Constraining the Emission Geometry and Mass of the White Dwarf Pulsar AR Sco using the Rotating Vector Model
Louis du Plessis, Zorawar Wadiasingh, Christo Venter, Alice K. Harding
aa r X i v : . [ a s t r o - ph . H E ] O c t Draft version October 17, 2019
Typeset using L A TEX twocolumn style in AASTeX63
Constraining the Emission Geometry and Mass of the White Dwarf Pulsar AR Sco using theRotating Vector Model
Louis du Plessis, Zorawar Wadiasingh,
2, 1
Christo Venter, and Alice K. Harding Centre for Space Research, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
ABSTRACTWe apply the standard radio pulsar rotating vector model to the white dwarf pulsar AR Sco’s opticalpolarization position angle swings folded at the white dwarf’s spin period as obtained by Buckley et al.(2017). Owing to the long duty cycle of spin pulsations with a good signal-to-noise ratio over the entirespin phase, in contrast to neutron star radio pulsars, we find well-constrained values for the magneticobliquity α and observer viewing direction ζ with respect to the spin axis. We find cos α = 0 . +0 . − . and cos ζ = 0 . +0 . − . , implying an orthogonal rotator with an observer angle ζ = 60 . ◦ +5 . ◦ − . ◦ . Thisorthogonal nature of the rotator is consistent with the optical light curve consisting of two pulses perspin period, separated by 180 ◦ in phase. Under the assumption that ζ ≈ i , where i is the orbitalinclination, and a Roche-lobe-filling companion M star, we obtain m WD = 1 . +0 . − . M ⊙ for the whitedwarf mass. These polarization modeling results suggest the that non-thermal emission arises froma dipolar white dwarf magnetosphere and close to the star, with synchrotron radiation (if non-zeropitch angles can be maintained) being the plausible loss mechanism, marking AR Sco as an exceptionalsystem for future theoretical and observational study. INTRODUCTIONAR Scorpii (AR Sco) is an intriguing binary systemcontaining a putative white dwarf (WD) with an M-dwarf companion. The system exhibits pulsed non-thermal radio, optical, and X-ray emission, likely ofsynchrotron radiation (SR) origin (Marsh et al. 2016;Buckley et al. 2017; Takata et al. 2018). AR Sco hasa binary orbital period of P b = 3 .
56 hours, a WDspin period of P = 1 .
95 minutes, and a beat periodof 1.97 minutes. The light cylinder radius and or-bital separation are R LC = c/ Ω ∼ × cm and a ∼ × cm, respectively, with Ω the spin angularfrequency. Thus, the M star is located within the WD’smagnetosphere at a ≈ . R LC . A change in spin pe-riod ˙ P = 3 . × − s s − was inferred by Marsh et al.(2016) but was disputed by Potter & Buckley (2018a)who argued that the observations were too sparse to de-rive an accurate spin-down time scale for the WD pulsar.With more extensive observations, Stiller et al. (2018)firmly established ˙ P = 7 . × − s s − that is al-most twice as large as the value reported in Marsh et al.(2016), but consistent with the constraints in Potter &Buckley (2018a). The concomitant spin-down power is˙ E rot = I WD Ω ˙Ω ≈ × erg s − for a fiducial WD mo-ment of inertia I WD = 3 × g cm . A lack of Doppler-broadened emission lines from accreting gas suggests the This is the radius where the corotation speed equals that of lightin vacuum. absence of an accretion disk . More recently, X-ray datahave established no evidence of an accretion column,and spectral analysis of the subdominant pulsed com-ponent suggests it is non-thermal (Takata et al. 2018).These two characteristics of observed spin-down and ab-sence of an accretion disk led Buckley et al. (2017) toattribute the observed non-thermal luminosity to mag-netic dipole radiation from the WD, conclusively estab-lishing AR Sco as the first known WD pulsar, analogousto rotation-powered neutron star pulsars.The WD is highly magnetized (with a polar surfacefield of B p ∼ × G estimated by setting ˙ E rot = ˙ E md ,with ˙ E md the energy loss due to a magnetic dipole ro-tating in vacuum) and its optical emission is stronglylinearly polarized (up to ∼ P of the WD. The polarization po-sition angle (PPA; ψ ) vs. time indicates clear periodicemission. In this paper, we model the linear polariza-tion signature with the rotating vector model (RVM;Radhakrishnan & Cooke 1969):tan( ψ − ψ ) = sin α sin( φ − φ )sin ζ cos α − cos ζ sin α cos( φ − φ ) , (1) This is supported by the fact that the X-ray luminosity is only 4%of the total, optically-dominated observed luminosity and is only ∼
1% of the X-ray luminosities of typical intermediate polars, andalso from the fact that all optical and ultraviolet emission linesoriginate from the irradiated face of the M-dwarf companion. with α the magnetic inclination angle of the magneticdipole moment µ and ζ the observer angle (observer’sline of sight), both measured with respect to the WD’srotation axis Ω , φ the rotational phase (WD spin), andthe parameters φ and ψ are used to define a fiducialplane. The light curve from Buckley et al. (2017) man-ifests double peaks with a more intense first peak fol-lowed by a dimmer second peak, exhibiting a peak sep-aration of ∼ ◦ . In addition, the PPA makes a 180 ◦ sweep. These facts imply that the WD may be an or-thogonal rotator if the emission originates close to itspolar caps (Geng et al. 2016; Buckley et al. 2017). Weshow that our solution for α using the RVM supportsthis conjecture.The observations by Marsh et al. (2016) indicate ir-radiation of the side of the WD facing the companion.This forms part of the observed sinusoidal radial ve-locity profile, suggesting that the two stars are tidallylocked (Buckley et al. 2017; Takata et al. 2017; Pot-ter & Buckley 2018b). To account for the observednon-thermal radiation, models invoke injection of rel-ativistic electrons by the companion along the magneticfield lines of the WD, where they are trapped and ac-celerated (Geng et al. 2016; Takata et al. 2017; Buck-ley et al. 2017). However, different proposed scenariosplace the non-thermal emission regions in different spa-tial locales. Geng et al. (2016) noted that the Goldreich-Julian charge number density (Goldreich & Julian 1969)of the WD is much lower than required by the observedSR spectrum and thus argued that the emission shouldoriginate closer to the companion (they suggest at an in-trabinary shock caused by the interacting stellar winds;Marsh et al. 2016; Geng et al. 2016) where the particlenumber density is higher. They noted that the spin-down luminosity of the WD is sufficient to power theemission of the system. On the other hand, upon aninjection of relativistic particles from the companion,emission may originate from near the magnetic poles ofthe WD by means of pulsar emission mechanisms (withthe emission from downward-moving particles being di-rected toward the WD; cf. Buckley et al. 2017; Takataet al. 2017; Potter & Buckley 2018b; Takata & Cheng2019). This is supported by the geometric model of Pot-ter & Buckley (2018b) that can reproduce the observedpolarization signatures if the emission location is takento be at the magnetic poles of the WD. In what follows,we demonstrate that the RVM provides an excellent fitto the PPA curve, thereby favoring the magnetosphericscenario, if particles sustain small pitch angles. How-ever, we note that this hypothesis may not be unique, It is important to note that the RVM is a purely geometricalmodel, and as such cannot make any statements about light curveshapes or spectra expected from AR Sco. Neither can we con-strain particle energies or injection rates or the acceleration pro-cess within this framework. We defer the construction of a fullemission model to a future paper. since Takata & Cheng (2019) claim that they can alsoreproduce the polarization properties using an indepen-dent emission model in which the particle pitch anglesevolve with time and become quite large at the emis-sion regions, which are incidentally also located rela-tively close to the WD polar caps.The structure of this article is as follows. In Section 2we discuss the folding of the data, code calibration andfits. We present our results in Section 3. Our discussionand conclusions follow in Section 4 and 5. METHOD2.1.
Folding of Data
We use the PPA data from Buckley et al. (2017), com-prising observations on 14 March 2016 in the 340 −
900 nm range. We obtain the minimum PPA of thedataset and define the corresponding time t as thestarting point to fold the dataset. A few data points( ∼ φ curve,but we note that the folding is affected by the choiceof t . We remedy this convention issue by generatingsmoothed PPA curves with a Kernel Density Estima-tion and a Gaussian kernel technique. By inspecting thedeviation between the smoothed curve and the foldeddata, we assign a new convention to the points with alarge deviation by shifting these points by 180 ◦ (whichis the intrinsic uncertainty of the convention assigned tothe PPA, i.e., there is an ambiguity in the parallel andanti-parallel directions). We then bin the data into 30rotational phase bins. The predicted values of ψ fromEq. (1) are discontinuous, since the arctangent functionis discontinuous at φ disc = arccos (tan ζ/ tan α ) . Usingthis expression, we locate the discontinuities and shiftthe predicted PPA by 360 ◦ at these phases to finallyobtain a smooth, continuous PPA model.2.2. Code Verification and Best Fit
We adopt the convention of Everett & Weisberg(2001), letting ψ increase in the counter-clockwise direc-tion. Thus we define ψ ′ = − ψ . A Bayesian likelihoodapproach is employed to constrain the RVM using theoptical PPA data. We define our “best fit” as the 50 th percentile or median in the posterior distribution of themodel parameters. We verified our code by obtainingand comparing independent RVM fits of radio pulsardata (Everett & Weisberg 2001), yielding consistent α and ζ values, within uncertainties.For the Bayesian analysis, we employ a Markov-ChainMonte Carlo technique (Foreman-Mackey et al. 2013)with 50 walkers, 40,000 steps and a step burn-in of14,000. We maximize the following likelihood function:ln p ( y | φ, α, β, φ , ψ , f ) = − . n [ ( y n − H ( φ, α, β, φ , ψ )) S n + ln(2 πS n )] , (2) R Sco RVM S n = σ n + f ( H ( φ, α, β, φ , ψ )) , y represents thedata values, σ the uncertainties of the data (assumed tofollow a Gaussian distribution) and H the model values.The factor f compensates for the case when the PPAuncertainties are underestimated. We assume uniformpriors on the cosine of α and ζ (rather than the anglesthemselves). For cos ζ , this is an appropriate choice fromCopernican arguments. For cos α , this convention is lessjustified, but nevertheless ultimately immaterial for thepresent context; we have verified there is no dramaticchange in resulting uncertainties with a uniform priorchoice on α rather than cos α . RESULTS3.1.
Constraints on WD Geometry
The red curve in Figure 1 depicts the best-fit RVM to the polarization data, with yellow curves associatedwith a random selection of parameters from the pos-terior distribution of cos α and cos ζ . The uncertain-ties for the best fit are taken to be the 68% probabilityaround the median, i.e., the 16 th and 84 th percentilevalues. From our best fits of these quantities, we ob-tain α = 86 . ◦ +3 . ◦ − . ◦ and ζ = 60 . ◦ +5 . ◦ − . ◦ . For the un-certainty parameter ln( f ), only the maximum of ln( f )is constrained, and f is quite small, thus we concludethe data are described well by the model without theinclusion of this term.The correlation or degeneracy seen between φ and ψ in Figure 2 owes to the fact that these parameterstranslate the model horizontally and vertically; thus, anatural degeneracy exists, since the model is cyclic and asimilar fit may be obtained for different choices of φ and ψ . This is also the reason why small, disconnected con-tours could be eliminated from Figure 2 by constrainingthe priors of the nuisance parameters φ and ψ . Thelarge duty cycle leads to relatively small uncertaintieson both α and ζ as compared to the case of known radiopulsars. The pulses of the latter typically have smallduty cycles and thus relatively large uncertainties on α (the impact angle β = ζ − α is typically better con-strained, given visibility requirements, but ζ may remainill-constrained).3.2. Constraints on WD Mass
Marsh et al. (2016) reported the mass ratio of AR Scoas 1 /q = M M /M WD ≥ .
35, assuming M WD = 0 . M ⊙ and M M = 0 . M ⊙ as a natural pairing for the systemthat is located at a distance of d = 116 ±
16 pc. They alsomeasured the radial velocity of the M star, K = 295 ± − and obtained the following mass function m WD sin i (cid:18) q q (cid:19) = P b K πG ≈ . +0 . − . M ⊙ . (3) See the Appendix for a parameter study that indicates the be-havior of the RVM for different choices of α and ζ . Figure 1.
The best-fit RVM solution (red line) for the PPAdata using an MCMC technique with the likelihood param-eter f included, also showing ensemble plots (possible fits)as yellow lines. The abscissa is chosen such that the ob-server crosses the WD fiducial plane, defined by the magneticmoment and spin vectors, at phase zero. Figure 2.
The best-fit for the model implementing cos α ,cos ζ and a likelihood parameter f . We found cos α =0 . +0 . − . , cos ζ = 0 . +0 . − . , φ = (19 . ◦ +9 . ◦ − . ◦ ), ψ =(135 . ◦ +5 . ◦ − . ◦ ), ln f = − . +2 . − . . The φ and ψ are nuisanceparameters. They report no significant evidence of non-sinusoidal ra-dial velocities, thus suggesting little overestimation ofthe true amplitude owing to irradiation. Based on theM star’s emission line velocities, Marsh et al. (2016) infer q . .
86, perhaps close to the cited maximum ( q ∼ . R M ≈ . R ⊙ using a mass of m M ≈ . M ⊙ via the volume-equivalent Roche radiusapproximation of Paczy´nski (1971),13 (cid:18) G m M P b π (cid:19) / ≈ R M ≈ . R ⊙ . (4)Finally, based on the stellar radius estimate and the stel-lar brightness, they quote a mass estimate of m M ≈ . d/
116 pc) M ⊙ . A more accurate parallax distanceof d = 117 . ± . m M ≈ . M ⊙ when inverting theabove mass-distance relation, suggesting general consis-tency with the estimate m M ∼ . M ⊙ . Substitution of m M ∼ . M ⊙ into Eq. (3) and demanding that m WD isbelow the Chandrasekhar mass limit of 1 . M ⊙ yieldsthe constraint i & ◦ .The mass m WD is generally insensitive to the valueof m M , as is readily apparent from the q ≫ R M and alsothe approximation employed in Eq. (4). Constraints on m WD may be obtained under the assumption ζ ∼ i ,which is generally expected from formation/evolutionand observed in other contexts (e.g., Albrecht et al. 2007;Watson et al. 2011). Note that the spin angular momen-tum of the WD is about two orders of magnitude inferiorto the total orbital angular momentum. Since it is likelythat the WD was spun up to its exceptionally low periodof P = 117 s via past accretion episodes, the transfer ofangular momentum would tend to align the WD spin tothe orbit angular momentum. Allowing m M be a free pa-rameter, while propagating normally-distributed uncer-tainties for K and cos ζ (see Figure 2) in Eq. (3) yields aband of allowable values of { m M , m WD } depicted in Fig-ure 3. Adopting R M ≈ . R ⊙ and solving the systemof equations Eq. (3) – (4) yields in m WD = 1 . +0 . − . M ⊙ for the median, with uncertainties for the 68% contain-ment region of probability. This yields q = 3 . +0 . − . that is somewhat in tension with Marsh et al. (2016)’squoted q . .
86 derived from the velocity amplitudes ofatomic emission lines relative to the M, but not mean- This choice of mass is due to the M5 spectral type of the M-dwarfand is also typical of donor-star masses in other systems havingsimilar orbital periods as AR Sco. This is true as long as q is in fact ≫
1. If q ∼ m WD issensitive on the value of q . For our assumed value of q ∼
3, thereis a small influence of m WD . Figure 3.
Various constraints on the the component masses { m M , m WD } . The gray band depicts the probability densityin the mass function Eq. (3) with the assumption ζ ≈ i con-structed by propagating uncertainties from the fitted cos ζ inthis paper and uncertainties of the radial velocity in Marshet al. (2016). The white dashed curve is the median of theprobability distribution of m WD while { red , blue } curves de-lineate regions of { , } probability. The pink bandis the constraint 1 < q < .
86 from Marsh et al. (2016),while the green band portrays m M ∼ . M ⊙ as inferred fromEq. (4). The Chandrasekhar limit of m WD < . M ⊙ istraced by the cyan dashed line. ingfully so, given the allowable range of uncertaintiesand systematics which may be present in the M starmass/radius estimate.3.3. Geodetic Precession
If we allow misalignment of the spin and orbital axes ζ = i , this may explain why the optical maximum doesnot occur at inferior conjunction of the WD (Buckleyet al. 2017). Katz (2017) proposed that this mismatchmay be explained either by dissipation in a bow wave orby misalignment between the WD spin axis and boththe orbital axis and the WD’s oblique magnetic mo-ment as well as potential oblateness of the WD, causingprecession of the spin axis. The latter would lead tovariable heating of the companion surface along with adrift in the phase of the optical maximum, with a pe-riod of decades. Peterson et al. (2019) analyzed a cen-tury’s worth of optical photometry on AR Sco but couldnot detect any precessional period as suggested by Katz(2017), although this period may be much longer for alarger angle of misalignment between the orbital planeand the WD spin axis, or a smaller oblateness.We suggest that effective spin-orbit interaction mayresult in the precession of the WD pulsar spin axis(Damour & Ruffini 1974; Esposito & Harrison 1975) ow-ing to the metric curvature of the companion, even if theWD is not oblate. While any precession in the systemmay be dominated by electromagnetic influences of thecompanion, the General Relativity (GR) rate calculatedbelow is the minimum rate in absence of any electro-magnetic torques. In the framework of classical GR, the R Sco RVM ω p is derived by Barker& O’Connell (1975a,b). The rate of precession is insen-sitive to m WD and may be estimated independent of theassumption ζ ≈ i , ω p = G / c (1 − ε ) (cid:18) πP b (cid:19) / m / q (1 + q ) / ∼ × − rad s − − ε ≈ . − − ε . (5)Since the orbital eccentricity ε is presumably close tozero, the numerical value in Eq. (5) constitutes a lowerlimit to the expected precession rate. This rate is sim-ilar to pulsar systems where such precession has beendetected (e.g., B1913+16 Weisberg et al. 1989; Cordeset al. 1990; Kramer 1998) over decade timescales. Theobservable scope of precession depends on the degree ofmisalignment; if ζ is significantly different from i , suchprecession (including that by oblateness of the WD) maybe imprinted on the pulses and PPA swings of AR Scoand therefore changes in ζ values may be detectable onsimilar decade timescales as in pulsar binary systems.That is, from long-term time evolution of pulses andpolarization data, the degree of misalignment may beestimated. Moreover, if well-characterized, such preces-sion also affords an independent constraint on the com-ponent masses.Takata & Cheng (2019) also note that a detailed com-parison between model and measured PPA sweeps mayconstrain the orientation of the spin axis of the WD. Infuture, we will study polarization data as a function oforbital phase φ b to constrain the effects of precessionthat a varying α or ζ vs. φ b may point to. In addition,predictions from a full emission model should lead topredictions of the PPA evolution that may be fit to datato constrain the WD spin axis alignment with the orbitalaxis and / or precession in the system. It is hoped thatcontraints on precession may teach us more about thesystem’s characteristics, analogous to the case of PSRJ1906+0746 where observations over several years of thisprecessing pulsar revealed the average structure of theradio beam (Desvignes et al. 2019). DISCUSSION4.1.
Assumptions Regarding the Particle Pitch Angle
Using our fits for α and ζ , we can constrain the geom-etry of the emission region for the optical radiation. Thederivation of the RVM (see Appendix of du Plessis et al.2019) assumes that the observer samples emission thatis tangent to local magnetic field lines and that the po-larization vector is in the poloidal plane. This is a goodapproximation in either SR or curvature radiation (CR)scenarios, provided that the particle momentum paral-lel to the field is relativistic and the relativistic particleshave small pitch angles.At first sight, this assumption of small pitch angle η seems plausible since, for a large Lorentz factor γ , the pitch angle is η ∼ θ γ ∼ /γ (see Eq. (10) where we esti-mate that the particles radiating optical emission have γ ∼ −
100 from the spectrum, depending on B and η , and that η ∼ − if γ ∼ γ and η evolve with distance as the particles move along the B -field lines. For example, Takata et al. (2017) solvean approximate form of the coupled set of equationsthat describe the evolution of γ and perpendicular mo-mentum p ⊥ , used earlier by Harding et al. (2005) in thecontext of neutron star pulsars. Takata et al. (2017,2018); Takata & Cheng (2019) assume that relativis-tic particles are injected from the companion and travelinto the WD magnetosphere (along closed B -field lines,an assumption based on α ∼ ◦ < ◦ ) before radiat-ing significantly, since the SR timescale is much longerthan the light crossing timescale r/c at the companionposition ( r ∼ a ). They then study a magnetic mirroreffect in which the first adiabatic invariant µ ∝ p ⊥ /B isconserved and find that if the initial pitch angle is largeenough (sin η > .
05, i.e. outside a loss cone set by ini-tial conditions), the mirror effect operates and the pitchangles increase to ∼ π/ B and η , before the particlesreturn outward.One can solve for the emission height at which the SRloss timescale equals the light crossing timescale r/c : ra ≈ . γ / µ / η / . ∼ . R LC , (6)with γ = γ/ µ = µ/ G cm , µ = 0 . B s R the magnetic moment, and η . = η/ . r ∼ . a , for γ = 50, µ = 6 . × G cm and η = 0 . γ = 150 and µ = 2 × G cm (which is closer to that impliedby the estimated surface magnetic field B s , see Eq. [7]),then Eq. (6) implies that r ∼ . a > . a . This sug-gests p ⊥ /B is no longer an invariant and the mirror ef-fect will not operate, but the particles will lose all their The set of equations used by Takata et al. (2017) are valid onlyfor γ ≫ η ≪
1, and assumes that the accelerating E -field is screened. The more general form of the equations usedby Harding et al. (2005) includes a term involving a non-zero E -field, but are subject to the drift approximation, where themotion of the guiding center is considered and the helical motionis averaged over gyrophase. We concur with Takata et al. (2017)that the local accelerating E -field (parallel to the local B -field)is probably screened – see Eq. (16) below. We do not addressthe details of the acceleration process in this paper, other thanto note it seems sufficient to accelerate particles to very large γ e , max ∼ eBR comp /m e c ∼ − . Such high factors areneeded if SR is to account for the observed X-ray emission, asnoted in Eq. (13). Figure 4.
A schematic diagram indicating an orthogonalrotator with a dipolar field, as well as four possible solutionsfor emission that is radiated tangent to the local magneticfield lines and is pointing toward a distant observer. The spinaxis is indicated by Ω , the magnetic moment by µ , and theobserver angle by ζ . The red arrows indicate inflowing whilethe blue arrows indicate outflowing particles, as defined withrespect to the nearest magnetic pole. energy abruptly suffering catastrophic SR losses. Thus,the pitch angle may never attain very large values.We thus note that small pitch angles (as assumed bythe RVM) are plausible in two cases: (i) There existsome parameter choices where t SR > t cross at a height r that is some substantial fraction of a , so that the parti-cles will radiate all their energy before undergoing mag-netic mirroring; (ii) particles with small enough initialpitch angles (e.g., sin η . .
05) will fall in the loss cone,and will not be impacted by the magnetic mirror. Acomplete model, solving the particle dynamics generally(for any η and γ , and not in the drift approximation)to predict the light curves, spectrum and polarizationproperties of AR Sco will be able to address this issuemore fully.4.2. Constraints on the Emission Geometry
In a dipolar magnetosphere, multiple locations satisfythe viewing constraints derivable using our α and ζ fitswithin the framework of the RVM, which assumes emis-sion to be tangential to the local B -field, irrespective ofaltitude, since a dipolar field is self-similar (additionalspectral information may help constrain the actual emis-sion heights, not just the emission directions). Depend-ing on the current system, any non-zero subset of theselocations could participate to produce the observed PPAcurve. Thus, since the RVM variation is seen over en- Figure 5.
Magnetic co-latitudes θ B and longitudes φ B (withrespect to µ ) which the observer samples during one full ro-tation in WD spin phase φ for a static dipole magnetosphere.As in Figure 4, the red and blue colors denote inflowing andoutflowing charges in a particular magnetic hemisphere (asdefined by φ B ), defined with respect to the closest magneticpole. The magnetic poles are located at θ B = 0 , π . tire spin rotation of the WD, at least one of the mul-tiple solutions must be realized (e.g., the model Pot-ter & Buckley 2018b that only assumes inflowing par-ticles would satisfy these constraints). We indicate thisschematically in Figure 4, where we show an orthogo-nal rotator and four different tangents pointing in theobserver direction for a particular ζ . Outflowing andinflowing particles with respect to the nearest magneticpole are indicated by blue and red arrows, respectively.Blue arrows thus occur in the top half of the meridionalslice (large magnetic longitude), and red ones in the bot-tom (small magnetic longitude). Dark and light colorsare used to further distinguish between solutions. Themeaning of colors remains identical in Figure 5, where weindicate constraints involving the magnetic co-latitude θ B and longitude φ B (both defined with respect to themagnetic dipole moment axis µ ). These constraints de-rive from the condition that the magnetic field tangentsare sampled by the observer, for α = 87 ◦ and ζ = 60 ◦ .For θ B , the solutions satisfy Eq. (34) and its reflectiongiven in Wadiasingh et al. (2018).The different panels in Figure 5 are 2D projections ofthe 3D plot in the leftmost corner. The top left panelindicates that the ‘red solutions’ are located around φ B ∼ φ B ∼ π (i.e., inmeridional and antimeriodional planes with respect to µ ). In the top right panel, the light and dark coloredlines coincide, indicating that the observer samples the R Sco RVM φ B for each color during the course of one rota-tion of the WD, independent of θ B . The red and bluesolutions are of similar functional form, but offset by afactor π in φ B , as previously. The bottom right projec-tion indicates that the light and dark red solutions havethe same form (offset by a factor ∼ π/ θ B , and thesame for the light and dark blue), but the red vs. blueones are mirror images of each other (being φ ∼ π out ofphase), since they originate close to opposite magneticpoles. Thus, the observer would sample the light blueand dark red solutions (or light red and dark blue ones),offset by half a rotation in spin phase φ , even thoughthey trace out the same range in θ B .The constrained spatial region that follows from ourRVM fits suggests that a large portion of the magneto-sphere, at multiple altitudes, must support relativisticcharges. Depending on scenarios of particle accelera-tion, charges may be either ingoing or outgoing relativeto the WD surface, halving the number of potential sitesof emission, unless there are counterstreaming beams. Ina more complete emission model beyond the geometricRVM, deriving constraints on the altitude of emissionshould be possible, thereby pinning down the precise lo-cation in the magnetosphere where emission arises atany phase.Interestingly, the estimated polar cap opening angle ismuch lower ( θ PC / π ∼ ∼ Constraining the Emission Mechanism
In order to constrain the emission mechanism thatmight be responsible for the polarized optical radia-tion, let us estimate several relevant quantities and com-pare pertinent timescales and lengthscales. The mag-netic field at the polar cap may be estimated assuming M WD = 1 . M ⊙ via B p ∼ × G × (cid:18) R WD × cm (cid:19) − (cid:18) P
117 s (cid:19) / ˙ P . × − s s − ! / , (7)from dipole spin-down, with R WD the WD radius. Genget al. (2016) calculates the magnetic field strength B x ata distance x = a − R WD ∼ × cm above the stellarsurface, where a is binary separation, obtaining B x = B p ( x/R WD ) − ∼ × G. We estimate the potentialdrop at the polar cap using (Goldreich & Julian 1969)Φ max ∼ × statvolt × (cid:18) R WD × cm (cid:19) (cid:18) B p × G (cid:19) (cid:18) P
117 s (cid:19) − . (8) Figure 6.
Acceleration, crossing and radiation (CR, SRand IC) loss timescales for ‘far’ ( B x ) and ‘near’ ( B p ) cases.The solid lines for SR represent a pitch angle of π/ . ∼ ◦ ). For thisfigure, we used a companion temperature of kT = 1 eV, i.e., T ∼ ,
000 K and radius of R comp = 0 . R ⊙ . This yields a maximum Lorentz factor of γ max , Φ ∼ × , although it is unlikely that the particle will be ableto tap the full potential.The corresponding SR, CR and inverse Compton (IC)timescales ( γ/ ˙ γ ) are indicated in Figure 6, along with anacceleration ( R L /c ) and crossing timescale ( a/c ), where R L is the Larmor radius. To calculate the IC loss ratewe used ˙ E IC = 4 σ T cU γ γ / γ + γ ) (Schlickeiser& Ruppel 2010), where γ KN = 3 √ m e c / πkT and U = 2 σ SB T R ⋆ /cR is the photon energy density. Here, m e is the electron mass, σ T the Thomson cross section, σ SB the Stefan-Boltzmann constant, k the Boltzmannconstant, R ⋆ the companion radius and R is the dis-tance from the companion’s center to the shock. Weconsider a ‘near’ ( B p ) and ‘far’ ( B x ) case. Since the ICcooling time is much larger than that of SR (even forsmall pitch angles), we can rule out IC.If we equate the SR and CR loss rates to solve for theaverage pitch angle η , this yields a maximum value ofsin η ∼ × − (cid:18) γ × (cid:19) (cid:18) B × G (cid:19) − , (9)where e is electron charge and ρ c ∼ R WD / θ pc thecurvature radius at the surface of the WD, with θ PC ∼ (Ω R WD /c ) / . Thus, we see that SR dominates for allreasonable values of γ and for non-zero pitch angles,with CR only becoming relevant for γ & γ max , Φ or forpitch angles η . − . Following Takata et al. (2018) and attributing the non-thermal pulsed X-ray emission to a power-law tail that isemitted by a power-law particle spectrum between γ min and γ max , we can infer constraints using the observedfrequencies that correspond to the peak in the spectralenergy density ( ν obs , min = 2 × Hz) and the highestpulsed X-ray photon ( ν obs , max & × Hz): γ B sin η ∼ × G (cid:18) ν obs , min × Hz (cid:19) , (10) γ B sin η & × G (cid:18) ν obs , max × Hz (cid:19) . (11)When enforcing that B < B p , we obtain γ min & (cid:18) ν obs , min × Hz (cid:19) (cid:18) sin η − (cid:19) − / , (12) γ max & × (cid:18) ν obs , max × Hz (cid:19) (cid:18) sin η − (cid:19) − / , (13)for a fiducial value of sin η ∼ − . (Imposing a lowerlimit of γ min ≈ η & × − .)These constraints are consistent with the assumptionsmade by Takata et al. (2017) who model the SR spec-trum assuming γ min = 50 and γ max = 5 × .Let us calculate a characteristic length scale for bothSR and CR: L SR ≈ c γ ˙ γ SR = 10 cm × (cid:18) ν obs , min × Hz (cid:19) − / (cid:18) B8 × G sin η − (cid:19) − / (14)and L CR = c γ ˙ γ CR = 10 cm (cid:18) ν obs , min × Hz (cid:19) − (cid:18) ρ c × cm (cid:19) . (15)Since L SR ≪ L CR , it is apparent that SR should domi-nate over CR, even for very small values for η (cf. Fig-ure 6).Observations by Buckley et al. (2017) indicate that thedegree of linear polarization of the optical data varieswith orbital phase, but may reach up to 40%. Thisconstraint should be exploited in an emission model,but is beyond the RVM’s capabilities. For example,Takata & Cheng (2019) argue that a polarization degreethat varies with orbital phase may be understood in theframework of different contributions of the SR and thethermal emission from the companion, since the lattermay depolarize the observed emission. We expect thata model that includes emission from different emissionheights that bunch in phase to form the peaks will notlikely overpredict the maximum observed polarizationdegree of 40%. SR from a single particle in an orderedfield may reach high values, but this will be lowered bycontributions from several particles within the popula-tion of radiating particles emitting at different spatial locales. This is similar to the findings of Harding &Kalapotharakos (2017) who found large PA swings anddeep depolarization dips during the light-curve peaks inall energy bands in their multiwavelength pulsar modelthat invoke caustic emission. Such a model may not beexactly applicable to the WD pulsar, although contribu-tions from emitting particles at different altitudes mayprovide a blended polarization signature, thus loweringthe polarization degree vs. that expected from a singleparticle in an ordered field. The detailed characteristicsdepend on emission position and mechanism.A constraint on B can be obtained by assuming thatthe cyclotron energy is below the SED peak (otherwisea break would be observable due to this threshold en-ergy). At the WD surface, the cyclotron energy is 9 eV.When ν cycl = ν obs , min , B . × G, implying thatthe emission site is at least ∼ R WD above the stellarsurface. Another constraint on B may be obtained byrequiring the ratio of electromagnetic to kinetic particleenergy density to be σ ≫ n GJ (Goldreich & Julian 1969): | n GJ | = Ω · B πec = 4 . × cm − (cid:18) P117 s (cid:19) − (cid:18) B8 × G (cid:19) = 1 . − (cid:18) P117 s (cid:19) − (cid:18) B2 × G (cid:19) , (16)with Ω the angular frequency. The number of par-ticles needed to explain the SR spectrum may be es-timated as follows. Let us focus on the peak fre-quency ν peak = 0 . ν c , with ν c = 3 eγ B sin η/ πm e c .We use F ( ν peak /ν c ) = F (0 .
29) = 0 .
924 and assumea delta distribution for the steady-state particle spec-trum dN/dE e = N δ ( E e − E ∗ e ), with E ∗ e = γm e c theenergy needed to explain the peak of the spectrum at0 . ν c . From Figure 5 of Takata & Cheng (2019), wetake νF obs ν = 8 × − erg/s/cm at an observed en-ergy of ∼ .
02 eV. Using P ν = √ e B sin ηm e c F ( ν peak /ν c ) (17)and a rough estimate of the emitting volume V ∼ π (1 − cos θ PC ) a / ∼ cm , we find N ∼ × (18) R Sco RVM n e ∼ N V ∼ cm − . (19)This is lower than the estimate of Geng et al. (2016) whofind n e ∼ × cm − , but still highly supra-Goldreich-Julian suggesting screening of E fields and electron-ionplasma sourced from the companion. On the other hand,since we have demonstrated that the PPA of the WDcan be modeled by the RVM, this could mean that theemission of the AR Sco system originates from a dipole-like magnetosphere surrounding the WD (Buckley et al.2017). Using observations of the linear flux, circular fluxand PPA observations, Potter & Buckley (2018b) showthat the polarization is coupled to the WD spin period,agreeing with our conclusion, while models that placethe emission site at the companion fail to explain thepolarization signatures that are clearly coupled to thespin period of the WD. The ‘missing’ particles noted byGeng et al. (2016) may either be supplied by the com-panion that may inject relativistic electrons into the WDmagnetosphere, or less likely by pair cascades (probablyneeding severely non-polar B-field structures) occurringin the WD magnetosphere, and does not per se argue foran emission location near a putative bow shock. More-over, freshly-injected particles from the companion maysolve the issue of needing non-zero pitch angles for SRto dominate. CONCLUSIONSWe applied the RVM to optical polarization data ofAR Sco. We obtained α ∼ ◦ confirming the valueexpected from light curve inspection and a 180 ◦ PPAswing with spin phase. Finding ζ ∼ ◦ also agreeswith the independent assumption by Potter & Buckley(2018b) who adopted ζ = 60 ◦ for their model to repro-duce the observed data. Using the result of ζ = 60 ◦ wethen found that m WD = 1 . +0 . − . M ⊙ , which is withinthe limits by Marsh et al. (2016). We next obtained q = 3 . +0 . − . , which is slightly larger compared to the value of q = 2 .
67 calculated by Marsh et al. (2016) whenadopting m WD = 0 .
8. Our fits of the PPA evolutionof the WD using the RVM could imply that the emis-sion of the AR Sco system originates from the dipole-likemagnetosphere of the WD, probably close to its polarcaps, while the particles are probably being acceleratedat and injected from the companion star. The issue ofpitch angle evolution should be addressed with detailedemission modelling.Future work includes fitting the RVM to PPA datain other energy bands, applying a general polarizationcalculation that includes special-relativistic correctionsand investigating the effect of different types of magneticfield structures on the predicted polarization signatures.We will also model the phase-resolved PPA to infer α and ζ for different orbital phases, which may constraineffects of spin precession in this system. This work high-lights the complementary constraints on system geome-try, emission locales and radiation physics that are sup-plied by adding polarization data to spectral and tem-poral data within a unified approach.ACKNOWLEDGMENTSWe thank the anonymous referee for insightful ques-tions, comments and suggestions. Z.W. thanks DemosKazanas for helpful discussions. This work is basedon the research supported wholly / in part by theNational Research Foundation of South Africa (NRF;Grant Number 99072). The Grantholder acknowledgesthat opinions, findings and conclusions or recommen-dations expressed in any publication generated by theNRF supported research is that of the author(s), andthat the NRF accepts no liability whatsoever in thisregard. Z.W. is thankful for support from the NASAPostdoctoral Program. This work has made use of theNASA Astrophysics Data System. Software: emcee (Foreman-Mackey et al. 2013)APPENDIX A. RVM ATLASIn Figure 7, we indicate the behavior of the RVM for different choices of α and ζ .REFERENCES One may use this rough estimate of particle number density aswell as assumptions for / limits on σ = B / (8 πn e γm e c ) atdifferent spatial positions, implying different values of B , to con-strain the average γ at those positions. E.g., at the companionposition, one would obtain h γ i . for σ > Albrecht, S., Reffert, S., Snellen, I., Quirrenbach, A., &Mitchell, D. S. 2007, A&A, 474, 565,doi: 10.1051/0004-6361:20077953 3.2Barker, B. M., & O’Connell, R. F. 1975a, PhRvD, 12, 329,doi: 10.1103/PhysRevD.12.329 3.3 Figure 7.
Atlas of RVM for different choices of α and ζ (in degrees) as indicated in the legend of each subplot.—. 1975b, ApJL, 199, L25, doi: 10.1086/181840 3.3Buckley, D. A. H., Meintjes, P. J., Potter, S. B., Marsh,T. R., & G¨ansicke, B. T. 2017, Nature Astronomy, 1,0029, doi: 10.1038/s41550-016-0029 (document), 1, 1,2.1, 3.3, 4.3, 4.3Cordes, J. M., Wasserman, I., & Blaskiewicz, M. 1990, ApJ,349, 546, doi: 10.1086/168341 3.3 Damour, T., & Ruffini, R. 1974, Academie des SciencesParis Comptes Rendus Serie Sciences Mathematiques,279, 971 3.3Desvignes, G., Kramer, M., Kejia, L., et al. 2019, Science,365, 1013 3.3du Plessis, L., Wadiasingh, Z., Venter, C., et al. 2019, arXive-prints, arXiv:1907.01311.https://arxiv.org/abs/1907.01311 4.1 R Sco RVM11