The Effect of Different Magnetospheric Structures on Predictions of Gamma-ray Pulsar Light Curves
aa r X i v : . [ a s t r o - ph . H E ] J a n The Effect of Different Magnetospheric Structures onPredictions of Gamma-ray Pulsar Light Curves
M Breed , C Venter , A K Harding , T J Johnson Centre for Space Research, North-West University, Potchefstroom Campus, Private BagX6001, Potchefstroom, 2520, South-Africa Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771,USA NRC Fellow, High-Energy Space Environment Branch, Naval Research LaboratoryE-mail:
Abstract.
The second pulsar catalogue of the
Fermi
Large Area Telescope (LAT) will containin excess of 100 gamma-ray pulsars. The light curves (LCs) of these pulsars exhibit a varietyof shapes, and also different relative phase lags with respect to their radio pulses, hinting atdistinct underlying emission properties (e.g., inclination and observer angles) for the individualpulsars. Detailed geometric modelling of the radio and gamma-ray LCs may provide constraintson the B-field structure and emission geometry. We used different B-field solutions, includingthe static vacuum dipole and the retarded vacuum dipole, in conjunction with an existinggeometric modelling code, and constructed radiation sky maps and LCs for several differentpulsar parameters. Standard emission geometries were assumed, namely the two-pole caustic(TPC) and outer gap (OG) models. The sky maps and LCs of the various B-field and radiationmodel combinations were compared to study their effect on the resulting LCs. As an application,we compared our model LCs with
Fermi
LAT data for the Vela pulsar, and inferred the mostprobable configuration in this case, thereby constraining Vela’s high-altitude magnetic structureand system geometry.
1. Introduction
Pulsars are considered to be cosmic lighthouses that rotate at tremendous rates and are highlymagnetized neutron stars (NS) [1]. The fact that pulsars are embedded in such extremeconditions make them valuable laboratories for studying a wide range of topics, including:nuclear physics, plasma physics, electrodynamics, magnetohydrodynamics (MHD), and generalrelativistic physics [2]. Pulsars emit radiation across the electromagnetic spectrum, includingradio, optical, X-ray and gamma ( γ ) rays [1]. We focus on γ -ray pulsars, specifically the Velapulsar, which was detected [3] by the Fermi
Large Area Telescope (LAT) [4]. The Vela pulsar isthe brightest persistent GeV source.
Fermi was launched in 2008 and has discovered in excessof 100 γ -ray pulsars. The LAT has a very large field of view of 2.4 sr which enables it to observe20% of the sky at any instant, scanning the entire sky in a time frame of a few hours. There are several models which can be used for the modelling of high-energy (HE) emission frompulsars. These models include the two-pole caustic (TPC) (the slot gap (SG) [5] model may igure 1.
A schematic representation of geometric pulsar models. The TPC emission regionextends from R NS (NS radius) up to R LC (light cylinder radius), OG region from R NCS (nullcharge surface radius) to R LC , and the PSPC from R NS to R LC [covering the full open volumeregion].be its physical representation) model [6], outer gap (OG) model [7],[8] and pair-starved polarcap (PSPC) model [9]. Figure 1 illustrates these geometric pulsar models, and their emissionregions [6].Consider an ( ~ Ω, ~µ ) plane, with ~µ (the magnetic moment) inclined by an angle α with respectto the rotation axis ~ Ω (the angular velocity). The observer’s viewing angle ζ is the angle betweenthe observer’s line of sight and the rotation axis. The gap region is defined as the region wherethe relativistic particles originate and particle acceleration takes place. The emissivity of HEphotons within this gap region is assumed to be uniform in the co-rotating frame and the γ -raysare expected to be emitted tangentially to the local magnetic field in this frame [10]. The gapregion for the TPC model extends from the surface of the NS along the entire length of thelast closed magnetic field lines, up to the light cylinder, as indicated by the dashed lines inFigure 1. For the OG model, the gap region extends from the null-charge surface (NCS), wherethe Goldreich-Julian charge density is ρ GJ “ The magnetospheric structures studied in this paper include the static [12] and retarded vacuumdipole [13]. The (aligned) static dipole is a special case of the retarded dipole and is describedby the following B-field equations in terms of spherical coordinates in the laboratory frame: st , r “ µr cos θ (1) B st ,θ “ µr sin θ. (2)The retarded dipole is described by the following B-field equations [14]: B ret , r “ µr r cos α cos θ ` sin α sin θ p r n sin λ ` cos λ qs (3) B ret ,φ “ ´ µr sin α rp r ´ q sin λ ` r n cos λ s (4) B ret ,θ “ µr p cos α sin θ ` sin α cos θ r´ r n sin λ ` p r ´ q cos λ sq (5) λ “ r n ` φ ´ Ω t (6) r n “ rR LC . (7)By setting r n equal to zero the retarded field simplifies to the general (non-aligned) staticdipole, with r the radial distance. The static dipole field is studied for numerous reasons. Twoof them are: (1) calculations are simpler for this B-field, and (2) when the results for the staticdipole are compared to those for the other B-fields, the importance of the near- R LC distortionsin the B-fields for predicted radiation characteristics can be gauged [15].In this paper we will study the impact of different magnetospheric structures on thepredictions of γ -ray pulsar LCs. The layout is as follows: § χ contour plots for the different combinations of the twoB-fields and two geometric models. Section 3 contains our results, § §
2. Method
We used an existing geometric modelling code [10] in which different B-field solutions andgeometric models are implemented. We constructed sky maps, which are defined as the intensityper solid angle as a function of phase and ζ , and LCs for the B-field and radiation model Table 1.
Best-fit ( α , ζ ) values for the Vela pulsar. Our model Reference fit [16] Radio polarization [17]Combination log χ α ( ) ζ ( ) α ( ) ζ ( ) α ( ) ζ ( ) Static Dipole :TPC 15.3 60 85OG 6.4 65 85
Retarded Dipole :TPC 15.7 70 55 62–68 64OG 1.3 80 70 75 64 53 59.5 ombinations, using a 5 resolution for α and ζ . LCs are obtained by making a constant- ζ cutthrough each sky map.After the preparation of the sky maps and LCs, a statistical method for finding the best fitsis applied. We used a χ method to compare our model LCs with Fermi
LAT data for the Velapulsar: χ “ N ÿ i “ p Y d , i ´ Y m , i q Y m , i , (8)with Y m , i the model (relative flux) value and Y d , i the measured number of counts (relativeunits) in each phase bin. First we lowered the model LCs resolution, so that both the modeland data have the same amount of bins N. Next, we smoothed the data using a Gaussian kerneldensity estimator (KDE). The data are treated as being cyclic. For computational efficiency, wealigned the maximum peaks of the model and data before calculating χ p α, ζ q . A contour plotof χ is shown in panel (b) of Figure 4.
3. ResultsFigure 2.
The sky maps (left) and LCs (right) as predicted from the TPC model using thestatic dipole field for different α and ζ values (deg).As an example, we show the sky maps and their corresponding LCs for the TPC model, forboth the static and retarded dipole fields (Figure 2 and 3). We used a maximum gap radius of R max “ . R LC for both the TPC and OG cases. There are different LCs on the right of eachsky map corresponding to different ζ -cuts. In both the figures there appear two dark circles, thePCs, followed by two sharp, bright regions near it, called the main caustics, on the sky map. igure 3. The sky maps (left) and LCs (right) as predicted from the TPC model using theretarded dipole field for different α and ζ values (deg). Figure 4.
Panel (a) indicates our best-fit LC for Vela (see Table 1). Panel (b) shows thecontour plot for χ p α, ζ q , indicating the best-fit solution.The caustic structure is qualitatively different between the two cases, leading to differences inthe resulting LCs. The caustics seem wider and more pronounced in the retarded dipole case.A thin line of emission, due to the ‘notch’ [14] is also visible in the latter case. For large α thecaustics extend over a larger range in ζ for the retarded case compared to the static case. TheOG models are not visible at all angle combinations and thus do not fill all phase space. Thisis due to emission that occurs below the null charge surface for the TPC model, but not for theOG model. The TPC model LCs also exhibit relatively more off-pulse emission. The LCs inthe OG models are due to emission from only one pole, while both poles are visible in the TPCmodel.he different model LCs are fitted to the Vela data of Fermi
LAT and for each model, weconstructed a χ contour plot which indicates the best possible fit. The white marker on thecontour plot (Figure 4, panel (b)) indicates the minimum value of log χ and these valuesare shown in Table 1. The first column shows our different combinations of magnetic field andgeometric model, the second indicates the minimum value of log χ , and the third and fourthcolumns indicate the best-fit α and ζ from our models. These are for the 5 resolution. We willestimate more rigorous errors on these values in future. The fifth column contains the derived ζ values from the pulsar wind nebula (PWN) torus fitting with α constrained [18]. The last twocolumns are derived from fits of the rotating vector model to the radio polarization angle (PA)versus phase φ [19]. The best-fit model LC to the Vela LC is shown in Figure 4, panel (a). Ourbest fit is close to values inferred from these independent studies.
4. Conclusions and future work
We have studied the effect of different magnetic fields on gamma-ray LC characteristics. Weutilized the static and retarded vacuum dipole solutions, in combination with the TPC and OGgeometries. It is evident that the magnetospheric structure and emission geometry determinethe pulsar visibility and also the γ -ray pulse shape. We applied our models to the Vela pulsarand found a best fit from the OG model using the retarded dipole field, for p α, ζ q “ p , q .This is reasonably close to the value of p α, ζ q “ p , q inferred by [16]. In future, we willimplement an additional magnetic field solution, the offset dipole [20] and study the effect ofthis solution on the predicted pulsar LCs. Acknowledgments
This work is supported by the South African National Research Foundation (NRF). A.K.H.acknowledges the support from the NASA Astrophysics Theory Program. C.V., T.J.J., andA.K.H. acknowledge support from the
Fermi
Guest Investigator Program.
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