Neutron Star Mass Determinations

I review attempts made to determine the properties of neutron stars. The focus is on the maximum mass that a neutron star can have, or, conversely, the minimum mass required for the formation of a black hole. There appears to be only one neutron star for which there is strong evidence that its mass is above the canonical 1.4M⊙, viz., Vela X-1, for which a mass close to 1.9M⊙ is found. Prospects for progress appear brightest for studies of systems in which the neutron star should have accreted substantial amounts of matter. 1 The Mimimum Mass Required to Form a Black Hole For the study of black holes, the relevance of neutron stars is mostly that below a certain maximum mass, degeneracy pressure due to nucleons is sufficient to prevent an object from becoming a black hole. Unfortunately, the equation of state of matter at densities above nuclear-matter density is rather uncertain and theoretical estimates of the minimum mass required to form a black hole range from just over 1.4M⊙ to about 2.5M⊙ [3,10]. Constraints on the equation of state can be obtained from a variety of observed properties of neutron stars. Among the more direct measurements that have been used or proposed are: (i) the maximum mass inferred from dynamical measurements in binaries; (ii) the minimum spin period among millisecond pulsars; (iii) identification of kHz quasi-periodic oscillations with the orbital frequency in the last stable orbit; (iv) identification of quasi-periodic oscillations with the Lens-Thirring precession period; (v) influence of gravitational light bending on X-ray light curves in radio pulsars; (vi) modelatmosphere fitting of X-ray lightcurves during X-ray bursts (resulting from to thermonuclear run-aways on the neutron-star surface); (vii) gravitational redshift from γ-ray spallation lines in accreting systems; (viii) gravitational redshift and surface gravity from model-atmosphere analysis of spectra of isolated neutron stars. Less direct measurements include: (ix) matching observed neutron-star temperatures and inferred ages to cooling curves; (x) neutrino light curves in supernova explosions; (xi) pulsar glitches; (xii) moment of inertia from accretion torques in combination with knowledge of the magnetic field strength from cyclotron lines; (xiii) comparing X-ray fluxes between states in which a neutron star is accreting and in which matter is stopped at the magnetosphere and “propelled” away. For references and somewhat more detail, see [20]. The strongest constraints on the equation of state are still set by dynamical mass measurements, so I will restrict myself to those below. 2 M. H. van Kerkwijk 2 Dynamical measurements Most mass determinations have come from radio timing studies of pulsars; see [16] for an excellent review. The most accurate ones are for pulsars that are in eccentric, short-period orbits with other neutron stars, in which several non-Keplerian effects on the orbit can be observed: the advance of periastron, the combined effect of variations in the second-order Doppler shift and gravitational redshift, the shape and amplitude of the Shapiro delay curve shown by the pulse arrival times as the pulsar passes behind its companion, and the decay of the orbit due to the emission of gravitational waves. The most famous of the double neutron-star binaries is the Hulse-Taylor pulsar, PSR B1913+16, for which recent measurements give MPSR = 1.4411± 0.0007M⊙ and Mcomp = 1.3874 ± 0.0007M⊙ [14,13]. Almost as accurate masses have been inferred for PSR B1534+12, for which the pulsar and its companion are found to have very similar mass: for both, M = 1.339± 0.003M⊙ [11]. Neutron-star masses can also be determined for some binaries containing an accreting X-ray pulsar, from the amplitudes of the X-ray pulse delay and optical radial-velocity curves in combination with constraints on the inclination (the latter usually from the duration of the X-ray eclipse, if present). This method has been applied to about half a dozen systems [6,8,18], but the masses are generally not very precise. So far, for all but one of the neutron stars, the masses are consistent with being in a surprisingly narrow range, which can be approximated with a Gaussian distribution with a standard deviation of only 0.04M⊙ [16]. The mean of the distribution is 1.35M⊙, close to the “canonical” value of 1.4M⊙. The one exception is the X-ray pulsar Vela X-1, which is in a 9-day orbit with the B0.5 Ib supergiant HD 77581. For this system, a rather higher mass of around 1.8M⊙ has consistently 1 been found ever since the first detailed study in the late seventies [21,19]. A problem with this system is that the measured radial velocities show strong deviations from a pure Keplerian radial-velocity curve, which are correlated within one night, but not from one night to another. A possible cause could be that the varying tidal force exerted by the neutron star in its eccentric orbit excites high-order pulsation modes in the optical star which interfere constructively for short time intervals. We were granted time at ESO to improve the mass determination of this possibly very massive neutron star from 200 new spectra, taken in as many nights. These cover more than 20 orbits, and make it possible to average out the velocity excursions and to constrain possible systematic effects with orbital phase. In combination with measurements from our old photographic plates, earlier CCD spectroscopy, and high resolution IUE spectra, 1 One study based on IUE spectra of HD 77581 appeared to find a lower neutronstar mass [12]. However, this was found to be due to a bug in the cross-correlation software used [1] Neutron Star Mass Determinations 3 Fig. 1. Constraints on the mass of Vela X-1 and its supergiant companion HD 77581 [1,2]. The constraint on the mass ratio from the X-ray pulse delay and optical radialvelocity curves is indicated by the solid line. The long and short-dashed lines next to it indicate the 95% and 99% confidence limits, respectively. The lines stop at the region excluded by the pulse-timing mass function (to go below it would require sin i > 1). The 95% and 99% confidence lower limits on the inclination derived from the duration of the X-ray eclipse are indicated by the two dotted lines we derived a 95% confidence constraint on the mass of the neutron star of MNS = 1.87 +0.23 −0.17 [1]. Our constraints are illustrated graphically in Fig. 1. One sees that even at 99% confidence, MNS > 1.6M⊙. It should be noted, however, that from the data it appears that while the excursions in radial velocity are mostly random, there is also a component that is systematic, locked to orbital phase. Since we do not understand these effects, it may be that our mass estimate is biased. In our trials with excluding the worst-affected phase ranges, however, we consistently found that the fitted mass became even higher [19,1].