Erratum: “Determining Neutron Star Masses and Radii Using Energy-resolved Waveforms of X-Ray Burst Oscillations” (2013, ApJ, 776, 19)

Abstract

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 ′

Constraining the Neutron Star Mass–Radius Relation and Dense Matter Equation of State with NICER. II. Emission from Hot Spots on a Rapidly Rotating Neutron Star

“…This allows us to check the normalizations of the outputs of the codes, as well as other results related to the sharp line tests (see Section 4.3). The results from all codes agree with the analytic expectations to high precision as described in Appendix C. Some of these tests, as well as others not listed here, are also discussed in Appendix A of Lo et al (2013).…”

“…The angular deflection of light rays from a static star and the propagation time relative to a radial ray given by the current code are in excellent agreement with the values computed using different algorithms by Lo et al (2013) and the values computed using Mathematica routines (see Lo et al 2013, sections A.1.1 and A.1.2). The angular deflection of a pencil beam from a small emitting spot on a rotating star is in excellent agreement with the value computed using a Mathematica routine (see Lo et al 2013, section A.1.6).…”

“…where dΩ is the solid angle subtended by the whole star. A spherical non-rotating star subtends an angle of (e.g., Lo et al 2013equation (A9), or Miller & Lamb 1993 (3))…”

Improved analytic modeling of neutron star interiors

“…One way to extract internal structure information is to measure the NS mass and radius independently [4][5][6], though current measurements may contain large systematic errors. An X-ray astrophysics payload NICER currently in operation at the International Space Station is expected to measure the stellar radius to ∼ 5% accuracy [7] with less systematics [8,9]. Another way to probe internal structure is to measure tidal deformabilities of neutron stars via gravitational waves.…”

Finite size effects on the light curves of slowly-rotating neutron stars

“…Moreover, they encode information of the spacetime curvature around the star and thereby of the bulk properties of the star. One of the main goals of the ongoing Neutron star Interior Composition ExploreR (NICER) [15][16][17] mission, is to measure light curves from a number of sources with unprecedented time-resolution, constraining their masses and radii within 5−10% accuracy in optimal cases [18][19][20] (see also e.g. [21][22][23]).…”