EX Draconis: Using Eclipses to Separate Outside-In and Inside-Out Outbursts
James M. C. Court, Simone Scaringi, Colin Littlefield, Noel Castro Segura, Knox S. Long, Thomas Maccarone, Diego Altamirano, Nathalie Degenaar, Rudy Wijnands, Tariq Shahbaz, Zhuchang Zhan
MMNRAS , 1–10 (2019) Preprint 16 April 2020 Compiled using MNRAS L A TEX style file v3.0
EX Draconis: Using Eclipses to Separate Outside-In andInside-Out Outbursts
J.M.C. Court (cid:63) , S. Scaringi , C. Littlefield , N. Castro Segura , K. S. Long ,T. Maccarone , D. Altamirano , N. Degenaar , R. Wijnands , T. Shahbaz ,Z. Zhan Department of Physics and Astronomy, Texas Tech University, PO Box 41051, Lubbock, TX 79409, USA Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA Anton Pannekoek Institute for Astronomy, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands Instituto de Astrof´ısica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain Department of Earth, Atmospheric, and Planetary Sciences, M.I.T., Cambridge, MA 02139, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present a study of the eclipses in the accreting white dwarf EX Dra during
TESS
Cycles 14 and 15. During both of the two outbursts present in this dataset, the eclipsesundergo a hysteretic loop in eclipse-depth/out-of-eclipse-flux space. In each case, thedirection in which the loops are executed strongly suggests an outburst which is trig-gered near the inner edge of the accretion disk and propagates outwards. This in turnsuggests that the outbursts in EX Dra are ‘Inside Out’ outbursts; events predicted byprevious hydrodynamic studies of dwarf nova accretion disks and confirmed spectro-scopically in a number of other accreting white dwarf systems. We therefore proposethat the direction of the loop executed in eclipse-depth/out-of-eclipse flux space beused as a test to phenomenologically distinguish between ‘inside out’ and ‘outside in’outbursts in other eclipsing dwarf novae; a reliable and purely photometric test todifferentiate between these phenomena.
Key words: accretion disks – cataclysmic variables – eclipses – stars: individual: EXDra
EX Draconis (Fiedler et al. 1997) (hereafter EX Dra) is anAccreting White Dwarf (AWD) which is accreting matterfrom a Roche-lobe overflowing M dwarf star (Knigge 2006).Photometric studies have shown that the system has an or-bital period of 5.04 hr, and that the system is nearly edge-on with an inclination of ∼ ◦ (Baptista et al. 2000). EXDra regularly undergoes outbursts, with a recurrence time of ∼ days, leading to this system being classified as a dwarfnova.The high inclination of EX Dra results in deep eclipsesof the white dwarf, the accretion disk and the hotspot (Jo-ergens et al. 2000), as well as shallower eclipses of the sec-ondary star offset in orbital phase by 0.5 (e.g. Golysheva (cid:63) E-mail: [email protected] et al. 2015; Khruzina et al. 2019). The large depths of theseeclipses make EX Dra a good candidate on which to performa study of eclipse variation over time, a technique which canshed light on the behaviour of the accretion disk (e.g. Smak1971) and help us better understand the nature of the out-bursts in these objects. Previous studies of EX Dra, such asthe study by Billington et al. (1996) shortly after the discov-ery of the source, have focused on the shape of the eclipsesand used them to place constraints on the parameters of thetwo objects in the binary. In particular these authors foundthat the flux at mid-eclipse during quiescence is consistentwith being entirely from the companion star, suggesting thatthe accretion disk is entirely eclipsed at this time and hencethe radius of the accretion disk is smaller than that of thedonor star.Dwarf nova outbursts such as those seen in EX Dracan be divided into two categories depending on the evo- c (cid:13) a r X i v : . [ a s t r o - ph . H E ] A p r J.M.C. Court et al. lution of the accretion disk throughout the event: so-called‘Outside-In’ outbursts, which begin partway through an ac-cretion disk at a substantial distance from the white dwarfand propagate inwards, and ‘Inside-Out’ outbursts, whichbegin near the disk’s inner edge and propagate out (e.g.Meyer & Meyer-Hofmeister 1984). Both types of outbursthave been modelled mathematically by Mineshige & Osaki(1985). They suggest that in an Outside-In outburst, thereis first a build-up of matter at the outer edge of the disk duea low disk viscosity which does not allow for efficient out-ward transfer of angular momentum. The surface densitythus increases at this radius until no stable configuration inthe ‘cold’ state exists, at which point this part of the outerdisk switches to the ‘hot’ state (see e.g. Ichimaru 1977; Mi-neshige & Osaki 1983). This triggers a heating wave whichpropagates inwards, until it either dissipates or the entiredisk is in the ‘hot’ state. While most AWDs preferentiallyshow outbursts of one type or the other, a number of sys-tems (e.g. NY Ser, Sklyanov et al. 2018) have been observedto undergo both Inside-Out and Outside-In outbursts.The models of Mineshige & Osaki (1985) show thatInside-Out outbursts should occur at lower accretion rate ˙ M (e.g. Mineshige & Osaki 1983). In this case, the masstransfer rate is low enough that material does not pile up atthe edge of the disk and is able to move inwards via viscousdiffusion, increasing surface density globally throughout thedisk. The critical surface density value above which no coldstate solution exists is smaller in the inner disk, and hencethe instability is first triggered relatively close to the inneredge of this disk. From this point, the heating wave propa-gates both inwards and outwards until the entire disk is inthe hot state.Although the models of Mineshige & Osaki (1985) pro-vide physically motivated explanations for how each type ofoutburst occurs, they make a number of predictions whichhave not been supported by subsequent observations (Buat-M´enard et al. 2001). If the type of outburst is determinedonly by the global accretion rate, for example, it would notbe expected that individual systems would show both typesof outburst without corresponding significant evolutionarychanges. Modifications to the models (e.g. Buat-M´enardet al. 2001) have so far been unable to explain the observedfact that both types of outburst seem to be able to occurin systems with both low and high accretion rate, meaningthat the trigger criteria behind Inside-Out and Outside-Inoutbursts remains an open question.Inside-Out and Outside-In outbursts can be distin-guished with high-quality two-colour observations (e.g. Ioan-nou et al. 1999; Webb et al. 1999). In the absence of multiplecolours, however, the common way to separate the two typesof outburst is with qualitative parameters such as the ‘shape’of each outburst. This method of separation outburst typeshas been noted as being inaccurate, subjective and heavilyaffected by human biases (e.g. Kato & Osaki 2013). As such,a new model-independent photometric test to differentiatethese two phenomena is required to build a robust sampleof each type of outburst in order to perform future studiesto better understand their physical differences.It has been shown (e.g. Smak 1971; Rutten et al. 1992)that, in eclipsing AWDs, the profile of the eclipses can alsobe used to distinguish between Inside-Out and Outside-Inoutbursts; as the secondary star passes in front of the accre- tion disk, different parts of the disk are eclipsed in sequence,and hence analysing the shape of the eclipses can indicatewhich radii in the disk are brighter than others. Baptistaet al. (2000) used this technique on a number of outburstsof EX Dra in 1995-1996 using a small sample ( ∼ ) of ob-served eclipses. They found that the outbursts in EX Drawere all consistent with being Inside-Out in nature. How-ever, this method of determining outburst type requires de-tailed modelling of individual eclipses, which in turn relieson high-quality lightcurve data with time resolutions on theorder of ∼ s, and hence the number of systems on whichsuch analysis can be performed is low.EX Draconis is among the objects currently being ob-served by TESS during its survey of transients in the north-ern hemisphere. Due to the high observation cadence andlong stare-time of this instrument, we now have an unprece-dented large sample of eclipses from this object, during bothoutbursts and quiescent periods, allowing population stud-ies of these events as well as studies of how they evolve overtime. In this paper we present such a study of the eclipses inthis object, and we show that a study of eclipses in a dwarfnova can be used to differentiate between the aforementionedOutside-In and Inside-Out outbursts in these systems with-out needing to assume complex eclipse profiles a priori.AWDs can also be studied in the timing domain. Thesesystems naturally show a strong Fourier peak correspondingto their orbital period, but many AWDs also show evidenceof a distinct signal at a slightly higher or lower frequency.This modulation, referred to as a ‘superhump’, is believedto be a beat frequency between the orbital period and someform of modulation in the accretion disk.Superhumps come in two different varieties. ‘Positive’superhumps, in which the period of the modulation is greaterthan the orbital period, are seen during unusually large‘super’-outbursts of a class of dwarf nova systems referredto as SU UMa-like AWDs. This type of superhump is be-lieved to be caused by the nodal precession of an elongatedaccretion disk (e.g. Horne 1984). Negative superhumps onthe other hand, in which the period of the modulation isshorter than the orbital period, are seen during quiescenceand smaller outbursts in a wider variety of AWDs (e.g. Har-vey et al. 1995). This type of superhump is generally inter-preted as being caused by the vertical precession of a tilted accretion disk (e.g. Bonnet-Bidaud et al. 1985).The geometric origins of superhumps mean that theirpresence, or absence, in a system can provide valuable in-formation as to the configuration of the accretion disk inthat system. As such we also probe the properties of thesuperhumps of EX Dra in this paper, giving us additionalinformation on the geometry of the system and helping toexplain how and why outbursts in this system evolve in theway that they do.
We make use of data from the Transiting Exoplanet SurveySatellite (
TESS , Ricker et al. 2009); a space-based observa-tory launched in 2018.
TESS ’s primary mission is to performa survey of optical transits in both the both the northern and
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MNRAS000 , 1–10 (2019) he Eclipses of EX Draconis southern ecliptic hemispheres. It does this by dividing thesky into a number of sectors, each of which is observed fora period of ∼ month. The entire sector is observed at acadence of 30 minutes and a number of pre-selected objects,including EX Dra, are each observed at a cadence of 2 min-utes. Both the short-cadence and long-cadence barycentreddata are freely available from the portal provided by theMikulski Archive for Space Telescopes (MAST ).EX Dra was observed by TESS during Sectors 14 &15, corresponding to the first two sectors in the survey ofthe northern ecliptic hemisphere; these in turn correspondto MJDs (Modified Julian Dates) 58682–58709 and 58710–58736 for a total of 53 days. We show the lightcurves ofthese observations in Figure 1. While it was being observed,EX Dra underwent two outbursts; once in Sector 14, begin-ning at MJD ∼ , and once in Sector 15, beginning atMJD ∼ . In both cases the beginning and the end ofthe outburst were observed, but a portion of the middle ofthe outburst fell into a ∼ day data gap caused by TESS telemetering data back to Earth.
The observations with
TESS in these two sectors include239 eclipses: 122 in Sector 14 and 117 in Sector 15. In orderto analyse the properties of the eclipses, we first isolatedthem using our own algorithm:(i) Create a ‘smoothed’ lightcurve by removing all oscil-lations at the orbital period:(a) For each datapoint f ( t ) in the original lightcurve,make a subset of all datapoints f ( x ) such that t − p ≤ x < t + p , where p is the orbital period of EX Dra.(b) Calculate the n th percentile rate value in f ( x ) ,where n corresponds to the approximate percentage ofeach orbital period in which the white dwarf and accretiondisk are at least partly eclipsed. Call this value q l For EXDra, we use n = .(c) Remove all datapoints with flux values less thanor equal to q l . As the main eclipse is always the faintestpart of each orbital period, this essentially removes theeclipse from the dataset, as well as the secondary eclipseif present. This in turn prevents a change in the depth ofan eclipse from artificially causing a change in the out-of-eclipse flux value measured in Step (iii).(d) Find the mean of the remaining datapoints in f ( x ) .Replace the flux value of f ( t ) with this value.(ii) Subtract the smoothed lightcurve from the originallightcurve to obtain the detrended lightcurve, or a lightcurvewhich only retains variability at the orbital period of thesystem.(iii) For each eclipse in the detrended lightcurve, select asmall time window around the eclipse minimum, such thatthe shape of lightcurve within does not include the ingress oregress of the hotspot. Fit a Gaussian to the lightcurve in thisperiod to extract values for eclipse depth and width. Usingthe same small time range, make a subset of data from the https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html smoothed lightcurve and take the mean flux of this subsetto be the out-of-eclipse flux.In Figure 2 we show how our algorithm decomposesthe lightcurve from Sector 14 into smoothed and detrendedcomponents. Note that the eclipse-isolating algorithm we usehere is different to the method used in Court et al. (2019) toisolate eclipses in the AWD Z Cha. In that paper, the authorsremove the eclipses from the lightcurve and fit splines acrossthe resulting data gaps to interpolate what the flux would beat each eclipse midpoint if no eclipse occured. This methodwas valid in in Z Cha because, aside from a weak and rela-tively constant-brightness hotspot during quiescence, therewas no significant modulation in the lightcurve on the orbitalperiod. By using splines to estimate the uneclipsed flux dur-ing data gaps caused by the removal of eclipses, the authorswere able to account for variability of the source flux overtimescales longer than an orbital period and hence more ac-curately estimate what the uneclipsed fluxes were at thesetimes. However in EX Dra the hotspot varies significantlyduring quiescence (e.g. Golysheva et al. 2015), as does thedepth of the secondary eclipse and the brightness of the fluxbetween the main and secondary eclipses, which would sig-nificantly contaminate our estimates for out-of-eclipse fluxfor each eclipse if we used the method of Court et al. (2019). We calculate an independent value for the orbital period ofEX Dra by first estimating a period from the largest peak ina Lomb-Scargle spectrum of the entire dataset from both ob-servations. We then folded the lightcurve over a range of pe-riods close to this estimate, choosing the period which gavethe lightcurve with the lowest dispersion. We iterated thisprocess for successively smaller ranges of periods until thechange in dispersion between periods was no longer signifi-cant. Using this method we found a period of 0.2099385(6)days, or 5.03852(1) hours. This is similar to but slightlylonger than previous periods reported for this object (e.g.0.20993698(1) d, Baptista et al. 2000). This discrepancy isconsistent with previously reported variations in the periodof this system, including sinusoidal variations on timescalesof years, and a long term trend towards higher ortbital pe-riod (Pilarˇc´ık et al. 2012; Baptista et al. 2000).The
TESS lightcurves of EX Dra show evidence of astrong negative superhump in this source during quiescencewith a period of ∼ . hr. These appear as diagonal ‘lines’in a flux-phase plot (shown in Figure 3), which indicate aperiodic signal offset slightly in frequency from the orbitalfrequency. We find no evidence of either positive or negativesuperhumps during either outburst.In order to test whether the superhump frequencychanges during the observations covered in this study, weused generalised Lomb-Scargle spectroscopy (Lomb 1976;Scargle 1982; Irwin et al. 1989) to calculate superhumpperiods separately from three different segments of thelightcurve: • Segment 1 between the start of the lightcurve and MJD2458694, corresponding to the quiescent period before thefirst outburst.
MNRAS , 1–10 (2019)
J.M.C. Court et al. R a t e ( e / s ) R a t e ( e / s ) Figure 1.
Lightcurves of EX Dra from
TESS
Sectors 14 (top) and 15 (bottom), showing one outburst in each. Inset in the lightcurveof Sector 14 is a zoom to the region highlighted in blue, showing the structure of the eclipses in this source. Grey periods indicate datagaps caused by telemetry constraints. BJD refers to Barycentred Julian Days. R a t e ( e − / s ) Origi al LightcurveSmoothed LightcurveDetre ded_Lightcurve
Figure 2.
The decomposition of the lightcurve of Sector 14 intosmoothed and detrended components, using the algorithm out-lined in Section 2.2. • Segment 2 between MJDs 2458706 and 2458719, corre-sponding to the quiescent period between the two outbursts. • Segment 3 between MJD 2458731 and the end of thelightcurve, corresponding to the quiescent period after thesecond outburst.In the Lomb-Scargle spectrum of each lightcurve seg-ment, we fit a Gaussian to the peak corresponding to thesuperhump frequency. We find peak superhump frequenciesof 4.9901(8), 4.9900(2) and 5.014(6) BJD − for Segments 1,2 and 3 respectively . The superhump frequencies calcu-lated for Segments 1 and 2 are consistent with being iden-tical. However, the frequency in Segment 3 is significantlyhigher. The Gaussian fit to the superhump frequency in Seg-ment 3 is also significantly broader, with a Gaussian widthof 6.369(9) BJD − , compared to widths of 3.1541(7) and2.9041(6) BJD − for Segments 1 & 2. This suggests that thesuperhumps in Segment 3 are somewhat less coherent than BJDs: Barycentred Julian Days MNRAS000
The decomposition of the lightcurve of Sector 14 intosmoothed and detrended components, using the algorithm out-lined in Section 2.2. • Segment 2 between MJDs 2458706 and 2458719, corre-sponding to the quiescent period between the two outbursts. • Segment 3 between MJD 2458731 and the end of thelightcurve, corresponding to the quiescent period after thesecond outburst.In the Lomb-Scargle spectrum of each lightcurve seg-ment, we fit a Gaussian to the peak corresponding to thesuperhump frequency. We find peak superhump frequenciesof 4.9901(8), 4.9900(2) and 5.014(6) BJD − for Segments 1,2 and 3 respectively . The superhump frequencies calcu-lated for Segments 1 and 2 are consistent with being iden-tical. However, the frequency in Segment 3 is significantlyhigher. The Gaussian fit to the superhump frequency in Seg-ment 3 is also significantly broader, with a Gaussian widthof 6.369(9) BJD − , compared to widths of 3.1541(7) and2.9041(6) BJD − for Segments 1 & 2. This suggests that thesuperhumps in Segment 3 are somewhat less coherent than BJDs: Barycentred Julian Days MNRAS000 , 1–10 (2019) he Eclipses of EX Draconis Figure 3.
A flux-phase diagram showing the
TESS lightcurve ofEX Dra folded over a period of 0.2099385 d, corresponding to theorbital period of the system. Eclipses can be seen as dark verticaltracks on this plot centred at phase 0, whereas the superhumpscan be seen as bright diagonal stripes. The count rate in eachorbital period has been normalised to better show the periodicbehaviour of the source rather than longer-term variability. in the other two segments, in turn suggesting that the sec-ond outburst (but not the first) disrupted the nodal diskprecession believed to give rise to negative superhumps (e.g.Harvey et al. 1995).
The luminosity L of an accretion disk depends on both thetemperature at each annulus in the disk ( T ( r ) ), which inturn depends on the local accretion rate ˙ m at each location.For a static disk, L can be given by: L ∝ (cid:90) R out R in T ( r ) r d r (1)where R in and R out are the inner and outer radii of the disk,and T ( r ) is given by: T ( r ) ∝ (cid:32) ˙ mr (cid:34) − (cid:114) R ∗ r (cid:35)(cid:33) (2)where R ∗ is the radius of the compact object.In eclipsing systems, a portion of the disk flux is ob-scured by the companion star during each eclipse. In the caseof a maximal eclipse, in which the companion star passes di-rectly in front of the compact object, the total unobscuredflux φ ecl from the system can be given by: Φ ecl ∝ Φ + k (cid:90) R out R ecl T ( r ) r d r if R ecl < R out Φ otherwise (3)where R ecl is the radius of the eclipsing companion star, Φ is the residual flux after the entire accretion disk and thecompact object itself are obscured, and k is a constant toconvert the luminosity of the disk into the flux from the diskas seen from Earth. Φ is assumed to be entirely from thecompanion star and constant. As can be seen from equations 1 & 3, measuring the flux from an eclipsing system both inand out of eclipse allows us to break the degeneracy betweenthe accretion rate in the system and the characteristic radiiof the disk.To find whether eclipses in EX Dra should cover a sig-nificant portion of the disk, we can compare the radius ofthe companion star to the ‘circularisation radius’ R circ of thedisk; the smallest value of R out corresponding to the lowestvalue of accretion rate that would form such a disk (e.g.Frank et al. 2002): R circ = a ( + q )( . − . q ) (4)where a is the semi-major axis of the binary orbit and q is the ratio of the donor mass to the accretor mass. Previ-ous studies (Barwig et al. 1994; Billington et al. 1996) haveestimated values of q = . ± . , leading to a circulariza-tion radius R circ ≈ . a (see also Baptista & Catal´an 2000;Joergens et al. 2000).There are a number of published methods to estimatethe effective radius R lobe of a Roche lobe (e.g. Kopal 1959;Paczy´nski 1971), and hence the radius of a Roche-lobe fillingstar as is present in AWDs. Eggleton (1983) showed that: R lobe = . q / a . q / + ln ( + q / ) (5)agrees with numerical calculations of Roche Lobe radii towithin 1% for all values of q . Using this equation and theapproximation of q found by Billington et al., we find that,in EX Dra, R lobe ≈ . a . As this is a factor of > largerthan R circ , we find that it would be possible for the disk tobe fully eclipsed in EX Dra.In Figure 4 we show plots of how the eclipse depth andthe out-of-eclipse flux of EX Dra vary over the course ofboth of the outbursts in this study. In both cases, we showlines with equations given by: d max = Φ − Φ (6)where d max is the maximum eclipse depth for a given un-eclipsed flux, Φ is the estimated uneclipsed flux at that time.To find Φ , we fit this function to the datapoints correspond-ing to eclipses which occurred during quiescence. If the ra-dius of the disk is smaller than the radius of the companionstar at this time (as found spectroscopically by Billingtonet al. 1996), then datapoints on this line will then corre-spond to eclipses in which the entire disk is obscured (e.g.Scaringi et al. 2013). As the response drift of TESS is notwell described, we perform this fit to find Φ independentlyfor each sector as recommended in the TESS
InstrumentHandbook .Notably in each sector we find eclipses above the lineof total eclipse when using our fit values of Φ ; this indi-cates non-physical fractional eclipse depths of greater than100%, in turn indicating that the accretion disk was not fully eclipsed during quiescence. As such, for further analy-sis, we redefine Φ in each sector as the largest value suchthat no rebinned eclipse has a depth greater than σ above https://archive.stsci.edu/missions/tess/doc/TESS_Instrument_Handbook_v0.1.pdf In order to reduce outlier effects on our results, we ‘rebin’ oureclipses by a factor 2, by finding the mean depth and uneclipsedflux values of each consecutive pair of eclipses in time.MNRAS , 1–10 (2019)
J.M.C. Court et al.
300 400 500 600 700 800 900Out of Eclipse Rate(e/s)100200300400500600700800 E c li p s e D e p t h ( e / s )
200 300 400 500 600 700 800 900 1000Out of Eclipse Rate(e/s)100200300400500600700800 E c li p s e D e p t h ( e / s ) Figure 4.
Plots of eclipse depth against out-of-eclipse flux for the eclipses in Sector 14 (left) and Sector 15 (right). To show the hystereticbehaviours in this parameter space, we show the line that would connect these eclipses as the source evolves in time. In both panels,arrows indicate the direction in which the outburst progressed. The darker dashed lines are the estimated lines of total eclipse we findby fitting a 1:1 line to the eclipses during quiescence in each sector. The paler dotted lines indicate the estimated lines of total eclipsewe find by asserting that no eclipse may have an eclipse depth that lies more than 1 σ above the line. The hysteretic behaviour of thesource in this parameter space can be more clearly seen when plotting fractional eclipse depth: see Figure 5. the line of total eclipse. This new value is by definition anupper limit on Φ , but decreasing this value further does notqualitatively change any of the results of this study.In both Sectors, the eclipses show significant deviationbelow the line of maximum eclipse during outburst, althoughthe magnitude of this deviation is significantly smaller thanthose seen in eclipses of the CVs KIS J192748.53+444724.5and Z Cha (Scaringi et al. 2013; Court et al. 2019). In orderto better quantify the hysteresis shown in this parameterspace, we can divide the depth of each eclipse by the pre-dicted maximum eclipse depth, thus obtaining the fractionaldepth ∆ of each eclipse: ∆ = d Φ − Φ (7)We show plots of fractional eclipse depth against out-ofeclipse flux in Figure 5; this figure better shows the pres-ence and direction of hysteretic loops in this parameter space(hereafter ∆ – Φ space), but the exact shapes of the loops inthis figure depend on the value we obtain for Φ . We showthat, in both Sectors, eclipses during the rise of the out-burst show an increase in fractional eclipse depth until theyapproach the line of total eclipse. In Sector 14, eclipses re-main close to the estimated line of total eclipse until nearthe peak of the outburst, at which point they decrease toan eclipse fraction of ∼ . . During the latter stages of theoutburst, the eclipses generally have lower eclipse fractionsthan during the outburst rise, resulting in a generally clock-wise hysteretic loop in the parameter space as defined here.Notably, this is different to the behaviour seen in Z Cha, inwhich eclipses execute a generally anticlockwise hystereticloop in the same parameter space over the course of an out-burst (Court et al. 2019). ‘Clockwise’ and ‘anticlockwise’ hysteretic loops in this studyrefer to the direction of a loop traced on a diagram with out-of-eclipse flux on the x -axis and eclipse depth on the y -axis. The hysteretic behaviour in ∆ – Φ space in Sector 15 isbroadly similar to that in Sector 14; the fractional eclipsedepth tends to be smaller during the fall of the eclipse thanduring the rise, again leading to a generally clockwise hys-teretic loop. There is some more complex behaviour near thepeak of the outburst in Sector 15 during which the eclipsestrace a smaller, anticlockwise hysteretic loop. This behaviouris likely linked to the more complex profile of the outburstin Sector 15 compared to Sector 14 (compare e.g. Figure 4). By studying the properties of eclipses in EX Dra, wehave discovered the existence of hysteretic loops in eclipse-depth/out-of-eclipse space ( ∆ – Φ space). We have also pre-sented evidence of negative superhumps during quiescencein this system. Both of these results place constraints on thegeometry of the EX Dra system and the physics of its accre-tion disk. In this section, we discuss these constraints, andcompare our findings in EX Dra to previous studies of simi-lar objects, as well as to models of Inside-Out and Outside-Inoutbursts in AWDs. The different hysteretic loop structure traced by the eclipsesin these outbursts highlight a number of significant differ-ences between EX Dra and Z Cha, another AWD on whichsimilar analysis has been performed. Court et al. (2019)found that eclipses in Z Cha underwent anticlockwise hys-teretic loops in ∆ – Φ space during the course of both an out-burst and a superoutburst. Court et al. used this behaviourto attempt to understand the behaviour of the accretiondisk during these events. In order to replicate the loop seenin data, they found that the outbursts in Z Cha must beginwith an increase in the physical size of the emitting region MNRAS000
Plots of eclipse depth against out-of-eclipse flux for the eclipses in Sector 14 (left) and Sector 15 (right). To show the hystereticbehaviours in this parameter space, we show the line that would connect these eclipses as the source evolves in time. In both panels,arrows indicate the direction in which the outburst progressed. The darker dashed lines are the estimated lines of total eclipse we findby fitting a 1:1 line to the eclipses during quiescence in each sector. The paler dotted lines indicate the estimated lines of total eclipsewe find by asserting that no eclipse may have an eclipse depth that lies more than 1 σ above the line. The hysteretic behaviour of thesource in this parameter space can be more clearly seen when plotting fractional eclipse depth: see Figure 5. the line of total eclipse. This new value is by definition anupper limit on Φ , but decreasing this value further does notqualitatively change any of the results of this study.In both Sectors, the eclipses show significant deviationbelow the line of maximum eclipse during outburst, althoughthe magnitude of this deviation is significantly smaller thanthose seen in eclipses of the CVs KIS J192748.53+444724.5and Z Cha (Scaringi et al. 2013; Court et al. 2019). In orderto better quantify the hysteresis shown in this parameterspace, we can divide the depth of each eclipse by the pre-dicted maximum eclipse depth, thus obtaining the fractionaldepth ∆ of each eclipse: ∆ = d Φ − Φ (7)We show plots of fractional eclipse depth against out-ofeclipse flux in Figure 5; this figure better shows the pres-ence and direction of hysteretic loops in this parameter space(hereafter ∆ – Φ space), but the exact shapes of the loops inthis figure depend on the value we obtain for Φ . We showthat, in both Sectors, eclipses during the rise of the out-burst show an increase in fractional eclipse depth until theyapproach the line of total eclipse. In Sector 14, eclipses re-main close to the estimated line of total eclipse until nearthe peak of the outburst, at which point they decrease toan eclipse fraction of ∼ . . During the latter stages of theoutburst, the eclipses generally have lower eclipse fractionsthan during the outburst rise, resulting in a generally clock-wise hysteretic loop in the parameter space as defined here.Notably, this is different to the behaviour seen in Z Cha, inwhich eclipses execute a generally anticlockwise hystereticloop in the same parameter space over the course of an out-burst (Court et al. 2019). ‘Clockwise’ and ‘anticlockwise’ hysteretic loops in this studyrefer to the direction of a loop traced on a diagram with out-of-eclipse flux on the x -axis and eclipse depth on the y -axis. The hysteretic behaviour in ∆ – Φ space in Sector 15 isbroadly similar to that in Sector 14; the fractional eclipsedepth tends to be smaller during the fall of the eclipse thanduring the rise, again leading to a generally clockwise hys-teretic loop. There is some more complex behaviour near thepeak of the outburst in Sector 15 during which the eclipsestrace a smaller, anticlockwise hysteretic loop. This behaviouris likely linked to the more complex profile of the outburstin Sector 15 compared to Sector 14 (compare e.g. Figure 4). By studying the properties of eclipses in EX Dra, wehave discovered the existence of hysteretic loops in eclipse-depth/out-of-eclipse space ( ∆ – Φ space). We have also pre-sented evidence of negative superhumps during quiescencein this system. Both of these results place constraints on thegeometry of the EX Dra system and the physics of its accre-tion disk. In this section, we discuss these constraints, andcompare our findings in EX Dra to previous studies of simi-lar objects, as well as to models of Inside-Out and Outside-Inoutbursts in AWDs. The different hysteretic loop structure traced by the eclipsesin these outbursts highlight a number of significant differ-ences between EX Dra and Z Cha, another AWD on whichsimilar analysis has been performed. Court et al. (2019)found that eclipses in Z Cha underwent anticlockwise hys-teretic loops in ∆ – Φ space during the course of both an out-burst and a superoutburst. Court et al. used this behaviourto attempt to understand the behaviour of the accretiondisk during these events. In order to replicate the loop seenin data, they found that the outbursts in Z Cha must beginwith an increase in the physical size of the emitting region MNRAS000 , 1–10 (2019) he Eclipses of EX Draconis
300 400 500 600 700 800 900Out of Eclipse Rate(e/s)0.800.850.900.951.001.051.10 F r a c t i o n a l E c li p s e D e p t h
300 400 500 600 700 800 900 1000Out of Eclipse Rate(e/s)0.750.800.850.900.951.001.05 F r a c t i o n a l E c li p s e D e p t h Figure 5.
Plots of fractional eclipse depth against out-of-eclipse flux for the eclipses in Sectors 14 (left) and 15 (right). Arrows indicatethe direction in which the hysteretic loops were executed over the course of the outburst in each Sector. To convert eclipse depth tofractional eclipse depth we assume Φ = . e − s − in Sector 14 and Φ = . e − s − in Sector 15, chosen to ensure that no eclipsehas an apparent fractional depth more than 1 σ above 1.0. of the disk followed, after some finite time delay, by an in-crease in the mass transfer rate through the disk and henceits surface brightness. The opposite direction of the loop wefind here (compare Figures 4 & 5 with Figures 12 & 13 inCourt et al. 2019) indicates that the order of events duringthe outburst in EX Dra is reversed; during both outburstscovered in this study, the disk first underwent an increase insurface brightness, followed by a radial increase in the sizeof the emitting region.EX Dra and Z Cha have significantly differing orbitalperiods; EX Dra has an orbital period of ∼ . hr (Baptistaet al. 2000), while Z Cha has an orbital period of ∼ . hr(e.g. McAllister et al. 2019). Notably, this places the twosystems on opposite sides of the so-called ‘period gap’; arange of orbital periods between ∼ . and ∼ . hr whichrelatively few AWDs are found to possess (e.g. Paczynski& Sienkiewicz 1983). A semi-empirical model of the evolu-tion of AWDs created by Knigge et al. (2011) suggests thatthe physics of mass transfer in these objects should differbetween systems above the period gap and systems belowit. In both cases, the accretion from the Roche-lobe fillingdonor star through L1 must be balanced by the shrinking ofthe orbital period of the AWD. In systems below the periodgap, Knigge et al. (2011) found that the system evolution isdominated by the emission of gravitational waves, leading toa slow orbital shrinkage and a corresponding low accretionrate ˙ M . Above the period gap the Roche Lobe is large enoughto contain a star with a radiative core, and the main evolu-tionary mechanism becomes magnetic braking (Rappaportet al. 1983). This method of angular momentum transfer issignificantly more efficient than gravitational wave emission,and hence Knigge et al. (2011) find that AWDs above the pe-riod gap should have higher ˙ M than systems below the gap.As such, we would expect EX Dra to have a significantlygreater ˙ M than Z Cha.Another notable phenomenological difference betweenZ Cha and EX Dra is the behaviour of the eclipses duringquiescence and at the start and end of each outburst. InZ Cha, Court et al. (2019) found that eclipses during quies- cence were consistent with having fractional depths of ,and that this remained the case during the initial stages ofeach outburst. In EX Dra, we have shown that the fractionaleclipse depth actually increases during the initial stage ofthe outburst, meaning that the eclipse fraction during qui-escence must have been < . The differences between ZCha and EX Dra, in terms of how their eclipses propertiesvary over the course of an outburst, suggest that the evo-lutions of the outbursts observed in each source evolve indifferent ways. The different ways in which an accretion disk evolves in bothan Inside-Out and an Outside-In outburst is reflected in thebehaviour of a source’s hysteretic behaviour in ∆ – Φ space.In an Outside-In outburst, the outburst onset is precededby a build-up of material at the outer edge of the disk, so itshould be expected that the outer radius of the disk increasesbefore its temperature. This is consistent with the resultsCourt et al. (2019) obtained by studying the eclipses in ZCha, suggesting that the outbursts in that source are indeedof the Outside-In type. We do not however see this behaviourin the eclipses of EX Dra, suggesting that the outbursts fromthis source are not Outside-In in nature.If we assume that the accretion disk in EX Dra is in-deed larger than the companion star during quiescence, thenan Inside-Out outburst can explain the increase in eclipsefraction during the outburst rise; Inside-Out outbursts havepreviously been identified in EX Dra during the 1990s (Bap-tista et al. 2000). At the onset of such an outburst, only theinner region of the disk experiences an increase in surfacebrightness, while the luminosity of the outer disk remainsunchanged (shown in stages A → B in Figure 6). Assumingthat the ‘extra luminosity’ L e due to some portion of thedisk going to the hot state is entirely concentrated in the MNRAS , 1–10 (2019)
J.M.C. Court et al.
A B C D FE
Time _ Total disk fluxFractional eclipse depth O u t - o f - E c li p s e R a t e ( e / s ) F r a c t i o n a l E c li p s e D e p t h Figure 6. Upper:
A diagram representing the evolution of an accretion disk during an Inside-Out outburst as described by Mineshige& Osaki (1985). In each image, portions of the disk in the ‘cold’ state are shaded in blue, portions of the disk in the ‘hot’ state are shadedin red and the radius r e corresponding to the radius of the eclipsing red dwarf is shown as a dashed line. In stage B a heating wave istriggered in the inner portion of the disk, within r e . This raises the fractional eclipse depth until stage C when the heating wave spreadsbeyond r e and the fractional eclipse depth begins to decrease again. In stage D a cooling wave is triggered at some radius outside that atwhich the heating wave was triggered, proagating throughout the disk ( E ) and eventually returning the disk to a quiescent state F . Theaccretion disks shown are not to scale. Middle: a plot, from a simple model based on this evolution and the disk temperature profilesolutions of Shakura & Sunyaev (1973), showing how the total disk luminosity and fractional eclipse depth would vary throughout suchan outburst.
Lower:
The out-of-eclipse rate and fractional eclipse depths which we infer for the outburst in Sector 14, to be comparedwith the predicted behaviour of these parameters in a simple inside-out outburst. The grey shaded region corresponds to the data gapin this sector. region of the disk which is eclipsed, we find: ∆ = d q + L e L q + L e (8)where d q and L q are the eclipse depth and out-of-eclipse diskflux when the disk is fully in the cold state during quies-cence. Assuming L q > d q , i.e. some part of the disk remainsuneclipsed during quiescence, we see that ∆ increases as thesurface brightness of the inner disk increases.As the outburst progresses, the outer radius of the hotregion of the disk increases and Equation 8 no longer holds(Stage C in Figure 6). As more of the uneclipsed portion ofthe disk also goes into the hot state, the eclipse fraction de-creases again, and the eclipses in a ∆ – Φ plot diverge fromline of maximum eclipse. The outwards heating wave willreach the outer edge of the disk or dissipate, marking thepeak of the outburst. It can be shown that a cooling wavefollowing behind the outwards heating wave will then beginto propagate inwards and outwards from a radius midwaythrough the disk (Stages D → F in Figure 6), extinguishing the outburst and returning the disk to the cool state (Mi-neshige & Osaki 1985). During this return to quiescence, theemission from the disk is less centrally peaked than duringthe outburst rise. This leads to eclipse fractions being gener-ally lower during the outburst decay than during the rise, inturn giving rise to a clockwise hysteretic loop in ∆ – Φ spacemuch as we see in Figure 5.Due to the similarities between the data and what wouldbe expected from the scenarios described above, we con-clude that the outbursts in both Sectors 14 and 15 wereof the Inside-Out type. The models of Mineshige & Osaki(1985) predict that Inside-Out outbursts should only occurin AWDs with a low accretion rate, in turn suggesting thatthe accretion rate in EX Dra is relatively low. This resultis at odds with the expectation that systems above the pe-riod gap, such as EX Dra, have significantly higher accretionrates (Knigge et al. 2011) than those below the period gap(such as Z Cha, a system which we interpret as undergoingOutside-In outbursts). Our result serves as further evidence MNRAS000
The out-of-eclipse rate and fractional eclipse depths which we infer for the outburst in Sector 14, to be comparedwith the predicted behaviour of these parameters in a simple inside-out outburst. The grey shaded region corresponds to the data gapin this sector. region of the disk which is eclipsed, we find: ∆ = d q + L e L q + L e (8)where d q and L q are the eclipse depth and out-of-eclipse diskflux when the disk is fully in the cold state during quies-cence. Assuming L q > d q , i.e. some part of the disk remainsuneclipsed during quiescence, we see that ∆ increases as thesurface brightness of the inner disk increases.As the outburst progresses, the outer radius of the hotregion of the disk increases and Equation 8 no longer holds(Stage C in Figure 6). As more of the uneclipsed portion ofthe disk also goes into the hot state, the eclipse fraction de-creases again, and the eclipses in a ∆ – Φ plot diverge fromline of maximum eclipse. The outwards heating wave willreach the outer edge of the disk or dissipate, marking thepeak of the outburst. It can be shown that a cooling wavefollowing behind the outwards heating wave will then beginto propagate inwards and outwards from a radius midwaythrough the disk (Stages D → F in Figure 6), extinguishing the outburst and returning the disk to the cool state (Mi-neshige & Osaki 1985). During this return to quiescence, theemission from the disk is less centrally peaked than duringthe outburst rise. This leads to eclipse fractions being gener-ally lower during the outburst decay than during the rise, inturn giving rise to a clockwise hysteretic loop in ∆ – Φ spacemuch as we see in Figure 5.Due to the similarities between the data and what wouldbe expected from the scenarios described above, we con-clude that the outbursts in both Sectors 14 and 15 wereof the Inside-Out type. The models of Mineshige & Osaki(1985) predict that Inside-Out outbursts should only occurin AWDs with a low accretion rate, in turn suggesting thatthe accretion rate in EX Dra is relatively low. This resultis at odds with the expectation that systems above the pe-riod gap, such as EX Dra, have significantly higher accretionrates (Knigge et al. 2011) than those below the period gap(such as Z Cha, a system which we interpret as undergoingOutside-In outbursts). Our result serves as further evidence MNRAS000 , 1–10 (2019) he Eclipses of EX Draconis that global ˙ M is not the sole criteria separating inside-outand Outside-In outbursts in AWDs (see also Buat-M´enardet al. 2001).The rise in ∆ during the onset of the outburst is de-pendent on the disk not being fully eclipsed during quies-cence. This contradicts the results of Billington et al. (1996)who found that the emission during eclipses in quiescenceis spectroscopically consistent with being entirely from thecompanion red dwarf. This disparity is likely due to the factthat the data used by Billington et al. was taken solely inthe H α band; in this band, the relatively red companionstar contributes a much higher fraction of the system’s flux,making it easier to reduce the disk flux to an undetectablefraction during eclipses. In any case, we find that the hys-teretic loop in ∆ – Φ space is not dependent on the quiescentradius of the accretion disk; an outburst as described aboveshould always show a clockwise loop in this parameter spaceregardless of whether the disk is smaller or larger than theeclipsing companion at the start of the outburst. This is astrong contrast with the anticlockwise hysteretic loop ob-served during Outside-In outbursts (e.g. Court et al. 2019).As such, we propose that the direction of a hysteretic loopin ∆ – Φ space can be used as a purely photometric test todifferentiate between inside-out and Outside-In outbursts ineclipsing AWDs. In systems with high accretion rates, matter gathers at theouter edge of the disk faster that it can be transferred in-wards, and the critical surface density is triggered at theouter edge of the disk, triggering Outside-In outbursts. Assuch, Outside-In outbursts are believed to be the more com-mon type of outburst in high accretion systems, with Inside-Out outbursts more common in low accretion rate systems(e.g. Mineshige & Osaki 1985). The accretion rate in EXDra is not known, but the well-constrained ∼ . hr pe-riod of the system places it firmly above the ‘period gap’,a range of periods between 2–3 hr in which relatively fewCVs are found. Above the period gap the inspiral rate ofthe binary, and hence the mass transfer rate ˙ M , is drivenby magnetic braking. However, below the period gap, it canbe shown that a Roche-lobe filling near-main sequence starshould be fully convective in nature, and hence not support asignificant magnetic field. In these systems, the inspiral rateis instead driven by the much slower process of the emit-ting gravitational radiation. As such it is expected that, ingeneral, AWDs above the period gap should have higher ˙ M than systems below it (Rappaport et al. 1982, 1983; Spruit& Ritter 1983).As EX Dra is above the period gap, the system is likelyto have a relatively high accretion rate. This is seemingly atodds with the fact that this sytem consistently undergoesInside-Out outbursts (e.g. Baptista et al. 2000, our discus-sion in Section 4.2). However, the models of Mineshige &Osaki (1985) are calculated for a smoothly accreting circulardisk in the orbital plane of the system, and a number of otherauthors have proposed ways to trigger Inside-Out outburstsin more complex high- ˙ M systems. For example, magneto-hydronamic simulations by Armitage & Livio (1996) haveshown that, in systems with very high ˙ M , a portion of theaccretion flow can ricochet off the outer edge of the disk and proceed in an arc to accrete directly onto the inner disk.However, it is unclear whether the ˙ M in EX Dra is largeenough for this process to occur.The presence of superhumps in EX Dra however doessuggest that this system is also more geometrically complexthan the systems modelled by Mineshige & Osaki (1985).Negative superhumps, such as we see in this system, are be-lieved to be caused by a beat between the orbital frequencyof a system and a slow vertical precession of a tilted accre-tion disk (e.g. Bonnet-Bidaud et al. 1985; Patterson et al.1993). As such the presence of negative superhumps, suchas we find in EX Dra, is evidence for such a tilted accretiondisk.Recent simulations by Kimura et al. (2019) have shownthat, in AWD systems containing a highly tilted accretiondisk, some of the matter donated from the companion starcan bypass the outer accretion disk and fall directly onto theinner regions of the disk. This causes the surface density inthe inner regions of the accretion disk to increase dispropor-tionately quickly compared to the outer regions. This in turnshould lead to inside-out outbursts being the dominant formof outbursts in such system. As such, the presence of nega-tive superhumps in EX Dra, as well as our identification ofinside-out outbursts in this source, provides direct evidencefor the accretion scenario proposed by Kimura et al.. Thisin turn can explain how inside-out outbursts can dominatein a system which likely has a high accretion rate. We have performed a study of the eclipses of the AWDEX Dra during the first two outbursts which were ob-served by
TESS . We have analysed how these eclipses evolvein eclipse-depth/out-of-eclipse-flux space ( ∆ – Φ space) overthe course of each outburst, finding significant hystereticbehaviour in both cases. This makes EX Dra the thirdAWD in which such hysteresis has been quantified, afterKIS J192748.53+444724.5 (Scaringi et al. 2013) and Z Cha(Court et al. 2019), strengthening the argument that this be-haviour is likely common to all eclipsing dwarf nova systems.Eclipse depth variability over the course of an outburst hasalso been reported in a number of other eclipsing CVs (e.g.V447 Lyr, Ramsay et al. 2012, V729 Sgr, Ramsay et al. 2017,and CRTS CRTS J035905.9+175034, Littlefield et al. 2018),suggesting that there are many more systems to which thesemethods can be applied.We find that during the initial stages of each outburst,the fractional eclipse depth rises, in turn implying that theaccretion disk cannot have been fully eclipsed during qui-escence. This is a direct contradiction to the results of aspectroscopic study performed by Billington et al. (1996),which is likely due to differences in the instrumentation andenergy bands used in our respective studies.We have found that the hysteretic loops in ∆ – Φ spacein EX Dra are executed in the opposite direction to thoseseen in Z Cha. We have shown that this difference wouldbe expected if the outbursts in EX Dra are of the ‘InsideOut’ type (outbursts that begin in the inner regions of theaccretion disk) as opposed to the ‘Outside In’ outbursts seenin Z Cha, despite EX Dra being expected to have a higheraccretion rate than Z Cha. As such, we have found that the MNRAS , 1–10 (2019) J.M.C. Court et al. direction of the hysteretic loops in ∆ – Φ space can be usedas a reliable and purely photometric method to phenomeno-logically distinguish between the two types of outburst.As analysis of hysteretic ∆ – Φ loops can be performedon any dwarf nova system which shows deep eclipses, thisnew method gives us a way to categorise historical dwarfnova outbursts in archival data, as well as any future out-bursts which occur in these systems. This will significantlyincrease the sample size of outbursts which are reliablyknown to belong to each type (‘Inside-Out’ or ‘Outside In’),allowing for population studies into how AWD phenomenol-ogy differs between these two types of outburst. As AWDsare oriented randomly in space, we can expect ∼ to be atangles of (cid:38) ◦ , and hence likely eclipsors, meaning that thetotal sample of AWDs on which our method can be appliedshould be high. Applying our method to identify inside-outoutbursts to additional AWDs which show negative super-humps will also be able to provide additional evidence forthe model of accretion proposed by Kimura et al. (2019),in which an accretion stream feeds directly into the innerportions of a highly tilted disk. ACKNOWLEDGEMENTS
In this study, we make use of the Numpy, Scipy (Joneset al. 2001) and Astropy (Astropy Collaboration et al.2013) libraries for Python. Figures in this paper were pro-duced using MatplotLib (Hunter 2007). This paper includesdata collected with the
TESS mission , obtained from theMAST data archive at the Space Telescope Science Insti-tute (STScI). Funding for the
TESS mission is provided bythe NASA Explorer Program. STScI is operated by the As-sociation of Universities for Research in Astronomy, Inc.,under NASA contract NAS 5–26555. SS and JMCC acknowl-edge support for this work from the TESS Guest Investigatorprogram under NASA grant 80NSSC19K1735. DA acknowl-edges support from the Royal Society. ND is supported by aVidi grant from the Netherlands Organization for ScientificResearch (NWO).
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