RESULTADOS FINALES

Some pulsars have two pulses and others have only one. This can also be explained by both the inner magnetosphere emission model and the outer magnetosphere emission model.

Inner Magnetosphere Case: Conical Beam

The explanation of various light curves is schematically shown in Fig 2.18. If the pulsed radiation comes from near the pulsar surface, the emission region should be conically beamed. Let me

54 2. Pulsars

define a imaginary sphere named “emission profile sphere” whose center is on the pulsar center and whose radius is infinitely long. By projecting the conical beam to the emission profile sphere and opening it like a Mercator chart, one can make a “emission profile map” as shown in the figure. The longitude of the map corresponds to the pulse phase while the latitude corresponds to the viewing angle, i.e. the angle between the rotation axis and direction to the observer (the Earth). Depending on the viewing angle, a light curve can have two peaks or one peak. The pulse width can also be any value. By adjusting the angle between the rotation axis and the magnetic dipole axis, any phase separation between two peaks can be explained as well.

It should be noted that because of the rotation of the neutron star, the magnetic field will not be a perfect dipole, especially near the light cylinder. This will cause an asymmetry in the polar cap shape when the rotation axis and the dipole axis are not aligned (see e.g. [53]), leading to different shapes between the two pulses, as shown in the right panel of Fig. 2.18.

a b c Rotation Axis Pulsar _PC+SG Emission Phase c a b Viewing Angle

Figure 2.18: Explanation for some light curves when the emission region is in the inner magnetosphere. The mission should be a conical beam. Defining an imaginary sphere “emission profile sphere” (left) and opening it to “emission profile map” (middle), different light curves can be explained by the different view angles. Near the light cylinder, dipole approximation is not valid. Therefore, if the rotation axis and the dipole axis are not parallel, then the polar cap shape is distorted, which may cause an asymmetry between the two pulses. The right panel shows the distorted polar cap shape when the dipole axis is inclined by 45 degrees. Figure adopted from [53].

Outer Magnetosphere Case: Fan Beam

If the pulsed radiation comes from the outer magnetosphere, the light curves are explained as follows (see the left panel of Fig 2.20). Particles move along the magnetic fields near the last closed field lines. The last closed field lines in a three-dimensional view taking into account the relativistic rotation effect are shown in Fig. 2.19. All synchrotron, curvature and inverse Compton radiations are strongly beamed to the direction of the particle motions by an angle 1= , where is the Lorentz factor of the particle. Therefore, the tangential lines of the closed lines could be projected to the “emission profile map” defined above. In the case of an emission from the outer magnetosphere, however, one has to take into account two corrections when the

2.7 Non-thermal Radiations in the Pulsar Magnetosphere 55

Figure 2.19: A three dimensional view of the last closed lines. The angle between the dipole and the rotation axis is 50 degrees. The effect of relativity near the light cylinder is taken into account, which makes the lines asymmetric. Figure adopted from [177].

emission profile map is created: the relativistic aberration and travel time. When the direction of the tangential lineu= (u

x ;u

y ;u

z

), (juj=1), wherez is the rotation axis direction andxis the azimuthal direction, the direction of the photon u

0 = (u 0 x ;u 0 y ;u 0 z ), (ju 0 j = 1) would aberrate as (see [177]) u 0 x = u x 1 u x ; u 0 y = u y (1 u x ) ; u 0 z = u z (1 u x ) (2.41) (2.42) where =jrj=and, =(1 2 ) 1

2. The correction for the travel time in the pulsar phase would be = ru 0 R L (2.43) The top right panel of Fig. 2.20 shows the emission profile map assuming that the angle between the rotation axis and the magnetic axis is 65 degrees and that the emission comes only from the last closed field lines (gap width is zero, see e.g. [53]). By choosing a viewing angle such as, for example, 82 degrees, one can make a light curve as shown in the bottom right panel of the figure. Assuming that emissivity is uniform over all the field lines, the intensity of the pulsation is proportional to the density of the lines. Depending on the viewing angle, the light curve can have two peaks or one peak.

56 2. Pulsars

Viewing Angle

Phase

Emission projection from a single closed field line

000 000 111 111 00 00 00 00 00 00 11 11 11 11 11 11 0000 0000 1111 1111 Tangential line

Last Closed Lines

Pulsar

Travel Time and abberation correction

Figure 2.20: Schematic explanation of the emission profile map. Top left: tangential lines of the last closed field lines are projected onto the emission profile sphere (red arrows). Then, relativistic aberration and travel time effects are corrected (green arrows). From one closed field line, one emission profile line is drawn in this way (green line). Bottom left: The emission profile line from a single last closed line on the emission profile map. Top right: Emission profile map composed of emission profile lines from all last closed field lines. The angle between the dipole and rotation axis is assumed to be 65 degrees. Figure adopted from [53]. Bottom right: Pulsar light curve based on the above emission profile map, assuming that the viewing angle is 82 degrees and that emissivity is uniform along the lines. Figure adopted from [53].