2 The TTI model

The equation of angular momentum conservation in a disc is (Hameury et al. 1998):

 

$$j \frac{\partial \Sigma}{\partial t} = - \frac{1}{r} \frac{\parti... ...\frac{\partial \dot{M}_{\rm tr}}{\partial r} - \frac{1}{2 \pi r} T_{\rm tid}(r)$$ (1)

where $\Sigma$ is the surface column density, $\dot{M}_{\rm tr}$is the rate at which mass is incorporated into the disc at radius r, $v_{\rm r}$ is the radial velocity in the disc, j = (G M₁ r)^1/2 is the specific angular momentum of material at radius r in the disc, $\Omega_{\rm K} = (G M_1/ r^3)^{1/2}$ is the Keplerian angular velocity (where M₁ is the primary mass), $\nu$ is the kinematic viscosity coefficient, and j₂ the specific angular momentum of the material transferred from the secondary. $T_{\rm tid}$ is the torque due to the tidal forces, for which we use the formula of Smak (1984), derived from the determination of tidal torques by Papaloizou & Pringle (1977):

 

$$T_{\rm tid} = c \omega r \nu \Sigma \left(\frac{r}{a}\right)^5$$ (2)

where $\omega$ is the angular velocity of the binary orbital motion, a the binary orbital separation and c a numerical constant taken so as to give an average disc radius equal to some chosen value.

In the TTI model, c is no longer a constant, but depends on whether the disc is eccentric or not. Ichikawa et al. (1993) reproduced supercycles assuming that the disc becomes eccentric when its radius reaches $r_{3:1} \sim 0.46 a$; c is then increased by a large factor. Afterward the disc starts shrinking and can no longer maintain its eccentricity when its radius becomes smaller than $r_{\rm crit0} \sim 0.35 a$, at which point c returns to its normal value. We choose to take a value c₀ when the disc is axisymmetric and a larger value c₁ when the disc is eccentric. This is our Prescription I for c, quite similar to that of Ichikawa et al. (1993) who extended the work of Osaki (1989) by taking into account a non zero tidal torque during quiescence.

We have included this variable tidal torque in the disc instability model used in Paper I, which is a modified version of the DIM described in Hameury et al. (1998) that includes heating by the stream impact and the tidal torque (we have corrected Eq. (1) of Paper I, in which the heating rate was overestimated by a factor $\sim$2). In the following, we also assume that $\alpha = \alpha_{\rm cold} = 0.04$ in quiescence and $\alpha = \alpha_{\rm hot} = 0.2$ in outburst. The inner radius is fixed at $r_{\rm in} = 10^9$ cm.

We obtain the cycle shown in Fig. 1. After a superoutburst, the disc mass and radius are small. The disc mass slowly increases, and several normal outbursts are triggered due to the thermal instability. During each outburst, only a small amount of mass is accreted by the white dwarf, so that the average disc mass increases; a significant disc expansion occurs during the rise to maximum, when the heat front reaches the outer edge of the disc. The disk then shrinks when a cooling front brings the disc back to a cool state, causing a significant reduction of the outward angular momentum flow. After the end of an outburst, the disc radius very slowly decreases as a result of mass addition and viscous diffusion, but the net balance is an increase of the radius from one outburst to the next one. Eventually, $r_{\rm out}$ becomes larger than r_3:1 during an outburst and the tidal torque is increased, as well as the associated energy dissipation. The disc is then maintained in the hot state until most of the mass has been accreted. During this phase the radius decreases until $r < r_{\rm crit0}$. The disc returns to a circular shape, the anomalous tidal dissipation stops, and a cooling wave starts from the disc outer edge, bringing the whole disc in a cool state.

Figure 1: Superoutburst cycle in the TTI model for SU UMa with $\dot{M}_{\rm tr} = 10^{16}$ g s-1. Top panel: disc radius; intermediate panel: disc mass; bottom panel: disc visual magnitude.

2.1 Radial dependence of c

There is, however, no compelling reason to assume that the whole disc is affected by the development of the tidal instability. We therefore use another prescription for c (hereafter Prescription II). We assume that when the disc is circular, c = c₀; when eccentric, c(r) = c₁ for $r > 0.9 \; r_{\rm crit0}$ and c = c₀ for $r < 0.8 \; r_{\rm crit0}$. A linear interpolation is assumed between $0.8 \; r_{\rm crit0}$ and $0.9 \; r_{\rm crit0}$. We compare and discuss the results obtained using both prescriptions in 3.1.

2.2 Time delays

Temporal variations of c(r) are not instantaneous. Observations show that normal superhumps appear within a few of days after the onset of a superoutburst, and that they develop within a day, so that it does not take long for the disc to become eccentric (Semeniuk 1980). However, it takes the disc at least a dynamical time to change its geometrical shape; we assumed that, when the tidal instability sets in, c increases linearly on a timescale $t_{\rm r} = 2 \times 10^4$ s. Note that in WZ Sge systems, the early superhumps that first appear seem to be of a different nature than the normal superhumps; Osaki & Meyer (2002) suggested that they could be related to the 2:1 resonance.

Similarly, the disc must dissipate its excess energy in order to return to a circular shape. The relative energy difference between a circular and an elliptic orbit with eccentricity e with the same angular momentum is $\Delta E/ E_{\rm circ} = e^2$. As the dissipation at radius r is $D \sim G M \Sigma \nu / r^3$, the time needed for the disc to return to a normal state is:

$$t_{\rm e-c} = \frac{\Delta E}{D} = e^2 \frac{r^2}{2 \nu} = e^2 t_\nu$$ (3)

where $t_\nu$ is the viscous time in the outer part of the disc. When the disc reverts to its normal state, the cooling wave has already started or will start with the end of tidal instability (see Sect. 3). $t_\nu$ therefore refers to the cool state, and is of the order of the recurrence time, since the outbursts obtained here are of the inside-out type. Typical values are tens of days. Numerical simulations of the development of the tidal instability show that e is of order of 0.1-0.3 (Murray 1996), which is confirmed by observations (Patterson et al. 2000), and $t_{\rm e-c}$ should be a few percent of $t_\nu$. We then assume that c linearly decreases when the disc returns to a circular shape on a time $t_{\rm f} = 2 \times 10^5$ s.

We use three different sets of parameters that are representative for SU UMa, a typical ER UMa star and a typical WZ Sge star respectively, shown in Table 1. $r_{\rm circ}$ is the circulation radius at which a particle leaving the Lagrangian point would stay in circular orbit if there were no accretion disc (see Paper I for details). One should note that $r_{\rm circ}$ is calculated taking into account the gravitational potential of the secondary, and therefore differs slightly from the Keplerian value. In the following section, we will use the term ultra-low mass-ratio (ULMR) stars when referring to simulations with ER UMa parameters.

 

 

Table 1: Parameters used for SU UMa subclasses simulations. M₂is the secondary mass and $r_{\rm circ}$ the circulation radius.

	SU UMa	ULMR Stars
		ER UMa type	WZ Sge type
$M_1 / M_\odot$	0.8	1.0	0.6
$M_2 / M_\odot$	0.15	0.1	0.1
$P_{\rm orb}$ (hr)	1.83	1.522	1.3752
a / 1010 cm	5.2	4.82	3.876
$r_{\rm circ} / 10^{10}$ cm	0.91	1.1	0.74