5 Irradiated discs with evaporation

Following the work of Menou et al. (2000), we now assume that the inner disc is gradually evaporated into a hot, radiatively inefficient accretion flow during the decay from outburst. The exact nature of the flow is not important as long as it does not participate in the dynamics of the thin accretion disc. A good assumption is that the inner disc is replaced by an advection-dominated accretion flow (ADAF). ADAF+thin disc models have been successful in explaining the spectral states of SXTs and in particular, the quiescent X-ray flux provided the mass accretion rate in the ADAF is large enough (e.g. Esin et al. 1997).

The accretion timescale in an ADAF becomes of the order of the thermal timescale so that the inner flow can be considered as quasi-steady and does not participate in the limit cycle. A good assumption in our dynamical study is therefore to treat evaporation as a variation of the inner radius of the thin disc we model. We take the same rate of evaporation as in Menou et al. (2000):

$$\dot M_{\rm ev}(R)=0.08 \dot M_{\rm Edd} \left[ \left( \frac{... ...l E}\left( \frac{R}{800 R_{\rm s}} \right)^2 \right]^{-1}\cdot$$ (14)

The inner radius of the disc is defined as:

$$\dot{M}_{\rm ev}(R_{\rm in})=\dot{M}_{\rm in}.$$ (15)

Below this radius all the matter evaporates. Because of the steep dependence of $\dot M_{\rm ev}$ on R, we can safely neglect evaporation above this radius. As noted in Menou et al. (2000), the detailed functional dependence of $\dot M_{\rm ev}(R)$, or of $R_{\rm in}(\dot M_{\rm in})$ has essentially no effect on the results; what matters is the value of the inner disc radius during quiescence.

Equation (14) is entirely ad hoc, and is not based on any particular physical mechanism. There are models that, in principle, allow for the determination of the evaporation rate as a function of radius. However, the variety of models shows that there is no general agreement on the physical cause of evaporation. For example, Meyer & Meyer-Hofmeister (1994) assume that electron conduction from the disc to the corona plays a major role, whereas Shaviv et al. (1999) consider a thermal instability related to the opacity law. In addition, even for a given model, the evaporation law depends on parameters which are not easily measured (as for example the magnetic field and its coherence in the case of electron conduction; Meyer et al. 2000). As all these models have been proposed to explain the existence of holes in observed accretion discs, they, by definition, tend to produce similar results, i.e. the truncation radius must be at a detectable distance from the compact objet. And of course, for the same reason, Eq. (14) would also reproduce these results. The latest calculations by Meyer et al. (2000) give evaporation rates of the order of $1.4 \times 10^{16}$ g s^-1 at 10¹⁰ cm from a $6~M_\odot$ black hole, and $1.6 \times 10^{17}$ g s^-1 at 10⁹cm. These are larger by a factor $\sim$15 and $\sim$2 respectively than the values we are using here. Note that the evaporation rate is so high in Meyer et al. (2000) that the quiescent disc would be truncated at too large a radius to be unstable.

The ADAF efficiency roughly scales as $\dot{M}$ (i.e. the luminosity scales as $\dot{M}^2$, Esin et al. 1997) . From Eq. (14), this is equivalent to $\epsilon \propto R^{-2}_{\rm in}$. In the model, we assume $\epsilon=0.1$ when $R_{\rm in}=R_{\rm min}$ (in outburst) and $\epsilon=0.1\times(R_{\rm min}/R_{\rm in})^{-2}$ when $R_{\rm in} \ge R_{\rm min}$ (in quiescence). Irradiation is usually negligible in quiescence so these assumptions have little importance there. Varying $\epsilon$ may change slightly the end of the outburst when evaporation becomes important.

We took $R_{\rm min}=5\times 10^{8}$ cm in all the following models. As discussed below in Sect. 5.1, the only time $R_{\rm min}$ is reached is in outburst where the inner disc is in thermal equilibrium and close to steady-state. The inner disc edge plays no role in this case.

In the following we discuss in some detail the same model as in Sect. 4 but including evaporation. The overall outburst time profile is shown in Fig. 10. A comparison with Fig. 3 shows the outburst is almost identical but that the disc spends much more time in quiescence ( $t_{\rm rec}\approx 21$ years). The maximum luminosity reached during outburst is also increased.

  
5.1 The outburst

After an outburst is triggered, the mass accretion rate at the inner edge gradually increases as in the previous model. The inner radius of the thin disc decreases until its minimum value is reached for $\dot{M}_{\rm in}=\dot{M}_{\rm ev}(R_{\rm min})$. A slight change of slope is associated with this (at the 4th dot in Fig. 11). The time $t_{\rm rise}$ the disc needs to reach $R_{\rm min}$ is a viscous timescale. Before the onset of the outburst, $R_{\rm in}\approx 6\times 10^9$ cm. Therefore, $t_{\rm rise}$is of the order of the difference between $t_{\rm vis}$ at $6\times 10^9$ cm and $5\times 10^8$ cm. The viscous timescale is given by

$$t_{\rm vis}= \frac{R^2}{\nu}=(GM_1R)^{1/2}\frac{\mu m_{\rm H}}{\alpha k T_{\rm c}}\cdot$$ (16)

For parameters appropriate to this case ( $T_{\rm c}\approx 10^5$ K, $\alpha=0.2$, $\mu=0.5$, M₁=7 $M_\odot$) the difference gives about 6 days which is in good agreement with the numerical calculation. Figure 11 shows the minimum radius is reached at $t\approx 5.5$ days, about 3 days after the onset of the outburst.

The subsequent evolution is identical to the model where the inner radius is kept fixed. The peak accretion rate and optical flux are higher in the model with evaporation because of the different quiescent histories of the discs. When the outburst starts, the discs do not have the same density distribution and total mass (Figs. 3, 10).

The decay from outburst is also identical to the previous model with a viscous and a linear decay (Fig. 12). The linear decay has a limited influence on the lightcurve since the cooling front only propagates to $8\times 10^{10}$ cm before evaporation sets in ( $t\approx 160$ days in Fig. 12). Irradiation then shuts off with the decreasing $\epsilon$ and the front cools the whole disc quickly. At the same time, the inner disc radius rises with the increasing evaporation which also quickens the cooling. In general, evaporation and/or changes in the irradiation efficiency $\epsilon$take place for $\dot{M}_{\rm in}\sim\dot{M}_{\rm tr}$ at which point the total disc mass is close to its minimum value (the disc starts filling in again) and there is little matter left to accrete. Therefore, the exact prescriptions for $\dot{M}_{\rm evap}$ and $\epsilon$ do not have a significant impact on the outburst lightcurve. For instance, these produce a difference of only a few days in the outburst lengths between the model of Sect. 4 and the model of this section.

Figure 10: Example of an outburst cycle when irradiation and evaporation are included. The parameters of the model are the same as in Fig. 3. From top to bottom: $\dot{M}_{\rm in}$ (full line) and $\dot{M}_{\rm irr}$ (dotted line); V magnitude; $R_{\rm out}$ (full line), $R_{\rm trans}$ (dotted line) and $R_{\rm in}$(dashed line), $M_{\rm disc}$. Details of the outburst and the evolution of the density and temperature profiles can be seen in Figs. 11-13.

  
5.2 Quiescence with evaporation

The whole disc becomes cold and enters quiescence when the cooling front reaches the varying inner radius $R_{\rm in}$. In Fig. 12 this happens at $t\approx 195$ days when $R_{\rm in}\approx 5\times 10^{10}$ cm and $\dot{M}_{\rm in}\approx 5\times 10^{15}$ g$\;$s^-1. The drift time of cold matter is long and the disc cannot maintain this high accretion rate: $\dot{M}_{\rm in}$ decreases as material is gradually accreted. This adjustment happens on a long timescale (cold material) which depends on the $\Sigma$ profile left after the outburst. With a fixed $R_{\rm in}$, $\dot{M}_{\rm in}$ is already very low (about 10¹² g$\;$s^-1, Fig. 7) when the whole disc becomes cold. This is actually a necessary condition for the disc at low radii to reach the cold branch. For such low accretion rates, the supply of material is enough for $\dot{M}_{\rm in}$ to increase steadily throughout quiescence.

This first stage lasts until material from the outer edge had enough time to drift to the inner edge and increase $\dot{M}_{\rm in}$. In Fig. 13 this happens at $t\approx 3000$ days. The first three $\Sigma$ radial profiles of Fig. 13 clearly show mass from the outer edge diffusing inwards and reaching the inner edge at $t\approx 2500$ days. The disc then steadily builds up mass until the combination of a decreasing $R_{\rm in}$ (hence $\Sigma _{\rm max}$) and increasing $\Sigma$ triggers an outburst (see also Cannizzo 1998b, Meyer & Meyer-Hofmeister 1999).

Under most circumstances (Sect. 4.6) the outburst will be of the inside-out type. The reason is that, because of the very large disc sizes of SXTs, the time it takes for matter to diffuse down the disc is shorter than the accumulation time at the outer disc. A truncated disc does not change this conclusion: evaporation does not prevent inside-out type B outbursts.

However, evaporation does suppress the small mini-outbursts which are found when the inner disc radius is small (see e.g. Hameury et al. 1998). With $R_{\rm in}$ fixed at 10⁹ cm, $\Sigma _{\rm max}$is small, of the order of 10 g$\;$cm^-2, and an outburst can be triggered easily as soon as matter diffuses in to the inner radii. This leads to sequences of mini-outbursts which are not observed. A truncated disc will need to build up more mass before an outburst can start. The critical density $\Sigma _{\rm max}$ varies with R and will be much higher when $R_{\rm in}$ is large, of the order of a 100 g$\;$cm^-2 at 10¹⁰ cm. This prevents small outbursts from being triggered.

A truncated disc at the onset of an outburst will be more massive than a non-truncated disc (compare Figs. 3 and 10). But since the disc in outburst cools under the same conditions (dictated by $T_{\rm irr}$), the mass of the disc at the end of the outburst will be roughly the same in both cases. If $\Delta M$ is the mass accreted during the outburst then we have $\Delta M_{\rm trunc} > \Delta M_{\rm no~trunc}$.

The quiescence time is the time it takes to replenish the mass lost during the outburst:

$$t_{\rm quiesc}=\frac{\Delta M}{\dot{M}_{\rm tr}-\dot{M}_{\rm in}}\cdot$$ (17)

Irradiation depletes the disc during the decay, increases $\Delta M$ and leads to longer $t_{\rm quiesc}$. In contrast to models in which the disc extends to low radii, truncated discs can have larger $\dot{M}_{\rm in}$ in quiescence implying longer $t_{\rm quiesc}$. Equation (17) also shows that the quiescence time depends on the mass transfer rate even for inside-out outbursts when $\dot{M}_{\rm in}$ in quiescence is a significant fraction of $\dot{M}_{\rm tr}$. In a disc with low quiescent $\dot{M}_{\rm in}$, $t_{\rm quiesc}$ for inside-out outbursts is of the order of $t_{\rm diff}$ which is independent of $\dot{M}_{\rm tr}$(Osaki 1996; Smak 1993).

Figure 11: The outburst rise for the model of Sect. 5 which includes irradiation and evaporation (see Sect. 5.1). Upper left panel shows $\dot{M}_{\rm in}$ and $\dot{M}_{\rm irr}$ (dotted line); bottom left panel shows the V magnitude. Right panels show radial profiles at the different times indicated by the black dots on the curves. Evaporation decreases during the first part of the rise with the thin disc extending to smaller radii. At $t\approx 5.5$ days the thin disc reaches the minimum possible inner radius of the model. The disc then behaves in exactly the same way as in Fig. 5.

Figure 12: The outburst decay for the model discussed in Sect. 5.1. The disc behaves in the same way as in Fig. 6 until evaporation sets in at $t\approx 170$ days ( $\dot{M}_{\rm in}=\dot{M}_{\rm evap}(R_{\rm min})$). This cuts off irradiation and the disc cools quickly. In contrast to Fig. 6, the irradiation cutoff happens before the cooling front could propagate through most of the disc, hence the irradiation-controlled linear decay ( $t\approx 80{-}170$ days) is less obvious in the time profile. $T_{\rm irr}$ (dotted line) is shown for the last temperature profile.

Figure 13: Quiescence for the model discussed in Sect. 5.2. The irradiation flux is negligible in quiescence. Overall, the evolution is the same as in the standard DIM. Mass transfer from the secondary is slow enough that matterdiffuses down the disc, gradually increasing $\dot{M}_{\rm in}$ (and lowering $R_{\rm in}$). The outburst is triggered at $R\approx 10^{10}$ 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 35 years).