Sharon M. Morsink

Light Curves for Rapidly Rotating Neutron Stars

We present ray-tracing computations for light emitted from the surface of a rapidly rotating neutron star in order to construct light curves for X-ray pulsars and bursters. These calculations are for realistic models of rapidly rotating neutron stars that take into account both the correct exterior metric and the oblate shape of the star. We find that the most important effect arising from rotation comes from the oblate shape of the rotating star. Approximating a rotating neutron star as a sphere introduces serious errors in fitted values of the stars radius and mass if the rotation rate is very large. However, in most cases acceptable fits to the ratio M/R can be obtained with the spherical approximation.

Colloquium : Measuring the neutron star equation of state using x-ray timing

One of the primary science goals of the next generation of hard x-ray timing instruments is to determine the equation of state of matter at supranuclear densities inside neutron stars by measuring the radius of neutron stars with different masses to accuracies of a few percent. Three main techniques can be used to achieve this goal. The first involves waveform modeling. The flux observed from a hotspot on the neutron star surface offset from the rotational pole will be modulated by the star’s rotation, and this periodic modulation at the spin frequency is called a pulsation. As the photons propagate through the curved spacetime of the star, information about mass and radius is encoded into the shape of the waveform (pulse profile) via special and general-relativistic effects. Using pulsations from known sources (which have hotspots that develop either during thermonuclear bursts or due to channeled accretion) it is possible to obtain tight constraints on mass and radius. The second technique involves characterizing the spin distribution of accreting neutron stars. A large collecting area enables highly sensitive searches for weak or intermittent pulsations (which yield spin) from the many accreting neutron stars whose spin rates are not yet known. The most rapidly rotating stars provide a clean constraint, since the limiting spin rate where the equatorial surface velocity is comparable to the local orbital velocity, at which mass shedding occurs, is a function of mass and radius. However, the overall spin distribution also provides a guide to the torque mechanisms in operation and the moment of inertia, both of which can depend sensitively on dense matter physics. The third technique is to search for quasiperiodic oscillations in x-ray flux associated with global seismic vibrations of magnetars (the most highly magnetized neutron stars), triggered by magnetic explosions. The vibrational frequencies depend on stellar parameters including the dense matter equation of state, and large-area x-ray timing instruments would provide much improved detection capability. An illustration is given of how these complementary x-ray timing techniques can be used to constrain the dense matter equation of state and the results that might be expected from a 10  m2 instrument are discussed. Also discussed are how the results from such a facility would compare to other astronomical investigations of neutron star properties.

CONSTRAINTS ON THE MASS AND RADIUS OF THE NEUTRON STAR XTE J1807-294

The accreting millisecond pulsar XTE J1807-294 is studied through a pulse-shape modeling analysis. The model includes blackbody and Comptonized emission from the one visible hot spot and makes use of the Oblate Schwarzschild approximation for ray-tracing. We include a scattered light contribution, which accounts for flux scattered off an equatorial accretion disk to the observer including time delays in the scattered light. We give limits to mass and radius for XTE J1807-294 and compare these to limits determined for SAX J1808-3658 and XTE J1814-334 previously determined using similar methods. The resulting allowed region for mass-radius curves is small but consistent with a mass-radius relation with nearly constant radius ({approx}12 km) for masses between 1 and 2.5 solar masses.

Pulse Shapes from Rapidly Rotating Neutron Stars: Equatorial Photon Orbits

We demonstrate that fitted values of the stellar radius obtained by fitting theoretical light curves to observations of millisecond-period X-ray pulsars can significantly depend on the method used to calculate the light curves. The worst-case errors in the fitted radius are evaluated by restricting ourselves to the case of light emitted and received in the equatorial plane of a rapidly rotating neutron star. First, using an approximate flux that is adapted to the one-dimensional nature of such an emission region, we show how pulse shapes can be constructed using an exact spacetime metric and fully accounting for time-delay effects. We compare this to a method that approximates the exterior spacetime of the star by the Schwarzschild metric, inserts special relativistic effects by hand, and neglects time-delay effects. By comparing these methods, we show that there are significant differences in these methods for some applications—for example, pulse timing and constraining the stellar radius. In the case of constraining the stellar radius, we show that fitting the approximate pulse shapes to the full calculation yields errors in the fitted radius of as much as approximately ±10%, depending on the rotation rate and size of the star as well as the details describing the emitting region. However, not all applications of pulse shape calculations suffer from significant errors; we also show that the calculation of the soft-hard phase lag for a 1 keV blackbody does not strongly depend on the method used for calculating the pulse shapes.

CONSTRAINTS ON THE PROPERTIES OF THE NEUTRON STAR XTE J1814–338 FROM PULSE-SHAPE MODELS

The accretion-powered (non-X-ray burst) pulsations of XTE J1814–338 are modeled to determine neutron star parameters and their uncertainties. The model is a rotating circular hot spot and includes: (1) an isotropic blackbody spectral component; (2) an anisotropic Comptonized spectral component; (3) relativistic time delays and light bending; and (4) the oblate shape of the star due to rotation. This model is the simplest possible model that is consistent with the data. The resulting best-fit parameters of the model favor stiff equations of state (EOS), as can be seen from the 3σ allowed regions in the mass-radius diagram. We analyzed all data combined from a 23 day period of the 2003 outburst, and separately analyzed data from two days of the outburst. The allowed mass-radius regions for both cases only allow EOS that are stiffer than the EOS of Akmal and colleagues, consistent with the large mass that has been inferred for the pulsar NGC 6440B. The stiff EOS inferred by this analysis is not compatible with the soft EOS inferred from a similar analysis of SAX J1808.

Rotational Broadening of Atomic Spectral Features from Neutron Stars

The discovery of the first gravitationally redshifted spectral line from a neutron star (NS) by Cottam et al. has triggered theoretical studies of the physics of atomic line formation in NS atmospheres. Chang et al. showed that the hydrogenic Fe Hα line formed above the photosphere of a bursting NS is intrinsically broad. We now include rotational broadening within general relativity and compare the resulting profile to that observed during type I bursts from EXO 0748-676. We show that the fine-structure splitting of the line precludes a meaningful constraint on the radius. Our fitting of the data show that the line-forming Fe column is log(NFe, n=2/cm-2) = 17.9 and gravitational redshift 1 + z = 1.345 with 95% confidence. We calculate the detectability of this spectral feature for a large range of spins and inclinations, assuming that the emission comes from the entire surface. We find that at 300 and 600 Hz only 10%-20% and 5%-10% of NSs would have spectral features as deep as those seen in EXO 0748-676.

The Late time singularity inside nonspherical black holes

It was long believed that the singularity inside a realistic, rotating black hole must be spacelike. However, studies of the internal geometry of black holes indicate a more complicated structure is typical. While it seems likely that an observer falling into a black hole with the collapsing star encounters a crushing spacelike singularity, an observer falling in at late times generally reaches a null singularity which is vastly different in character to the standard Belinsky, Khalatnikov and Lifschitz (BKL) spacelike singularity. In the spirit of the classic work of BKL we present an asymptotic analysis of the null singularity inside a realistic black hole. Motivated by current understanding of spherical models, we argue that the Einstein equations reduce to a simple form in the neighborhood of the null singularity. The main results arising from this approach are demonstrated using an almost plane symmetric model. The analysis shows that the null singularity results from the blueshift of the late-time gravitational wave tail; the amplitude of these gravitational waves is taken to decay as an inverse power of advanced time as suggested by perturbation theory. The divergence of the Weyl curvature at the null singularity is dominated by the propagating modes of the gravitational field. The null singularity is weak in the sense that tidal distortion remains bounded along timelike geodesics crossing the Cauchy horizon. These results are in agreement with previous analyses of black hole interiors. We briefly discuss some outstanding problems which must be resolved before the picture of the generic black hole interior is complete.

MEASURING NEUTRON STAR RADII VIA PULSE PROFILE MODELING WITH NICER

The Neutron-star Interior Composition Explorer (NICER) is an X-ray astrophysics payload that will be placed on the International Space Station. Its primary science goal is to measure with high accuracy the pulse profiles that arise from the non-uniform thermal surface emission of rotation-powered pulsars. Modeling general relativistic effects on the profiles will lead to measuring the radii of these neutron stars and to constraining their equation of state. Achieving this goal will depend, among other things, on accurate knowledge of the source, sky, and instrument backgrounds. We use here simple analytic estimates to quantify the level at which these backgrounds need to be known in order for the upcoming measurements to provide significant constraints on the properties of neutron stars. We show that, even in the minimal-information scenario, knowledge of the background at a few percent level for a background-to-source countrate ratio of 0.2 allows for a measurement of the neutron star compactness to better than 10% uncertainty for most of the parameter space. These constraints improve further when more realistic assumptions are made about the neutron star emission and spin, and when additional information about the source itself, such as its mass or distance, are incorporated.

Rotational Corrections to Neutron-Star Radius Measurements from Thermal Spectra

We calculate the rotational broadening in the observed thermal spectra of neutron stars spinning at moderate rates in the Hartle-Thorne approximation. These calculations accurately account for the effects of the second-order Doppler boosts as well as for the oblate shapes and the quadrupole moments of the neutron stars. We find that fitting the spectra and inferring the bolometric fluxes under the assumption that a star is not rotating causes an underestimate of the inferred fluxes and, thus, radii. The correction depends on the stellar spin, radius, and observers inclination. For a 10 km neutron star spinning at 600 Hz, the rotational correction to the flux is ~1-4%, while for a 15 km neutron star with the same spin period, the correction ranges from 2% for pole-on sources to 12% for edge-on sources. We calculate the inclination-averaged corrections to inferred radii as a function of the neutron-star radius and mass and provide an empirical formula for the corrections. For realistic neutron star parameters (1.4 M

THE IMPACT OF SURFACE TEMPERATURE INHOMOGENEITIES ON QUIESCENT NEUTRON STAR RADIUS MEASUREMENTS

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