Enhanced Mass Transfer in U Geminorum During Its Outbursts and Superoutburst

ACTA ASTRONOMICA Vol. 55 (2005) pp. 315–320

Enhanced Mass Transfer in U Geminorum during its Outbursts and Superoutburst

by J. Smak

N. Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland e-mail: [email protected]

Received August 24, 2005

ABSTRACT

The mass transfer rate in U Gem at quiescence, estimated to be M˙ ≈ 1.3−2.0×1016 g/s, is used to calculate the amount of mass ∆Mtr transfered to the disk during quiescence. Light curves of U Gem are used to estimate the amounts of mass ∆Maccr accreted during its three types of outbursts. In the case of wide outbursts and the 1985 superoutburst ∆Maccr are much larger than ∆Mtr , indicating signiﬁcant enhancement in the mass transfer by a factor of f ≈ 20−50. There is no evidence for comparable enhancement during narrow outbursts. Key words: Accretion, accretion disks – navae, cataclysmic variables – Stars: dwarf novae – Stars: individual: U Gem

1. Introduction

U Gem is a classicaldwarf nova with well determined system parameters (Smak 2001) and a well covered light curve (Mattei et al. 1987, 1991, 1996). It appears therefore to be a perfect case for detecting and studying the effects of the enhanced mass transfer during outbursts. Our analysis, which is essentially similar to that applied earlier to the case of SU UMa stars (Smak 2004), consists of the following steps. The observational data for the narrow and wide outbursts of U Gem, as well as for its 1985 superoutburst, collected in Section 2, are used in Section 3 to determine the amount of mass accreted onto the white dwarf during three different types of outbursts. In Section 4 we use the observed luminosity of the hot spot to determine the lower limit to the mass transfer rate at quiescence. These data are then used (Section 5) to estimate the mass transfer rates during outbursts. Results are discussed in Section 6. 316 A. A.

2. Observational Data

Our analysis will be based on AAVSO light curves of 10 narrow and 10 wide outbursts of U Gem taken from Mattei et al. (1987, 1991, 1996). They were chosen in an essentially random way, the only condition being their good observational coverage. For each outburst we determine the magnitude at maximum Vmax , the width of outburst W1 at a level one magnitude below maximum, and the length of cycle ∆t (counted from the previous outburst). These parameters are listed in Table 1, where the ﬁrst column gives the approximate JD for the mid-rise. Listed in the last row are data for the 1985 superoutburst (Smak and Waagen 2004). The unusually long duration of the preceding cycle (∆t = 290 d) is simply the time interval between the last recorded normal outburst and the superoutburst. In fact, however, it is highly probable that another outburst could have occurred somewhere during the interval JD 2446215-300 not covered by observations. If so, ∆t would be much closer to mean value of 110 days.

3. Mass Accreted during Outbursts

The amount of mass accreted onto the white dwarf during outburst (or superoutburst) is calculated as

∆M = M˙ (Mv) dt (1) accr Z W1 where the M˙ = f (Mv) relation is obtained using system parameters of U Gem (including inclination) and assuming stationary accretion (for details see Smak 2002b). This requires three comments. First, that when calculating Mv = f (M˙ ) we assume that the radius of the disk during outburst is equal to the tidal radius: rd = rtid = 0.85rRoche . Secondly, that the assumption of stationary accretion may not be applicable to the situation during outbursts (particularly the narrow ones). To check this point we apply Eq. (1) to model light curves and ﬁnd that values resulting from Eq. (1) are actually lower limits to ∆Maccr . Thirdly, that our M˙ = f (Mv) relation includes the contribution from the spot. Therefore, in a situation with no enhancement in the mass transfer rate during outburst, this relation will give lower limits to M˙ and ∆Maccr . The resulting values of ∆Maccr are listed in the ﬁfth column of Table 1.

4. The Mass Transfer Rate at Quiescence

Needed in our further analysis will be the mass transfer rate in U Gem during quiescence. It was estimated earlier by several authors (e.g.,by Paczynski´ and Schwarzenberg-Czerny 1980). Using new data we redetermine it as follows. The Vol. 55 317

Table1 U Gem Outburst Parameters

JD Vmax W1 ∆t ∆Maccr ∆Mtr hM˙ enhi 2400000+ days days /1023g /1023g /1017g/s

Narrow outbursts 40657 9.4 3.5 81 1.29 0.88 1.36 42837 9.2 3.5 131 1.93 1.43 1.68 43563 9.3 5.0 129 2.58 1.40 2.72 43795 9.1 3.5 123 2.39 1.34 3.47 44238 9.6 4.5 97 1.38 1.06 0.84 44637 9.2 3.5 116 2.32 1.26 3.51 45043 9.3 3.0 99 1.47 1.08 1.49 45712 9.3 3.5 105 1.98 1.14 2.76 45952 9.5 3.5 103 1.22 1.12 0.31 47594 9.4 3.5 93 1.41 1.01 1.30 mean 9.3 3.7 108 1.80 1.17 1.95 Wide outbursts 39212 9.1 9.5 103 6.05 1.12 6.01 39579 9.1 10.5 111 6.56 1.21 5.90 42447 9.2 10.0 100 5.85 1.09 5.51 43170 9.2 9.5 115 5.09 1.25 4.68 43435 8.9 12.5 141 8.20 1.53 6.17 43956 9.0 12.5 161 8.60 1.75 6.34 44706 9.3 9.5 69 4.73 0.75 4.85 44944 9.0 10.5 89 8.85 0.97 8.68 45393 9.1 12.0 115 7.28 1.25 5.82 46056 9.2 10.0 105 5.71 1.14 5.28 mean 9.1 10.6 111 6.69 1.21 5.96 1985 Superoutbursts 46345 9.0 37.0 290 16.40 3.16 4.14 bolometric luminosity of the hot spot can be written as π 2 1 2 1 2 2 A Ls = η ∆V M˙ = η ∆v M˙ (2) 2 2 P where ∆V is the “impact velocity”, i.e.,the velocity of the stream relative to the outer parts of the disk; accordingly 1/2 ∆V 2 is the energy dissipation per 1 gram 2 of the stream material and ∆v = f (µ,r/rRoche) – its dimensionless equivalent (see Appendix in Smak 2002a); P is the orbital period and A – the radius of the or- bit. To calculate Ls as a function of M˙ we adopt system parameters of U Gem 2 (Smak 2001). In particular, for ∆v we adopt rd = 0.7rRoche , i.e.,a value which corresponds to the quiescence. 318 A. A.

Included in Eq. (2) is the “efﬁciency” factor η ≤ 1 which represents the fraction of the available energy responsible for the luminosity of the spot. This factor ac- counts, in particular, for the effects of stream overﬂow (cf. Smak 2001). Adopting η = 1 we obtain an upper limit to Ls . The bolometric luminosity of the spot can be written also as

σ 4 Ls = S Ts (3) where S is the surface of the hot spot and Ts – its effective temperature. The 2 spot’s surface is calculated as S = πsasbA , where sa = 0.04 and sb = 0.009 are the horizontal and vertical dimensions of the spot as determined from its eclipses (Smak 2001). With Ls = f (M˙ ) from Eq. (2) we now also have Ts = f (M˙ ). Using it, together with ﬂuxes from Kurucz (1993) model atmospheres, we can calculate the luminosity of the spot in the V -band. Radiation from the hot spot is strongly non-isotropic. Using formulae given by Paczynski´ and Schwarzenberg-Czerny (1980) we have for the spot luminosity corresponding to the “hot spot hump” observed at inclination i 12 L (i,max) = (1 − u + usini)sinihL i (4) s 3 − u s where hLsi is the mean luminosity (discussed above) and u is the coefﬁcient of limb darkening. In the case of U Gem with i=69◦ (Smak 2001) and u=0.6 Eq. (4) reduces to Ls(i,max) = 4.48hLsi. Using results obtained above we now calculate the absolute visual magnitude corresponding to the “hot spot hump” Mv(i,max) = f (M˙ ). Paczynski´ and Schwarzenberg-Czerny (1980), using photometric data from Paczynski´ (1965), obtained for the spot: V(i,max) = 14.76. With distance modu- lus (m − M) = 4.91 (Smak 2001) this gives Mv(i,max) = 9.85. Using this value and our theoretical relation Mv(i,max) = f (M˙ ) we obtain for the mass transfer rate: M˙ = 1.26 × 1016 g/s. We also obtain the effective temperature of the spot Ts ≈ 12000 K, i.e.,a value which looks quite reasonable. In further analysis we are going to use the value of M˙ given above but we must keep in mind that it was obtained with η = 1 and that – consequently – it is actually a lower limit to the mass transfer rate during quiescence.

5. Enhanced Mass Transfer during Outbursts

The values of ∆Maccr obtained in Section 3, together with the mass transfer rate during quiescence (Section 4) can now be used to estimate the mass transfer rates during outbursts. Assuming that all material accumulated in the disk during quiescence ∆Mtr = M˙ ∆t is accreted onto the white dwarf we have:

∆Maccr = M˙ ∆t + hM˙ enhiW1 (5) Vol. 55 319 where hM˙ enhi is the enhanced mass transfer rate during outburst averaged over the outburst duration W1 . Eq. (5) can be rewritten as

hM˙ enhi = ∆Maccr − M˙ ∆t /W1. (6) The resulting values of ∆Mtr and hM˙ enhi are listed in the last two columns of Table 1. The consequences of using the lower limit to M˙ will be discussed below.

6. Discussion

In the case of narrow outbursts our results (Table 1), when taken at their face values, could imply that the mass transfer rate is enhanced. Remembering, however, that the value of M˙ used in our calculations was actually a lower limit and taking into account all other uncertainties we do not consider this result as signif- icant. Turning the problem around we can use Eq. (5) to determine the value of M˙ which would produce no enhancement. As a result we get M˙ ≈ 2 × 1016 g/s, which is only by a factor of ≈ 2 larger than our lower limit. For comparison we can calculate the critical accretion rate. Using the revised value of the critical tem- ◦ ˙ perature log Tcrit = 3.65 (Smak 2002b) and adopting rd = 0.7rRoche we get Mcrit = 4.17 × 1016 g/s. In our further discussion we are going to use M˙ = 2 × 1016 g/s as an upper limit to M˙ . Results listed in Table 1 for the wide outbursts show that the mass transfer rate 17 16 is enhanced up to hM˙ enhi = 6.0×10 g/s. With M˙ = 2×10 we get slightly lower 17 value hM˙ enhi = 5.2 × 10 . These values compare favorably with an independent 17 estimate by Froning et al. (2001) giving M˙ = 4 × 10 . Our two values of hM˙ enhi imply that the mass transfer rate is enhanced by a factor of – respectively – f ≈ 47 and f ≈ 26. Taking into account all uncertainties we adopt f ≈ 20−50. Finally, turning to the 1985 superoutburst we may ﬁrst note that our estimate 24 of ∆Maccr = 1.64 × 10 g compares favorably with an independent estimate of 24 17 ≈ 10 g by Cannizzo et al. (2002). The value of hM˙ enhi = 4.14 × 10 , listed in Table 1, implies that the mass transfer rate was enhanced by a factor of f ≈ 33. 16 17 With M˙ = 2 × 10 we get hM˙ enhi = 3.56 × 10 and f ≈ 18. On the other hand, 16 17 with M˙ = 1.26 × 10 and ∆t = 110 we get hM˙ enhi = 4.75 × 10 and f ≈ 38. Taking into account all uncertainties we adopt f ≈ 20−40.

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