Simultaneous, precise measurements of the mass M and radius R of neutron stars can yield uniquely valuable information about the still uncertain properties of cold matter at several times the density of nuclear matter. One method that could be used to measure M and R is to analyze the energy-dependent waveforms of the X-ray flux oscillations seen during some thermonuclear bursts from some neutron stars. These oscillations are thought to be produced by X-ray emission from hot regions on the surface of the star that are rotating at or near the spin frequency of the star. Here we explore how well M and R could be determined by analyzing energy-resolved X-ray data obtained using a future space mission having 2-30 keV energy coverage and an effective area of 10 m 2 , such as the proposed LOFT or AXTAR missions. We do this by generating energy-dependent synthetic observed waveforms for a variety of neutron star and hot spot properties and then using a Bayesian approach and Markov chain Monte Carlo sampling methods to determine the joint posterior probability distributions of the parameters in our waveform model, given each synthetic waveform. We use the resulting posterior distributions to determine Bayesian confidence regions in the M -R plane by marginalizing the other parameters in our model. We explore how the sizes and positions of these confidence regions depend on the inclinations of the hot spot and the observer, the background count rate, and deviations in the actual shape of the hot spot, radiation beaming pattern, and spectrum from those assumed in the waveform model. We also explore the effect on the confidence regions if the distance to the star or the inclination of the observer are known from other measurements, if a resonance scattering line is observed in the burst oscillation spectrum, or if the properties of the background are independently known. We assume that about 10 6 counts are collected from the hot spot and that all sources of background contribute about 0.3 × 10 6 , 10 6 , or 9 × 10 6 counts.We find that the uncertainties in the measured values of M and R depend strongly on the inclination of the hot spot relative to the spin axis. If the hot spot is within 10 • of the rotation equator, both M and R can usually be determined with an uncertainty of about 10%. If instead the spot is within 20 • of the rotation pole, the uncertainties are so large that waveform measurements alone provide no useful constraints on M and R. Observation of an identifiable atomic line in the hot-spot emission always tightly constrains M/R; it can also tightly constrain M and R individually if the spot is within about 30 • of the Following the discovery of thermonuclear X-ray bursts from neutron stars in the mid-1970s (Grindlay et al. 1976Lewin et al. 1976; for a review, see Lewin et al. 1993) and the discovery two decades later that some of these bursts produce X-ray flux oscillations at or near the star's spin frequency (Strohmayer et al. 1996; for a review, see Watts 2012), several approaches have been proposed f...
We investigate further a model of the accreting millisecond X-ray pulsars we proposed earlier. In this model, the X-ray-emitting regions of these pulsars are near their spin axes but move. This is to be expected if the magnetic poles of these stars are close to their spin axes, so that accreting gas is channeled there. As the accretion rate and the structure of the inner disk vary, gas is channeled along different field lines to different locations on the stellar surface, causing the X-ray-emitting areas to move. We show that this "nearly aligned moving spot model" can explain many properties of the accreting millisecond X-ray pulsars, including their generally low oscillation amplitudes and nearly sinusoidal waveforms; the variability of their pulse amplitudes, shapes, and phases; the correlations in this variability; and the similarity of the accretion-and nuclear-powered pulse shapes and phases in some. It may also explain why accretion-powered millisecond pulsars are difficult to detect, why some are intermittent, and why all detected so far are transients. This model can be tested by comparing with observations the waveform changes it predicts, including the changes with accretion rate.
We have shown previously that many of the properties of persistent accretion-powered millisecond pulsars can be understood if their X-ray emitting areas are near their spin axes and move as the accretion rate and structure of the inner disk vary. Here we show that this "nearly aligned moving spot model" may also explain the intermittent accretion-powered pulsations that have been detected in three weakly magnetic accreting neutron stars. We show that movement of the emitting area from very close to the spin axis to ∼ 10 • away can increase the fractional rms amplitude from 0.5%, which is usually undetectable with current instruments, to a few percent, which is easily detectable. The second harmonic of the spin frequency usually would not be detected, in agreement with observations. The model produces intermittently detectable oscillations for a range of emitting area sizes and beaming patterns, stellar masses and radii, and viewing directions. Intermittent oscillations are more likely in stars that are more compact. In addition to explaining the sudden appearance of accretion-powered millisecond oscillations in some neutron stars with millisecond spin periods, the model explains why accretion-powered millisecond oscillations are relatively rare and predicts that the persistent accretionpowered millisecond oscillations of other stars may become undetectable for brief intervals. It suggests why millisecond oscillations are frequently detected during the X-ray bursts of some neutron stars but not others and suggests mechanisms that could explain the occasional temporal association of intermittent accretion-powered oscillations with thermonuclear X-ray bursts.
The physical properties of the stellar atmosphere that appear in the radiation transport and other equations that would determine the properties of the hot spots considered in this work are defined in a local inertial frame that is momentarily comoving with the rotating atmosphere of the star and naturally yield the sizes, shapes, temperature distributions, and other properties of the hot spots as functions of linear (rather than angular) dimensions in the atmosphere, when measured in the frame comoving with the atmosphere. For example, computations of the evolution of the heated region produced by a thermonuclear X-ray burst yield the linear dimensions of the heated region as a function of time when measured in the comoving frame.In this work, the star was assumed to be uniformly rotating and spherical, the exterior spacetime was assumed to be the Schwarzschild spacetime, and for simplicity, the dimensions of the phenomenological hot spots that were considered were defined in angular rather than linear coordinates. We describe physics within the stellar atmosphere using a spherical polar coordinate system that corotates with the star and define the colatitude θ ′ in this coordinate system as the colatitude θ in the Schwarzschild coordinate system. Angular separations are unchanged when one transforms from the rotating frame to the static frame of the Schwarzschild spacetime, so the angular separations dθ and dφ of two points on the stellar surface when measured in the Schwarzschild coordinate system are the same as the angular separations dθ ′ and dφ ′ of the same two points on the stellar surface when measured in the rotating frame.In a differential time dt ′ , a burning or heating front that advances with speed v f in the local comoving frame will cover a linear differential distance dx ′ = v f dt ′ as measured in the comoving frame. The expression for the differential area dS ′ defined by the locally orthogonal differential linear coordinate intervals dx ′ and dy ′ on the stellar surface is simply dS ′ = dx ′ dy ′ when measured in the rotating frame by observers who use centrally synchronized clocks (see § IV of Kassner 2012). In contrast, the expression for the differential surface area dS ′ defined by the differential angular coordinate intervals dθ ′ and dφ ′ is dS ′ = R 2 sin θ ′ dθ ′ dφ ′ γ(θ ′ ) when measured in the rotating frame (again see § IV of Kassner 2012). Here R is the radius of the star in the corotating frame and γ(θ ′ ) ≡ 1/[1 − (v φ (θ ′ )/c) 2 ] 1/2 is the Lorentz gamma factor for the linear azimuthal speed v φ (θ ′ ) of the gas in the atmosphere at colatitude θ ′ as measured in the static frame, where v φ (θ ′ ) = Ω ′ R sin θ ′
Precise and accurate measurements of neutron star masses and radii would provide valuable information about the still uncertain properties of cold matter at supranuclear densities. One promising approach to making such measurements involves analysis of the X-ray flux oscillations often seen during thermonuclear (type 1) X-ray bursts. These oscillations are almost certainly produced by emission from hotter regions on the stellar surface modulated by the rotation of the star. One consequence of the rotation is that the oscillation should appear earlier at higher photon energies than at lower energies. Ford (1999) found compelling evidence for such a hard lead in the tail oscillations of one type 1 burst from Aql X-1. Subsequently, Muno,Özel & Chakrabarty (2003) analyzed oscillations in the tails of type 1 bursts observed using RXTE. They found significant evidence for variation of the oscillation phase with energy in 13 of the 51 oscillation trains they analyzed and an apparent linear trend of the phase with energy in six of nine average oscillation profiles produced by folding the energy-resolved oscillation waveforms from five stars and then averaging them in groups. In four of these nine averaged energy-resolved profiles, the oscillation appeared to arrive earlier at lower energies than at higher energies. Such a trend is inconsistent with a simple rotating hot spot model of the burst oscillations and, if confirmed, would mean that this model cannot be used to constrain the masses and radii of these stars and would raise questions about its applicability to other stars. We have therefore re-analyzed individually the oscillations observed in the tails of the four type 1 bursts from 4U 1636−536 that, when averaged, provided the strongest evidence for a soft lead in the analysis by Muno et al. (2003). We have also analyzed the oscillation observed during the superburst from this star. We find that the data from these five bursts, treated both individually and jointly, are fully consistent with a rotating hot spot model. Unfortunately, the uncertainties in these data are too large to provide interesting constraints on the mass and radius of this star.
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