4 Irradiated discs

In this section we study the effect of including self-irradiation to the standard DIM. We present only one model for this purpose (other may be found in Dubus 2000) with M₁=7 $M_\odot$, $\dot{M}_{\rm tr}=10^{16}{\rm ~g}\;{\rm s}^{-1}$, $\alpha _{\rm h}=0.2$, $\alpha _{\rm c}=0.02$, $<R_{\rm out}>\,=10^{11}$ cm and a fixed inner radius at $R_{\rm in}=10^9$ cm (this is further discussed below in Sect. 4.6). Irradiation is included using Eq. (8) with $\epsilon=0.1$ when $\dot{M}_{\rm in}>10^{16}$ g$\;$s^-1 and $\epsilon=0.1 (\dot{M}_{\rm in}/10^{16}{\rm ~g~s}^{-1})^6$ below. This amounts to a quick cutoff of irradiation below $10^{16}{\rm ~g~s}^{-1}$. The exact form of $\epsilon$ matters little as long as it ensures that irradiation is only important during the outburst (a reasonable assumption). The resulting outburst time profile is shown in Fig. 3.

Figure 3: Example of an outburst cycle when irradiation is included. From top to bottom: the mass accretion rate at the inner edge $\dot{M}_{\rm in}$ (full line) and $\dot{M}_{\rm irr}=\epsilon\dot{M}_{\rm in}$ (dotted line), the V magnitude, the outer disc radius $R_{\rm out}$ (full line) and the transition radius $R_{\rm trans}$ between the hot and the cold regions (dotted line), the mass of the disc $M_{\rm disc}$. The parameters are M1=7 $M_\odot$, $\dot{M}_{\rm tr}=10^{16}{\rm ~g}\;{\rm s}^{-1}$, $\alpha _{\rm h}=0.2$, $\alpha _{\rm c}=0.02$, $<R_{\rm out}>\,=10^{11}$ cm and a fixed inner radius at $R_{\rm in}=10^9$ cm. Details of the outbursts and the evolution of the density and temperature profiles can be seen in Figs. 5-7.

  
4.1 Rise

We start with a cold, quiescent disc in which most of the matter has been accreted in a preceding outburst. During quiescence (see below), mass transfer from the secondary replenishes the flow until the density becomes high enough and an annulus arrives at the thermally unstable branch of the $\mathit{S}$-curve. The ring cannot maintain thermal equilibrium and undergoes a transition to the hot branch.

As in the standard DIM, two heating fronts are formed which propagate inwards and outwards from the ignition radius (Menou et al. 1999a and references therein). In "inside-out'' (type B) outbursts the ignition radius is small and the propagation time of the inward bound front is very small compared to the other one. In "outside-in'' (type A) outbursts, the opposite is true. In the standard DIM, "inside-out'' fronts can stall rather easily leading to short, low-amplitude outbursts. The density profile in quiescence is generally not far from $\Sigma_{\rm max}\propto R$ so that, as the outward front progresses through the disc, it encounters regions of higher densities and $\Sigma _{\rm max}$ (the critical density needed to raise a ring to the hot branch). If the front does not transport enough matter to raise the density at some radius above $\Sigma _{\rm max}$, it stalls and a cooling front develops. "Outside-in'' fronts always progress through regions of decreasing $\Sigma _{\rm max}$ and therefore always heat the whole disc. Another consequence is that inside-out fronts propagate slowly, leading to slow rise times. This is particularly evident in Fig. 8 of Menou et al. (2000).

Irradiation does not change the structure of the heating front (compare Fig. 4 with Fig. 1 of Menou et al. 1999a) but does change the maximum radius to which an inside-out outburst can propagate. Since we do not consider self-screening, the outer disc always sees the irradiation flux from the central regions. As an inside-out front propagates, $\dot{M}_{\rm in}$ rises and the outer cold disc is increasingly irradiated (see Fig. 5). Irradiation heating reduces the critical density needed to reach the hot branch (Eq. (3)), easing front propagation. Obviously, a larger hot region implies a greater optical flux and irradiation always lowers the peak optical magnitude.

In an inside-out outburst, the outer regions the front has not yet reached are frozen on the timescale on which $\dot{M}_{\rm in}$evolves (see Fig. 5). If $\dot{M}_{\rm in}$ (hence $\dot{M}_{\rm irr}$) increases on timescales shorter than the thermal timescale in the cold disc, this can lead to situations where $T_{\rm irr}>T_{\rm c}$ at some radii. The vertical heat flux changes sign and would require negative values of $\alpha$ in our treatment of thermal imbalance i.e. the assumptions of the code break down. We assumed that such an annulus is dominated by irradiation i.e. is isothermal at $T_{\rm c}$ and that irradiation contributes an additional heating term to the radial thermal equation (Eq. (7)) $Q^+_{\rm add}=\sigma (T_{\rm irr}^4-T_{\rm c}^4)$to reflect the imbalance at the photosphere between the outgoing flux $\sigma T^4_{\rm c}$ and the incoming flux $\sigma T^4_{\rm irr}$. This, or other assumptions, actually had very little influence on the rise-to-outburst lightcurve. Further studies of this phenomenon would require a detailed model of irradiation in SXTs where the geometry, the exact flux and spectrum of irradiating photons are properly set out. Note also that the absence of any back loop in the equations to prevent this situation may suggest that some additional physics is needed. One possibility is that rapidly increasing irradiation would evaporate the upper layers of the disc (Begelman et al. 1983; Hoshi 1984; Idan & Shaviv 1996; de Kool & Wickramasinghe 1999).

Figure 5 shows the evolution of the $\Sigma$ and $T_{\rm c}$ radial profiles during the outburst rise. As the front reaches the outer edge of the disc, the profiles in the hot region converge to those of a steady disc with constant $\dot{M}(R)$. This is because the viscous timescale, which is inversely proportional to $\alpha T_{\rm c}$, becomes short enough to equilibrate the mass flow in the hot region (e.g. Menou et al. 1999a, and references therein). Irradiation has little influence on the actual vertical structure in this region as discussed in Dubus et al. (1999) and we find $T_{\rm c}\propto\Sigma\propto R^{-3/4}$ as in a non-irradiated steady disc. Only in the outermost disc regions does the vertical structure become irradiation-dominated, i.e. isothermal.

The peak accretion rate (and optical magnitudes) is rather difficult to estimate analytically since it depends on the maximum radius to which the heat front can propagate. Qualitatively, this is set mostly by the conditions in the disc at the onset of the outburst. The principle factor is the total disc mass when the outburst starts $M_{\rm max}$ which is constrained by $\Sigma _{\rm max}$. Parameter studies (Sect. 6) show that any changes which result in a lower $\Sigma _{\rm max}$, hence in a lower $M_{\rm disc}$ do lead to smaller outburst peaks. Increasing M₁, $\alpha _{\rm c}$ or decreasing $R_{\rm out}$give smaller $\dot{M}_{\rm peak}$. Increasing the mass transfer rate leads to higher disc masses in quiescence and higher peaks.

In any case, $\dot{M}_{\rm peak}$ should be lower than or of order of the critical mass accretion rate $\dot{M}_{\rm crit}$ corresponding to $\Sigma _{\rm min}$ at $R_{\rm out}$, as the disc is close to steady state at maximum if it has been brought entirely to a hot state. If the heating front has not been able to reach the outer edge, $\dot{M}_{\rm peak}$ will be lower.

Figure 4: An "inside-out'' heating front in an irradiated disc. This is the front at $t\approx 1.5$ days in Fig. 5. $\Sigma$is in g$\;$cm-2, temperatures are in K, $\dot{M}$ is in units of 1016 g$\;$s-1. The cold outer disc is almost isothermal with $T_{\rm c}\approx T_{\rm irr}$.

4.2 Decay from outburst maximum

The outburst decay can generally be divided into three parts:

• First, outer disc X-ray irradiation inhibits the cooling-front propagation. Since more mass is lost than gained from the secondary ( $\dot{M}_{\rm in}\gg\dot{M}_{\rm tr}$) the disc is drained by viscous accretion of matter (King & Ritter 1998);

• Second, the accretion rate becomes low enough that the X-ray irradiation is unable to prevent the cooling from propagating. The propagation speed of this front is, however, controlled by irradiation;

• Irradiation plays no role and the cooling front switches off the outburst on a local thermal time.

We now discuss these three phases in more detail.

  
4.2.1 Viscous "exponential'' decay

In the model presented here, the disc becomes fully ionized at the outburst peak. The disc then evolves with $\dot{M}(R)$ almost constant so that $\nu \Sigma \sim \dot{M}_{\rm in}(t)/3\pi$; the total mass in the disc is thus

$$M_{\rm d}=\int 2\pi R\Sigma {\rm d}R \propto \dot{M}_{\rm in} \int {2\over 3} {R \over \nu} {\rm d}R$$ (10)

and one can consider that the disc decays through a sequence of quasi-stationary states. In irradiation dominated discs as well as in standard Shakura-Sunyaev discs, $\nu \propto T \propto \dot{M}^{\beta/\left(1+\beta\right)}$, with $\beta = 3/7$ in hot Shakura-Sunyaev discs, and $\beta$ = 1/3 in irradiation dominated discs. The outer disc radius does not vary much so the time evolution of the disc mass is:

$$\frac{{\rm d} M_{\rm d}}{{\rm d} t}=-\dot{M}_{\rm in}\propto M_{\rm d}^{1+\beta},$$ (11)

showing that $\dot{M}_{\rm in}$ evolves almost exponentially, as long as $\dot{M}_{\rm in}^\beta$ can be considered as constant (i.e. over about a decade in $\dot{M}_{\rm in}$, as found by King & Ritter 1998 for irradiated discs, and by Mineshige et al. 1993 for non-irradiated discs with angular momentum removal). Power-laws with indices close to -1 are found in discs with constant angular momentum, i.e. in which there are no tidal torques preventing the outer disc radius to expand indefinitely (Lyubarskii & Shakura 1987; Cannizzo et al. 1990; Mineshige et al. 1993), and are of no practical interest here.

The decay is viscous as long as thermal equilibrium can be maintained. Incidentally, this makes it difficult, if not impossible, to have a viscous decay in the standard DIM without additional assumptions. In outside-in outbursts, matter accumulates at the outer disc edge and the resulting density excess decreases viscously, producing "flat-top'' light curves. Thermal equilibrium requires the whole disc to be kept in the hot state. If the outer disc is cold then there will be a thermally unstable region evolving rapidly (i.e. non-viscously, see next paragraph). Keeping a disc hot can be achieved by additional mass transfer during the outburst or non-standard sources of heating (e.g. tidal heating as in Buat-Ménard et al. 2001). A smaller $\alpha _{\rm c}$ increases the difference between $\Sigma _{\rm max}$ and $\Sigma _{\rm min}$ and hence can also help keep the disc in the hot state longer (Menou et al. 2000).

Irradiation heating not only leads to the exponential decay but also provides a natural way to keep the disc hot. An irradiation temperature at the outer edge above 10⁴ K ensures hydrogen is everywhere ionized. For such temperatures, the disc is thermally stable whatever the local density $\Sigma$ and accretion rate. Put differently, the lower branch of the $\mathit{S}$-curve disappears when $T_{\rm irr}\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13E$ }}10^4$ K (see Figs. 4-5 of Dubus et al. 1999) or $\Sigma_{\rm min}=\Sigma_{\rm max}$ at $T_{\rm irr}\approx 10^4$ K (and are undefined below: see Eqs. (3), (5)).

One should remember, however, that many SXT light-curves are not exponential, despite their discs being irradiated. Clearly additional physical processes have to be taken into account if one wishes to explain such "non-typical'' behaviour (see e.g. Esin et al. 2000a).

Figure 5: The outburst rise for the model of Sect. 4 (irradiated, with a fixed $R_{\rm in}$). The upper left panel shows $\dot{M}_{\rm in}$ and $\dot{M}_{\rm irr}$ (dotted line); the bottom left panel shows the V magnitude. The spike at $t\approx 1$ day corresponds to the arrival at $R_{\rm in}$ of the inward propagating front. The rise is dominated by the outward front (inside-out outburst). Each dot corresponds to one of the $\Sigma$ and $T_{\rm c}$ profiles in the right panels. The disc expands during the outburst to transport the angular momentum of the material being accreted. The profiles close to the peak are those of a steady-state disc ( $\Sigma \propto T_{\rm c}\propto R^{-3/4}$).

Figure 6: The outburst decay for the model discussed in Sects. 4.2.1-4.3. The fully ionized disc decays viscously (exponential lightcurve) until $t\approx 70$ d where $T_{\rm irr}(R_{\rm out})\approx 10^4$ K. Thereafter the outer disc can fall on the cold branch of the $\mathit{S}$-curve and a cooling front appears, propagating inward only as far as $T_{\rm irr}\approx 10^4$ K (irradiation-controlled linear decay). At $t\approx 190$ d irradiation shuts off and the disc cools quickly. The dashed line is $\Sigma _{\rm min}$ for $\alpha _{\rm h}$ without irradiation, showing the post-front $\Sigma$ are very low when irradiation is included. $\Sigma _{\rm min, max}(\alpha ,T_{\rm irr})$, and $T_{\rm irr}$ are shown for the last $\Sigma$ and $T_{\rm c}$ profiles (dotted lines).

Figure 7: Quiescence for the model discussed in Sect. 4.6. The irradiation flux is negligible in quiescence and the overall evolution is the same as in the standard DIM. Mass transfer from the secondary is slow enough that matter diffuses down the disc, gradually increasing $\dot{M}_{\rm in}$. The outburst is triggered inside at $R\approx 2\times 10^9$ cm when $\Sigma$ reaches $\Sigma _{\rm max}$ (dotted line). Lower densities at the beginning of the quiescent state (due to the irradiation-controlled outburst decay) lead to a long recurrence time (about 10 years).

  
4.3 Irradiation-controlled linear decay

The second stage of the decay starts when a disc ring cannot maintain thermal equilibrium and switches to the cool lower branch of the $\mathit{S}$-curve. The disc outer edge always becomes unstable first with $\Sigma<\Sigma_{\rm min}$ since $\Sigma\propto R^{-3/4}$ (see above) while $\Sigma_{\rm min}\propto R$. In an irradiated disc this happens when the central object does not produce enough X-ray flux to keep the $T_{\rm irr}(R_{\rm out})$ above 10⁴ K so that the low state is again possible. The material in the low state has a lower viscosity and piles up, leading to the appearance of a cooling front with a width $w\propto H$ (Papaloizou & Pringle 1985; Meyer 1986; Fig. 4 of Menou et al. 1999a). The front propagates through the disc at a speed $V_{\rm front}\approx \alpha_{\rm h} c_{\rm s}$ (Meyer 1984; Vishniac & Wheeler 1996). In the standard DIM, the sound speed depends on the temperature at the transition between the hot and cold regions and is almost constant as verified by numerical calculations (Menou et al. 1999a).

In an irradiated disc, however, the transition between the hot and cold regions is set by $T_{\rm irr}$ since the cold branch only exists for $T_{\rm irr}\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13C$ }}10^4$ K. Unlike a non-irradiated disc, the cooling front can propagate only as far as the radius at which $T_{\rm irr}\approx 10^4$ K, i.e. as far as there is a cold branch to fall onto. Here also irradiation controls the decay. The hot region stays close to steady-state but with a shrinking size $R_{\rm hot}\sim \dot{M}_{\rm in}^{1/2}$ (Eq. (8) with $T_{\rm irr}(R_{\rm hot})= {\rm const.}$). The mass redistribution during front propagation is complex with the hot region losing mass both through inflow and outflow and our models show steeper declines than what some analytic approximations predict (e.g. King 1998).

The linear decay is illustrated in Fig. 6. The irradiation temperature is lower than 10⁴ K at the outer edge for t>70 days where the decay becomes steeper. The cooling front (see also Fig. 8) appears as a depression propagating inwards in the surface density profile. The irradiation temperature and the critical densities $\Sigma_{\rm min,max}$ are plotted for the last profile at $t\approx 190$ days, showing the cooling front is at the radius for which $T_{\rm irr}\approx 10^4$ K and that $\Sigma_{\rm min,max}$ are undefined for lower radii.

4.4 Final thermal decay

The quick decay after t>190 days is due to vanishing irradiation as $\epsilon$ becomes very small for $\dot{M}_{\rm in}<10^{16}$ g$\;$s^-1. The cooling front thereafter propagates freely inwards, on a thermal time scale.

Figure 8: The cooling front in an irradiated disc. This is the front at $t\approx 190$ days in Fig. 6. $\Sigma$ is in g$\;$cm-2, temperatures are in K, $\dot{M}$ is in units of 1016 g$\;$s-1. The front is at the position for which $T_{\rm irr}$ (dotted line) is ${\approx } 10^4$ K. At this point $\Sigma _{\rm min}\approx \Sigma _{\rm max}$ since there is no cold branch for higher $T_{\rm irr}$ (smaller R). As $T_{\rm irr}$decreases the two critical $\Sigma$ separate and converge to their non-irradiated values (this can be seen in Fig. 6; note the values of $\Sigma _{\rm max}$ and $\Sigma _{\rm min}$ shown here are only accurate to the precision of the fits given by Eq. (3), (5)). The cold outer disc is almost isothermal with $T_{\rm c}\approx T_{\rm irr}$. The negative values of $\dot{M}$ show the cooling front transports matter to the outer disc.

  
4.5 Mini-reflares

As discussed above, the edge of the hot zone is at the stability limit set by $T_{\rm irr}\approx 10^4$ K. At this point, one has $\Sigma\approx\Sigma_{\rm min}\approx\Sigma_{\rm max}$ because irradiation modifies the critical surface densities. Further away from the edge of the hot zone, the irradiation temperature decreases rapidly ( $T^4_{\rm irr}\propto R^{-2}$) and $\Sigma _{\rm max}$gradually becomes greater than $\Sigma _{\rm min}$. In the non-irradiated limit, the ratio $\Sigma_{\rm max}/\Sigma_{\rm min}$ is a constant depending only on the ratio $\alpha_{\rm c}/\alpha_{\rm h}$(see e.g. Fig. 2).

The region immediately behind the cooling front where $\Sigma _{\rm max}$ is close to $\Sigma _{\rm min}$ is clearly very unstable. Slight variations of $\Sigma$ may suffice to have $\Sigma>\Sigma_{\rm max}$behind the cooling front and therefore start a reflare just as in a non-irradiated disc. This depends on the numerical details of the model, such as the functional of $\alpha$ with temperature (Sect. 2.4).

These mini-reflares are unimportant. The critical density $\Sigma _{\rm min}$ increases quickly behind the cooling front (see Fig. 8) because the irradiation flux decreases with R^-2. Therefore, the heating front in a mini-reflare reaches almost immediately a radius where $\Sigma<\Sigma_{\rm min}$ and cooling resumes. Furthermore, irradiation seriously depletes the disc and the front finds very little matter to fuel its propagation. $\Sigma _{\rm min}$ is much lower than its non-irradiated value (Fig. 6). For instance, Fig. 4 of Dubus et al. (1999) shows $\Sigma_{\rm min}(T_{\rm irr}\approx 10^4{\rm K})\approx30$ g $\;{\rm cm}^{-2}$ instead of $\Sigma_{\rm min}(T_{\rm irr}=0{\rm K})\approx150$ g $\;{\rm cm}^{-2}$. The much lower post-cooling front densities prevent large reflares as in the standard DIM. In practice, mini-reflares in irradiated discs have no influence on the lightcurve (very low amplitudes and cycles, see Fig. 9) but can be a numerical nuisance.

Figure 9: Example of mini-reflares in an irradiated disc. The region close to $T_{\rm irr}\approx 10^4$ K is highly unstable with $\Sigma _{\rm min}\approx \Sigma _{\rm max}$ (Fig. 8). This can lead to a succession of heat fronts propagating quickly back and forth in a small region (bottom panel). The exact details depend on the model assumptions (e.g. $\alpha$). This has no influence on either $\dot{M}_{\rm in}$ (top panel) or the optical flux (not shown). See Sect. 4.5 for details.

  
4.6 Quiescence

As discussed above, irradiation dramatically slows down the cooling process. The decay lasts much longer and, accordingly, the total mass accreted during an outburst is much larger. The densities after propagation of the cooling front are low so the total mass of the disc as it enters quiescence is much less than in the non-irradiated case.

The low quiescent surface densities imply low midplane temperatures. The ($\Sigma$,$T_{\rm c}$) $\mathit{S}$-curves show that for low $\Sigma$ and no irradiation (as appropriate once the disc enters quiescence) the lower branch is flat with $T_{\rm c}\approx 2000$ K whatever R or M₁ (see e.g. in Figs. 4, 5 of Dubus et al. 1999). In contrast, the standard DIM leads to higher $\Sigma$, close to the non-irradiated $\Sigma _{\rm min}$ and thus to $T_{\rm c}\approx 4000$ K (e.g. Osaki 1996; Menou et al. 2000).

The critical density $\Sigma _{\rm max}$ that the disc must reach before an outburst can start will be the same as in the standard DIM if irradiation in quiescence is negligible. This is in general true since $\dot{M}_{\rm in} \propto T^4_{\rm irr}$ is very small. Since the quiescent disc after an irradiation-controlled decay is always less massive, the time to build up the critical density will be longer.

In the model presented in this section, $T_{\rm irr}=0$ in quiescence. The evolution of the $\Sigma$ and $T_{\rm c}$ profiles (Fig. 7) shows that matter arriving from the secondary at the outer edge diffuses down the cold disc. The disc contracts due to addition of this lower angular-momentum material. The temperature is almost constant in most of the disc with $T_{\rm c}\approx 2000$ K. This is not true in the later stages of quiescence where $T_{\rm c}$ rises in the inner disc due to the build up of matter there: the inner rings move up along the lower branch of the ($\Sigma$,$T_{\rm c}$) $\mathit{S}$-curve and out of the flat $T_{\rm c}\approx 2000$ K region discussed above. In the last profile, $\Sigma$ is close to $\Sigma _{\rm max}$ (dotted line) and the outburst is ignited at $R\approx 2\times 10^9$ cm.

The ignition radius depends essentially on the mass transfer rate from the secondary and the disc's size (Smak 1984). For low $\dot{M}_{\rm tr}$, matter drifts down the cold disc and $\Sigma _{\rm max}$ is reached in the inner disc (inside-out, type B outburst). As $\dot{M}_{\rm tr}$ increases, the mass accumulation time at the outer radius can become lower than the drift time, triggering an outburst in the outer disc (an outside-in, type A outburst). The accumulation time is (Ichikawa & Osaki 1994; see also Osaki 1995,1996 and Lasota 2001):

$$t_{\rm acc}={2\pi R_{\rm out}\Sigma_{\rm max}(R_{\rm out}) \over \dot{M}_{\rm tr}}\Delta R.$$ (12)

A ring of material spreads viscously over $\Delta R\approx (\nu t_{\rm acc})^{1/2}$ during $t_{\rm acc}$. Setting $t_{\rm acc}=t_{\rm drift}$gives an estimate of the accretion rate for which the transition from type B to A outbursts is expected. Using Eq. (3) and $t_{\rm drift}=\delta t_{\rm vis}$ ($\delta$ is a numerical correction factor introduced by Osaki 1996) outside-in outbursts are therefore expected when

$$\dot{M}_{\rm tr} \mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13E$ }} 2\pi R_{\rm out}\delta^{-1/2} \nu \Sigma_{\rm max}(R_{\rm out})$$

$$\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13E$ }} 3.3\times 10^{16} \left({\alpha \over 0.02}\right)^{0.2} \left({R... ...ot}\right)^{-0.9} \left({T_{\rm c} \over 2000\;{\rm K}}\right) {\rm\ g\ s}^{-1}$$ (13)

with $\delta=1$. This is smaller than the mass accretion rate needed to stabilize an irradiated disc with these parameters ( $\dot{M}_{\rm tr}\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13E$ }}10^{17}$ g s^-1, Eq. (30) of Dubus et al. 1999). In principle type A outbursts are thus possible in SXTs, but numerical calculations show that real outside-in outbursts starting far out in a large disc are difficult to obtain. Osaki (1996) finds $\delta\approx 0.05$ provides an adequate fit to his calculations. The quiescence time of our model (about 4000 days) implies $t_{\rm drift}\approx 0.1 t_{\rm vis}(R_{\rm out})$ so that the required $\dot{M}_{\rm tr}$ for a type A outburst is above the accretion rate for which the disc is stable.

The inner edge of the accretion disc plays a crucial role in quiescence. For low accretion rates $\Sigma _{\rm max}$ is reached in the inner disc. Since $\Sigma_{\rm max}\propto R$ the amount of mass needed to trigger an outburst is reduced when the disc inner radius is smaller: the higher $R_{\rm in}$ in quiescence, the longer the recurrence time. Even with $R_{\rm in}=10^9$ cm the model is still short of providing

• the long recurrence timescale ( $t_{\rm rec} \approx$ 10 years compared to the tens of years inferred in SXTs) and

• the quiescent X-ray luminosities: the predicted quiescent accretion rate, in the range 10^12-13 g s^-1 is still too small to account for observed X-ray luminosities exceeding 10³²erg s^-1.

The next section will show that including disc evaporation into a hot, low-density, accretion flow can solve these problems.