Mem. S.A.It. Vol. 83, 619c© SAIt 2012 Memorie della

Dwarf novae in the shortest orbital periodregime

M. Uemura1, T. Kato2, T. Ohshima2, D. Nogami3, and H. Maehara3

1 Hiroshima Astrophysical Science Center, Hiroshima University, Kagamiyama 1-3-1,Higashi-Hiroshima 739-8526, Japane-mail: uemuram@hiroshima-u.ac.jp

2 Department of Astronomy, Faculty of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

3 Kwasan Observatory, Kyoto University, 17 Ohmine-cho Kita Kazan, Yamashina-ku,Kyoto 607-8471, Japan

Abstract. Dwarf novae (DNe) having very short orbital periods (Porb) are interesting ob-jects in terms of two points of view: the binary evolution and the physics of accretion disks.They are considered as one of the final evolutionary stages of low-mass binaries. It is wellknown that the observed Porb distribution of cataclysmic variables is inconsistent with thatexpected from population synthesis studies. We evaluate the intrinsic population of low ac-tivity DNe in the shortest Porb regime, which could reconcile the discrepancy between theobservation and theory. In the view point of the physics of accretion disks, short Porb DNe,in particular, WZ Sge stars, have received attention because they exhibit unique variations,like early superhumps. We have recently developed a method to reconstruct the structure ofdisks using multi-band light curves of early superhumps. Here, we introduce the results ofthis method using the data of the dwarf nova, V455 And.

Key words. Stars: evolution – Stars: novae, cataclysmic variables

1. Introduction

Dwarf novae (DNe) having very short orbitalperiods (Porb) have received attention becauseof their unique nature in terms of both thebinary evolution and the physics of accretiondisks. It is widely accepted that they are one ofthe final evolutionary stages of low-mass bina-ries. WZ Sge stars form a sub-group of DNe inthe shortest Porb regime. They were originallydefined by a long recurrence time of super-outbursts (& 10 yr) and large outburst ampli-

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tudes. Later observations have revealed severalunique features, such as early superhumps andrebrightenings (e.g. Howell et al. 1995; Katoet al. 2001).

It is well known that the observed Porb dis-tribution of cataclysmic variables (CVs) is in-consistent with that expected from populationsynthesis studies (e.g. Howell et al. 1997; Kolb1993). The theoretical studies predict that mostCVs would evolve and accumulated to the min-imum Porb (Pmin), which is estimated to be∼ 70 min. However, the observed Porb distri-bution of CVs shows no such an accumulationof objects near Pmin, and the observed Pmin is

620 Uemura: Dwarf Novae in the Shortest Orbital Period Regime

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Fig. 1. Outburst detection probability, Doutb as afunction of Porb − Pmin. The filled circles indicateSU UMa stars whose Ts is known (Uemura 2008).The dotted lines indicate the models (for details, seeUemura et al. 2010).

∼ 80 min. In § 2, we evaluate the intrinsic pop-ulation of low activity DNe in the shortest Porbregime, like WZ Sge stars, which could rec-oncile the discrepancy between the observationand theory.

In the view point of the physics of accretiondisks, WZ Sge stars are noteworthy objectsbecause they exhibit unique variations, likeearly superhumps. Recent observations havesuggested that early superhumps are caused bythe rotation effect of asymmetrically expandeddisks (Kato 2002; Maehara et al. 2007; Matsuiet al. 2009). It indicates that the light curves ofearly superhumps have information of the spa-tial structure of accretion disks. We have re-cently developed a method to reconstruct thedisk structure using multi-band light curves ofearly superhumps. In § 3, we introduce the re-sults of this method using the data of the dwarfnova, V455 And.

2. WZ Sge stars as the missingpopulation near Pmin

We can consider that the detectability of super-outbursts, Doutb is inversely proportional to thesupercycle, Ts. The Doutb which are calculatedfrom the observed Ts are shown in figure 1, inwhich they are normalized by an average Tsof 470 d in Porb > 85 min. As can be seenin the figure, Doutb decreases toward Pmin, in

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Fig. 2. The distribution of Porb of our DNe sample.The filled and open histograms indicate the ASASand VSNET samples, respectively. The dashed lineindicates the best model derived from our analysis.

other words, superoutbursts are less frequentin shorter Porb systems. Therefore, a numberof undiscovered WZ Sge stars could be presentin the shortest Porb regime.

In Uemura et al. (2010), we estimate theintrinsic Porb distribution of DNe. We assumethat the observed distribution, Q(Porb) can beexpressed with the detectability of superout-bursts, D(Porb), and the intrinsic distribution,I(Porb), as Q ∝ DI. D is constructed by Doutband the dependency of the absolute magnitudeat the supermaximum on Porb. The sample,Q is taken as DNe whose superoutbursts aredetected with VSNET or ASAS database be-tween 2003 January and 2007 December (Katoet al. 2004; Pojmanski 2002). Figure 2 showsthe Porb distributions of the two sets of sam-ples. The intrinsic distribution, I, is expressedwith two parameters, α and Pmin as follows:

I(p) =

{p−αe−α/p/AI (p ≥ 1)0 (p < 1) (1)

p = Porb − Pmin + 1 (min). (2)

Here, α is a parameter for the degree of theconcentration. A larger α yields a distributionwith a stronger spike near Pmin. The distribu-tion becomes flat in the case of α = 0. AI is anormalization factor of I. While this formulaitself has no physical meaning, it can repro-duce the profile of the CV distribution belowthe period gap predicted by population synthe-sis studies, with high α.

Uemura: Dwarf Novae in the Shortest Orbital Period Regime 621

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Fig. 3. Intrinsic Porb distribution of DNe estimatedfrom the observed sample of DNe.

We can regard the Porb distributions of oursamples as a probability density distributions.Then, we can estimate I in the framework ofthe Bayesian statistics. We calculated poste-rior probability distributions of α and Pmin us-ing the Markov Chain Monte Carlo method(MCMC). Figure 3 shows the intrinsic Porb dis-tribution of DNe with the best parameters ofα = 0.88±0.30 and Pmin = 70.5+2.6

−4.6 min for theASAS sample. The prominent “period spike”feature can be seen. The estimated Pmin is sig-nificantly shorter than the observed one. Thesefeatures are in agreement with those obtainedfrom the VSNET sample.

This experimental approach indicates thatlong Ts DNe, like WZ Sge stars, could be re-sponsible for the missing population near Pmin.Another formulation of I and samples shouldbe tested in this model to validate the presentconclusion. Details of the model and calcula-tion can be found in Uemura et al. (2010).

3. Early superhump mapping usingmulti-band light curves

Figure 4 shows the multi-band light curvesof early superhumps observed in V455 And.Based on these data, Matsui et al. (2009) re-port that the object was bluest at the hump min-imum, and the color of the hump componentwas totally red. It supports the scenario thatearly superhumps are caused by the geomet-rical effect of accretion disks which are verti-cally deformed non-axisymmetrically. Several

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Fig. 4. Light curves of early superhumps ofV455 And. From top to bottom, the panels show g,V , Rc, Ic, and J-band light curves. The magnitudesare differential ones from the average magnitudesfor each band.

models have been proposed for the mechanismof such a deformation of disks, for example, atidal deformation scenario (Kato 2002) and the2:1 resonance scenario (Osaki, Meyer 2002).

The geometrical structure of disks can bereconstructed from multi-band light curves ofearly superhumps because its information ofthe azimuthal direction can be obtained fromthe time-series data, and that of the radial onefrom color variations. We have developed amodel to reconstruct a height map of disks with

622 Uemura: Dwarf Novae in the Shortest Orbital Period Regime

the following key assumptions: 1) the observedflux is all from the disk surface, which emitsmulti-temperature blackbody radiation. 2) Thetemperature is given as T = Tin(r/rin)−3/4,where Tin and rin denote the innermost tem-perature and radius. 3) The outermost radiusof the disk is set to be r = 0.6a, where a isa binary separation, corresponding to the 2:1resonance radius. In addition, the eclipse bythe secondary star and self-occultation effectin the disk are considered. Giving Tin, a, mass-ratio, and inclination angle, we can calculatethe multi-band light curves by rotating the diskhaving a height map, h(r, θ). Our model is aBayesian model, which estimates h(r, θ) fromobserved light curves. Details of the model andcalculation will be published in our forthcom-ing paper (Uemura, et al. 2011, in prep.).

Figure 5 shows the reconstructed heightmap of the disk of V455 And, which was cal-culated from the light curves in figure 4. Themodel light curves are indicated by the dottedlines in figure 4. We can confirm that the ob-served light-curves are well reproduced by themodel. In the height map, we can see two out-ermost flaring parts, which are responsible forthe primary and secondary maxima of the lightcurves. In addition to those two major parts,we can see arm-like patterns elongated to rela-tively inner parts of the disk. They are empha-sized in the lower panel of figure 5.

The two outermost flaring parts and theright-lower arm-like feature are apparentlysimilar to the disk structure which is expectedfrom the tidal deformation scenario (Ogilvie2002; Kato 2002). On the other hand, it is diffi-cult to explain the left-upper arm-like feature,which is responsible for the secondary mini-mum at phase ∼ 0.3 of the light curve.

Thus, our new tomography method willprovide a new insight into the physics of ac-cretion disks.

4. Summary

DNe with shorter Porb tend to have longer re-currence time of superoutburst in the shortestperiod regime of Porb . 85 min. It impliesthat a number of DNe is left undiscovered nearPmin. Our Bayesian estimation of their intrin-

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Fig. 5. Height map for the data of V455 And. Theupper and lower panels show the maps of h and h/r,respectively. Both the contours and color-maps rep-resent the same h or h/r. The upper panel draws thecontours of h = 0.00–0.15 with an interval of 0.01.The lower panel draws the contours of h/r = 0.00–0.24 with an interval of 0.02. The secondary star islocated at (x, y) = (1.0, 0.0).

sic population reproduces a period spike fea-ture at Pmin ∼ 70 min in the Porb distribution,as expected from population synthesis stud-ies. Our new tomography method enables usto reconstruct the structure of accretion disksfrom multi-band light curves of early super-humps. The reconstructed disk-structure fromthe data of V455 And is analogous to that ex-pected from tidal deformation, while the arm-like structure at phase ∼ 0.3 is an unexpectedfeature.

Uemura: Dwarf Novae in the Shortest Orbital Period Regime 623

5. Discussion

ELENA PAVLENKO: Did you compare col-ors of early superhumps and ordinary super-humps also?MAKOTO UEMURA: Yes. The color vari-ations of ordinary superhumps were distinctfrom those in early superhumps. In particular,a clear bluing trend was associated with the ris-ing phase of ordinary superhumps. I interpret itas a sign of viscous heating. No sign of viscousheating was seen in early superhumps.VITALY NEUSTROEV: Can the proposedmethod of multicolour tomography be used notonly for superhumps study, but also for otherorbital variabilities?MAKOTO UEMURA: Generally speaking -yes. However, our approach is restricted byblackbody approximation.CHRISTIAN KNIGGE: How do you recon-cile the very short Pmin you derive with themuch longer Pmin derives by Gansicke et al. forthe SDSS sample?MAKOTO UEMURA: The short Pmin ispartly due to our model assumption. Since ourmodel assumes that the recurrence time of su-peroutbursts reaches infinity at a real Pmin, Pminis estimated to be shorter than the observedone. The validity of our model assumptionshould be tested, for example, by a search forsuperoutbursts of DNe having Porb ∼ 70 min.

Acknowledgements. This work was partly sup-ported by a Grand-in-Aid from the Ministryof Education, Culture, Sports, Science, andTechnology of Japan (19740104 and 22540252).

References

Howell, S. B., Rappaport, S., & Politano, M.1997, MNRAS, 287, 929

Howell, S. B., Szkody, P., & Cannizzo, J. K.1995, ApJ, 439, 337

Kato, T. 2002, PASJ, 54, L11Kato, T., Uemura, M., Ishioka, R., Nogami, D.,

Kunjaya, C., Baba, H., & Yamaoka, H. 2004,PASJ, 56, 1

Kato, T., Sekine, Y., & Hirata, R. 2001, PASJ,53, 1191

Kolb, U. 1993, A&A, 271, 149Maehara, H., Hachisu, I., & Nakajima, K.

2007, PASJ, 59, 227Matsui, R., Uemura, M., Arai, A., Sasada, A.,

Ohsugi, T., Yamashita, T., Kawabata, K.,Fukazawa, Y., et al. 2009, PASJ, 61, 1081

Ogilvie, G. I. 2002, MNRAS, 330, 937Osaki, Y. & Meyer, F. 2002, A&A, 383, 574Pojmanski, G. 2002, Acta Astron., 52, 397Uemura, M. et. al. 2008, PASJ, 62, 187Uemura, M., Kato, T., Nogami, D., & Ohsugi,

T. 2010, PASJ, 62, 613