Mass Ratio

We used the method described in Section 2.4.8 to perform a K-correction and estimate the mass ratio of the systems. For the models, we usedi=60ofor QZ Vir andi=45ofor the ones with no known inclination constraints. Figures 5.24 to 5.26 show the derived dynamical constraints for each system. The full black circle represents our best solution while the full square represents the superhump solution. The other constraints are the same as in Figure 3.15.

5.5. DOPPLER TOMOGRAPHY 127

Figure 5.23: Histograms ofKy (top) andKx (bottom) at which the lowest residual values were found for the 500 bootstrap samples. The average of both components is used to estimateK1.

128 CHAPTER 5. ORBITAL PARAMETERS OF FOUR SUPERHUMPING CVS System K1[km/s] Line AQ Eri K1=31±7 Ca II QZ Vir K1=26±3 Ca II SDSS0137 K1=43±19 Hα RZ Leo K1=68±24 Hα

Table 5.4: Primary velocity of the four systems calculated using the centre of symmetry tech- nique. 0.00 0.05 0.10 0.15 0.20 q −50 0 50 100 150 200 K1 (km/s)

Figure 5.24: K1,q plane for AQ Eri. Kem >KL1 is the top diagonal solid black line with dashed

errors.Kem<K2is the bottom diagonal line with dashed errors. The vertical solid line represents

the maximum value ofq. The grey diamonds are the solutions from our model. The black point with error bars represents our best solution for the value ofK1found from the centre of symmetry

5.6. DISCUSSIONS AND CONCLUSIONS 129 0.00 0.05 0.10 0.15 0.20 q −20 0 20 40 60 80 100 120 140 160 K1 (km/s)

Figure 5.25:K1-qplane for QZ Vir, all markers as indicated in Figure 5.24.

For AQ Eri, the superhump solution isqsh =0.126±0.08, givingK1sh =47±1.6 km/s, K2sh=376 km/s. Our best solution isq=0.085±0.02,K1=31±7 km/s andK2=363±41 km/s,

only marginally consistent with the superhump solution.

The superhump solution for QZ Vir:qsh=0.108±0.005, leads to component velocities of K1sh=24.7±0.9 km/s,K2sh=229±1 km/s. Using our best estimates ofK1andKem, we obtained: q=0.11±0.01,K1=26±3 km/s,K2=230±25 km/s. These solutions are in very good agreement.

As for SDSS0137, the superhump solution:q=0.116±0.010,K1sh=46±4 km/s andK2sh= 396±37 km/s is also in agreement with our best solution:q=0.109±0.049,K1=43±19 km/s and

K2=393±41 km/s. Considering that the correction fromKabtoK2is negligible, we calculated the

value ofqfromKab, plotted with the black diamond in Figure 5.26, obtainingqab=0.13±0.06, in agreement with the previous solutions.

5.6 Discussions and Conclusions

SDSS0137 is the only CV in our sample with a detectable absorption line from the secondary star, which is likely a late type star. A M-dwarf star would present not only Na I absorption lines, but, at least K I. The absence of any other features in the spectrum of SDSS0137 and the very broad nature of the Na I doublet ( 300 km/s) might be indicative of a secondary later and cooler than a

130 CHAPTER 5. ORBITAL PARAMETERS OF FOUR SUPERHUMPING CVS 0.00 0.05 0.10 0.15 0.20 0.25 0.30 q −50 0 50 100 150 200 250 K1 (km/s)

Figure 5.26:K1-qplane for SDSS0137, markers as indicated in Figure 5.24. Black diamond with

error bars is theKab solution.

M type star. Despite that measuring the radial velocity lines of the secondary’s absorption lines has proven to be a reliable way to findK2 and q (e.g. Steeghs & Jonker 2007), we find that our

radial velocity measurements (Kab=337±8 km/s) seem low in comparison withKem=317±20 (if we consider that a typical correction forKem is around∆K∼80 km/s and thatKab>K2). To

better understand this, we calculated the projected rotational velocity of the secondary, using the expressionvsin(i)/K2=0.46[(1+q)2q]1/3(Wade & Horne, 1988), finding that it is of the order of

85 km/s. This value is not comparable to the line widths, which further indicates another source of broadening like the one found in latter type (Díaz et al., 2007) and cooler (Tripicchio et al., 1997) stars. We will investigate this issue and its implications on the calculation ofvandKabin a future paper, although discrepancies between velocities calculated from de Na I doublet has been reported before (see (Bleach et al., 2002) and references within).

One of the most notorious features of these four systems is the presence of enhanced features in the Doppler maps that are not in the location of the bright spot. We have seen this kind of emission before in the second quadrant of the Doppler maps of V2051 Oph. At the time, these features were identified by Papadaki et al. (2008) as the superhump light source. We found the same feature in our quiescence-spectra Doppler maps for V2051 Oph. The Doppler maps of the four systems studied in this chapter presented notorious emission in “anomalous locations”.

5.6. DISCUSSIONS AND CONCLUSIONS 131

Despite having been observed during quiescence, AQ Eri and QZ Vir were observed∼15 days after superoutburst, but in the case of RZ Leo and SDSS0137, the observations were∼a year from any superoutburst before or after. As the four systems are in different stages of quiescence -hence their discs are not elliptical- we cannot support the superhump light source hypothesis of Papadaki et al. (2008) to explain all of these.

In the particular case of QZ Vir, Szkody et al. (2001) reported an enhanced emission in Hα at the same position where we encountered it. Given that the anomalous located bright spots were indeed the superhump bright source, one would expect them to move with the precession of the elliptical disc. It is also noticeable that Hαdoes not present any sign of a bright spot in the “normal” position, and is the only line in which we find this feature.

Independently of the origin of these anomalously located bright spots, our method to estimate the radial velocity of the primary star,K1, can successfully avoid most asymmetries of

the accretion disc. We believe that the value ofK1would not be affected by these anomalously

located bright spots, hence the lower velocities derived from our method in comparison with the average solution of the diagnostic diagram.

If we revise the case presented by Wolf et al. (1998) with the CV IP Peg in quiescence, an enhanced emission in an “anomalous” position (centre-right of the Doppler map) is identified as evidence of spiral structure in the accretion disc, found by Steeghs et al. (1997) in the system during outburst. As in the case of the quiescence maps of IP Peg by Wolf et al. (1998), neither of the Doppler maps of our four targets seem to present clear signs of spiral shocks, other than double bright spots. Despite that no evidence of spiral structure has been found in short period SU Uma stars in outburst (see Harlaftis et al. (1999) and references within), Baba et al. (2002) present evidence of spiral-like structure in WZ Sge in superoutburst. WZ Sge is a sub class of SU Uma type CVs, with a period below the period gap, opening the possibility that spiral shocks could occur during superoutburst in other short period systems. However, we would not ex- pect such tidally excited features in quiescence. Furthermore, if we look the Doppler maps of other short period CVs, like SDSS 123813.73-033933.0 in figure 4 of Aviles et al. (2010), we can clearly see an extended bright region opposite to the bright spot. We find the same features in the Doppler maps of SDSS J080434.20+510349.2 if we look at figure 9 of Zharikov et al. (2013). Also, for SDSS J103533.02+055158.3, presented in figure 11 of Southworth et al. (2006), looking at the constructed map (left side) no anomalous bright spot is detected, but after subtracting the symmetric part (right hand), the familiar asymmetry is revealed. Zharikov et al. (2013) proposed the following: In low mass ratio (q.0.06) systems, given that the WD is massive enough, we can assume that it will have a huge primary Roche lobe. If this is the case, the accretion disc could extend to the 2:1 resonance radius, as proposed by Kunze & Speith (2005). Reaching this reso- nance, a two arms pattern would appear at the disc rim, which (according to the model proposed by Zharikov et al. 2013) will be seen as the enhanced double emission in the Doppler maps of the

132 CHAPTER 5. ORBITAL PARAMETERS OF FOUR SUPERHUMPING CVS

short period CVs in quiescence and it could be a signal of a bounce back CV.

To try and set our systems on the pre- or post-bounce sequence, we used the parameters derived for the CVs donnor sequence given a typicalM1=0.75M¯, prsented in table 4 of Knigge et al. (2011), to estimate the solutions for our systems.

For AQ Eri, the pre-bounce solution is q =0.112 and the post-bounce solution is q =

0.057. Given that our best solution for AQ Eri,q =0.085±0.02, is between both values, we can not use it to further clarify its nature. Nevertheless, the superhump solutionqsh=0.126±0.08 would be indicative of a pre-bounce system. For QZ Vir, the results are: for pre-bounceq=0.102 and for the post-bounce caseq=0.064. Our best solution isq=0.11±0.01, also favouring a pre- bounce system. The superhump solution,qsh=0.108±0.005, agrees with this conclusion. In the case of SDSS0137, Knigge et al. (2011) derived solutions for systems with orbital periods down to 82 minutes, so it is not included in their models. We did calculate bothKem=317±20 km/s with the Hαemission line andKab=337±8 km/s for the Na I absorption line. AsKem<Kab, and given that the absorption line showed a broad profile in comparison with the emission lines, we believe that the absorption lines could be generated from a large area on the back side of the secondary star, farther away from the centre of mass, while it is believed that Ca II emission lines are generated on the inner face of the donor near theL1point. As the true value ofK2should

be betweenKemandKab, our estimated valueK2=393±41 km/s is rather large in comparison,

This could be indicative of our method overestimating the value ofK2, but we should consider

that in the case of SDSS0137, we calculated the values ofKemusing the secondary feature inHα and not the optimal feature Ca II. The intrinsic broadening of the Balmer lines, as proposed by Marsh & Dhillon (1997), could have led to a saturated secondary signal, filling the theoretical Roche lobe and hence increasing the measured radial velocityKem. Another possibility is the one mentioned above, given that the Na I profile is too broad to be explained solely by rotational broadening, the absorption may come from another source. This would imply an incorrectKab, since the absorption lines would not be coming from the donor’s surface.

We have derived solutions for the mass ratio of three super-humping short period CVs, independent from the superhump relation. In general, our independent results agree with the superhump solution. Given the reduced number of calibrators for the superhump period-mass ratio relation on the short-period systems end, studies such as the one presented here, can im- prove the current model.

5.7 Summary

In this chapter, we have used the techniques tested with eclipsing systems in Chapter 3, to derive dynamical solutions for a selection of short period CVs with known superhump periods, but oth- erwise poorly constrained parameters. We have compared our best solutions against the mass

5.7. SUMMARY 133

ratio calculated with the observationally derived relation between superhump period and mass ratio. We found our best solutions to be in agreement with the superhump-derived solution. We found that the Doppler maps for these systems share similar features, namely, enhanced emis- sion at the location of the bright spot and on the opposite side of the disc. These features have been found before in several post-bounce candidate systems and it is believed to be a sign of outer disc structure caused by tidal interactions in systems with evolved secondary stars. We used the empirical relation between secondary mass and orbital period to determine the posi- tion of the systems in the evolutionary track, finding a favourable pre-bounce solution for two systems. Better constraints onK1would be required to reduce our uncertainty onq before we

Six

Discussions and Conclusions

6.1 Dynamic constraints with Doppler tomography

Doppler map based methods can provide strong constraints on the orbital parameters of short period CVs and LMXB. We compared these methods against classic double Gaussian methods finding advantages from the former over the classic ones.

Our method to constrain the value ofγ, although following the same logic as Casares et al. (2003), has the advantage of being able to determine the optimal value with uncertainties lower than 50 km/s, significantly smaller than the error derived from the difference between the FWHM of the secondary for differentγ, which is often only detected for values&30 km/s apart. Using Doppler tomography instead of other radial velocities techniques also has advantage over the uncertainties induced by disc asymmetries, giving more reliable results.

We have shown that using the donor emission in Doppler tomography is an easy and reliable way to measureKemand to determine the phase zero point. As the emission of the donor originates from the inner Lagrangian point, we need to apply a K-correction to our measuredKem to find the true radial velocity of the secondary,K2, which we have done using model spectra to

find the best solution for a given set of parameters. We found that this K-correction is quite well constrained and is not the limiting factor in any of the data sets we considered.

We have automatised the centre of symmetry technique to find the optimal value ofK1,

giving it meaningful uncertainties with the bootstrap method while avoiding obvious asymme- tries.

Our Doppler tomography based methods give good estimates for the values ofK1 and

K2, but our relatively large error on the smallK1, expcted in low q systems, sets the accuracy of

our derived mass ratios. We have showed the advantages of the method, delivering individual systems with spectroscopic parameters and provided calibrators to vindicate the reliability of superhump inferred mass ratios.

136 CHAPTER 6. DISCUSSIONS AND CONCLUSIONS