), Kv – observed semi-amplitude of line of sight velocity of the normal star (in km/s), P – orbital period (in days), e – orbital eccentricity, i – orbital inclination (the angle between the orbital plane and line of sight). One can see that the mass function is the lower limit for the mass of a compact star. The mass of a compact object can be calculated as: So, to derive the mass it is necessary to know (besides the line of sight velocity) independently two more parameters: mass ration q=mx/mv, and orbital inclination i. Some mass estimates ArXiv: 0707.2802 More measurements 1101.2465 Six X-ray binary systems. All are eclipsing pulsars. Mass-radius diagram and constraints astro-ph/0608345, 0608360 Unfortunately, there are no good data on independent measurements of masses and radii of NSs. Still, it is possible to put important constraints. Most of recent observations favour stiff EoS. Useful analytical estimates for EoS can be found in 1310.0049. Observations vs. data 1205.6871 Some newer results by the same group are presented in 1305.3242 Mass and radius for a pulsar! 1211.6113 PSR J0437–4715 NS+WD The nearest known mPSR 155-158 pc XMM-Newton observations showed thermal emission. H-atmosphere model fits. Hot caps are non-antipodal. Combination of different methods Ozel astro-ph/0605106 EXO 0748-676 Radius determination in bursters See, for example, Joss, Rappaport 1984, Haberl, Titarchuk 1995 Explosion with a ~ Eddington liminosity. Modeling of the burst spectrum and its evolution. http://www.astro.washington.edu/ben/a510/NSTARS.new.html Limits on the EoS from EXO 0748-676 Ozel astro-ph/0605106 Stiff EoS are better. Many EoS for strange matter are rejected. But no all! (see discussion in Nature). X- hydrogen fraction in the accreted material Some optimistic estimates 1002.3825 4U 1820-30 1002.3153 4U 1608−248 EXO 1745−248 4U 1820−30 Pessimistic estimates 1004.4871 1002.3153 1005.0811 It seems that Ozel et al. underestimate different uncertainties and make additional assumptions. Atmospheric uncertainties 1301.3768 qLMXB in M13 Hydrogene Helium Pulse profile constraints 1303.0317 The idea is that: sharp pulses are possible only in the case of a large star. Based on Bogdanov, Grindlay 2009 Green – excluded region PSR J0030+0451 Hot spots and pulse profiles 1602.01081 As the neutron star rotates, emission from a surface hotspot generates a pulsation. The figure shows observer inclination i, and hotspot inclination α. The invisible surface is smaller than a hemisphere due to relativistic light-bending. Detailed model description in 2104.06928. NICER’s mPSRs 1912.05707, 1912.05706 Four near-by millisecond radio pulsars: J0437−4715 J0030+0451 J1231−1411 J2124−3358 Results from NICER. PSR J0030+0451 1912.05702 For the ST-PST model Single temperature+Protruding single temp. No antipodal symmetry. But several other tried models are not ruled out. For example, in the ST-CST model Results from NICER. PSR J0030+0451 1912.05703 Two types of EoS models are considered: - PP (piecewise-polytropic); - CS (speed of sound). Results from NICER. PSR J0030+0451 1912.05705 Three oval spots model. Non-trivial field structure. Results from NICER. PSR J0740+6620 Without XMM data the radius is smaller. Model: two circular spots, pure hydrogen model atmospheres that allow for the possibility of partial ionization. 2105.06979 PSR J0740+6620 and EoS 2105.06979 Altogether now 2105.06981 Joint constraints based on NICER, GW, etc. New analysis for PSR J0740+6620 2209.12840 Geometry of hot regions: non-centered and (non-)dipole 2209.12840 Astroseismology 1602.01081 M – R diagram showing the seismological constraints for the soft gamma-ray repeater SGR 1806–20 using the relativistic torsional crust oscillation model of Samuelsson and Andersson (2007), in which the 29 Hz QPO is identified as the fundamental and the 625 Hz QPO as the first radial overtone. The neutron star lies in the box where the constraints from the two frequency bands overlap. This is a simplified model. Fe K lines from accretion discs [Cackett et al. arXiv: 0708.3615] Measurements of the inner disc radius provide upper limits on the NS radius. Ser X-1 <15.9+/-1 4U 1820-30 <13.8+2.9-1.4 GX 349+2 <16.5+/-0.8 (all estimates for 1.4 solar mass NS) Suzaku observationsSee also Papito et al. arXiv: 0812.1149, a review in Cackett et al. 0908.1098, and theory in 1109.2068. Fits from QPOs 2010.08291 Inner radius of the accretion disc, from fits to the energy spectra, as a function of the frequency of the lower kHz QPO, from fits to the power spectra, in 4U 1608–52 Limits on the moment of inertia Spin-orbital interaction PSR J0737-3039 (see Lattimer, Schutz astro-ph/0411470) The band refers to a hypothetical 10% error. This limit, hopefully, can be reached in several years of observ. See a more detailed discussion in 1006.3758 Most rapidly rotating PSR 716-Hz eclipsing binary radio pulsar in the globular cluster Terzan 5 Jason W.T. Hessels et al. astro-ph/0601337 Previous record (642-Hz pulsar B1937+21) survived for more than 20 years. Rotation starts to be important from periods ~3 msec. QPO and rapid rotation XTE J1739-285 1122 Hz P. Kaaret et al. astro-ph/0611716 Miller astro-ph/0312449 1330 Hz – one of the highest QPO frequency The line corresponds to the interpretation, that the frequency is that of the last stable orbit, 6GM/c2 Rotation and composition (Weber et al. arXiv: 0705.2708) Computed for a particular model: density dependent relativistic Brueckner-Hartree-Fock (DD-RBHF) (equatorial) (polar) Detailed study of the influence of rotation onto structure and composition is given in 1307.1103 Rotation and composition (Weber et al. arXiv: 0705.2708) 1.4 solar mass NS (when non-rotating) hyperon quark-hybrid quark-hybrid (quarks in CFL) GW170817: deformability Λ Many papers are published based on detection of GW signal from GW170817: 1803.00549, 1804.08583, 1805.09371, 1805.11579, 1805.11581, 1901.04138. k2~β-1 Λ~β-6 Collapse to a BH after ~1 sec? (1901.04138) Low spin priors Solid – theoretical EoS Colored – limits (dashed 50%, solid 90%) for four waveform models 1805.11579 GW170817: M-R 1805.11581 Microlensing and weak lensing In the future (maybe already with Gaia) it can possible to determine NS mass with lensing. Different techniques can be discussed: photometric (normal) microlensing (1009.0005), astrometric microlensing, weak lensing (1209.2249). 1209.2249 See recent studies in 2107.13697, 2107.13701 ATHENA 1912.01608 Using only spectra M and R can be determined within 3-10% and 2-8%, respectively. References • Observational Constraints on Neutron Star Masses and Radii 1604.03894 The review is about X-ray systems • Mass, radii and equation of state of neutron stars 1603.02698 The review about different kinds of measurements, including radio pulsars. Recent lists of mass measurements for different NSs. • Measuring the neutron star equation of state using X-ray timing 1602.01081 The review about EoS and X-ray measurements • The masses and spins of neutron stars and stellar-mass black holes 1408.4145 The review covers several topics. Good brief description of radio pulsar mass measurements. • Properties of DNS systems. 1706.09438 The review covers all aspects of observations, formation and evolution. • Testing the equation of state of neutron stars with electromagnetic observations. 1806.02833 The BIG review describes observational tests of the EoS. • Birth events, masses and the maximum mass of Compact Stars. 2011.08157 The review covers mass measurements and birth rates.