Mass‐Radius Constraints from a Pulse Shape Model for Hercules X‐1

Abstract

ABSTRACTAn accretion column model is applied to reproduce the pulse shape of Her X-1. The motivation for the modelis the observed 35 day sequence of pulse shape changes, which imply a pencil beam from the near pole and a fanbeam from the far pole. The pulse shape depends on the beam pattern, which is calculated for an emitting surfacein the shape of a filled cone. The calculation has a parameterized emissivity function and includes gravitationallight-bending and shadowing effects. Least-squares fitting of the model to the observed pulse profile yields best-fit parametersandstatistical uncertaintiesforthe viewinggeometry,thegeometry oftheaccretioncolumn,andthemass-to-radius ratio of the neutron star. Thederived mass-to-radius ratio is 0.121–0.128M km 1 (3 statisticalrange).CombinedwiththemassdeterminedfromanalysisoftheorbitofHerX-1,severalrecentequationsofstatefor dense matter are excluded for the neutron star in Her X-1.Subject headings: binaries: close — stars: individual (Hercules X-1) — stars: neutron — X-rays: binaries1. INTRODUCTIONHerX-1/HZHerisaneclipsingsystemthatcontainsa1.24speriod pulsar in a 1.7 day circular orbit with its optical com-panion HZ Her. The system also displays a 35 day cycle (Scott& Leahy 1999), which consists of high and low X-ray fluxstates lasting about 10 and 5 days, respectively, separated by10 day long low states. X-ray pulsations are visible duringthe high states but cease during the low states. The 35 daycycle is caused by a counterprecessing, tilted, twisted accre-tion disk. The disk is responsible for the evolution of thepulse profile (Scott et al. 2000), for reprocessing much ofthe emission during the short high state (Leahy 2000), for the35 day X-ray light curve (Leahy 2002), and for shadowingthe X-ray illumination of the companion star HZ Her (Leahy& Marshall 1999; Leahy et al. 2000). The main conclusion,derived from the change in pulse shape due to occultation bythe inner-disk edge (Scott et al. 2000), was that the neutronstaremitsapencilbeamfromitsnearpoleandafanbeamfromits far pole.The purpose of this paper is to present the results ofleast-squares fitting a quantitative pulse shape model to theobserved pulse shape of Her X-1. Hollow-cone models foraccretion columns are discussed in Kraus (2001) and Leahy(2003). Several authors (e.g., Leahy & Li 1995) have dis-cussed flat emission regions. Leahy (2004, hereafter Paper I)showed that only a filled-cone model could fit the observedpulse shape of Her X-1 and gave parameters for one filled-cone model that reproduced the pulse shape. Here the calcu-lation is extended and used to explore the full parameter spaceof filled-cone models. This allows an improved pulse shapemodel to be constructed and allows confidence limits to beplaced on the parameters, including limits on the accretioncolumn parameters and on the mass-to-radius ratio of theneutron star.2. OBSERVED PULSE SHAPE OF HER X-1Her X-1 is obscured much of the time during its 35 daycycle by its accretion disk (Scott et al. 2000; Scott & Leahy1999) but not during the early part of main high state. Thepulse shape from early main high in the energy band 9–14keVwasderivedfromRXTE PCAobservationsduring1997September for Paper I. That pulse profile consists of 64 phasebins, has three characteristic peaks, and is used for the mod-eling here. In addition to the 9–14 keV pulse profile, pulseprofileswerederivedforthe2–5,5–9,14–20,and20–60keVenergy bands. Poisson statistics was assumed for the statisticaluncertainty in count rate in each bin.3. THE EMISSION REGION MODELThere are two accretion columns (referred to as cone 1 andcone 2), one for each magnetic pole, each emitting from thetop and sides. The top is spherical and produces a pencilbeam; the side is conical and produces a fan beam. Figure 1shows a wire-frame image of the two columns on the neutronstar, and Table 1 lists the parameters. Beam patterns and pulseprofiles from the model emission regions were calculated asdescribed in Paper I, with some differences described below.The observed flux at each rotation phase is calculated by in-tegrating differential flux, dF( ) ¼ I( )d ,overanimagesurface far from the neutron star. The intensityI( ) and solidangle element d are related to quantities at the emittingsurface by using ray tracing in the Schwarzchild geometry.Effects of blockage of the emission surface by the neutron staror by other parts of the emission surface are included. Thedetails are described in Leahy (2003).Paper I showed that the two columns are offset from therotation axis by independent angles, but