[vsnet 17] a preprint on a theoretical model of PG0943+521

[osaki@dept.astron.s.u-tokyo.ac.jp]

Dear Colleages;
						January 27, 1995

Following is a preprint of my paper on a theoretical model of PG0943+521,
which was submitted to PASJ (Letter) in November 1994 and it was 
recently accepted for publication (to appear PASJ(Letter) Vol. 47, No. 2, 1995).

Yoji OSAKI
Department of Astronomy, School of Science
University of Tokyo, Bunkyo-ku, Tokyo 113 


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\centerline{\bf A Model for a Peculiar SU Ursa Majoris-Type Dwarf Nova 
PG 0943+521}
\vskip 20mm
\centerline{Yoji OSAKI}
\vskip 5mm
\centerline{Department of Astronomy, School of Science}
\centerline{University of Tokyo, Bunkyo-ku, Tokyo 113} 
\centerline{E-mail (YO) osaki@dept.astron.s.u-tokyo.ac.jp }
\vskip 40mm

\centerline{(Received 1994 November 25; 
             accepted 1995 January 19)}
\vskip 20mm
\noindent{ PASJ (Letter), {\bf 47}, No. 2, 1995 (in press)} 
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\centerline{\bf Abstract} \vskip 5mm
\par
A cataclysmic variable star PG~0943+521 was recently 
discovered by Kato and Kunjaya (1994) to be an SU UMa-type dwarf nova 
having an extremely short superoutburst cycle of 43 days. 
Here we present light curve simulations of this 
star based on the thermal-tidal instability model for the SU UMa-type dwarf 
nova proposed by the present 
author (Osaki 1989). It is demonstrated that the observed light curve of 
this star is very well reproduced as the case of an SU UMa star having 
high mass transfer-rate: near the borderline between nova-like stars 
and SU UMa-type dwarf novae.

\par
Key words: Accretion disk--- Dwarf novae--- stars: SU UMa stars--- 
Stars: individual (PG~0943+521)

\beginsection{\bf 1. Introduction}

An ultraviolet excess object PG~0943+521 is a very interesting cataclysmic 
variable star (abbreviated as CV). It exhibits a recurrent light variation 
with a period of about 43 days 
in which the star spends a half of this cycle in a high state (about 20 days) 
and it spends the remaining half in a dwarf nova-type outbursting state with 
a very short recurrence cycle of about 4 days (Kato, Kunjaya 1994; Honeycutt 
1994). It thus resembles the Z Cam-type dwarf nova having a short 
"standstill". Honeycutt (1994) interpreted this star as a nova-like star 
showing secular variation in mass 
transfer rate from the secondary star with a recurrence cycle of 43 days 
and that dwarf nova-type outbursts would be produced when the mass transfer 
rate is decreased below the critical mass transfer rate for the thermal 
instability [see, e.g., a review by Cannizzo (1993) on the thermal 
instability of dwarf novae]. 

However, Kato and Kunjaya (1994) have recently discovered that this star 
exhibits periodic humps with a period of 0.065 days during a high state.
These authors have interpreted that the periodic humps discovered 
are essentially the 
same phenomenon as the superhumps of SU UMa stars and that the high state of 
this star is accordingly a "superoutburst", that is, 
the star PG~0943+521 is an extreme SU UMa star having an extremely short 
superoutburst cycle. The superhump phenomenon is now well understood as a 
development of the eccentric disk due to the tidal instability in an 
accretion disk (Whitehurst 1988; Hirose, Osaki 1990).

If we accept this interpretation, then the star PG~0943+521 has  
the following characteristics as an SU UMa star (Kato, Kunjaya 1994): 
(1) Its superoutburst cycle ("supercycle") of about 43 days is very short; 
it is the shortest one in SU UMa stars so far observed. 
(2) The duty cycle of the superoutburst is very long; it amounts to 
as long as about a half cycle (20 days). 
(3) The recurrence time of normal outbursts is extremely short, as short as 
4 days. The decline rate in normal outburst is also very short with a 
maximum decline rate of 0.7 mag d$^{-1}$.
(4) There are some short outbursts as bright as the superoutbursts.
(5) The full amplitude of light variation is about 3.0 mag, which is 
exceptionally small for SU UMa-type dwarf novae. 

In this letter we present light curve simulations for this star based on 
the thermal-tidal instability model for SU UMa-type dwarf novae proposed by the
present author (Osaki 1989, here-after referred as Paper I), and we try to 
explain all these observational 
characteristics of the star PG~0943+521 as an extreme SU UMa star having a 
rather high mass transfer rate.


\beginsection{\bf 2. Light Curve Simulations Based on the Thermal-Tidal 
Instability Model}

The thermal-tidal instability model proposed by the present author is based 
on the disk instability model in which the mass transfer rate is assumed to be 
constant and all outburst activities are caused by intrinsic instabilities 
within accretion disks. In this model, both the normal outburst 
and superoutburst are caused by the thermal instability in the accretion 
disk. In the early phase of the supercycle, the disk is 
compact and the thermal instability produces quasi-periodic episodes of 
accretion which are observed as normal outbursts but accreted mass in each 
normal outburst is smaller than that transferred during quiescence because of 
inefficient tidal removal of angular momentum from the disk.  
Both mass and angular momentum of the disk are gradually built up. 
The disk radius expands 
further with each successive outburst until it eventually exceeds the critical 
radius for the tidal instability and this final normal outburst triggers 
the tidal instability, producing the precessing eccentric disk (observed as 
"superhumps"). The resulting outburst clear the disk mass greatly 
(producing "superoutburst") because of greatly enhanced tidal torques 
due to the eccentric disk. After the end of 
the superoutburst, the disk returns back to the starting compact 
state. This is the basic idea of the thermal-tidal instability model 
for SU UMa stars. 

The present author (Osaki 1994, 1995) further examined the dependence of 
the outburst behavior of SU UMa-type dwarf novae on the mass transfer rate. 
It was demonstrated there that the supercycle length depends on mass 
transfer rate in the sense that the lower is the mass transfer rate, 
the longer is the supercycle length. This suggests that the observed 
characteristics of PG~0943+521 will most naturally be explained 
if the mass transfer rate of this star is higher than those of 
the typical SU UMa stars; that is, near the borderline case 
between nova-like stars and dwarf novae, i.e., corresponding to a ``Z Cam" 
counterpart of CVs below the period gap.  
This possibility was already suggested by Kato and Kunjaya (1994).


To confirm this conjecture, we perform light curve simulations 
based on the simplifying model used in the original proposal of the 
thermal-tidal instability model by the present author (Osaki 1989) 
in which light curves of two SU UMa stars, VW Hyi and Z Cha, were 
simulated (Paper I). To do so, we need to specify the input 
parameters. As for the binary parameters, since nothing is known for 
PG~0943+521 except for the superhump period, we simply adopt the same 
values used in Paper I. That is, we adopt $M_1=1M_{\odot}$ for 
the mass of the primary star, the critical disk radius, $R_{\rm crit}$, 
for 3:1 resonance as $R_{\rm crit}=0.46A$ and the circularization radius 
$R_{\rm LS}$ (or the Lubow-Shu radius) as $R_{\rm LS}=0.194A$ where $A$ is 
the binary separation. On the other hand, since the superhump period of 
PG~0943+521 is shorter than those of VW Hyi and Z Cha, we assume a shorter 
orbital period and thus a smaller binary separation $A$ for PG~0943+521. 
Here we take $A=4.87\times 10^{10}$cm for PG~0943+521 instead of 
$A=5.43\times 10^{10}$cm of Paper I for VW Hyi and Z Cha by assuming a 
scaling for the binary separation of the form 
$A\propto P_{\rm superhump}^{2/3}$. 

Furthermore we need to specify four more parameters for outburst simulations, 
and they are the mass 
transfer rate, $\dot M$, and the viscosity parameter in hot state, 
$\alpha_{\rm hot}$, and two other parameters, $\beta$ and $R_0$. Here 
the parameter $\beta$ stands for the critical torus mass in dimensionless 
unit above which no cold state exists, that is, the critical torus mass for 
ignition of the normal outburst, while the parameter $R_0$ stands for the 
disk radius at the end of the superoutburst which represents the strength of 
the tidal torques of the eccentric disk during the superoutburst (see Paper 
I for more details). Here we adopt $\alpha_{\rm hot}=0.3$, $\beta=0.6$, 
and $R_0=0.35A$.  On the other hand, we choose the mass 
transfer rate as a free parameter and we examine the dependence of the 
cycle length on the mass transfer rate.

To begin with, we first note that the critical mass transfer rate, which 
separates the nova-like systems from the dwarf nova systems, 
is given  in the disk instability model by (see, Smak 1983)
$$\dot M_{\rm crit}={8\pi\over 3}\sigma T^4_{\rm eff,crit}{R^3_d\over GM_1},
\eqno(1)$$
where $\sigma$ and $G$ are the Stefan-Boltzmann constant and the 
gravitational constant, respectively, 
$R_d$ is the disk radius and $T_{\rm eff,crit}$ is the critical 
effective temperature of an accretion disk below which no hot state exists.
Here we use $\log T_{\rm eff,crit}=3.9-0.1 \log R_{d,10}$ where $R_{d,10}=
R_d/10^{10}{\rm cm}$.
We then find for our binary parameters used here 
$$\dot M_{\rm crit}\simeq 5.6\times 10^{16}{\rm gs}^{-1}. \eqno(2)$$

We have calculated light curves for various mass transfer rates 
and the results are summarized in figure 1 in which 
the recurrence time (i.e., the supercycle length) is shown 
as a function of mass transfer rate from the secondary. 
As seen from figure 1, the superoutburst recurrence time, $t_{\rm S}$, 
is inversely proportional to the mass transfer rate if the mass transfer rate 
is low. Its asymptotic relation is given by
$$t_{\rm S}\simeq 100~{\rm days}/\dot M_{16}.\eqno(3)$$ 
where $\dot M_{16}$ stands for the mass transfer rate in units of 
$10^{16}{\rm gs}^{-1}$. 
On the other hand, when the mass transfer rate is increased further, 
the supercycle length exhibits a broad minimum around 40 days.  
It then begins to increase rapidly if it approaches to the critical 
mass transfer rate given in equation (2), and it becomes infinity 
above the critical mass transfer rate.

Figure 2 illustrates the light curve and the disk radius variation for a 
case of mass transfer rate, $\dot M=4\times 10^{16}{\rm gs}^{-1}$, which is 
supposed to simulate the star PG~0943+521 most well.
As seen from figure 2, we find in this case that (1) the superoutburst 
recurrence time is 44 days, (2) the duration of a superoutburst is 
about 20 days (i.e., the duty cycle of the superoutburst is about a half 
cycle), (3) the recurrence time of short (normal) outbursts is about 
$4\sim 5$ days, 
(4) rather small outburst amplitude of the present case (about 4 mag in 
bolometric luminosity) as compared with 
those of VW Hyi and Z Cha in Paper I. All of these characteristics are 
in a good agreement with Kato and Kunjaya's (1994) observations 
of PG~0943+521. It must, however, be noted that the mass transfer rate 
inferred for PG~0943+521 is higher by a factor 10 than that expected from 
the CV evolutionary scenario based on the gravitational wave radiation 
theory, and this problem is discussed in the next section.   

For comparison we show in figure 3 two more 
light curves for slightly higher mass transfer rates. As seen in the upper 
panel of figure 3, the light curve for 
a mass transfer rate of $\dot M=5\times 10^{16}{\rm gs}^{-1}$ is very 
similar to figure 2 but the recurrence time and the duty cycle of the 
superoutburst are slightly longer in this case. However, if we proceed to 
a case of mass transfer rate, $\dot M=6\times 10^{16}{\rm gs}^{-1}$, which 
is above the critical value given 
in equation (2), we find that the star eventually settles down to 
a steady state as seen in the lower panel of figure 3. This model 
corresponds to a nova-like star which shows no outburst but still exhibits the 
permanent superhumps (Skillman, Patterson 1993).

\beginsection{\bf 3. Discussions}

In this letter we have successfully simulated the light curve of 
PG~0943+521 based on the thermal-tidal instability model proposed by the 
present author (Paper I). By putting it in the other way around, our 
success gives a strong support to the thermal-tidal instability theory for 
the SU UMa stars. However, since our simulation method used here is rather
rudimental, it would be desirable to confirm the present results
by using a more sophisticated method such as used by Ichikawa, Hirose,
and Osaki (1993).

In our model the star PG~0943+521 is identified to an SU UMa star having a 
high mass transfer rate and its value inferred from our 
simulation is higher by a factor ten than that expected 
in the standard CV evolutionary scenario for CVs below the period gap 
based on the gravitational-wave radiation theory.  In fact, Patterson 
and his group (see, e.g., Skillman, Patterson 1993) have 
shown that there are some nova-like stars exhibiting ``permanent superhumps"
and this suggests that these stars should have much high mass transfer rate 
as well. It is most likely that the instantaneous mass transfer rate 
of CVs may not be determined uniquely by the binary parameters but it 
may vary secularly such as in the hibernation scenario of nova/dwarf nova 
alternation (see, Livio 1992). 

\bigskip

I would like to thank Taichi Kato for calling my attention to this 
interesting object and for sending me a preprint by Kato and Kunjaya.

\beginsection{\bf References}  

\ref Cannizzo J.K. 1993, in Accretion Disks in Compact Stellar Systems 
 ed J. C. Wheeler (World Scientific Publishing, Singapore), p6

\ref Hirose M., Osaki Y. 1990, PASJ 42, 135

\ref Honeycutt, R. K. 1994, a talk presented in Padova-conference on 
Cataclysmic Variables: Inter Class Relations 
 
\ref Ichikawa S., Hirose M., Osaki Y.  1993, PASJ 45, 243

\ref Kato, T. and Kunjaya, C. 1994, submitted to PASJ 

\ref Livio M. 1992, in Vina del Mar Workshop on Cataclysmic Variable
Stars, ed N. Vogt, ASP Conference Series, 29, 269
  
\ref Osaki Y. 1989, PASJ 41, 1005 (Paper I)

\ref Osaki Y. 1994, in Theory of Accretion Disks-2 
ed W. Duschl, et al. (Kluwer Academic Publishers, Dordrecht), 93

\ref Osaki, Y. 1995, PASJ in press

\ref Skillman D. R. and Patterson, J. 1993, ApJ 417, 298

\ref Smak J. 1983, ApJ 272, 234

\ref Whitehurst R. 1988, MNRAS 232, 35
 
\vfill\eject


\beginsection{\bf Figure Captions}

\bigskip
  
\ref Fig. 1. Superoutburst recurrence time, $t_{\rm S}$, in units of days 
as a function of the mass transfer rate, $\dot M_{16}$.

\ref Fig. 2. Bolometric light curve (upper panel) and disk radius variation 
(lower panel) simulated for the star PG~0943+521 based on the
simplifying model used by Osaki (1989). Parameters used are; the mass
transfer rate $\dot M=4.0\times 10^{16}{\rm g~s}^{-1}$; 
the binary separation $A=4.87\times 10^{10}$cm;
the viscosity parameters in the hot state, $\alpha_{\rm H}=0.3$; the
critical torus mass in dimensionless unit $\beta=0.6$, 
the terminal disk radius of the superoutburst $R_0=0.35A$. 

\ref Fig. 3. Bolometric light curves for the same parameters as figure 2   
except for two different mass transfer rates 
$\dot M=5.0\times 10^{16}{\rm g~s}^{-1}$ (upper panel), 
and $\dot M=6.0\times 10^{16}{\rm g~s}^{-1}$ (lower panel). 
       
\bye

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