Ciudadela Rio Cauca

where γ0 = γγb(1 − ββb) and K1 is the modified Bessel function of the first order.

4.1.1

Scattering process

Once the magnetospheric structure is specified, providing in input the polar value Bpol of the surface magnetic field, the twist angle ∆φN−S, the bulk velocity βb and

the temperature Tel of the electrons, the code follows the propagation of photons, as

they interact with the magnetospheric charges. For the sake of simplicity, general relativistic effects (described in section 3.2) are not accounted for in this case; in fact, typical computing times for each run are of about 30 min for processing ∼ 106 photons on an Intel core i7 2.30 GHz processor, and the full treatment that includes photon propagation along null geodesics would significantly increase the computing time, without adding much physical insight.

Photons are emitted from the cooling star surface with an assumed, isotropic black- boby distribution. The surface is divided into discrete, equal-area angular patches, each of them labelled with the values of the cosine of the magnetic colatitude µ = cos θ and the magnetic azimuth φ that identify its centre. The number of surface patches, their temperature and the number of photon emitted by each of them are inputs of the code. For the sake of generality, in the following I take an uniform temperature T = 0.5 keV on the whole surface, as well as an equal number of photons emitted from each patch. Also the intrinsic polarization fraction can be controlled patch by patch, choosing if photons are emitted all polarized in the X- or in the O-mode (Πe

L = 1), or if the number

of ordinary and extraordinary photons is equal (Πe

L = 0). For the reasons discussed in

the previous chapters, also in this case I assume that seed photons are 100% polarized in the X-mode for all the surface patches.

After the emission, the code evaluates the typical distance l traveled by a photon between two consecutive scatterings in the region where condition (1.24) for which RCS occurs is met. This is done integrating the infinitesimal optical depth

dτi−j = dl

Z βmax

βmin

dβ ¯neγ3(1 − β cos θBk)σi−jfe, (4.2)

where i, j = O, X and the first (second) index refers to the photon polarization mode before (after) scattering. Furthermore, in the previous equation, [βmin, βmax] is the

charge velocity spread, ¯ne is the particle density (1.23) integrated over the velocities,

σi−jare given by equations (1.29) and fe= γ−3n−1e dne/dβ (see equation 4.1) is the mo-

mentum distribution function (Nobili, Turolla & Zane, 2008a). Introducing an uniform deviate U , the value of l is fixed by the condition

τi =

Z l

0

dτi = − ln U , (4.3)

4.1. MONTE CARLO CODE 71 Integral (4.2) is performed by using a stepwise, fourth-order Runge-Kutta method. The integration is stopped as τi≥ − ln U , and l is then obtained by extrapolating the

values in the last two integration steps. Besides a reduction of the computational time required for a run, this approach is convenient also because it allows to control at each step if the resonant condition ω = ωD holds (see equation 1.24). This happens when

the two roots β1,2 = 1 cos2_θ Bk+ (ωB/ω)2 h cos θBk± ωB ω p (ωB/ω)2+ cos2θBk− 1 i (4.4)

obtained solving the resonant condition for β, are real, i.e. for (ωB/ω)2 ≥ sin2θBk.

If at a certain step this condition is found to be not satisfied, the code verifies if the photon moves towards the region where RCS is allowed or not; this is made possible by calculating numerically the tangent to the photon trajectory at (ωB/ω)2 ' sin2θBk.

If yes, the procedure is repeated, otherwise the photon is left to freely propagate up to infinity, without any further interaction, and the code proceeds with the emission of a new photon (up to reach the maximum photon number set in input). Actually, to avoid that photons remain trapped in the magnetosphere, a maximum number of scatterings Nmax = 1000 is fixed, after which the photon is considered to be re-absorbed by the

star surface.

Energy and direction of photons that scatter onto magnetospheric particles can be obtained from the cross sections (1.29). However, since the latter depend on the polarization modes of the incident and scattered photons, it is necessary to determine in what polarization state a photon initially polarized in the mode i emerges after the interaction. To this end a new uniform deviate U1 is defined: if it results

U1 >

σi−i

σi−i+ σi−j

(4.5) the photon will change its original polarization state. It has to be noticed that, since σO−O/(σO−O + σO−X) = 1/4 and σX−X/(σX−X + σX−O) = 3/4 (see equations 1.29),

it is more likely for an ordinary photon to change its polarization mode, while an extraordinary photon tends to maintain its original state. Given that the seed radiation is assumed to be polarized 100% in the X-mode, the number of O-mode photons that are expected to arise due to RCS is quite small (see Nobili, Turolla & Zane, 2008a). In doing this calculation, the photon polarization mode between two consecutive scatterings has been kept fixed: the validity of this approximation will be verified below.

The scattering electron velocity is decided between the two values β1 and β2 given

by equation (4.4) through a similar method, by definining the random number U2 and

comparing it with the ratios RO = S1(β1) S1(β1) + S1(β2) RX= S2(β1) S2(β1) + S2(β2) , (4.6) where S1(β1,2) = | cos θBk− β1,2| 1 − cos θBkβ1,2 fe(β1,2) S2(β1,2) = 1 − cos θBkβ1,2 | cos θBk− β1,2| fe(β1,2) . (4.7)

72 CHAPTER 4. POLARIZED EMISSION FROM MAGNETARS So, when U2 < Ri (U2 > Ri) the i-mode photon scatters onto an electron with velocity

β1 (β2). The photon frequency ω0 after the interaction is then given by

ω0 = γ_1,22 ω(1 − β1,2cos θBk)(1 + β1,2cos θ0) , (4.8)

where, according to the angular dependence of the RCS cross sections (see Nobili, Turolla & Zane, 2008a), the magnetic colatitude θ0 and azimuth φ0 of the scattered photon are given by cos θ0 = 2U3 − 1 and φ0 = 2πU4, respectively, with U3, U4 two

further uniform deviates.