6 Parameter study

  
6.1 Changing the irradiation strength $\cal C$

We show in Fig. 14 the effects of a stronger or weaker irradiation on the outbursts. The strength of irradiation is set by the value of $\cal C$ and we show models for ${\cal C}=10^{-3}{\rm ,\ }5\times10^{-3}{\rm ,\ }10^{-2}{\rm ,\ }10^{-1}$. As irradiation grows stronger, $\Sigma _{\rm min}$ becomes lower (see Eq. (5)) and the first effect is to increase the length of the outburst: more mass gets accreted before $\Sigma _{\rm min}$ is reached. Thus, the relative amount of matter accreted during outburst, $\Delta M/M$, increases with $\cal C$. The disc spends more time accreting in quiescence before it can build up enough density somewhere to reach $\Sigma _{\rm max}$.

Like $\Sigma _{\rm min}$, $\Sigma _{\rm max}$ also decreases with increasing irradiation (see Eq. (3)). Therefore, one would also expect $\Sigma _{\rm max}$ to be reached more easily, hence to have shorter quiescent times. Usually the irradiation flux is low at the end of quiescence so only large values of $\cal C$ can heat the disc enough to change $\Sigma _{\rm max}$ (see Fig. 15).

There are two competing effects: (i) the increased amount of matter accreted during an outburst lengthens the quiescence time; (ii) unusually strong irradiation in quiescence lowers the critical density needed to reach an outburst, which reduces the quiescence time. This happens when $\cal C$ is large, in which case irradiation is no longer negligible in quiescence (in contrast to the models of Sects. 4-5). The numerical models show that there is a ${\cal C}_{\rm max}$for which the recurrence time is greatest, ${\cal C}_{\rm max}\approx 5\times10^{-3}$, i.e. the "standard'' value for which persistent low-mass X-ray binaries are stabilized (Dubus et al. 1999). This is probably no more than a coincidence as ${\cal C}_{\rm max}$ certainly changes as a function of M₁, $R_{\rm out}$, etc.

The average mass in the disc <$M_{\rm d}$> decreases with stronger irradiation. ${<}M_{\rm d}$> depends mostly on the mass of the disc in quiescence. Since $\Sigma<\Sigma_{\rm max}$ in quiescence and $\Sigma _{\rm max}$ decreases with irradiation, <$M_{\rm d}$>decreases as well. Another consequence of stronger irradiation are the higher optical fluxes in outburst due to the higher outer disc temperatures.

   
6.2 Changing the viscosity $\alpha$

The effect of changing $\alpha _{\rm h}$ and $\alpha _{\rm c}$ is straightforward. The $\alpha$ parameter sets the viscous time of the accretion flow (Eq. (16)). A disc with a low viscosity in quiescence diffuses mass slowly to the inner edge. Therefore, the time to reach $\Sigma _{\rm max}$ is longer when $\alpha _{\rm c}$ is lower (see top panel of Fig. 16). This also modifies the peak accretion rate of the outburst. Similarly, a disc with a higher viscosity in outburst accretes mass more quickly and the decay time is shorter (Eqs. (10)-(11)). The conditions for the disc to enter quiescence and the subsequent evolution are independent of $\alpha _{\rm h}$. Therefore, the models shown in the top panel of Fig. 16 which have different $\alpha _{\rm h}$ but the same $\alpha _{\rm c}$ have the same recurrence time.

Figure 14: Effect of changing the irradiation parameter $\cal C$ on the outburst time evolution and cycle (see Sect. 6.1 for details). ${\cal C} / 10^{-3}=0$ (full line), 1 (dash-dotted), 5 (dashed), 100 (dotted). All other parameters are identical to those of the model described in Sect. 4. The non-irradiated curve shows no reflares in this particular case. This is in general not the case, even with evaporation: the same model with M1=6 $M_\odot$ shows them.

Figure 15: Density profiles $\Sigma (R)$ in quiescent evaporated discs before the onset of an outburst. The dotted line is $\Sigma _{\rm max}$. For ${\cal C}=5\times 10^{-3}$ irradiation is negligible in quiescence and the critical $\Sigma _{\rm max}$ is the same as for a non-irradiated disc. For larger values of $\cal C$ irradiation becomes important, lowering $\Sigma _{\rm max}$. The critical density the disc must reach before an outburst is significantly less and the recurrence time between outbursts is decreased. See Sect. 6.1 for details.

Figure 16: Effect of changing the viscosity parameters on the outburst time profiles. Bottom: changing $\alpha _{\rm c}$ modifies the recurrence time (full line $\alpha _{\rm c}=0.01$, dashed line $\alpha _{\rm c}=0.02$, both with $\alpha _{\rm h}=0.1$). Top: changing $\alpha _{\rm h}$ modifies the outburst time evolution (full line $\alpha _{\rm h}=0.2$, dashed line $\alpha _{\rm h}=0.4$, both with $\alpha _{\rm c}=0.02$. The recurrence time is the same $t_{\rm rec}\approx 20$ years). See Sect. 6.2 for details.

   
6.3 Changing the mass transfer rate $\mathsfsl{\dot{M}_{\mathsf{tr}}}$

Next, we explore the effect of changing the mass transfer rate $\dot{M}_{\rm tr}$, keeping all other parameters identical. For inside-out outbursts, the time spent in quiescence with fixed $R_{\rm in}$ is the diffusion time of matter from the outer to the inner edge, which does not depend on $\dot{M}_{\rm tr}$. However, models including evaporation show a dependence on $\dot{M}_{\rm tr}$ when $\dot{M}_{\rm in}$ in quiescence is not negligible compared to $\dot{M}_{\rm tr}$(Eq. (17)). Figure 17 shows the quiescence time shortens with increasing $\dot{M}_{\rm tr}$.

Higher $\dot{M}_{\rm tr}$ result in longer outbursts since (i) the companion provides more material during the outburst and (ii) the total disc mass at the onset of the outburst increases with $\dot{M}_{\rm tr}$. The conditions in the disc are the same at the end of the outburst, regardless of the mass transfer rate (except, of course, at the outer edge) because cooling happens in the same way for discs of the same size. A ring at a given radius can only cool when $T_{\rm irr}$ reaches 10⁴ K and the surface density behind the front is roughly the same in all cases. Thus, even with different $\dot{M}_{\rm tr}$, the discs have almost the same total mass as they enter quiescence (Fig. 17).

In a truncated disc the drift of material in quiescence will change with $\dot{M}_{\rm tr}$ because the distance it has to travel is rather short. Therefore, the conditions at the beginning of the outburst depend on the mass-transfer rate. In particular, the ignition radius increases with $\dot{M}_{\rm tr}$ as could be expected: a larger amount of mass from the outer edge diffuses down the disc and can trigger the outburst at a larger radius (but as mentioned earlier outbursts starting at the outer disc edge do not occur). This also increases the total amount of mass in the disc at the onset of the outburst and hence the peak accretion rate (see Sect. 4.1).

For the high mass transfer rates the disc stays fully ionized for a significant fraction of the outburst, leading to a viscous decay. Our model with the highest $\dot{M}_{\rm tr}$ show this is accompanied by significant variations of the outer radius which produce deviations from the expected exponential (lower panel of Fig. 17).

Figure 17: Effect of changing $\dot{M}_{\rm tr}$ on the outburst cycles. Top two panels: $\dot{M}_{\rm in}$ and $M_{\rm disc}$ for $\dot{M}_{\rm tr}/10^{16}{\rm \ g}\;{\rm s}^{-1}=$, 0.5 (full line), 5 (dashed line) and 50 (dotted line). The bottom panels zoom on the outburst for the same models, showing $\dot{M}_{\rm in}$ (full line) and $\dot{M}_{\rm irr}$ (dotted line), $R_{\rm out}$, $R_{\rm in}$ and the transition radius between the hot and cold regions (dashed line). The other parameters of the models are M1=7 $M_\odot$, $R_{\rm out}\approx 2.5\times 10^{11}$ cm, $\alpha _{\rm h}=0.2$, $\alpha _{\rm c}=0.02$ and ${\cal C}=5\times 10^{-3}$. See Sect. 6.3 for details.

  
6.4 Changing the disc size $\mathsfsl{R_{\mathsf{out}}}$

The disc size in the model depends on the strength of the tidal truncation term in the angular momentum conservation equation and on the circularization radius. The ratios $R_{\rm circ}/a$ and $R_{\rm out}/a$ depend only on the mass ratio q=M₂/M₁. Assuming $q\approx 0.1$, which is reasonable for black hole SXTs, we derive $R_{\rm circ}/R_{\rm out} \approx 0.5$ (Papaloizou & Pringle 1977). In the following, we changed the mean outer disc radius, keeping the above ratio constant. This is equivalent to simulating binaries with the same components but increasing orbital period $P_{\rm orb}\approx 4.6\ (R_{\rm out}/10^{11}{\rm ~cm})^{3/2} M_1^{-1/2}$ hours (q=0.1).

Figure 18: Effect of changing $R_{\rm out}$ on the outburst cycles. Top two panels: $\dot{M}_{\rm in}$ and $M_{\rm disc}$ for < $R_{\rm out}$> $/10^{10}{\rm ~cm}=$ 9.3 (dotted line), 25.7 (dashed line), 107 (full line). The bottom panels zoom on the outburst for the same models, showing $\dot{M}_{\rm in}$ (full line) and $\dot{M}_{\rm irr}$(dotted line), $R_{\rm out}$, $R_{\rm in}$ and the transition radius between the hot and cold regions (dashed line). The other parameters of the models are M1=7 $M_\odot$, $\dot{M}=5\times 10^{16}$ g$\;$s-1, $\alpha _{\rm h}=0.2$, $\alpha _{\rm c}=0.02$and ${\cal C}=5\times 10^{-3}$. See Sect. 6.4 for details.

Increasing the disc size makes it more difficult for the heat front to reach the outer disc radius and fully ionize the disc. In small discs the decay is entirely viscous while larger discs show only irradiation-controlled linear decays (Fig. 18). In the case of a viscous exponential decay the decay rate increases with smaller disc radius in accordance with Eqs. (10)-(11).

The ignition radius of the outburst increases for smaller disc sizes since the diffusion timescale is shorter. Smaller discs are also less massive and, therefore, take less time to replenish than large discs for the same mass transfer rate: the time spent in quiescence is shorter. This also leads to smaller outburst peaks (Sect. 4.1).

   
6.5 Changing M₁

Finally, we vary the mass of the accreting object M₁. Irradiation is implicitely assumed to be the same since the prescription for $T_{\rm irr}$ (Eq. (8)) does not depend on M₁. $\Sigma _{\rm max}$ and $\Sigma _{\rm min}$ have some dependence on M₁(Eqs. (3) and (5)) but their influence on the outburst is negligible compared to the changes induced by evaporation. The strength of evaporation depends strongly on M₁ through $\dot{M}_{\rm Edd}$ and $R_{\rm S}$ in Eq. (14): $\dot{M}_{\rm ev}\propto M_1^3$ (this cannot be seen in Menou et al. 2000 where by error the evaporation formula for neutron-star SXT outbursts uses $M_1= 6~M_{\odot}$). In outburst, this modifies the accretion rate at which evaporation sets in which in turn reduces the length of the outburst for lower M₁ (see also Meyer & Meyer-Hofmeister 2000 who studied extensively the M₁ dependence of the instability cycle with evaporation, but without irradiation, and reached conclusions similar to those of Menou et al. 1999).

In quiescence, the disc around a lower mass compact object is truncated at smaller radii which reduces the inner accretion rate $\dot{M}_{\rm in}$. Mass is accumulated more easily (Eq. (17)), leading to shorter recurrence time (Fig. 19). The lower accretion rates in the evaporated disc in quiescence do not necessarily translate into lower quiescent luminosities. In our assumptions, the increase in accretion efficiency implied by a lower $R_{\rm in}$ ( $\epsilon \propto R_{\rm in}^{-2}$) compensates for the lower $\dot{M}_{\rm in}$. Plotting $\dot{M}_{\rm irr}=\epsilon\dot{M}_{\rm in}$ instead of only $\dot{M}_{\rm in}$ as in Fig. 19, we find $\dot{M}_{\rm irr}\approx 10^{10-11}$ g$\;$s^-1 for both M₁=1.4 $M_\odot$ and 7 $M_\odot$. The actual efficiency of a $1.4~M_\odot$ neutron star may be much higher in quiescence than what we assume here. It might actually be more correct to assume $\epsilon$ is always 0.1 for the neutron star since the accreted matter will radiate at the surface. In this case, the luminosity in quiescence deduced from $\dot{M}_{\rm in}$ will be higher than for a black hole primary. However, a propeller effect might significantly reduce the actual mass accretion rate onto a neutron star in quiescence (Menou et al. 1999c) making the distinction between black holes and neutron stars less obvious.

Figure 19: Effect of changing M1 on the outburst cycles of the inner mass accretion rate $\dot{M}_{\rm in}$. The model of Sect. 5 is shown by the full line (M1=7 $M_\odot$). The same model but with M1=1.4 $M_\odot$ is shown by the dotted line. See Sect. 6.5 for details.