The Synchrotron Emission Pattern of IntraBinary Shocks
DDraft version May 30, 2019
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The Synchrotron Emission Pattern of IntraBinary Shocks
D. Kandel, Roger W. Romani, and Hongjun An Department of Physics, Stanford University, Stanford, CA, 94305, USA Department of Astronomy and Space Science, Chungbuk National University, Cheongju, 28644, Republic of Korea
Submitted to ApJABSTRACTWe model millisecond pulsars winds colliding with radiatively-driven companion winds in blackwidow and redback systems. For the redbacks, the geometry of this intrabinary shock (IBS) is quitesensitive to the expected equatorial concentration in the pulsar outflow. We thus analytically extendIBS thin-shock models to ∼ sin n θ pulsar winds. We compute the synchrotron emission from suchshocks, including the build-up and cooling of the particle population as it accelerates along the IBS. Forreasonable parameters, this IBS flux dominates the binary emission in the X-ray band. The modelingshows subtle variation in spectrum across the IBS peak, accessible to sensitive X-ray studies. Asexample applications, we fit archival CXO / XMM data from the black widow pulsar J1959+2048 andthe redback PSR J2339-0533, finding that the model reproduces well the orbital light curve profiles andenergy spectra. The results show a very hard injected electron spectrum, indicating likely dominanceby reconnection. The light curve fitting is sensitive to the geometric parameters, including the veryimportant orbital inclination i . Coupled with optical fits of the companion star, such IBS X-ray lightcurve modeling can strongly constrain the binary geometry and the energetics of the MSP wind. Keywords: gamma rays: stars - pulsars: individual (J1959+2048, J2339-0533) INTRODUCTIONIn the so-called ’spider’ binaries, millisecond pulsarheating drives winds from the non-degenerate compan-ions which occlude the pulsed radio signal; accordingly,few such objects were found in classical radio pulsarsurveys. However, the penetrating γ -ray emission fromsuch pulsars makes them prominent in the Fermi
LATsky and dedicated searches of LAT unidentified sourceshave turned up many such objects, leading to a renais-sance in the study of this intense interaction phase. Theobjects are generally divided into ‘redbacks’ (hereafterRB) with M c ≈ . − . M (cid:12) stellar mass companionsand ‘black widows’ (BW) orbited by M c ≤ . M (cid:12) sub-stellar objects. One may identify sub-classes of the RB:‘Transitioning MSP’ (Tr) when the companion’s Roche-lobe overflow drives it to an intermittent accretion phaseand ‘Huntsman’ (Hu) MSP when the companion is inthe giant phase. For the BW, a ‘Tidarren’ (Ti) subclass Corresponding author: R.W. [email protected] identifies those exceptionally short P B , low M c systemswhere the companion is hydrogen-stripped; these extenddown to short-period pulsar-planet MSP.While the GeV γ -ray signal dominates the total pho-ton output of these systems, in the optical, the com-panion, often with a strongly heated face, dominates.In the X-rays, emission is seen from the pulsar mag-netosphere and the heated companion surface, but formany systems, especially for RB, the orbital light curvepresents strong peaks of hard-spectrum Γ ≈ − β = ˙ M w v w c/ ˙ E PSR (1)controls the shock geometry (Romani & Sanchez 2016,hereafter RS16). In this paper we further explore thesynchrotron spectrum of this IBS component. a r X i v : . [ a s t r o - ph . H E ] M a y Kandel et al. zx r Dφ = constant θθ ∗ θ ∆ φr Figure 1.
Intrabinary shock geometry. The shocked windsare assumed to flow along φ slices of infinitesimal width ∆ φ ;the momentum is integrated along these slices. Here θ ∗ de-notes the polar angle of the (possibly anisotropic) wind.2. ANISOTROPIC PULSAR WIND – STELLARWIND INTERACTIONNumerical simulations of pulsar magnetospheres(e.g. Tchekhovskoy et al. 2016) suggest that the en-ergy/momentum flux of the pulsar wind is equatoriallyconcentrated as ∝ sin n θ ∗ with n = 2 or even n = 4 foran oblique rotator ( θ ∗ is the angle from the pulsar spinaxis, assumed perpendicular to the orbital plane). Inthe RS16 implementation of IBS shock emission in the ICARUS binary light curve modeling code, we allowedthe pulsar power to be distributed ∝ sin n θ ∗ , but thispower illuminates an IBS shock computed from the con-tact discontinuity (CD) shape for the collision of twocold isotropic winds described by Canto et al. (1996).This is a good approximation when the pulsar winddominates ( β (cid:28)
1, the ‘black widow’ BW case), butthere are significant shape distortions when β > n θ .For clarity, we draw the β > θ fromthe line of centers between the stars and φ measuredfrom the orbit normal. Such a point subtends an angle θ from the line of centers as measured from the com-panion. To connect with the anisotropic pulsar wind, wenote that cos θ ∗ = sin θ sin φ . The shocked winds flowalong the CD, where momentum conservation requiresthe total momentum flux to be the vector sum of thetwo wind momentum fluxes, integrated along the flowlines of constant φ (Wilkin 2000). For the pulsar wind, we have incident energy, linear-and angular- momentum fluxes asd ˙ E P ( θ (cid:48) , φ ) = ˙ E PSR π sin n θ ∗ sin θ (cid:48) d θ (cid:48) , (2)˙Π P z = (cid:90) θ cos θ d ˙ E P c . (3)˙Π P r = (cid:90) θ sin θ d ˙ E P c , (4)and ˙ J P θ = 0 . (5)For the case when both winds are isotropic, the cor-responding equations are given in Canto et al. (1996).Following their treatment, the shape of the IBS, a purelygeometrical result, is given by r ( θ, φ ) = D sin θ csc( θ + θ ) , (6)where D is the separation between the stars. From themomentum balance condition and fluxes (2)-(5) we canrelate θ and θ by θ cot θ = 1 + 2 β − csc θ (cid:90) θ (1 − sin φ sin θ (cid:48) ) n sin( θ (cid:48) − θ ) sin θ (cid:48) d θ (cid:48) . (7)where β is the ratio of momentum flux from the com-panion to that of the pulsar (Eq. 1). For n = 0, Eq.(7) recovers the standard result for two isotropic winds,whereas for n = 1, we have θ cot θ = 1 + β − θ cot θ (5 + 3 cos 2 φ ) − θ ) sin φ ] (8)and for n = 2, we have θ cot θ = 1 + β − (cid:20) θ cot θ (5 cos 4 φ + 28 cos 2 φ + 31)+(cos 4 θ −
14 cos 2 θ −
47) sin φ +24(cos 2 θ + 5) sin φ − (cid:21) . (9)For β < θ at a CD location becomes theangle measured with respect to the line of centers fromthat star while θ is now the angle from the pulsar. Wealso re-assign cos θ ∗ = sin θ sin φ to describe the pulsarwind asymmetry. Then, the IBS geometry is describedby exchanging θ ↔ θ and β − → β in Eqs. (8) and BS Synchrotron Emission Figure 2.
Shape of the CD for interaction of an anisotropic sin θ ∗ pulsar wind with an isotropic companion wind. Left: β = 10(RB case, shock wraps around pulsar). Right: β = 0 . (9). For n (cid:54) = 0, the two cases give quite different geome-tries. When β (cid:28) n and closeto the isotropic result. In contrast, for β (cid:38)
1, the shocksurrounds the pulsar and the anisotropy increases with n , with the IBS increasingly flattened at the poles. Theresult is an ‘hourglass’ cross-section (Fig. 2). IBS MODEL FOR SYNCHROTRON RADIATIONThe pulsar outflow is a cold wind of relativistic( γ W (cid:29)
1) electrons and positrons of number den-sity n embedded in a strong magnetic field with σ w = B / (4 πγ w n c ) (cid:29)
1. Models of the pulsar magne-tosphere (e.g. Tchekhovskoy et al. 2016, Lyutikov et al.2018) suggest that the wind likely has a sector structurewith the power concentrated in the equatorial plane.Recent PIC simulations (Philippov & Spitkovsky 2018)suggest that the magnetization σ may also vary withlatitude and that ions may be a significant componentof the outflow along some streamlines. Such variationmay be important for the axially symmetric structures(equatorial tori and polar jets) seen in a number of iso-lated pulsar wind nebulae (PWNe). Lacking a detailedprescription for such variation, we focus here on the bulkenergetics of the outflow. Beyond the light cylinder, wecan assume a toroidal structure with B ∼ /r untilthe wind shocks. At the IBS, we expect shocks and/orreconnection to convert a fraction η of the bulk energyto a power-law e ± (hereafter electron) distribution withspectral index p and significant pitch angle to the rem-nant embedded B . This results in synchrotron emissionfrom the accelerated particles. To study this emissionand its dependence on shock and viewing geometry, wehave further extended the ICARUS IBS code (Bretonet al. 2012, RS16), including cooling of and synchrotronradiation from these electrons.We assume that at each point on the IBS, the imping-ing pulsar wind converts a fraction η of its power into an electron spectrum with energy γ min < γ e < γ max distributed as N ( γ e )d γ e = N γ − pe d γ e (10)and N normalizes the sky-integrated power to η ˙ E . Theindex p depends on the nature of the particle accel-eration. If dominated by diffusive shock acceleration(DSA), we expect p ≥
2, with lower values possible foroblique shocks. However, if reconnection and magneticturbulence dominate the acceleration, we may expect avery hard powerlaw extending up to γ max ∼ σ w γ w andindices as hard as p ∼ σ w (cid:29)
1. In fact, most IBSX-ray spectra are very hard suggesting that reconnec-tion dominates. Numerical simulations of such reconnec-tion confirm the general trend toward harder spectrumat high magnetization (e.g. Sironi & Spitkovsky 2014;Werner et al. 2016) and many more recent simulationsare exploring this process. Our present bulk energet-ics treatment of the shock radiation will not be able toaddress the predictions of such simulations, although itis to be hoped that, as our spectral measurements andphenomenological understanding of the IBS improve, wecan compare with the particle spectral index predictedby high fidelity simulations. Note that the latitudinalvariations in particle content and σ noted above couldallow the accelerated spectrum to vary across the shock;we also ignore such effects in the present treatment.If we assume that the pulsar wind’s embedded mag-netic field B is toroidal outside of the light cylinder r LC = c/ Ω, then we expect B ( r ) = B r r , (11)where B = B LC ( r LC /r ) = ( − I Ω ˙Ω / cr ) / and r are the magnetic field at the nose of the IBS and theshock standoff distance, respectively ( I , Ω and ˙Ω arethe pulsar moment of inertia, angular frequency andspin-down rate). The motion of relativistic electrons inthis magnetic field leads to synchrotron radiation whose Kandel et al. power per unit angular frequency ω for a single electronin its rest frame is given by (Rybicki & Lightman 1979) P ω = √ q B sin αmc F (cid:18) ωω c (cid:19) , (12)where ω c = 3 γ e qB sin α mc , (13)and F ( x ) ≡ x (cid:82) ∞ x K / ( ξ )d ξ .Of course, synchrotron cooling depletes the electrons’energy as they travel downstream in the IBS. Thus, at agiven point on the IBS, the local electron spectrum andradiated photon spectrum depend on both the freshlyaccelerated electrons and the upstream injection andsubsequent cooling. The energy loss rate of the rela-tivistic electron is given by (Rybicki & Lightman 1979) γ e = γ e, (1 + Aγ e, τ ) , (14)where τ is the cooling time and A = 2 e B sin α m c . (15)To model the IBS electrons, we set up a grid of IBSzones and a logarithmic grid of energy bins. We thenfollow the injection of fresh electrons (Eq. 10) and thetransfer of the electrons to lower energy bins via cooling(Eq. 14) as the population flows along the IBS surfacewith bulk motion Γ.We expect Γ to slowly increase along the IBS, as thehot shocked pulsar wind expands downstream. This ac-celeration should be slower in the RB case, where theshocked relativistic flow is confined by the massive com-panion wind. In this paper, we take the variation of bulkgamma along the shock asΓ( r ) = Γ (cid:18) k sr (cid:19) , (16)where Γ is the bulk Γ at the nose with standoff distance r , s is the arc length from the nose to a position ofinterest along the IBS, and k is a scaling coefficient.Examining the relativistic hydrodynamic simulations ofBogovalov et al. (2012) for the BW geometry and ofDubus et al. (2015); Bogovalov et al. (2008) for the RBcase, we can estimate Γ = 1 . k ∼ . = 1 . k ∼ . s scaling is better forRB than the pulsar distance scaling used by RS16, whichwas adequate for BW. These parameters are estimatedfor the bulk of the shocked wind flow along the CD.The simulations do, however, include the finite width ofthe shocked flow and there appears to be some variation in bulk Γ across this width, which may be captured infuture more detailed models.For each IBS grid zone of length (cid:96) grid we compute theresidence time as τ res = (cid:96) grid c (cid:113) − / Γ . (17)Then electrons spend equal time (have equal weight) inthis zone over the range t = (0 , τ res ). We apply Eq. (14)to determine the re-partition of each zone’s energy binsinto the bins of the downstream zones. For such cooling,the number spectrum is approximately a broken power-law, with break energy ∼ /A , although the detailedshape of this break and the spectrum, especially at highenergies, depend on the history of the electron popu-lation as set by the upstream IBS shape and magneticfield.The electrons flow downstream along the IBS, whosegeometry is given by Eqs. (6), (7), and the equilib-rium spectrum of electrons in each grid zone is deter-mined by summing electrons freshly from the pulsarand cooled electrons flowing from upstream grids. Toapproximate the flow pattern on our numerically re-alized (triangularly tiled) IBS surface, we identify thetwo triangles adjacent to a given zone and partitionthe original zone’s (cooled) electrons to these daugh-ters. Such partitioning needs to capture the flow down-stream and conserve the electron number at the sametime. To achieve these, we start at the triangular tileat the nose and use a graph search algorithm to findits triangular neighbors, each sharing an edge with thenose tile. The parent tile is then marked as visited, andits cooled equilibrium electron population is distributedevenly to its three neighbors. The same procedure is re-peated for next-generation tiles, but these three tiles andtheir subsequent generation will only have two unvisitedneighbors. Therefore, we distribute cooled electron fluxevenly among the two daughter tiles. This gives theequilibrium number spectrum of relativistic electrons ineach grid zone along the IBS.The total power per unit frequency from the IBS syn-chrotron emission can finally be obtained by summingover emission from all electrons as L ω = (cid:90) d γ e N ( γ e ) P ω ( γ e ) . (18)The above expression is valid in the rest frame of elec-trons. In order to obtain observed power, one needs toaccount for Doppler boosting and beaming. Doing so,we calculate observed L ω from each triangular grid at agiven sky angle, and sum over the entire CD to obtainthe flux at each sky position. This can be represented BS Synchrotron Emission L ω = (cid:88) j j (cid:18) − (cid:113) − j cos θ j (cid:19) × (cid:90) d γ e N j ( γ e ) P ω j ( γ e ) , (19)where index j represents each triangular grid zone of theCD, θ j the angle between flow direction on a triangulargrid and the sky direction, and ω j is plasma frame pho-ton frequency which is Doppler shifted to the observedfrequency ω by ω = ω j Γ j (cid:18) − (cid:113) − j cos θ j (cid:19) . (20)The first factor inside the sum in Eq. (19) follows fromthe Lorentz invariance of phase space density of pho-tons during transformation between plasma frame andobserver’s frame.Finally, Eq. (19) can be used to produce energy-resolved sky-maps of the synchrotron emission from theIBS. In turn, for a given viewing angle at binary incli-nation i , we can extract the observable phase-resolvedspectrum (or energy-dependent light curves). PROPERTIES OF IBS SYNCHROTRONEMISSIONFig. 3 shows sky maps of emission from the IBS fora RB geometry for isotropic, sin θ ∗ and sin θ ∗ winds.The general geometry is a ring of emission surround-ing pulsar inferior conjunction φ B = 0 .
75. As the Earthline-of-sight cuts through this ring near the orbital planeone sees a light curve with two caustic peaks bracketinga fainter ‘bridge’ region. A more polar view can havea single peak or no peaks. For a finite velocity com-panion wind, the peaks can be delayed and asymmetric(RS16). In general, the brightest emission comes fromdownstream on the CD, where the bulk Γ is large, in-creasing beaming and Doppler boosting, and the IBSsurface is flatter, leading to brighter caustic peaks in thelight curves. The colors show that the regions furtherdownstream are most important at the inner edges ofthe peaks. Since these downstream regions have experi-enced the most cooling, this can lead to (subtle) energydependence in the synchrotron light curves. We see thatthe peak-bridge flux ratio is largest in the isotropic case.This is because for the anisotropic case, the IBS down-stream from the nose is flatter and more tangential tothe pulsar wind so that the electron power away from thenose is smaller. Thus, the broadly distributed emissionfrom low Γ nose regions is more important; such zonesradiate to more of the sky, adding flux to the bridge. Since synchrotron electrons are cooled as they flowdownstream, there is a break in the spectrum. Inter-estingly, for typical parameters the break occurs in theX-ray band. The peak flux includes emission from far-ther along the CD, with larger Doppler factor. Thus,in the observer’s frame, the peak emission has a higherbreak energy, as shown in Fig. 4. However, the separa-tion depends on the cooling and hence the IBS B field.While the solid curves in Fig. 4 are for a field swept upfrom the pulsar wind, one might also imagine a shock-generated field increasing along the IBS. For this case(dashed lines), the cooling rate grows so that bridge andpeak are from very similar regions with similar breakenergies. For typical parameters, both peak and bridgebreaks are accessible if one includes hard X-rays. Evenin the soft X-ray band, this difference may be discernedby small changes in the effective photon index.These spectral changes induce energy dependence inthe light curves (Fig. 5, top). These light curves arealso sensitive to the growth of the bulk Doppler factoralong the IBS. For black widows, the acceleration of Bo-govalov et al. (2012) generates narrow peaks and faintbridge, while the slower Γ growth expected for RB leaveslarger bridge emission. The finite width of the shock alsosegregates cooled electrons from fresh injection; this canlead to additional energy dependence, which we describein a future communication. APPLICATION OF THE MODEL5.1.
PSR J1959+2048
PSR J1959+2048 = PSR B1957+20 (hereafter J1959)is a BW system with a P = 1 . E = 1 . × erg s − millisecond pulsar in a P b = 9 . ∼ . M (cid:12) companion. Modeling the optical and X-rayorbital variations can provide important information onthe system masses and the wind physics.For the optical data, we used the BVRIK s magni-tudes from Reynolds et al. (2007) and assumed an ex-tinction A V = 0 .
82. As shown in Fig. 6, the directheating model provides a reasonable fit to the data,and results in an inclination i ∼ . ◦ ± . ◦ , a dis-tance 2 . ± .
02 kpc. This angle is consistent withthe findings of van Kerkwijk et al. (2011) and for theirmeasured radial velocity implies a neutron star mass ∼ . M (cid:12) . The distance is somewhat smaller than the3.3 kpc estimated in SR17; it is now consistent withthe largest DM estimates, but the required direct heat-ing, a sky-integrated 5 × erg s − , is still substan-tially larger than inferred from the observed (mostlyGeV) photon flux, which gives an isotropic equivalentluminosity 1 . × erg s − . It is also a large frac-tion of the Shklovskii-corrected spindown luminosity of Kandel et al.
Figure 3.
Sky map of emission from the IBS. Left is for isotropic, center for sin θ ∗ and right for sin θ ∗ distribution of theenergy flux from the pulsar. Red corresponds to emission from near the nose of the CD, i.e. θ < ◦ , green from 30 ◦ < θ < ◦ and blue from θ > ◦ . In all cases, the peak emission comes mainly from downstream CD regions, while off-pulse emissioncomes mainly from the nose. Figure 4.
Spectrum of emission from the peak and thebridge for a RB model, assuming electrons with p = 0 . B = 100 G. This B either follows a toroidal pat-tern (solid) or increases along the CD (dashed). The coolingbreak at the peak is higher than at the bridge, as can beprobed with X-ray measurements. This difference is smallerfor B increasing with s . . × I erg s − ; we thus conclude that the compan-ion sees more pulsar flux than directed at Earth and,likely, the moment of inertia is larger than the default10 g cm .Archival X-ray observations of this object include 212ks of CXO ACIS-S exposure (Obs ID 9088, 1911) and31.5 ks of XMM exposure (Obs ID 0204910201). Unfor-tunately, the XMM observation was made in PN timingmode (with high background) so that only the MOSdata were useful for this orbital study. Also, the obser-vation did not cover the ‘Br’ phase between the peaks,severely limiting the effective XMM exposure. Never-theless, we employed these data to extract a combined Figure 5.
Top: Light curves for different energy bands of0.5 - 10 keV, 10 - 25 keV and 25 - 100 keV for the toroidal B model of Fig. 4. The peak to bridge contrast is largerat high energies. Bottom: lightcurves for BW-type and RB-type bulk gamma growth. light curve and spectra, using HEAsoft , CIAO and
SAS .Spectra were measured in four orbital phase bins (Ta-ble 1) with the analysis showing hard power law emissionwith (low significance) variation between phases. As de-scribed in Sec. 4, this may be understood if the coolingbreaks falls within the X-ray range.The light-curve shows a nice double-peaked structurecentered at φ B = 0 .
25 (Fig. 6). To model the IBSemission, we estimate the magnetic field strength by as-suming a toroidal field outside of the light-cylinder and I ∼ g cm together with the orbital parameters,which give B ∼
20 G. Fitting this lightcurve with ourmodel (for a sin θ ∗ wind) gives i ∼ . ◦ ± . ◦ (see Ta-ble 2), which is higher than that inferred for the directheating model (e.g. van Kerkwijk et al. 2011, Sanchez BS Synchrotron Emission Figure 6.
Left: BVRIKs Optical lightcurves of J1959, compared with the direct heating model from the best-fit parameters.Right: Combined
CXO and
XMM lightcurve of J1959 for energy range of 0 . − Table 1.
Spectral fit results
Phase J1959 φ J1959 Γ J2339 φ J2339 ΓOff 0.55-0.05 1 . ± .
19 0.05-0.45 1 . ± . . ± .
44 0.45-0.65 0 . ± . . ± .
89 0.65-0.85 0 . ± . . ± .
35 0.85-0.05 0 . ± . Note —Γ represents the IBS PL index on top of a fixed ther-mal (NS surface) plus power-law (NS magnetosphere, largescale PWN) background from the Off interval. & Romani 2017). With such inclination, the mass ofneutron star would be ∼ . M (cid:12) , a value much smallerthan inferred through optical fit. In a combined fit, theoptical points with small errors dominate, giving an in-clination of 64 . ◦ ± . ◦ , close to the optical-only fit.The x-ray and combined fit suggest β ∼ .
12, signifyinga very strong pulsar wind.We do note some X-ray peak asymmetry which (inhigher S/N data) may be used to constrain sweepbackdue to the finite speed of the companion wind via theparameter f v in RS16. This is interesting as, togetherwith β , one can infer the mass loss rate to see if completecompanion evaporation is expected.5.2. PSR J2339-0533
PSR J2339 − P s = 2 . E = 1 . × erg s − ‘redback’ MSP in a 4.6 h orbit about a ∼ . M (cid:12) companion. The companion wind seems par- ticularly strong, with nearly continuous radio eclipses.This wind may also be associated with the large orbitalperiod instabilities (Pletsch & Clark 2015). With a mod-est ˙ E and a strong companion wind, it is not surprisingthat the observed X-ray IBS peaks bracket φ B = 0 . β > θ ∗ ). These regions,away from the equatorial plane, may have a large σ w and small particle content (see Philippov & Spitkovsky2018). We can speculate that at these IBS positions, theX-ray spectrum is affected by large γ max ≈ σ W γ w withthe local Maxwellian peak leading to a particularly hardspectrum. It seems unlikely that p < Chandra (20 ks ObsID 11791),
Swift (49.4 kscumulative),
XMM-Newton (182 ks ObsID 721130101,790800101),
Suzaku (104 ks ObsID 406007010) and
NuS-TAR (163 ks ObsID 30202020002). We note that the sec-ond
XMM-Newton observation (rev. 3121) is contempo-raneous with
NuSTAR one for simultaneous broadbandX-ray coverage. Data were analyzed with the appro-priate response functions and fitted simultaneously us-ing
XSPEC . For this relatively bright system, we havesufficient counts to examine the orbital light curve in
Kandel et al.
Table 2.
Model fit results
Parameters J X J O J O + X J X ˙ E IBS (10 erg / s) 8 . ± .
40 4 . ± .
10 6 . ± .
13 4 . ± . i ( ◦ ) 75 . ± . . ± . . ± .
35 54 . ± . β . ± .
015 - 0 . ± .
018 5 . ± . T N (K) - 2743 ±
29 2748 ±
33 - d kpc . ± .
02 2 . ± .
02 1.25 f - 0 . ± .
007 0 . ± .
008 - M NS ( M (cid:12) ) 1.86 2.37 2.33 2.16 χ /ν Note — M NS from fit parameters and literature companion radial velocity. Figure 7.
Combined
CXO , XMM and NuSTAR light curvesfor RB J2339. Higher energy curves are offset in y. several X-ray bands. To do this, we made effectivearea/exposure-weighted count light curves, which aredisplayed with higher energy bands offset in rate in Fig.7. The double peak bracketing φ B = 0 .
75 with sepa-ration ∆ φ B = 0 .
35 is quite prominent, with a strongbridge, especially at low energy. There may be a slighttrend toward decreasing ∆ φ B with energy. These be-haviors are as expected from § >
40 keV (chanceprobability for a constant light curve being p ≈ − ),but at E >
15 keV the double peaks disappear anda new, single or tight double component arises cen-tered at φ B = 0 .
75. While the statistics are insuffi-cient for a strong conclusion, the double peak cut-offappears steeper than a cooling break, suggesting thatwe may be probing their electron γ max in the hard X- Figure 8.
Combined
CXO , XMM and NuSTAR 3 −
15 keVlight curve for J2339, compared with the best-fit IBS model. ray band. Similarly, the high energy ‘bridge’ peak ap-pears abruptly suggesting a very hard spectrum for thiscomponent.Fitting simple absorbed power-laws to the combineddata sets confirms a very hard Γ ∼ −
15 keV light curve in Fig. 8. Themagnetic field B ∼
30 G used in the IBS model is esti-mated by using typical pulsar I ∼ gm cm togetherwith the orbital parameters. The model fitting gives i ∼ . ◦ ± . ◦ . This low inclination is consistent withprevious findings of Romani & Shaw (2011) and Yatsuet al. (2015), and yields an inferred neutron star massof ∼ . M (cid:12) . BS Synchrotron Emission CONCLUSIONSWe have examined the intrabinary shock emis-sion from colliding winds in ‘spider’-type companion-evaporating pulsars, finding interesting sensitivity tothe anisotropy of the pulsar wind. This leads to im-portant differences in the contact discontinuity shapebetween the ‘redback’ (RB) case where the companionwind’s momentum dominates and the pulsar-dominated‘black widow’ (BW) case. For both, the shock producesa very hard X-ray spectrum, implying that reconnectionprovides the principal energization of e ± in the shock ofthe high σ pulsar wind.By following the accumulation and cooling of theseelectrons along the IBS, we see that the resulting syn-chrotron emission is a useful probe of the shock struc- ture. The caustic peaks from the mildly relativistic flowalong the CD are sensitive to the adiabatic accelerationof the shocked wind and the cooling break (and possiblythe upper energy cut-off) are sensitive to the magneticfield and the flow speed.We have applied these models (for an assumed sin θ ∗ wind) to a BW (PSR J1959+2048) and a RB (PSRJ2339 − β ) and the viewing geometry. These es-timates can, at least for the BW cases which seem tobe dominated by direct heating, be compared with val-ues inferred by fits to the companion optical data. ForJ1959, the X-rays prefer somewhat higher inclination(and lower neutron star mass), but higher S/N X-raydata will be needed for a very constraining comparison.For the RB J2339, direct heating does not appear toexplain the optical light curves and so models includingmagnetic ducting or other non-axisymmetric heat dis-tribution are needed. However, in both cases, IBS fitsprovide information on the pulsar and companion windsbeyond that obtainable from optical data alone.This work was supported in part by NASA grant80NSSC17K0024. Software: