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Astron. Astrophys. 352, 239-247 (1999)

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4. Discussion

4.1. A moving hot spot?

Any reasonable model for the photometric variability of RZ Leo should reproduce the non-coherent humps and their amplitude variations.

It is currently assumed that the humps reflect the release of gravitational energy when the gas stream hits the accretion disk. The disk's luminosity is produced by the same process when disk gas slowly spirals towards the central white dwarf (e.g. Warner 1995a).

An explanation for the varying humps could be a hotpsot moving along the outer disk rim. A bright spot co-rotating with the binary should reflect the binary orbital period, but random translations of the hot spot along the outer disk rim should produce a non-coherent signal. Support for this view arises from the evidence of moving hot spots in some dwarf novae, e.g. KT Per (Ratering et al. 1993) and WZ Sge (Neustroev 1998).

The large scatter observed in the [FORMULA] diagram of RZ Leo (up to 0.4 cycles) is atypical for dwarf novae. For example, U Gem (Eason et al. 1983), IP Peg (Wolf et al. 1993) and V 2051 Oph (Echevarría & Alvarez 1993) show quasi-cyclic period variations, of small amplitude, on time scale of years. In these cases, the [FORMULA] residuals are always lower than 0.02 cycles. The interpretation of the changes observed in the above stars is still controversial.

4.2. RZ Leonis in the context of WZ Sge stars

4.2.1. Evidence for a normal [FORMULA] disk

In this section we analyze the events of early 1998. The observed correlation between the hump amplitude and mean brightness might provide important clues on the numerical value of the disk viscosity.

The hot spot and disk bolometric luminosity can be approximated by (Warner 1995a, Eqs. 2.21a and 2.22a):



where [FORMULA] and [FORMULA] are the mass and radius of the primary, [FORMULA] the disk's radius and [FORMULA] and [FORMULA] the mass transfer and mass accretion rates, respectively.

In the following we assume that [FORMULA] is proportional to the hump's peak luminosity and [FORMULA] is proportional to the cycle mean luminosity. The disk luminosity so defined includes some contribution from the hot spot, but it is difficult to exclude the wide and long-lasting humps from the analysis. It is apparent from Figs. 9 and 10 that the increase of hot spot luminosity is followed by an increase of the disk's luminosity. This effect seems to be true, and not simply a consequence of the hump rising.

Accordingly to Eqs. 3 and 4, the events of early 1998 may be interpreted as follows: a mass transfer burst starts at the secondary in January 1998 and then continues with increasing [FORMULA], until March 1998. The burst, evidenced in the rising of hump's luminosity in Fig. 9 triggers an increase of mass accretion rate inside the disk, as observed in the rising of the total systemic luminosity.

The time scale for matter diffusion across the accretion disk, is called the viscous time scale (Pringle 1981):


where the viscosity is given by the Shakura & Sunyaev (1973) ansatz :


with H the half-thickness of the disk and [FORMULA] the sound velocity. Replacing Eq. 6 in 5 and using typical parameters [FORMULA] = 0.01, [FORMULA] cm, [FORMULA] cm s-1 we obtain


For the diffusion process observed in RZ Leo [FORMULA] [FORMULA] 6.0 [FORMULA] 106 s (70 days), we find [FORMULA] = 0.08, a common value among dwarf novae (Verbunt 1982). This value contrasts with the low [FORMULA] ([FORMULA] 0.01) invoked to explain the long recurrence times and large amplitude outbursts of some dwarf novae, in particular WZ Sge (Meyer-Hofmeister et al. 1998). Since our observations indicate a rather normal [FORMULA], the long recurrence time must be explained by another cause. In this context it is worthy to mention the hypothesis of inner disk depletion.

The removal of the inner disk by the influence of a magnetosphere (Livio & Pringle 1992) or the effect of mass flow via a vertically extended hot corona above the cool disk (also referred as "coronal evaporation", Meyer & Meyer-Hofmeister 1994, Liu et al. 1997, Mineshige et al. 1998) naturally explains the long recurrence times. Spectroscopic evidence indicates that the inner disk depletion might be a common phenomenon in SU UMa stars (Mennickent & Arenas 1998, Mennickent 1999).

4.2.2. Evidence for a main sequence like secondary

It has been suggested that many large amplitude dwarf novae have bounced off from the orbital period minimum (at [FORMULA] 80 min) and are evolving to longer orbital periods with very old, brown-dwarf like secondaries (Howell et al. 1997). This view is supported by the finding of undermassive secondaries in WZ Sge (Ciardi et al. 1998) and V 592 Her (van Teeseling et al. 1999) and the suspection - based on the "superhump" mass ratio - of this kind of objects in AL Com and EG Cnc (Patterson 1998). In principle, the large amplitude and long cycle length of RZ Leo suggest that this star is an ideal candidate for a post-period minimum system and therefore, for an undermassive secondary. Since superhumps have not been yet detected in this star, the only way to investigate this view is analyzing the flux distribution. We have compiled data from different sources. They are generally non-simultaneous, and may contain possibly significant variations in the emission of the CVs. However, to minimize this effect, we have excluded data taken during outburst, and we have considered data from as few sources as possible and as close together in time as possible.

The flux distributions of RZ Leo and other dwarf noave with recognized brown-dwarf like secondaries (and available photometric data) are compared in Fig. 11. The optical-IR flux of a steady disk, scaled to fit the UBV data of RZ Leo, is also shown. 2

[FIGURE] Fig. 11. The flux distribution of three large-amplitude, long cycle-length dwarf novae. Fluxes of WZ Sge are from Ciardi et al. (1998) and references therein. Those for V592 Her are based on photometry published by van Teeseling et al. (1999) and Howell et al. (1991). UBV data for RZ Leo is from this paper and JK data from Sproats et al. (1996) and Szkody (1987). The scaled flux of a steady, optically thick, accretion disk is given by the dotted line.

We find that, in contrast with that observed in the objects with undermassive secondaries, the flux distribution of RZ Leo does not drop in the red wavelengths, but rises with respect to the disk's contribution. This is expected if the secondary were a main-sequence red dwarf. In fact, the [FORMULA] color of RZ Leo (viz. 3.65, Sproats el at. 1996) is representative of a main sequence M0 star (Bessell & Brett, 1988). This is consistent with the finding that most secondaries stars for cataclysmic variables with [FORMULA] [FORMULA] 3 h are close to the solar abundance main sequence defined by single field stars (Beuermann et al. 1998). The above arguments probably rule out the possibility of an undermassive secondary in RZ Leo.

Our results indicate that large amplitude - long cycle length - dwarf novae might not necessarily correspond to objects in the same evolutive stage. We have shown that, in spite of the extreme cycle length and outburst amplitude, RZ Leo cannot be properly named a WZ Sge like star , as suggested in the Ritter & Kolb catalogue (1998).

4.3. Anti-humps

The ratio between hot spot and disk luminosity, for the case of an optically thick, steady state accretion disk and a simple planar bright spot on the edge of the disk, is (Warner 1995a, Eq. 2.71):


where i is the systemic inclination, f an efficience factor [FORMULA] 1 and [FORMULA] and [FORMULA] are the bolometric corrections ([FORMULA] 0) for the spot and disk respectively.

Roche-lobe geometry and the assumption of a disk radius 70% the Roche lobe radius (an usually good approximation for dwarf novae), yield to:


In addition, hot spot and disk temperatures inferred for dwarf novae indicate [FORMULA] [FORMULA] [FORMULA] (Warner 1995a's discussion after Eq. 2.73 and references therein). Therefore we obtain:


where [FORMULA] is a function with a numerical value in the range 0.03-0.3 for most practical purposes. The condition of "anti-humps" is given by:


The above equations suggest that the apparition of "anti-humps" in a single system depends on the relative values of [FORMULA] and [FORMULA]. In particular, for RZ Leo, assuming the orbital and photometric periods equals, a mass ratio of 0.15, i.e. representative for dwarf novae below the period gap (e.g. Mennickent et al. 1999), and a moderate inclination angle of 65o, this occurs when [FORMULA] (assuming f =1). The rarity of the phenomenon indicates that [FORMULA] is a condition rarely fulfilled among dwarf novae and that [FORMULA] is probably always larger or equal than [FORMULA]. Systems with large amplitude humps are candidates for [FORMULA] [FORMULA] [FORMULA] whereas high inclination systems with no prominent humps (e.g. WX Cet, Mennickent 1994) are candidates for [FORMULA] [FORMULA] [FORMULA].

We can estimate the mass accretion rate from the recurrence time:


(Eq. 37 by Warner 1995b). Using a supercycle length [FORMULA] 2 yr we obtain [FORMULA] g s-1. This implies that [FORMULA] g s-1 is required to develop "anti-humps". This condition is easily satisfied if the mass transfer rate is driven by gravitational radiation, as expected for a dwarf novae below the period gap. In this case, using the system parameters given above, we estimate:


from Eq. 9.20 by Warner (1995a).

Since the mass accretion rate [FORMULA] is proportional to the viscosity (e.g. Cannizzo et al. 1998), an extremely low [FORMULA] disk is not a good site for developing "anti-humps". The reason is that, in this case, the condition imposed on the mass transfer rate to satisfy Eq. 11 is too strong, requiring, probably, unrealistic low [FORMULA] values. Therefore the presence of "anti-humps" in RZ Leo is consistent with the normal [FORMULA] found in the previous section.

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Online publication: November 23, 1999