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Volume 532, August 2011
Article Number A76
Number of page(s) 14
Section Stellar structure and evolution
Published online 26 July 2011

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Appendix A: Calculation of the observed flux from an accretion column

We present a relativistic ray-tracing computations of the beam pattern emitted from the accretion column surface of a slowly rotating neutron star to model the phase lags observed in the X-ray pulse profiles of 4U 0115 + 63. We consider the geometrical properties of the column and the relativistic effects, i.e., light bending and the lensing effect in a Schwarzschild metric. Since the NS is slowly rotating, the relativistic travel time delay and the Doppler boosting have not been taken into account. The observer is located at positive infinity on the z-axis in the non-rotating observer frame with polar coordinates (r,θ,φ). The accretion column reference frame has the polar coordinates (r′,θ′,φ′) with the z′-axis coincident with the accretion column axis. See Fig. A.1 for the geometry of the systems. The deflection angle of a photon emitted from the accretion column surface is ψ, while α indicates the angle between the local radial direction and the photon direction at the emission point.

A.1. Light bending

The geodetic equation for an emitting position of a photon depends on the impact parameter, b (Misner et al. 1973), which is defined as (A.1)where r is the radial coordinate of the emitting surface element, and rs = 2GM/c2 is defined as the Schwarzschild radius.

The relation between α and the photon deflection angle, ψ, i.e., the light bending, can be directly obtained using the approximated null geodetic equation, strictly valid for α < π/2 (see Beloborodov 2002, for more details): (A.2)For each column surface element at a given column orientation, ψobs, (see Eq. (2)), we determine the angle, α, and ,therefore, ψ, at which the emitted radiation reaches the observer (the only trajectories considered for the pulse profile modeling10).

thumbnail Fig. A.1

The geometry of the systems (not in scale). The dashed line shows a photon trajectory from the accretion column surface to the observer located at infinity, the dot-dashed line is the rotational axis of the NS.

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For photons emitted towards the NS surface (α > π/2), the allowed α values can be estimated by imposing that the trajectory not be swallowed by the relativistic photosphere of the compact object. This translates into the relation π/2 < α < αmax, where (A.3)and r is the radial coordinate at which the photon is emitted from the column (e.g., point B in Fig. A.2). These trajectories have a “turning point” and thus possess a periastron; it is then convenient to express all relevant quantities in terms of the periastron distance, p. For αmax > αB > π/2, we first compute the impact parameter using Eq. (A.1), and then estimate the value p of the periastron by taking the largest real solution of the equation p3 − b2p + b2rs = 0. This equation is obtained by setting α = π/2 and r = p in Eq. (A.1), as αp = π/2 at the periastron. Using Eq. (A.2), we find ψp = cos-1(1 − 1/[1 − rs/p]) . The photon trajectory again reaches the radial coordinate r at point A (see Fig. A.2), where the angle formed by the photon with the local radial direction is αA = π − αB. The angle ψA can now be computed from Eq. (A.2). Finally, from simple symmetry considerations, (A.4)This characterizes the considered trajectory fully, since the impact parameter for ψA has the same value as for the deflection angle at the emission point, ψB.

thumbnail Fig. A.2

Schematic representation of a photon trajectory (dashed line) from the accretion column surface to the distant observer for trajectories with a turning point (not in scale). The emission point is B and the angle to be computed is ψB. A is the point at which the trajectory reaches for the second time the radial coordinate at which it originated. p indicates the location of the trajectory’s periastron.

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For each photon-emitting point we can now follow the photon trajectory to verify if it is absorbed by the NS or accretion column surface. For α < π/2, the trajectory can be directly computed using the following equation (Beloborodov 2002): (A.5)where values are in the range (0,ψ).

For α > π/2, we use Eq. (A.1) to link to along the trajectory (the impact parameter b is constant in each geodesic). The corresponding value of is found with the method outlined above. The trajectories that hit the NS or intersect one of the optically thick accretion columns are excluded from the computation of the source flux measured at the observer’s location.

A.2. Lensing and red shift

We now compute the flux emitted by the column and measured by a distant observer. The flux emitted by a surface element from the accretion column, when it reaches the observed plane normal to the direction of the line of sight, is given by dFν = IνdΩ, where the solid angle can be expressed in terms of the impact parameter (Misner et al. 1973): (A.6)Here, φ is the azimuthal angle around the line of sight of the plane containing the trajectory of the photon, and Iν(b,φ) is the differential intensity of the radiation emitted by the surface element when it reaches the observed plane at a distance D.

The impact factor b depends on r and α, but not on φ. The angle α can be expressed as a function of r and ψ, as shown in the previous section. We further note that the differential in Eq. (A.6) can be expressed as db = (db/dr)ψdr + (db/dψ)rdψ, and is computed along a line on a geometrical cone at constant φ, which is by definition a radial line. Therefore we can simplify the differential and write db = (db/dr)ψdr, since ψ is constant along a radial line.

Up to this point we expressed the equations in the observer’s reference frame defined above. To integrate the emission on the column surface, we transform the integration variables to the column’s frame to use quantities directly related to the accretion column. The Jacobian of the coordinate transformation involves only the variables r (constant) and φ, and is: |dφ/dφ′|; therefore, Eq. (A.6) can be written as (A.7)We still need to express Iν in the system of reference of the column to account for the gravitational red-shift. Since I/ν3 is a Lorentz invariant, we can write (A.8)As already described in Sect. 4, knowing the exact functional shape of I0,ν0(r,α) would require a detailed treatment of the cyclotron scattering process in a strong magnetic field (B ~ 1012 G) and of the relativistic beaming caused by the bulk motion of the plasma in the accretion column above the surface of the NS. This is outside the scope of the present paper. Instead, we aim at reproducing the beam pattern in the energy range of interest (>10 keV) found by using the pulse decomposition method described in Sasaki et al. (2011). For this purpose, we use a combination of two Gaussians, one pointed downwards and one upwards characterized by different amplitudes: (A.9)\newpageIn the equation above, α1 = 150°, α2 = 0°, σ1 = 45°, σ2 = 45°, N1 = 100, and N2 assumed values from 0 to 100 to mimic a variable contribution from the cyclotron scattered radiation (See also Sect. 4).

The total flux emitted by the column is finally estimated from Eqs. (A.7)–(A.9) performing a Monte-Carlo integration with φ′ ∈ (0,2π), and r ∈ (rNS,rmax). The maximum height for the accretion column is equal to rmax − rNS. We did not include in the total flux the contribution of the photons that hit the NS surface and/or are absorbed by the column.

To verify the consistency of our results with similar calculations published before, we show in Fig. A.3 the beam patterns obtained from an isotropic emission on the lateral surface of an accretion column with the same configuration adopted in Fig. 3 of Riffert & Meszaros (1988). We found fairly good agreement between our results and those published by these authors. There is only a negligible discrepancy in the estimated fluxes of an accretion column extending up to 3.9 gravitational radii above the NS surface. Exact calculations predict that the critical height of a column for its total obscuration in the antipodal position should be 3.94rs. With the approximation used in our calculations, we obtained a value of 3.85rs. This discrepancy can only be relevant for a height of the column that is close to this critical value and for angles close to the antipodal position. Our approximation thus gives a satisfiable result for the accuracy required in the present work.

thumbnail Fig. A.3

Column beam pattern seen by a distant observer and calculated for an isotropic emission in the case of a NS with a radius r = 3rS. The accretion column is assumed to have an half-aperture of 5 degrees, and an heights of (from bottom to top) 3.03, 3.15, 3.3, 3.6, 3.9, 4.2, and 4.6 rS. The angle ψ is measured from the line of sight to the axis of the column.

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© ESO, 2011

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