An array of n photons is generated for the simulation with some initial location xi
in the magnetosphere and some energy E. Because the precise nature of VHE emission from pulsars is not well-understood, few detailed assumptions are made about the emission mechanism(s) producing the photons for the simulation. We do assume that VHE photons are emitted through some relativistic process (e.g. curvature radiation or inverse Compton scattering), and thus photons are subject to relativistic beaming. As discussed in Chapter 2, the various possible VHE emission mechanisms are limited to occurring only within a few defined theoretical gap regions within the pulsar magnetosphere. As such, the photons generated have emission locations randomly distributed throughout the slot gap (SG) and outer gap (OG) regions, each considered as a separate case in the simulation studies.
The SG region (depicted in Figure 6.3) begins at the polar cap of the neutron star. The region then extends to higher altitudes in the magnetosphere along the outside of the last closed field line, sin2θ/r = 1/Rlc for an aligned dipole, and has some thickness . This
emission region extends all the way out to the light cylinder Rlc. In many SG models the
thickness of the SG region becomes smaller at higher altitudes. This condition is relaxed for our simulations and the thickness of the region matches that of the OG region.
The OG region is bounded at low altitudes on the inside by the null surface, where Ω·B = 0 and on the outside by the light cylinder. The region extends out to the light cylinder along the outside of the last closed field line with some thickness that is limited physically by pair creation in the region. In this simulation the thickness is set to an arbitrary value. The OG region used is depicted in Fig. 6.2. To ensure the simulation code is working properly the initial positions of photons generated are plotted and shown in Fig. 6.2 for comparison and the x, y, z position distributions are shown in Fig. 6.4.
The emitted phase angle value φi is given based on the geometric emission location
and φ is the azimuthal angle measured in the xy-plane from the x−axis to the +y−axis with φ defined to be along the +ˆx direction. The observer lies in the x − z plane with an azimuthal angle φ = 0. The phase angle is given in units of 2π such that one full rotation of the neutron star goes from φ = 0 to φ = 1. Given the periodic nature of pulsars, one might naively expect a light curve to be symmetric as is the one shown in the top of Fig. 6.5(a). Typically observed light curves are not perfectly symmetric and this is due, in part, to the difference in times of flight required for photons emitted at different locations in the magnetosphere. This difference in time of flight for photons is accounted for by subtracting a phase-correction factor φc = −~r · ˆk/Rlc from the photons emission phase location φi to give
the observed phase φ.
Figure 6.2: Outer gap region used for simulation shown in orange shaded region with the boundaries of the light cylinder shown. Photons are given randomly seeded initial positions throughout the OG region.
Figure 6.3: Slot gap region used for simulation shown in orange shaded region with the boundaries of the light cylinder shown. Photons are given randomly seeded initial positions throughout the SG region.
(a) Distributions of photon emission locations separated into x, y, z coordinates.
(b) Plot of photon positions. Each photon is represented by an orange point.
Figure 6.4: Example of the distributions of photon emission locations for one simulation with ξ = π/4.
(a) (b)
Figure 6.5: (a) shows the light curves from one simulation with ξ = π/3 and ζ = π/4 for both the SG model (black) and the OG model (blue). The top plot is the light curve before any time of flight corrections are applied to the emission phase φi. The bottom plot in (a)
is the light curve for the same set of propagated photons with phase-corrections φc applied.
(b) is the distribution of values for phase-corrections applied to the propagated photons.
where ~k is the photon wave vector and we consider photons where ˆk points to the observer.
Fig. 6.5(b) shows the distribution of phase-correction factors applied to one simulation and the bottom plot in Fig. 6.5(a) shows the phase-corrected light curve.
While the emission mechanisms and emission regions are not fully understood, the energies of photons produced by pulsars can be defined using functions fit to observational data, with the caveat that the already observed spectrum includes the effects of pair absorption. For emission from pulsars in the 10 GeV to 300 GeV range, the spectral energy distribution is well-fit using a smoothly-broken power law of the form:
F (E) = A E E0 α" 1 + E E0 α−β#−1 (6.6)
using the parameters A = 1.45, E0 = 4, α = −1.96, and β = −3.52 (VERITAS Collaboration
et al., 2011). Photon energies are assigned using this distribution function for the range 100 GeV to 10 TeV. Frequency shifts due to relativistic effects are not considered. The distribution of energies for one simulation of 100000 photons is shown in Fig. 6.6. VHE
Figure 6.6: Distribution of photon energies used in simulations. Energies range from 100 GeV to 10 TeV. Photon energies are generated using a smoothly-broken power law function and parameters that have been fit to Fermi-LAT and VERITAS observations of the Crab Pulsar from VERITAS Collaboration et al. (2011).
photons in pulsar magnetospheres are not isotropically emitted. Due to the relativistic nature of the various particle acceleration processes that lead to the emission of VHE photons, these emitted photons are subject to relativistic beaming. This puts a constraint on the regions from which photons may reach an observer for various inclination angles ξ and observer angles ζ. It is assumed that emitted photons are beamed forward within a conical region centered along the magnetic field line at the photon’s emission location. If this conical region overlaps with the unit vector to the observer then a photon may be observed. The precise physical processes that determine the opening angle (e.g. turbulent variations in the field about the mean dipole field) are unknown; we therefore treat the opening angle as a
free parameter of the model to be determined empirically by matching the observed light curves. For the purposes of this study and partially due to limits of computational power, a wide beaming angle cut is applied, much larger than the naive 1/γ angle about the B−field direction.