Although only 22 millisecond pulsars (MSPs) are currently known to exist in
the globular cluster (GC) 47 Tucanae, this cluster may harbor 30-60 MSPs, or
even up to ~200. In this Letter, we model the pulsed curvature radiation (CR)
gamma-ray flux expected from a population of MSPs in 47 Tucanae. These MSPs
produce gamma-rays in their magnetospheres via accelerated electron primaries
which are moving along curved magnetic field lines. A GC like 47 Tucanae
containing a large number of MSPs provides the opportunity to study a
randomized set of pulsar geometries. Geometry-averaged spectra make the testing
of the underlying pulsar model more reliable, since in this case the relative
flux uncertainty is reduced by one order of magnitude relative to the variation
expected for individual pulsars (if the number of visible pulsars N=100). Our
predicted spectra violate the EGRET upper limit at 1 GeV, constraining the
product of the number of visible pulsars N and the average integral flux above
1 GeV per pulsar. GLAST/LAT should place even more stringent constraints on
this product, and may also limit the maximum average accelerating potential by
probing the CR spectral tail. For N=22-200, a GLAST/LAT non-detection will lead
to the constraints that the average integral flux per pulsar should be lower by
factors 0.03-0.003 than current model predictions.Comment: 10 pages, 2 figures, to appear in the Astrophysical Journal Letter

de Jager, O. C.

Venter, C.

English

arXiv

Although only 22 millisecond pulsars (MSPs) are currently known to exist in the globular cluster (GC) 47 Tucanae, this cluster may harbor 30-60 MSPs, or even up to ∼200. In this Letter, we model the pulsed curvature radiation (CR) gamma-ray flux expected from a population of MSPs in 47 Tucanae. These MSPs produce gamma rays in their magnetospheres via accelerated electron primaries which are moving along curved magnetic field lines. A GC such as 47 Tucanae containing a large number of MSPs provides the opportunity to study a randomized set of pulsar geometries. Geometry-averaged spectra make the testing of the underlying pulsar model more reliable, since in this case the relative flux uncertainty is reduced by 1 order of magnitude relative to the variation expected for individual pulsars (if the number of visible pulsars N = 100). Our predicted spectra violate the EGRET upper limit at 1 GeV, constraining the product of the number of visible pulsars N and the average integral flux above 1 GeV per pulsar. GLAST LAT should place even more stringent constraints on this product, and may also limit the maximum average accelerating potential by probing the CR spectral tail. For N = 22-200, a GLAST LAT nondetection will lead to the constraints that the average integral flux per pulsar should be lower by factors of 0.03-0.003 than current model predictions

De Jager, O.C.

NWU Institutional Repository  (North-West University)

L125The Astrophysical Journal, 680: L125–L128, 2008 June 20 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.CONSTRAINING A GENERAL-RELATIVISTIC FRAME-DRAGGING MODEL FOR PULSED RADIATIONFROM A POPULATION OF MILLISECOND PULSARS IN 47 TUCANAE USING GLAST LATC. Venter1,2 and O. C. de Jager1,2,3Received 2007 December 11; accepted 2008 May 12; published 2008 June 3ABSTRACTAlthough only 22 millisecond pulsars (MSPs) are currently known to exist in the globular cluster (GC) 47Tucanae, this cluster may harbor 30–60 MSPs, or even up to ∼200. In this Letter, we model the pulsed curvatureradiation (CR) gamma-ray flux expected from a population of MSPs in 47 Tucanae. These MSPs produce gammarays in their magnetospheres via accelerated electron primaries which are moving along curved magnetic fieldlines. A GC such as 47 Tucanae containing a large number of MSPs provides the opportunity to study a randomizedset of pulsar geometries. Geometry-averaged spectra make the testing of the underlying pulsar model morereliable, since in this case the relative flux uncertainty is reduced by 1 order of magnitude relative to the variationexpected for individual pulsars (if the number of visible pulsars ). Our predicted spectra violate theN p 100EGRET upper limit at 1 GeV, constraining the product of the number of visible pulsars N and the average integralflux above 1 GeV per pulsar. GLAST LAT should place even more stringent constraints on this product, and mayalso limit the maximum average accelerating potential by probing the CR spectral tail. For N p 22–200, a GLASTLAT nondetection will lead to the constraints that the average integral flux per pulsar should be lower by factorsof 0.03–0.003 than current model predictions.Subject headings: gamma rays: theory — globular clusters: individual (47 Tucanae) — pulsars: general —radiation mechanisms: nonthermal1. INTRODUCTIONMillisecond pulsars (MSPs) have been detected in severalglobular clusters (GCs), including the nearby 47 Tucanae (Cam-ilo et al. 2000; Freire et al. 2003). Although only 22 MSPs arecurrently known to exist in 47 Tucanae (for a recent review,see Lorimer et al. 2003), it is expected that this cluster mayharbor 30–60 MSPs (Camilo & Rasio 2005; Heinke et al.2005), or even up to ∼200 (Ivanova et al. 2005).In this Letter, we model the pulsed curvature radiation (CR)gamma-ray flux expected from a population of MSPs in 47Tucanae (§ 2), and compare it with the GLAST LAT sensitivityand EGRET upper limits. Our approach differs from similarrecent studies (Bednarek & Sitarek 2007; Harding et al. 2005[hereafter HUM05]; Cheng & Taam 2003) in a number ofrespects. First, we use a fully 3D general-relativistic (GR) polarcap (PC) pulsar model (see, e.g., Venter & de Jager 2005),sampling nonthermal gamma-ray radiation from a range ofmagnetic field lines above the PC. Next, our study involves apopulation of MSPs, and we sample randomly from this en-semble. We are therefore able to calculate absolute cumulativefluxes expected from the MSPs in 47 Tucanae, without needingto use approximate spectra or scale up single MSP predictionsor single particle spectra. We are also specifically interested inthe range of pulsed spectra expected due to the uncertain ge-ometries and spread in values of the period P and its derivativeof these pulsars.ṖX-rays from the MSPs in 47 Tucanae are believed to be ofthermal origin, and are likely produced by a return currentheating the pulsars’ PCs (Bogdanov et al. 2006; Harding &Muslimov 2002; Cheng & Taam 2003). These X-rays may be1 Unit for Space Physics, North-West University, Potchefstroom Campus,Private Bag X6001, Potchefstroom 2520, South Africa; christo.venter@nwu.ac.za, okkie.dejager@nwu.ac.za.2 Centre for High Performance Computing, CSIR Campus, 15 Lower HopeStreet, Rosebank, Cape Town, South Africa.3 South African Department of Science and Technology, and National Re-search Foundation Research Chair: Astrophysics and Space Science.Compton upscattered to high-energy gamma rays. However,this inverse Compton scattering (ICS) component is believedto be dominated by the CR component (see, e.g., Fig. 3 ofBulik et al. 2000), and we therefore omitted it from our cal-culations below.We lastly make some conclusions regarding the possibility ofconstraining a model of gamma-ray radiation from GCs (§ 4),since an ensemble of MSPs in a GC cluster provides a uniqueopportunity for specifically constraining the GR frame-drag-ging pulsar model (e.g., Harding & Muslimov 1998) in a ge-ometry-independent way (§ 3).2. PULSED GAMMA-RAY FLUXThe measured values of for GC pulsars are affected byṖtheir acceleration in the gravitational potential of the GC, andtherefore need to be corrected to obtain intrinsic period deriv-atives . This was done for 47 Tucanae by Bogdanov et al.Ṗint(2006) who derived spin-down luminosities assuming aĖrotKing model. From their Table 4, we selected 13 MSPs andcalculated their corresponding using their periods P as givenṖintby Freire et al. (2003) and the expression Ė prot, with the moment of inertia.2 3˙4p I P /P INS int NSThe details of the implementation of our isolated MSPmodel, using the GR-framework of Harding, Muslimov, andTsygan (e.g., Harding & Muslimov 1998), may be found inVenter & de Jager (2005), where we discuss the essential linkbetween pulsar visibility and the assumed geometry, i.e., mag-netic inclination angle x and observer angle z (both measuredwith respect to the spin axis ).QHUM05 found that most MSPs are inefficiently screened byCR and ICS pairs. Screening caused by electron-positron pairswill lower the accelerating potential, leading to a lower pulsedCR spectral cutoff. Higher numbers of energetic electrons mayhowever boost the unpulsed gamma-ray spectrum. The vastmajority of MSPs in our population lie below the critical spin-L126 VENTER & DE JAGER Vol. 680Fig. 1.—(a) Mean and (b) standard deviation j of (in units of2E dN /dEg gergs s1 cm2) vs. , with the number of values of used to2N N E dN /dEt t g gcalculate the mean and j-values. Line types correspond to different energiesas shown in the legend. Thick lines are for values of summed over2E dN /dEg grandomly chosen visible pulsars, while thin lines are for values ofN p 100for single ( ) randomly chosen pulsars. The latter values were2E dN /dE N p 1g gscaled by a factor 100 in (a), and a factor in (b). Results converge100 p 10beyond .4N ∼ 10tdown luminosity above which screening is expected to occur(Harding et al. 2002),1/7P34 1Ė ≈ 1.4 # 10 ergs s . (1)rot,break ( )2B12Here, G and P is the pulsar period in seconds.12B p B/1012(Note the positive sign of the power of P.) This condition maybe rewritten so that screening occurs when˙log P 1 2.625 log P  12.934, (2)10 10assuming , stellar radius cm, and pulsar2 6I p 0.4MR R p 10NSmass . We used an unscreened electrical potentialM p 1.4 M,for the bulk of the population, and used the approximation ofa screened electric field given by Dyks & Rudak (2000) forMSPs with spin-down powers greater than this critical spin-down power (which was only the case for 47 Tuc U).For each pulsar i in our population, we calculated phase-averaged CR spectra via (see also eq. [12] of Venter & de Jager2005),idN 1g (E , x, z ) p #g 2dE 2pd sin z dz dEgzdz 2p iI(E , E  dE ) dL (f , x, z, E )g g g g L g df dz dE ,  L g[ ]cE df dz dEz 0 g L g(3)where is the phase angle and picks out thef I(E , E  dE )L g g ggamma-ray energies in the range . Primary elec-(E , E  dE )g g gtrons leave a stellar surface patch dS relativistically (b p0) at a rate of , where is the charge˙v /c ∼ 1 dN p r dSb c/e re e 0 e0density. These primaries radiate CR at a position above(r, v, f)the PC with a characteristic energy of ,c 3 2 1E p 1.5l g m c rg c e cwhere is the Compton wavelength andl p /(m c) r (r, v,c e cis the curvature radius of the GR-corrected dipolar magneticf)field. The CR is associated with an incremental gamma-rayluminosity of , which is radiated in a time dt.˙ ˙dL p dN E dtg e CRThe CR loss rate is given by , where2 4 2Ė ≈ 2e cg /(3r ) g(r,CR cis the Lorentz factor of the electron primaries. We cal-v, f)culated CR photon spectra for x p z p 10,idN /dE(E , x, z )g g20, …, 80 and , 2, …, 13, assuming cm,6i p 1 R p 10, and . All spectra (2M p 1.4 M I p 0.4MR 13 # 8 #, NS) were scaled to a distance of kpc (Gratton et8 p 832 d p 5al. 2003).We next randomly chose visible MSPs (with ran-N p 100dom x and z), and summed their pulsed spectra to obtain amillion cumulative spectra from the Monte Carlo process:Np100j kdN dNg gp (E , x, z ), (4) g( )dE dEkp1cumwhere for each index , 2, …, , a total of6j p 1 N p 10 N ptspectra (randomly sampled from the above-mentioned 832100spectra) were summed. Therefore, j corresponds to a specificchoice of x, z, and k-values when oversampling from 13 to100 pulsar spectra. This procedure was repeated for 1 MSPinstead of 100, i.e., setting . Upon comparison of theseN p 1results, one would expect that the mean values of the singleMSP differential spectra would scale with N, and the standarddeviations with . In Figure 1a the mean values ofNat three different photon energies are shown as2E dN /dE Eg g ga function of the number of values used to calculate theNtmean (with a maximum of ). The thick lines indicate6N p 10tresults when , while the thin lines indicate results forN p 100scaled by a factor 100. Similarly, Figure 1b indicatesN p 1,the standard deviation of at three different photon2E dN /dEg genergies versus . The thick lines again indicate results whenNt, while the thin lines indicate results for scaledN p 100 N p 1by a factor 10. It is clear that the mean and j-values convergebeyond , for both and . The4N ∼ 10 N p 1 N p 100 N p 1tresults are more erratic at small values. The relative errorNton the mean values are therefore a factor 10 lower when usinga population of 100 MSPs, as opposed to the much larger errorobtained for single pulsars. Figure 2 depicts average cumulativepulsed differential spectra for and2AE dN /dES N p 100g g cum, as well as 2 j bands. Also shown are the GLAST6N p 10tLAT sensitivity curve (HUM05) and EGRET upper limits(Fierro et al. 1995), as well as other predictions of pulsedgamma-ray radiation from 47 Tucanae (HUM05).3. DISCUSSIONThe gamma-ray visibility of an isolated MSP depends ex-tremely sensitively on the assumed pulsar geometry. Inclinationand observer angles x and z are usually estimated from radiopolarization measurements (e.g., Manchester & Johnston 1995),involving the rotating vector model. In our MSP model, theelectric potential boundary condition of is usedF(y p 1) p 0(Muslimov & Harding 1997) as the last open magnetic fieldlines are treated as equipotentials. In addition, the(y p 1)No. 2, 2008 PULSED GAMMA RAYS FROM MSPs IN 47 TUCANAE L127Fig. 2.—Average differential pulsed gamma-ray spectrum (solid line) with2 j error bands (shaded band between dashed lines) for andN p 100 N pt, with smoothed high-energy tails. Furthermore we indicate predictions of610HUM05 (lower square for 15 MSPs, upper square scaled to 100 MSPs). Thepredictions from HUM05 involve estimates of single-particle spectra scaledusing the Goldreich-Julian PC current (Goldreich & Julian 1969) and arbitrarybeaming angles. Our spectra include variations across the PC (sampling dif-ferent magnetic field lines) as well as variations due to various inclinationangles and observer geometries (with beaming angles calculated implicitly).The GLAST LAT sensitivity (HUM05) as well as EGRET upper limits (dia-monds; Fierro et al. 1995) are also shown (the EGRET upper limits and HUM05results were converted to differential values assuming an spectrum).2Espace-charge-limited nature of this model requires that the ac-celerating electric field parallel to the magnetic field (Ek) shouldbe zero at the stellar surface . This means that on-beam(r p R)radiation (when ) will have a spectral cutoff whichx ∼ zroughly scales with the maximum potential . Off-beamFmaxradiation will however rapidly decrease as the impact angleincreases, because this radiation is produced alongb p z  xfield lines where the electric potential F, and hence Ek , hasdropped significantly.An ensemble of MSPs provides the unique opportunity tosidestep the issue of pulsar geometry (which is crucial whenmodeling a single MSP) by averaging over numerous spectraand geometries, which results in a “geometry averaged” spec-trum with relatively small uncertainty. We also exploit thisscenario in the sense that we use results from the corotatingframes of the MSPs, since we expect that the effect of Lorentztransformations to the observer frame will average out over theensemble. We find that the relative spread of ensemble valuesfor the bolometric gamma-ray and electron luminosities as wellas the spin-down power and average gamma-ray spectra aretypically one order smaller than for the same quantities av-eraged as single MSP quantities (i.e., they scale with ).1/2NThis illustrates the importance of testing pulsar models by ob-serving GCs. (This conclusion follows when using a constantequation of state [EOS]. Variations in the EOS have a muchsmaller effect than that of differing pulsar geometries and aretherefore not included in this discussion; see § 4.)Due to the lower relative ensemble errors, one should there-fore expect to be able to constrain average pulsar parametersusing average spectra and their errors shown in Figure 2. Thiswould not be the case if ensemble values did not converge(Fig. 1) and had relatively small errors. Furthermore, the factthat the ratios of average cumulative luminosities and spin-down powers to their single MSP counterparts are very closeto gives us confidence in our sampling algorithm.N p 100In our model, the predicted average single pulsar efficienciesfor electron and gamma-ray production are typically ∼2% and∼7%, depending on the population of MSPs used. For our 47Tucanae MSP population (with ergs s1),34˙AE S p 2.2 # 10rotwe found 0.74% and 6.1%, respectively. This should be com-pared to the estimate of Bednarek & Sitarek (2007) of 1% forthe first value, and to that of Harding et al. (2002) of ∼10%for the latter quantity. Venter & de Jager (2005) quoted valuesof 1%–2.5% and 2%–9% for these quantities for the case ofthe pulsar PSR J04374715.In Figure 2 we converted integral EGRET upper limits(Fierro et al. 1995) and model predictions of HUM05 to dif-ferential values assuming a spectrum. Our predicted pulsed2Egamma-ray flux at 100 MeV is below the differential EGRETupper limit, much lower than that of HUM05. Their predictionof the integral flux above 100 MeV exceeds the EGRET upperlimit by a factor for 15 MSPs, and for 100 MSPs.f ∼ 19 f ∼ 125(This seemingly overprediction of integral flux was also foundin the case of PSR J04374715 [Venter & de Jager 2005].HUM05’s prediction is a factor above the EGRET upperf ∼ 35limit for this MSP.) However, our summed pulsed spectra vi-olate the EGRET upper limit at 1 GeV (Fierro et al. 1995),providing constraints on the low-energy tail of the averagepulsed spectrum. GLAST LAT observations will however con-strain the predicted spectrum at all energies of relevance, whichshould provide a meaningful constraint on the product of visiblepulsars N, and the average integral flux above 1 GeV per pulsar.4. CONCLUSIONSIn this Letter, we modeled the pulsed gamma-ray flux ex-pected from a population of MSPs in 47 Tucanae. We chosea number of visible members, but the spectra canN p 100easily be (linearly) scaled to any other reasonable number. Wewere especially interested in obtaining errors on the pulsedgamma-ray spectrum which would reflect the uncertainty inthe geometries, as well as the spread in P and , of the indi-Ṗvidual pulsars. In this way, we wanted to see whether it wouldbe possible to constrain the underlying pulsar model indepen-dently of the assumed pulsar geometry. It remains to be shownwhether the spread of the 13 selected pulsars represent the trueinherent spread of MSPs in 47 Tucanae.For the sake of completeness, we furthermore investigatedthe effect of varying the EOS ( , and I). By choosingM, Rand letting M and R vary between2I p 0.4MR M/M pNS ,, 1.5, 1.6 and , 1.2, 1.4, 1.6, we found61.4 R p R/10 cm p 1.06that the maximum integral flux did not vary by more than ∼4%(although it increases with both M and R; the spectral cutoffenergy furthermore did not vary by more than ∼13%). Sincethis variation will not significantly influence our constraintsderived below, we used fixed values of andM p 1.4 M,throughout.R p 16Our average pulsed spectrum, including errors, exceed theEGRET upper limit at 1 GeV (for the average integral fluxspectrum, by a factor of ∼3.9). Assuming that this model pre-dicts the gamma-ray flux correctly, this EGRET upper limitconstrains the number of GC pulsars to . Gen-100/3.9 ≈ 25L128 VENTER & DE JAGER Vol. 680erally, the constraints derived may be summarized as follows:NAF S 25obs ≤ , (5)AF S qmodelwith and the average observed and predicted in-AF S AF Sobs modeltegral flux above 1 GeV per pulsar, and q a factor indicatingtelescope sensitivity. For EGRET, , while forq p 1 q p 40GLAST LAT. For example, setting , 100, and 200, weN p 22find that , 0.25, 0.13 for EGRET. This in-AF S/AF S ≤ 1.1obs modeldicates that the EGRET limit is obeyed for , but thatN p 22larger values of N imply a reduction in the model-predictedintegral flux (by, e.g., factors of 4 and 8 for and 200).N p 100In the case of a GLAST LAT nondetection, we find, 0.006, 0.003 for , 100, and 200.AF S/AF S ≤ 0.03 N p 22obs modelGLAST LAT might lastly even constrain the maximum averageelectric potential per pulsar via the CR cutoff energy.AF SmaxIt should also in principle be possible to look for the radioperiods corresponding to the 22 known MSPs in 47 Tucanaein the gamma-ray data (or detect new gamma-ray periods ifbeaming properties are different), should GLAST LAT detectthe cumulative pulsed CR spectrum.This publication is based on research supported by the SouthAfrican National Research Foundation and the SA Centre forHigh Performance Computing.REFERENCESBednarek, W., & Sitarek, J. 2007, MNRAS, 377, 920Bogdanov, S., Grindlay, J. E., Heinke, C. O., Camilo, F., Freire, P. C. 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The Astrophysical Journal

Constraining a general-relativistic frame-dragging model for pulsed radiation from a population of millisecond pulsars in 47 Tucanae using GLAST LAT

C. Venter

O. C. de Jager

MUCC (Crossref)

Constraining A General-Relativistic Frame-Dragging Model for Pulsed Radiation from a Population of Millisecond Pulsars in 47 Tucanae usingGLASTLAT

Constraining A General-Relativistic Frame-Dragging Model for Pulsed
  Radiation from a Population of Millisecond Pulsars in 47 Tucanae using
  GLAST/LAT

Constraining A General-Relativistic Frame-Dragging Model for Pulsed Radiation from a Population of Millisecond Pulsars in 47 Tucanae using GLAST/LAT

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