Low bounds for pulsar γ-ray radiation altitudes

K. J. L 2 3 4 Y. J. Du 1 H. G. Wang 0 G. J. Qiao 2 R. X. Xu 2 J. L. Han 1 0 Center for Astrophysics, Guangzhou University , Guangzhou 510400, China 1 National Astronomical Observatories, Chinese Academy of Sciences , 20A Datun Road, Chaoyang, Beijing 100012, China 2 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University , Beijing 100871, China 3 Max-Planck-Institut fur Radioastronomie , Auf dem Hugel 69, 53121 Bonn, Germany 4 The University of Manchester, School of Physics and Astronomy, Jodrell Bank Centre for Astrophysics , Alan Turing Building, Manchester M13 9PL A B S T R A C T The observational determination of radiation locations can constrain pulsar radiation models. The -B process in a strong magnetic field is one of the fundamental physical processes contributing to pulsar radiation mechanisms. Photons generated near a pulsar surface with sufficient energy will be absorbed in the magnetosphere. Considering aberrational, rotational and general relativistic effects, we calculate the -B optical depth for -ray photons, and we use the derived optical depth to determine the lower bounds of the radiation altitude for photons with given energies. As a case study, we obtain the low bounds of radiation altitude for the Crab pulsar for photons with energies of 5 GeV-1 TeV. 1 I N T R O D U C T I O N 2.1 Analytical approach for simple geometry cm1. res,[cm] = 2.6 104B02,/[512][MeV]p[s]1/5W 2/5 2/5 which has the asymptotic approximation (1.1 105 1.9 103 ln th)B0,[12][MeV]p[s]1/5. 2/5 2/5 Among these effects, the curved spacetime effect and the frame-dragging effect are of higher order compared with other effects, which is consistent with the results of Gonthier & Harding (1994). Therefore, we only need consider the aberration and magnetic field rotation effects to correct for geometrical effects to the order of 102 for a short-period pulsar (p 0.1 s) and higher altitude absorption (r100 km). Thus, we can simply use a flat spacetime geometry (including photon direction and magnetic field direction) to calculate the geometrical parameters (e.g. i). However, gravitational effects also play two other important roles in B processes. T ij 0 = n,z nB,z where matrix L is defined as vx = rs cos s sin + cos cos s sin s . g = + 2[6 cos s drs2 2(3h + f sin )] sin s sin s , 2gb n,z = f cos 2b3h tan , (i) For a radiation source located at {rs, s, s}, we calculate the photon-propagating direction n with equation (7). (ii) We calculate the photon position after time t using x(t) = rs + n t. (iii) We solve equation (10) to calculate the photon position in magnetic polar coordinates (r, , ) from the coordinates of x(t). (iv) We use the photon position (r, , ) to calculate the magnetic field strength according to equation (9). We then calculate B using B = B B ( B n )2. (v) We use (rs) = 0 (t ; rs) dt to calculate the optical depth for a photon coming from position rs. is calculated from equation (1). (vi) We solve (res) = th with respect to res to determine the low bounds for radiation altitude, given the required th. There is no analytical solution to (res) = 1 with respect to res. We use a bisection method to solve it numerically, while the integration is performed using an adaptive integration method to refine a preset logarithmic mesh of t to achieve the necessary numerical precision. The results are shown in Fig. 3. We also obtain the results for th = 3 and th = 10; these plots are very similar to Fig. 3 due to the logarithmic dependence of th. 2.3 Radiation geometry and phase-resolved low bounds for radiation altitudes Here, we determine the radiation location for different longitudinal phases. The details of radiation geometry can be found in Gil, Gronkowski & Rudnicki (1984), Lyne & Manchester (1988) and Lee et al. (2006). We omit aberration effects here, because they constitute a second-order effect when calculating the pulse phase. Given the pulse-profile longitude (see Fig. 2 for details) and the view angle , the half angular beamwidth for the radiation beam can be solved by (Gil et al. 1984; Lyne & Manchester 1988) = sin2 s = arccos sin4 10 sin2 + 9 sin2 2.4 Application to the Crab pulsar 3 D I S C U S S I O N A N D C O N C L U S I O N S 1 Rankin (1990) obtained = 86. AC K N OW L E D G M E N T S R E F E R E N C E S This paper answers a different question compared with Baring (2004). We are discussing the altitude low bound for observed photons with a specific energy, while Baring (2004) discussed the spectral cut-off for collective photons. Nevertheless it is interesting to see the difference between the results of Baring (2004) and our results. R3 c B B sin i Bc/rcur B0 rs3 rcur due to the curvature of the magnetic field. The absorption criterion is therefore which is just Bcr B0 rs R0 p 0.4p If we take c/ = 17, we get the results of Baring (2004): 1/2 3 This paper has been typeset from a TEX/LATEX file prepared by the author. (...truncated)