Periodic Variations in the Residual Eclipse Flux and Eclipse Timings of Asynchronous Polar V1432 Aql: Evidence of a Shifting Threading Region
Colin Littlefield, Koji Mukai, Ryan Cain, Raymond Mumme, Katrina C. Magno, Taylor Corpuz, Davis Sandefur, David Boyd, Michael Cook, Joseph Ulowetz, Luis Martinez
aa r X i v : . [ a s t r o - ph . S R ] J a n Mon. Not. R. Astron. Soc. , 1–15 (2015) Printed 27 March 2019 (MN L A TEX style file v2.2)
Periodic Eclipse Variations in Asynchronous Polar V1432Aql: Evidence of a Shifting Threading Region
Colin Littlefield, , ⋆ Koji Mukai, , Raymond Mumme, Ryan Cain, Katrina C. Magno, Taylor Corpuz, Davis Sandefur, David Boyd, Michael Cook, Joseph Ulowetz, Luis Martinez Department of Physics, University of Notre Dame, Notre Dame, IN 46556 Department of Astronomy, Wesleyan University, Middletown, CT 06459 CRESST and X-ray Astrophysics Laboratory, NASA/GCFC, Greenbelt MD 20771, USA Department of Physics, University of Maryland, Baltimore, MD 21250, USA CBA Oxford, 5 Silver Lane, West Challow, Wantage, OX12 9TX, United Kingdom CBA Ontario, Newcastle Observatory, 9 Laking Drive, Newcastle, Ontario, Canada L1B 1M5 CBA Illinois, 855 Fair Lane, Northbrook, IL 60062 Lenomiya Observatory, Casa Grande, Arizona
Accepted —. Received —; in original form —
ABSTRACT
We report the results of a twenty-eight-month photometric campaign studying V1432Aql, the only known eclipsing, asynchronous polar. Our data show that both theresidual eclipse flux and eclipse O − C timings vary strongly as a function of the spin-orbit beat period. Relying upon a new model of the system, we show that cyclicalchanges in the location of the threading region along the ballistic trajectory of theaccretion stream could produce both effects. This model predicts that the threadingradius is variable, in contrast to previous studies which have assumed a constantthreading radius. Additionally, we identify a very strong photometric maximum whichis only visible for half of the beat cycle. The exact cause of this maximum is unclear,but we consider the possibility that it is the optical counterpart of the third accretingpolecap proposed by Rana et al. (2005). Finally, the rate of change of the white dwarf’sspin period is consistent with it being proportional to the difference between the spinand orbital periods, implying that the spin period is approaching the orbital periodasymptotically.
Key words: accretion, accretion disks — binaries: eclipsing — novae, cataclysmicvariables — stars: individual (V1432 Aql, RX J1940.1-1025) — stars: magnetic field— white dwarfs
Cataclysmic variables (CVs) are interacting binary systemsin which a low-mass star—usually a red dwarf—overfills itsRoche lobe and transfers mass onto a white dwarf (WD).Warner (1995) and Hellier (2001) offer excellent overviewsof these intriguing systems. In a subset of CVs known aspolars, the exceptionally strong magnetic field ( ∼ tens ofMG) of the WD synchronizes the WD’s spin period withthe orbital period of the binary (see Cropper (1990) for acomprehensive review of polars specifically). The accretionstream from the secondary star follows a ballistic trajec-tory toward the WD until the magnetic pressure matchesthe stream’s ram pressure. When this occurs, a threading ⋆ E-mail: [email protected]. region forms in which the accretion stream couples onto theWD’s magnetic field lines, and the captured material is thenchanneled onto one or more accretion regions near the WD’smagnetic poles. The impact of the stream creates a shock inwhich the plasma is heated to X-ray-emitting temperatures,so polars can be significantly brighter in X-ray wavelengthsthan ordinary non-magnetic CVs. In addition to X-rays, theaccretion region produces polarized cyclotron emission inthe optical and in the infrared, the detection of which is adefining characteristic of polars.Eclipses of the WD have provided great insight into po-lars. Because a polar has no accretion disk, an eclipsing polarwill generally exhibit a two-step eclipse: a very sharp eclipseof the compact ( ∼ white dwarf radius) cyclotron-emittingregion, followed by a much more gradual eclipse of the ex-tended accretion stream (see, e.g. , Harrop-Allin et al. (1999) c (cid:13) Littlefield et al. for an eclipse-mapping study of HU Aqr). When the accre-tion rate is high, the WD photosphere makes only a modestcontribution to the overall optical flux, overshadowed by thetwo accretion-powered components mentioned above.Eclipsing polars also make it possible to determine theorientation of the magnetic axis with respect to the sec-ondary. In HU Aqr, the orientation of the dominant mag-netic pole leads the line of centers of the binary by about 45 ◦ (Harrop-Allin et al. 1999), while in DP Leo, another eclips-ing polar, the equilibrium orientation leads the line of centersby 7 ◦ ± ◦ but with a long-term oscillation with an amplitudeof ∼ ◦ (Beuermann et al. 2014).In at least four polars, the WD’s spin period differsfrom the orbital period by as much as several percent. Inthese asynchronous polars, the WD’s magnetic field is gradu-ally synchronizing the spin period with the orbital period ontimescales of centuries. For example, Schmidt & Stockman(1991) detected a derivative in the WD spin period in V1500Cyg and estimated that the system would approach resyn-chronization about 150 years after the publication of theirstudy.Because the prototype asynchronous polar, V1500 Cyg,was almost certainly desynchronized during its 1975 novaeruption, the canonical view is that these systems arebyproducts of nova eruptions which break the synchronousrotation by causing the primary to lose mass and to inter-act with the secondary (Stockman, Schmidt, & Lamb 1988).However, Warner (2002) combined the fraction of asyn-chronous systems among all known polars with their es-timated synchronization timescales and estimated an un-expectedly short nova recurrence time of a few thousandyears for polars—far more rapid than the expected recur-rence time of ∼ × years. Every aspect of Warner’sdeduction ought to be explored, including the possibility ofan additional channel for desynchronizing polars, selectioneffects that might alter the fraction of asynchronous polars,and methods of calculating the synchronization time scale.Interestingly, in each of the four confirmed asyn-chronous polars, the threading process is inefficient in com-parison to fully synchronous systems. In synchronous sys-tems, the accretion stream is fully captured not long after itleaves the L1 point, well before it can travel around the WD(e.g. Schwope, Mantel, & Horne 1997). In none of the asyn-chronous systems is this efficient threading seen. For exam-ple, Doppler tomography by Schwope (2001) of V1432 Aqlshowed an azimuthally extended accretion curtain, a findingwhich is possible only if the accretion stream can travel sig-nificantly around the WD. X-ray observations of V1432 Aqlalso indicate that the accretion stream travels most of theway around the WD before it is fully threaded onto the mag-netic field lines (Mukai et al. 2003). Likewise, in the otherthree systems, there is mounting evidence that the accre-tion flow can significantly extend around the WD. In CDInd, the accretion stream appears to thread onto the samemagnetic field line throughout the beat cycle, requiring that In addition to the subject of this study (V1432 Aql), threeother polars are incontrovertibly asynchronous: BY Cam, V1500Cyg, and CD Ind. At the time of writing, there are at least twocandidate systems: V4633 Sgr (Lipkin & Leibowitz 2008) and CPPup (Bianchini et al. 2012). the stream be able to travel around the WD (Ramsay et al.1999). With regard to V1500 Cyg, Schmidt & Stockman(1991) argued that the smooth sinusoidal variation of thepolarization curve was consistent with the infalling streamforming a thin accretion ring around the WD. More re-cently, Litvinchova, Pavlenko, & Shugarov (2010) detectedevidence that this accretion ring is fragmented, periodicallyreducing the irradiation of the donor star by the hot WD. Inthe remaining system, BY Cam, Doppler tomograms showthat the accretion curtain extends over ∼ ◦ in azimutharound the WD, requiring a similar extent of the accretionstream (Schwarz et al. 2005). Although a sample size of fouris small, it is remarkable that in each of the confirmed asyn-chronous polars, the threading process is so inefficient thatthe accretion stream can travel much of the way around theWD. V1432 Aql (= RX J1940.1-1025) is the only known eclips-ing, asynchronous polar and was identified as such byPatterson et al. (1995) and Friedrich et al. (1996). There aretwo stable periodicities in optical photometry of V1432 Aql.The first (12116 seconds) is the orbital period, which is easilymeasured from the timings of the eclipses of the WD by thesecondary. Initially, the nature of the eclipses was unclear;Patterson et al. (1995) argued that the secondary was theocculting body, but Watson et al. (1995) contended that adense portion of the accretion stream was the culprit. Muchof the confusion was attributable to the presence of residualemission lines and X-rays throughout the eclipses, as well asthe variable eclipse depth. Since X-rays in polars originateon or just above the WD’s surface, the apparent X-ray signalthroughout the eclipse was inconsistent with occultations bythe donor star. Additionally, there was considerable scatterin the eclipse timings, and the system’s eclipse light curvesdid not show the rapid ingresses and egresses characteris-tic of synchronous polars (Watson et al. 1995). However,Mukai et al. (2003) resolved the dispute with high-qualityX-ray observations which showed that the donor actuallyeclipses the WD and that the residual X-ray flux previouslyattributed to V1432 Aql was actually contamination from anearby Seyfert galaxy.The second periodicity ( ∼ φ sp = 0 .
0, the WD is occulted by material accreting onto oneof the magnetic poles, producing a broad “spin minimum”(Friedrich et al. 1996). Analyses of the spin minima haverevealed several fascinating insights into V1432 Aql. Forexample, Geckeler & Staubert (1997) undertook an O − Cstudy of the timing residuals of the spin minima and man-aged to detect a decrease in the WD spin period, indicatingthat the system is resynchronizing itself. They also mea-sured a cyclical variation in the timings of the spin minima,caused by (1) a longitudinal offset between the magneticpole and its corresponding accretion region on the WD’ssurface and (2) the accretion stream threading onto differ-ent magnetic field lines throughout the spin-orbit beat pe-riod ( P − beat = | P − orb − P − sp | ). Using these timings and a dipoleaccretion model, the authors managed to constrain the com- c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql bined effect of the threading radius and the colatitude ofthe magnetic axis on the WD, but they could not constrainthese parameters individually. Staubert et al. (2003) appliedthe methodology of Geckeler & Staubert (1997) to a largerdataset and refined the results of the earlier paper.A critical concept which emerges from the literature isthe beat period between the spin and orbital periods. Thebeat period is simply the amount of time that it takes forthe WD (and its magnetic field) to rotate once as seen fromthe perspective of the donor star. As Geckeler & Staubert(1997) first demonstrated, the accretion stream will interactwith different magnetic field lines as the system progressesthrough its beat period, a foundational principle which in-forms our analysis throughout this paper.V1432 Aql is especially suitable for long-term study be-cause its long-term brightness has remained constant notonly in our own observations but also in data from the Amer-ican Association of Variable Star Observers dating back to2002. Similarly, the Catalina Sky Survey (Drake et al. 2009)does not show any low states in the system since coverage ofV1432 Aql began in 2005. While many polars alternate un-predictably between bright and faint states due to changesin the mass-transfer rate, V1432 Aql has not been observedto do so.We supplement these previous studies by reporting thedetection of stable periodicities in both the residual eclipseflux and the O − C timing residuals of the eclipses. Thesephenomena occur at the beat period, and we use a modelto show that our observations are consistent with a thread-ing radius whose position with respect to the WD variesthroughout the beat cycle.In response to this study’s observational findings, oneof us (DB) followed up by analyzing a different set of obser-vations obtained by the Center for Backyard Astrophysics over a much longer timespan. His group’s analysis providesconfirmation of the residual-flux and timing variations de-scribed in this paper while also reporting additional beat-cycle-related phenomena (Boyd et al. 2014). As part of a twenty-eight-month effort to study V1432Aql’s behavior at different beat phases, six of us (CL, RM,RC, KCM, TC, and DS) obtained unfiltered, time-resolvedphotometry using the University of Notre Dame’s 28-cmSchmidt-Cassegrain telescope and SBIG ST-8XME CCDcamera between July 2012 and July 2014. The exposuretime was 30 seconds for each individual image, with an over-head time of 8 seconds per image. A total of 76 light curves,consisting of over 17,500 individual measurements, were ob-tained with this instrument. These observations constitutethe bulk of our dataset, and their uniformity avoids the in-troduction of errors caused by combining unfiltered obser-vations from different telescope-CCD combinations. Becauseof their homogeneity, we use these data for all three parts ofour analysis: studying the eclipse O-C variations, measuringthe mid-eclipse magnitude, and for constructing phase plotsof the system at different beat phases. http://cbastro.org/ Figure 1.
Two representative eclipses of V1432 Aql. The datarepresented in black were obtained at φ beat = 0 .
89, and the datain gray at φ beat = 0 .
54. The solid lines are the best-fit poly-nomials for each dataset. The polynomials satisfactorily modelthe asymmetric eclipses while smoothing noisy, possibly spuriousfeatures in the light curves.
We also obtained a number of light curves with othertelescopes, but since these instruments have different spec-tral responses, we only used this supplemental data to ex-plore eclipse O − C variations. CL obtained four unfilteredtime series in July 2014 using the University of Notre Dame’s80-cm Sarah L. Krizmanich Telescope and two more withWesleyan University’s 60-cm Perkin Telescope in Septem-ber 2014. The data obtained with the Krizmanich andPerkin Telescopes have much higher time resolution (expo-sure times between 5 and 7 seconds, each with a ∼ V filter, a 23-cm Schmidt-Cassegrain and QSI-583wsCCD camera, a 25-cm Newtonian with an unfiltered SXV-H9 CCD camera, and a 28-cm Schmidt-Cassegrain equippedwith an STT-1603 CCD camera. With the exception of LM,who used 45-second exposures, each of them used an expo-sure time of 60 seconds.To compensate for light-travel delays caused byEarth’s orbital motion, the timestamp for each ob-servation was corrected to the BJD (TDB) standard(Eastman, Siverd, & Gaudi 2010).With unfiltered photometry of a CV, it is possible to in-fer the approximate V -band magnitude of the CV by select-ing a same-color comparison star and using its V magnitudewhen calculating the magnitude of the CV. Since polars tendto be quite blue, we relied upon AAVSO field photometryto select two relatively comparison blue stars; we utilized These stars are labeled 117 and 120 in AAVSO chart13643GMF, and they have B − V colors of 0.20 and 0.43, respec-tively, according to the APASS photometric survey (Henden et al.2012).c (cid:13) , 1–15 Littlefield et al. these comparison stars for all photometry used in the anal-yses of mid-eclipse magnitude and the spin modulation atdifferent beat phases.One of the most obvious phenomena in the photome-try is the highly variable magnitude of the system at mid-eclipse, which ranges from V ∼ V ∼ We used χ minimization to determine the best-fit ephemerides for the spin and orbital periods us-ing our data in conjunction with the published opti-cal eclipse and spin-minima timings in Patterson et al.(1995), Geckeler & Staubert (1997), Staubert et al. (2003),and Mukai et al. (2003). Some of the timings fromthese studies lacked uncertainties; for those observa-tions, we adopted the average uncertainty of all measure-ments which did have error estimates. Furthermore, bothAndronov, Baklanov, & Burwitz (2006) and Bonnardeau(2012) have made their photometry of V1432 Aql avail-able electronically, and while their time resolution was too Table 1.
Observed Times of Minimum Eclipse FluxBJD a φ beat φ sp BJD φ beat φ sp a low for inclusion in our eclipse analysis, it was adequatefor measuring the spin minima. In the interest of uni-formity of analysis, we measured the spin minima in theAndronov, Baklanov, & Burwitz (2006) and Bonnardeau(2012) datasets ourselves instead of using their publishedtimings. The original preprint of this paper used the timings reported inBonnardeau (2012) without reanalyzing their photometry. Usingour timing measurements of their spin minima resulted in a sig-c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql Figure 2. O − C timing residuals for the spin minima as a func-tion of φ beat . The black dataset represents the new timings whichwe report in Table 2, while the gray datapoints are from previ-ously published studies as described in the text. The data arerepeated for clarity. Our lack of timings from 0 . < φ beat < . The best-fit linear eclipse ephemeris is T ecl [ HJD ] = T ,ecl + P orb E ecl , (1)with T ,ecl = 2454289 . ± . P orb =0 . ± . The spin ephemeris of Bonnardeau (2012) fits our datavery well, and we offer only a modestly refined cubic spinephemeris of T min,sp [ HJD ] = T ,sp + P sp, E sp + ˙ P E sp + ¨ P E sp , (2)where T min is the midpoint of the spin minimum, T ,sp =2449638 . ± . , P sp, = 0 . ± . P / − . ± . × − d cycle − , and ¨ P / − . ± . × − d cycle − . The uncertainties on theseparameters were determined by bootstrapping the data. Wedo not have enough observations to meaningfully searchfor higher-order period derivatives like those reported byBoyd et al. (2014), but these values are within the errorbounds of those reported by Bonnardeau (2012). nificantly lower values of values of χ red for our spin ephemeridesin Section 4.1.2. While a polynomial fit accurately models the existingdata, P sp will likely approach P orb asymptotically over thesynchronization timescale (P. Garnavich, private communi-cation). If this is correct, then ˙ P is probably proportional tothe difference between P sp and P orb so that˙ P ≡ dP sp dE sp = k ( P sp − P orb ) . (3)Integrating the solution to this differential equation yieldsan ephemeris of T min,sp = P sp, − P orb k ( e kE sp −
1) + P orb E sp + T ,sp , (4)where P orb is the measured value and the threefree parameters are k = − . ± . × − cycles − , P sp, = 0 . ± . T =2449638 . ± . χ red = 2 . ∼ χ red .With this caveat in mind, the comparable values of χ red for each ephemeris lead us to conclude that they model thedata equally well as could be expected. Though we use thecubic ephemeris for the sake of simplicity when calculat-ing the beat phase, the exponential spin ephemeris is atleast grounded in a physical theory of the resynchronizationprocess. Moreover, in principle, the only parameter whichshould need to be updated in the future is the constant k .By contrast, a polynomial ephemeris could require an un-gainly number of terms in order to attain a satisfactory fit. Because there are several non-trivial steps in calculating thesystem’s beat phase, the beat ephemeris is too unwieldy tolist here. Nevertheless, to facilitate future studies, we havewritten a Python script which calculates the system’s beatphase at a user-specified Heliocentric Julian Date using theprocedure outlined in Appendix A. Additionally, it calcu-lates future dates at which the system will reach a user-specified beat phase. This script is available for download assupplemental online material and may also be obtained viae-mail from CL.
As defined by Schmidt & Stockman (1991), a first-orderapproximation of an asynchronous polar’s synchronizationtimescale is given by τ s = P orb − P sp ˙ P . (5) c (cid:13) , 1–15 Littlefield et al.
If one assumes rather unrealistically that ˙ P will remainconstant until resynchronization, this formula provides avery rough estimate of when resynchronization will occur.If Equation 3 is substituted for ˙ P in Equation 5, this equa-tion simplifies to τ s = − k − . Since k is essentially a decayrate, this formula yields the amount of time necessary for theinitial value (in this context, the asynchronism at T , givenby P spin, − P orb ) to be reduced by a factor of e − . Because − k − = 237700 spin cycles, τ s = 71 . ± . ∼ ∼ P orb . While this estimate of τ s is obviously notan estimate of when resynchronization will actually occur, itis slightly less than the values in Geckeler & Staubert (1997)and Andronov, Baklanov, & Burwitz (2006) and consider-ably less than Staubert et al. (2003).It is unclear how long an exponential spin ephemerismight remain valid, but if ours were to hold true indef-initely, it predicts that P sp will approach P orb to withinone second in the year ∼ ∼ ∼ P sp − P orb < ∼ ∼ − C In a conference abstract, Geckeler & Staubert (1999) firstreported the discovery of a 200-second O − C shift in V1432
Table 2.
Observed Times of Spin MinimaBJD a φ beat φ orb BJD φ beat φ orb a Aql’s eclipse timings. We followed up on this periodicity byperforming an O − C analysis on all eclipse timings listedin Table 1. We calculated both the O − C timing residualand the beat cycle count ( C beat ; see Appendix A) for eacheclipse and then used the analysis-of-variance (ANOVA)technique (Schwarzenberg-Czerny 1996) to generate severalperiodograms, with C beat serving as the abscissa.The first periodogram used all of the eclipse timingsreported in Table 1, and it showed a moderately strong sig-nal at 1.00 ± ± φ beat ∼ .
5, which is when theresidual eclipse flux is strongest (see Section 4.3.1). Boththe periodogram and waveform are shown in Figure 3. Be-tween ∼ . < φ beat < ∼ .
85, the eclipses occur ∼ φ beat ∼ .
85, the eclipses beginoccurring later, and by φ beat ∼ .
0, the eclipses are oc-curring ∼
120 seconds late. Although the 240-second O − Cjump at φ beat ∼ . − Cplot, there is a 120-second jump towards earlier eclipses at φ beat ∼ .
0. Considering the gradual eclipse ingresses andegresses, the WD must be surrounded by an extended emis-sion region, so these eclipse timings track the centroid ofemission rather than the actual position of the WD.
Given the asynchronous nature of the system and the abil-ity of the stream to travel most of the way around the WD(Mukai et al. 2003), we hypothesize that cyclical changes inthe location of the threading region are responsible for the c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql Figure 3.
From left to right: the power spectrum of the timing residuals of the combined dataset described in section 4.2, and thewaveform of the combined dataset when phased at the beat period. Black data points represent our data as listed in Table 1, while graydata points indicate previously published data as described in Section 4.2. O − C variation. In an asynchronous system, the position ofthe threading region can vary because the WD rotates withrespect to the accretion stream, causing the amount of mag-netic pressure at a given point along the stream to varyduring the beat period. Threading occurs when the mag-netic pressure ( ∝ r − ) balances the stream’s ram pressure( ∝ v ). For a magnetic dipole, the magnetic flux density B has a radial dependence of ∝ r − , but with an additionaldependence on the magnetic latitude; the magnetic pressurewill be even greater by a factor of 4 near a magnetic poleas opposed to the magnetic equator. An additional consid-eration is that the stream’s diameter is large enough thatthe magnetic pressure varies appreciably across the stream’scross section (Mukai 1988).KM modeled this scenario using a program which pre-dicts times of eclipse ingresses and egresses of a point givenits x, y , and z coordinates within the corotating frame of thebinary. The physical parameters used in the program are P orb = 3 . M WD = 0.88M ⊙ , M donor =0.31M ⊙ , R donor = 2 . × cm, i = 76 . ◦ , and binaryseparation a = 8 . × cm (Mukai et al. 2003). The codetreats the donor star as a sphere for simplicity, but sincewe do not attempt to comprehensively model the systemin this paper, the errors introduced by this approximationshould be minimal. For instance, as a result of this approxi-mation, we had to decrease i by 0.9 ◦ compared to the valuefrom Mukai et al. (2003) in order to reproduce the observedeclipse length.We first calculated the ballistic trajectory of the accre-tion stream and arbitrarily selected four candidate thread-ing regions along the stream (P1, P2, P3, and P4) underthe assumption that the stream will follow its ballistic tra-jectory until captured by the magnetic field (Mukai 1988).The eclipse-prediction program then returned the phases ofingress and egress for each of the four points given their x and y coordinates within the corotating frame of the binary.We selected these four points arbitrarily in order to demon-strate the effects that a changing threading region wouldhave on eclipse O − C timings; we do not claim that thread-
Figure 4.
A schematic diagram of the system as used in ourmodel, viewed from above the binary rest frame. The WD is restat the origin, and the black curved line is the accretion streamtrajectory, which originates at the L1 point near the right edgeof the diagram. P1, P2, P3, and P4 are illustrative threadingregions, and the cross indicates the location of the stream’s closestapproach to the WD. Since P sp > P orb , the WD rotates clockwisein this figure. ing necessarily occurs at these positions or that this processis confined to a discrete point in the x, y plane. Figure 4shows a schematic diagram of this model.Once threading occurs, the captured material will fol-low the WD’s magnetic field lines until it accretes ontothe WD. To simulate the magnetically channeled portionof the stream, we assumed that captured material travels in c (cid:13) , 1–15 Littlefield et al.
Figure 5.
A sketch indicating the general positions of the accretion spots at different beat phases as seen from the donor star. Theblack crosses represent accretion spots visible from the donor, and the vertical line is the WD’s spin axis. Section 4.2.3 explains how weinferred the positions of the two magnetic poles. a straight line in the x, y plane from the threading regionto the WD while curving in the z direction, where z is theelevation above or below the x, y plane. This is another sim-plification since the magnetic portion of the stream mightbe curved in the x, y plane, but presumably, this approxi-mation is reasonable. Since the magnetic field lines will liftthe captured material out of the orbital plane, we calculatedthe x, y coordinates of the midpoint between each thread-ing region and the WD and computed its ingress and egressphases at several different values of z . We reiterate that thisis not a comprehensive model, but as we explain shortly, itis sufficiently robust to offer an explanation for the observedO − C variations.
Before this model is applied to the observations, it is help-ful to determine the orientations of the poles at differentpoints in the beat cycle. We assume that there are two mag-netic poles which are roughly opposite each other on theWD (Mukai et al. 2003). Since i = 90 ◦ , one hemisphere ofthe WD is preferentially tilted toward Earth, and we refer tothe magnetic pole in that hemisphere as the upper pole. Thelower pole is the magnetic pole in the hemisphere which isless favorably viewed from Earth. In isolation, our observa-tions do not unambiguously distinguish between these twopoles, but since the midpoint of the spin minimum ( i.e. , φ sp = 0 .
0) corresponds with the transit of the accretionregion across the meridian of the WD (e.g. Staubert et al.2003), we can estimate when the poles face the donor star.When φ beat ∼ .
15, the spin minimum coincides with theorbital eclipse, so one of the poles is approximately orientedtowards the secondary at that beat phase. At φ beat ∼ . ∼ P sp > P orb , the accretion spots will increasinglylag behind the orbital motion of the donor star with eachsubsequent orbit. Consequently, the orientation of the ac-cretion regions with respect to the donor star will continu-ously change across the beat cycle. When viewed throughoutthe beat period at the phase of eclipse, the accretion spotsappear to slowly move across the face of the WD, therebycausing detectable changes in the times of X-ray ingress andegress.Critically, at some point during the beat cycle, each ac-cretion region will have rotated out of view at the phase ofeclipse, resulting in a jump in either the ingress or egresstimings, depending on which pole has disappeared. TheMukai et al. (2003) model predicts that when the upper poleis aimed in the general direction of P4, the X-ray egresseswill undergo a shift to later phases as the upper polecap ro-tates behind the left limb of the WD as seen at egress (seetheir Figure 15). Likewise, the disappearance of the lowerpole behind the left limb at the phase of ingress results ina shift toward later phases in the ingress timings. Basedon data in Table 5 of Mukai et al. (2003), the egress jumpoccurs near φ beat ∼ .
9, so at that beat phase, the upperpole should be pointed toward the P3-P4 region. The egressjump is more distinct than the ingress jump, so we base ouridentification of the poles on the egress jump only.Our identification of the upper and lower poles is aninference and should not be viewed as a definitive claim.For our method to be reliable, it would be necessary forthe accretion geometry to repeat itself almost perfectly inboth 1998 (when Mukai et al. (2003) observed) and the 28-month span from 2012-2014 when we observed V1432 Aql. c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql Even though the accretion geometry does seem to repeatitself on a timescale of two decades (see, e.g. , Section 4.4),this may not always be the case, as is evidenced by an ap-parent discontinuity in the timings of the of the spin minimain 2002 (Boyd et al. 2014). If the accretion rate during ourobservations was different than it was in 1998, there wouldbe changes in the location and size of the X-ray-emitting ac-cretion regions (Mukai 1988). Moreover, Mukai et al. (2003)cautioned that their model was a simplification because theaccretion geometry was poorly constrained. For example,they noted that their model did not account for the offsetbetween the accretion region and the corresponding mag-netic pole.If the upper pole is aimed towards P3-P4 near φ beat ∼ .
9, then the upper pole would face the donor at φ beat ∼ . φ beat ∼ .
15. We provide asketch of the system in Figure 5 which shows the inferredpositions of the polecaps throughout the beat cycle.
Even though the four P points were arbitrarily selected, theresults of the eclipse-timing program provide testable pre-dictions concerning the O − C variations. In our model, theemission from the accretion curtain and the threading re-gion result in a moving centroid which is responsible for anO − C shift with a half-amplitude of about ±
120 seconds (seeFig. 4.2). When the centroid of emission is in the + y regionin Figure 4, the O − C would be positive, and if it were inthe − y half of the plot, the O − C would be negative. Accord-ing to calculations using the model, eclipses of point sourcesat P1, P2, P3, and P4 would result in O − C values of 289seconds, 204 seconds, 0 seconds, and −
533 seconds, respec-tively. As for the midpoints between each of those four pointsand the WD, the O − C values would be 122 seconds, 103 sec-onds, 0 seconds, and −
289 seconds for the P1, P2, P3, andP4 midpoints, respectively. The O − C values for the mid-points have a negligible dependence on the height above theorbital plane (provided that the secondary can still eclipsethat point). Since the actual O − C variation does not exceed ±
120 seconds, it is clear that the actual O − C timings areinconsistent with a centroid near P1, P2, and P4. However,centroids near the midpoints for P1, P2, and P3 would beconsistent with the observed O − C timings.It makes sense that the centroid of the emission regionwould have a less dramatic O − C value than the candidatethreading points. Because we expect that the magnetically-channeled part of the stream travels from the threading re-gion to the WD, the light from this accretion curtain wouldshift the projected centroid of emission towards the WD. Inaddition, since the threading region likely subtends a wideazimuthal range, the ability of the projected centroid to de-viate dramatically from the WD’s position would be limited.With these considerations in mind, the consistency of thetheoretical O − C values for the P1, P2, and P3 midpointswith the observed O − C variations indicates that our modeloffers a plausible explanation of the O − C timings.The sudden jump to early eclipses near φ beat ∼ . Figure 6.
Two eclipses observed on consecutive nights with the80-cm Krizmanich Telescope. Note the different vertical scale forthe two panels. The vertical dashed lines indicate the expectedphases of the WD’s ingress and egress. On the first night (PanelA), the eclipse is very deep and begins with the WD’s disap-pearance, but on the second night (Panel B), the eclipse startsbefore the occultation of the WD. These light curves are consis-tent with the appearance of a new threading region near P3-P4in our model, indicating that this process requires less than 24hours to take place. increased magnetic pressure on that part of the stream isable to balance the decreasing ram pressure, resulting in aluminous threading region. Since the P3-P4 vicinity is inthe − y half of Figure 4, an emission region there would re-sult in an early ingress. In all likelihood, the centroid ofthat threading region does not approach P4 or its midpointbecause the theoretical O − C values do not agree with theobserved values. However, a centroid closer to P3 would re-sult in a less-early eclipse which would be more consistentwith the observations.As the WD slowly rotates clockwise in Figure 4, thecorresponding changes in the magnetic pressure along thestream’s ballistic trajectory would move the position ofthe threading region within the binary rest frame, and theeclipses would gradually shift to later phases. Half a beatcycle after the φ beat ∼ . − C timings, the lowerpole would be oriented in the general direction of P2 andthe upper pole towards P4. As the upper pole’s magnetic c (cid:13) , 1–15 Littlefield et al. pressure increases on the stream in the P3-P4 vicinity, anew threading region would form there, producing the O − Cjump observed near φ beat ∼ .
0. In short, our model predictsthe two distinct O − C jumps and explains why they are fromlate eclipses to earlier eclipses.Our observations provide circumstantial evidence of thebrief, simultaneous presence of two separate emission regionsas the system undergoes its O − C jump near φ beat ∼ . − C jump, the time of minimumeclipse flux had an O − C of ∼
140 seconds, but on the verynext night, there were two distinct minima within the sameeclipse. Separated by a prominent increase in brightness, oneminimum had an O − C of −
80 seconds, while the other hadan O − C of 240 seconds, consistent with the presence of dis-crete emission regions in the − y and + y halves of the plotin Figure 4. Moreover, assuming a WD eclipse duration of700 seconds (Mukai et al. 2003) centered upon orbital phase0.0, the optical eclipse on the first night commenced whenthe donor occulted the WD, implying a lack of emission inthe − y region. However, the egress of that eclipse continuedwell after the reappearance of the WD, as one would expectif there were considerable emission in the + y area. Indeed, acentroid of emission near the P1 midpoint would account forthe observed O − C value. On the ensuing night, by contrast,the eclipse began before the disappearance of the WD, andended almost exactly when the WD reappeared. The im-plication of these two light curves is that within a 24-hourspan between φ beat ∼ . − .
48, the locations of the emis-sion regions changed dramatically. Figure 6 shows these lightcurves and indicates in both of them the times of anticipatedWD ingress and egress. Further observations are necessaryto determine whether this behavior recurs during each beatcycle.
Our hypothesis that the location of the threading radius isvariable has ramifications for previous works. In particular,Geckeler & Staubert (1997) and Staubert et al. (2003) usedthe timing residuals of the spin minima to track the accre-tion spot as it traced an ellipse around one of the magneticpoles. One of their assumptions was that the threading ra-dius is constant, but this is inconsistent with the conclusionswe infer from our observations and model of the system. Avariable threading radius would change the size and shape ofthe path of the accretion spot (Mukai 1988)—and therefore,of the waveform of the spin minima timings used in thosestudies to constrain the accretion geometry.Additionally, the agreement between the model and ourobservations provides compelling evidence which substan-tiates previous claims (see Section 1) that the accretionstream in V1432 Aql is able to travel around the WD, as isalso observed in the other asynchronous polars. The ineffi-cient threading in asynchronous systems could be indicativeof a relatively weak magnetic field or a high mass-transferrate. For example, Schwarz et al. (2005) found that if theaccretion rate in the asynchronous polar BY Cam were 10-20 times higher than normal accretion rates in polars, thestream could punch deeply enough into the WD’s magne-tosphere to reproduce the observed azimuthal extent of theaccretion curtain. Although it is at least conceivable that the asynchronism itself causes the inefficient threading, it isnot immediately apparent why this would be so when P sp and P orb are so close to each other.Regarding the possibility of a high mass-transfer rate,previous works (e.g., Kovetz, Prialnik, & Shara 1988) haveproposed that irradiation by a nova can temporarily in-duce an elevated mass-transfer rate which persists for manydecades after the eruption has ended. In line with this the-ory, Patterson et al. (2013) proposed that CVs with con-sistently elevated mass-transfer rates—specifically, nova-likeand ER UMa systems—exist fleetingly while the donor starcools after having been extensively irradiated by a nova. Ifall asynchronous polars are recent novae, as is commonly be-lieved, this theory would predict that the same nova whichdesynchronizes the system also triggers a sustained, height-ened mass-transfer rate as a result of irradiation. The in-creased ram pressure of the accretion stream would enableit to penetrate deeply into the WD’s magnetosphere, therebyoffering a plausible explanation as to why all four confirmedasynchronous polars show strong observational evidence ofinefficient threading. However, this would not resolve theproblem of the short nova-recurrence time in polars (Warner2002, discussed in Section 1). The WD is invisible during eclipse, leaving two possiblecauses for the variation in residual eclipse flux: the donorstar and the accretion stream. The magnetic field lines of theWD can carry captured material above the orbital plane ofthe system, so depending on projection effects, some of theaccretion flow could remain visible throughout the WD’seclipse. Therefore, as the accretion flow threads onto dif-ferent magnetic field lines throughout the beat period, theresulting variations in the accretion flow’s trajectory couldcause the residual eclipse flux to vary as a function of φ beat .After we calculated the beat cycle count ( C beat ) foreach eclipse observation, we generated a power spectrumusing the ANOVA method with C beat as the abscissa andthe minimum magnitude as the ordinate. For this particularperiodogram, we used only the 71 eclipses observed with the28-cm Notre Dame telescope due to the difficulty of combin-ing unfiltered data obtained with different equipment. Thestrongest signal in the resulting power spectrum has a fre-quency of 0 . ± .
012 cycles per beat period. Figure 7shows both the periodogram and the corresponding phaseplot, with two unequal maxima per beat cycle.While a double-wave sinusoid provides an excellentoverall fit to the residual-flux variations, the observed mid-eclipse magnitude deviated strongly from the double sinu-soid near φ beat ∼ .
47 in at least three beat cycles. Twoeclipses observed on consecutive nights in high-cadence pho-tometry with the 80-cm Krizmanich telescope provide thebest example of this variation. On JD 2456842, the systemplummeted to V ∼ . φ beat = 0 . While there are sporadic departures from the double-sinusoid,none is as dramatic as the behavior near φ beat ∼ .
47 or showsevidence of persistence across multiple beat cycles.c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql Figure 7.
The power spectrum of the residual flux and a phase plot showing the waveform of the signal at the beat period. Spanning11.8 beat cycles, these plots use only the observations made with the 28-cm Notre Dame telescope. The double-wave sinusoid in thephase plot is meant to assist with visualizing the data and does not represent an actual theoretical model of the system.
24 hours later, the mid-eclipse magnitude had surged to V ∼ . φ beat = 0 . φ beat ∼ .
47 wereobserved during two additional beat cycles (one in 2013 andanother in 2014), so there is at least some evidence thatthe residual flux might be consistently lower near this beatphase. Unfortunately, gaps in our data coverage make it im-possible to ascertain whether the mid-eclipse magnitude al-ways fluctuates near φ beat ∼ .
47, so confirmation of thisenigmatic variation is necessary.
We propose that the overall variation in mid-eclipse fluxis the signature of an accretion curtain whose vertical ex-tent varies as a function of the threading radius. When thethreading region is farther from the WD, the stream cancouple onto magnetic field lines which achieve such a highaltitude above the orbital plane that the donor star cannotfully eclipse them. By contrast, when the threading region iscloser to the WD, the corresponding magnetic field lines aremore compact, producing a smaller accretion curtain whichthe donor occults more fully. The schematic diagram in Fig-ure 8 offers a visualization of this scenario.While it is conceivable that the residual flux varia-tion is caused by material within the orbital plane, theavailable evidence disfavors this possibility. In particu-lar, Schmidt & Stockman (2001) saw no diminution in thestrength of high-excitation UV emission lines during aneclipse with considerable residual flux at φ beat = 0 .
58. Ifthese emission lines originated within the orbital plane, theywould have faded during the eclipse. Furthermore, if thesource of the residual flux were in the orbital plane, theeclipse width would likely correlate with the mid-eclipsemagnitude. The eclipses with high levels of residual flux would be long, while the deeper eclipses would be short.We do not see this pattern in our data, and Figure 12 inBoyd et al. (2014) does not show such a correlation, either.Our model from Section 4.2.2 predicts that the thread-ing radius will vary by a factor of ∼ . − C jump, this hypothesis predicts that theamount of residual flux would be greatest near those jumpsand lowest between them, as is observed in a comparison ofFigures 3 and 7. In the case of a magnetic stream originatingfrom a threading region between P2-P4, the midpoint of thestream would be visible if it achieves a minimum altitudeof z ∼ . a above the orbital plane, where a is the binaryseparation. At P4, this is only one-quarter the predictedthreading radius, but at P2 and P3, this is three-quarters ofthe predicted threading radius.This hypothesis also explains why some spectra ofV1432 Aql during mid-eclipse show intense emission lines(e.g. Watson et al. 1995; Schmidt & Stockman 2001), whileothers show only weak emission (e.g. Patterson et al. 1995).For each of these previously published spectroscopic obser-vations, we calculated φ beat and found that the ones show-ing strong emission lines were obtained when the predictedresidual flux was near one of its maxima in Figure 7. By con-trast, the spectra containing weak emission were obtainedwhen the expected residual flux was approaching one of itsminima. If our hypothesis is correct, then the variation inthe emission lines is simply the result of the changing visi-bility of the accretion curtain during eclipse. Watson et al.(1995) suggested a somewhat related scenario to account forthe presence of emission lines throughout the eclipse, butthey disfavored this possibility largely because of the appar-ent residual flux at X-ray wavelengths. (As mentioned previ- c (cid:13) , 1–15 Littlefield et al.
Figure 8.
Two schematic diagrams providing a simplified illus-tration of our explanation for the residual flux variations at mid-eclipse. In both panels, the captured material travels in both di-rections along an illustrative magnetic field line. The secondaryis the gray sphere eclipsing the WD, and the threading point isshown as a large +. The inclination of the magnetic axis withrespect to the rotational axis was arbitrarily chosen as 30 ◦ . Theportion of the magnetic stream which travels upward and which isvisible at mideclipse is highlighted. The threading point in PanelA is near P4, and its threading radius is 3.6 times larger thanthat of the threading point in Panel B, when the threading pointis near the stream’s closest approach to the WD. ously, Mukai et al. (2003) later demonstrated that the resid-ual X-ray flux was contamination from a nearby galaxy.)An excellent way to test our theory would be to obtainDoppler tomograms near the times of maximum and mini-mum residual eclipse flux. Schwarz et al. (2005) showed thatthis technique is capable of revealing the azimuthal extent ofthe accretion curtain in BY Cam, and it would likely proveto be equally effective with V1432 Aql.We do not have enough data to consider why the resid-ual flux can vary by as much as ∼ φ beat = 0 .
47 would be anecessary first step in this analysis.
As the WD slowly spins with respect to the secondary, theaccretion stream will couple to different magnetic field lines,meaning that the spin modulation will gradually changethroughout the beat cycle. To explore this variation, weconstructed non-overlapping, binned phase plots of the spinmodulation in ten equal segments of the beat cycle ( e.g. ,between 0 . < φ beat < . P sp , and we did not calculate binsif they consisted of fewer than five individual observations.Figure 9 shows these ten phase plots, and several fea-tures are particularly striking. For example, the spin mini-mum near spin phase 0.0 is highly variable. Conspicuous be-tween 0 . < φ beat < .
0, it becomes feeble and ill-defined formost of the other half of the beat cycle. Sometimes, the spinminimum is quite smooth and symmetric, as it is between0 . < φ beat < .
8, but it is highly asymmetric in other partsof the beat cycle, such as 0 . < φ beat < .
6. Additionally,there is a striking difference between the phase plots imme-diately before and after the O − C jump near φ beat ∼ .
5, asone would expect if the O − C jump marks a drastic changein the accretion geometry.There is also a stable photometric maximum near spinphase ∼ . φ sp ∼ . . < φ beat < .
5. Since this feature shares the WD’sspin period, we refer to it as the second spin maximum.The second spin maximum can be exceptionally prominentin photometry, attaining a peak brightness of V ∼ . ∼ . φ beat ∼ . . < φ beat < .
3. It then weakens considerably as φ beat approaches 0.5, and after the O − C jump near φ beat ∼ . . < φ beat < .
5, it vanished in a matterof hours on JD 2456842 ( φ beat ∼ . c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql Figure 9.
Binned phase plots of the spin modulation at different beat phases, with each bin representing 0.01 spin cycles. Gaps in thelight curves are due to eclipses. The second spin maximum ( φ sp ∼ .
4) is strongest in panel C.c (cid:13) , 1–15 Littlefield et al.
Just 24 hours later, it was again visible in two successivespin cycles. This unexpected behavior coincides with theapproximate beat phase at which we would expect the dom-inant threading region to shift to the P3-P4 region in ourmodel. Nevertheless, our lack of observations near this beatphase precludes a more rigorous examination of this partic-ular variation.The second spin maximum is very apparent in somepreviously published light curves of V1432 Aql from as farback as two decades ago. For example, Watson et al. (1995)presented light curves of V1432 Aql obtained in 1993 whichshowcase the gradual growth of the second spin maximum(see Panels B-G of their Figure 2). Using our method ofdetermining the beat phase, we extrapolate a beat phaseof 0.96 for the light curve shown in their Panel B and abeat phase of 0.12 for the light curve in their Panel G. Theincreasing strength of the second spin maximum in theirlight curves agrees with the behavior that we observed atthose beat phases (see our Figure 9). Likewise, Figure 1 inPatterson et al. (1995) shows the second spin maximum atthe expected beat phases. These considerations suggest thatthe second spin maximum is a stable, recurring feature inoptical photometry of V1432 Aql.The overall predictability of the second spin maximumdoes not answer the more fundamental question of whatcauses it. One possibility is that it is the result of an ele-vated accretion rate on one pole for half of the beat cycle.The apparent gap between the two spin maxima, therefore,might simply be the consequence of an absorption dip su-perimposed on the photometric maximum or a cyclotronbeaming effect, splitting the spin maximum into two.A more interesting scenario is that the second spin max-imum could be the optical counterpart to the possible thirdpolecap detected by Rana et al. (2005) in X-ray and polari-metric data. In that study, Rana et al. (2005) detected threedistinct maxima in X-ray light curves as well as negative cir-cular polarization at spin phase 0.45, which is the approx-imate spin phase of the second spin maximum in opticalphotometry. They also measured positive circular polariza-tion at spin phases 0.1 and 0.7, which correspond with thespin minimum and the primary spin maximum, respectively.Quite fortuitously, the authors obtained their polarimetricobservations within several days of the photometric detec-tion of the second spin maximum by Patterson et al. (1995).Thus, it is reasonable to conclude that the circular polariza-tion feature near spin phase 0.45 is related to the secondspin maximum, consistent with a third accreting polecap.The conclusions of Rana et al. (2005), coupled with ouridentification of a second spin maximum, suggest that V1432Aql might have at least three accreting polecaps—and there-fore, a complex magnetic field. However, the available evi-dence is inconclusive, and follow-up polarimetry across thebeat cycle could clarify the ambiguity concerning the WD’smagnetic field structure.
We have presented the results of a two-year photomet-ric study of V1432 Aql’s beat cycle. We have confirmedand analyzed the eclipse O − C variations first reported byGeckeler & Staubert (1999), and we found that the residual mid-eclipse flux is modulated at the system’s beat period.We interpret these variations as evidence that the threadingregion’s location within the binary rest frame varies appre-ciably as a function of beat phase. Doppler tomography ofthe system at different beat phases could reveal any changesin the azimuthal extent of the accretion curtain, thereby pro-viding a direct observational test of our model of the system.Our observations provide circumstantial evidence thatthe mid-eclipse magnitude undergoes high-amplitude varia-tions on a timescale of less than a day near φ beat ∼ . ∼ φ beat ∼ .
47 fluctuations in mid-eclipse magnitude. Amateur astronomers are ideally suitedto undertake such an investigation, especially when one con-siders that our residual-flux analysis utilized a small tele-scope and commercially available CCD camera. Moreover,observers with larger telescopes could also obtain relativelyhigh-cadence photometry to study whether double-minimaeclipses consistently appear near this beat phase.In addition, we report a second photometric spin max-imum which appears for only about half of the beat cycle.This phenomenon might be evidence of a complex magneticfield, but a careful polarimetric study of the beat cycle wouldbe necessary to investigate this possibility in additional de-tail. We also offer updated ephemerides of the orbital andspin periods (see Sec. 4.1), as well as a Python script whichcalculates V1432 Aql’s beat phase at a given time and whichalso predicts when the system will reach a user-specifiedbeat phase. An exponential spin ephemeris models the dataas well as a polynomial ephemeris and is consistent with anasymptotic approach of the spin period toward the orbitalperiod. According to the exponential ephemeris, the rateof change of the spin period is proportional to the level ofasynchronism in the system; consequently, if the exponen-tial ephemeris were to remain valid indefinitely, the resyn-chronization process in V1432 Aql would take considerablylonger than previous estimates.Finally, while a comprehensive theoretical model ofV1432 Aql is beyond the scope of this paper, such an analy-sis could refine our description of the system and shed addi-tional light on V1432 Aql’s unusual threading mechanisms.
ACKNOWLEDGMENTS
We thank Peter Garnavich and Joe Patterson for their help-ful comments, as well as the anonymous referee, whose sug-gestions greatly improved the paper.This study made use of observations in the AAVSO In-ternational Database, which consists of variable star obser-vations contributed by a worldwide network of observers.The Sarah L. Krizmanich Telescope was generously do-nated to the University of Notre Dame in memory of its c (cid:13) , 1–15 eriodic Eclipse Variations in V1432 Aql namesake by the Krizmanich family. This is the first publi-cation to make use of data obtained with this instrument.DB, MC, and JU participate in the Center for BackyardAstrophysics collaboration, which utilizes a global team ofprofessional and amateur astronomers to study cataclysmicvariable stars. REFERENCES
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APPENDIX A: DETERMINING THE BEATPHASE
The spin-orbit beat cycle is the key to making sense of V1432Aql’s behavior. To calculate the beat phase ( φ beat ) of anobservation is to determine the relative orientation of theWD’s magnetic field at that time. However, since the WDspin period is variable, the beat period ( P beat ) changes, too.For example, Patterson et al. (1995) measured P sp = 12150seconds, leading to a P beat ∼
50 days. But by 2013, the spinperiod had decreased, leading to a beat period of ∼
62 days.This Appendix outlines the procedure that we employ in ourPython script to calculate φ beat given the time of observation( T ). Since P beat is given by P − beat = | P − orb − P − sp | , (A1)one solution is to determine the average length of the spinperiod ( ¯ P sp ) between T and T . The first step in determin-ing ¯ P sp is to differentiate the cubic spin ephemeris from Sec-tion 4.1 with respect to the spin epoch E sp , yielding a for-mula for the instantaneous spin period. ¯ P sp is given by¯ P sp = 1 E T Z E T ( P + ˙ P E sp + 12 ¨ P E sp ) dE, (A2)where E T is the number of spin cycles between T and T . E T , in turn, is found by applying the cubic formula to thespin ephemeris in order to express E as a function of T .Once known, ¯ P sp may be used in conjunction with P orb in Equation A1 to determine the average length of the beatperiod ( ¯ P beat ) between T and T . Thus, the number of beatcycles since T is C beat = T − T ¯ P beat , (A3)the decimal portion of which is φ beat . In our beat-phase cal-culations, we arbitrarily selected T = 2449638 . φ beat = 0 .
0, the spin phaseis 0.0 and the orbital phase is 0.86.This paper has been typeset from a TEX/ L A TEX file preparedby the author. c (cid:13)000