4 Toward more realistic models

The models presented so far have some clear deficiencies when compared with characteristic features relevant to protostellar discs. The first deficiency concerns the relative amount of matter accreted onto the central object compared with what goes into the wind. A typical figure from the models presented above was that as much as 30% of all the matter released in the disc goes into the wind and only about 70% is accreted by the central object. Earlier estimates (e.g., Pelletier & Pudritz 1992) indicate that only about 10% of the matter joins the wind. Another possible deficiency is the fact that in the models presented so far the low-temperature region extends all the way to the stellar surface whilst in reality the cool disc breaks up close to the star because of the stellar magnetic field. Finally, the overall temperature of the disc is generally too high compared with real protostellar discs which are known to have typical temperatures of about a few thousand Kelvin.

The aim of this section is to assess the significance of various improvements in the model related to the above mentioned characteristics. We consider the effect of each of them separately by improving the model step by step.

   
4.1 A larger sink

As discussed in Sects. 3.1 and 3.2, properties of the outflow, especially of the nonmagnetic ones, are sensitive to the parameters of the central sink. It is clear that a very efficient sink would completely inhibit the outflow near the axis. On the other hand, a magnetosphere of the central object can affect the sink efficiency by channelling the flow along the stellar magnetic field (Shu et al. 1994; see also Fendt & Elstner 1999, 2000). Simulating a magnetosphere turned out to be a difficult task and some preliminary attempts proved unsatisfactory.

Figure 21: As in Fig. 7, but with a mass sink at the centre which is five times larger, r0 = 0.25, and $\tau _{\rm star}=0.112$. Note the absence of outflows along the axis and a larger opening angle of the conical shell, which crosses $z=\pm 1$ at $\varpi \approx 1.2$. Averaged over times $t=500 \dots 530$, $\beta =0.1$.

Instead, we have considered a model with a geometrically larger sink, and illustrate it in Fig. 21. Here, r₀=0.25 in Eqs. (5) and (23), which is five times larger than the sink used in our reference model of Fig. 7. The relaxation time of the sink in Eq. (6), $\tau_{\rm star}$, was rescaled in proportion to the free-fall time at r₀ as $\tau_{\rm star}\simeq r_0/v_{\rm ff}\propto r_0^{3/2}$, which yields $\tau _{\rm star}=0.112$. The resulting mass loss rate into the wind is $\dot M_{\rm wind}\approx0.8$, which corresponds to $1.6\times10^{-7}~M_\odot~{\rm yr}^{-1}$ in dimensional units. Although this is indeed smaller than the value for the reference model ( $6\times10^{-7}~M_\odot~{\rm yr}^{-1}$), the overall mass released from the disc is also smaller, resulting in a larger fraction of about 40% of matter that goes into the wind; only about 60% is accreted by the central object. As we show below, however, larger accretion fractions can more easily be achieved by making the disc cooler. We conclude that even a sink as large as almost twice the disc half-thickness does not destroy the outflow outside the inner cone. However, the outflow along the axis is nearly completely suppressed.

   
4.2 Introducing a gap between the disc and the sink

In real accretion discs around protostars the disc terminates at some distance away from the star. It would therefore be unrealistic to let the disc extend all the way to the centre. Dynamo action in all our models is restricted to $\varpi\geq0.2$, and now we introduce an inner disc boundary for the region of lowered entropy as well. In Fig. 22 we present such a model where the  $\xi_{\rm disc}(\vec{r})$ profile is terminated at $\varpi =0.25$. This inner disc radius affects then not only the region of lowered entropy, but also the distributions of $\alpha$, $\eta_{\rm t}$, $\nu_{\rm t}$ and $q_\varrho ^{\rm disc}$. The radius of the mass sink was chosen to be r₀=0.15, i.e. equal to the disc semi-thickness. The correspondingly adjusted value of $\tau_{\rm star}$ is 0.052. The entropy in the sink is kept as low as that in the disc.

As can be seen from Fig. 22, the gap in the disc translates directly into a corresponding gap in the resulting outflow pattern. At the same time, however, only about 20% of the released disc material is accreted by the sink and the rest is ejected into the wind. A small fraction of the mass that accretes toward the sink reaches the region near the axis at some distance away from the origin and can then still be ejected as in the models without the gap. Note that our figures (e.g., Fig. 22) show the velocity rather than the azimuthally integrated mass flux density (cf. Fig. 13); the relative magnitude of the latter is much smaller, and so the mass flux from the sink region is not significant.

4.3 A cooler disc

We now turn to the discussion of the case with a cooler disc. Numerical constraints prevent us from making the entropy gradient between disc and corona too steep. Nevertheless, we were able to reduce the value of $\beta$down to 0.005, which is 20 times smaller than the value used for the reference model in Fig. 7. The $\beta$ value for the sink was reduced to 0.02; smaller values proved numerically difficult.

Figure 22: As in Fig. 7, but with a gap between the sink and the disc with the disc inner radius at $\varpi =0.25$. The radius of the stellar mass sink is three times larger, r0 = 0.15, and $\tau _{\rm star}=0.052$. Outflow is absent along field lines passing through the gap. Averaged over times $t=400 \dots 660$, $\beta =0.1$ in both the disc and the sink.

Figure 23: As in Fig. 22, but with $\beta = 0.005$ in the disc and $\beta =0.02$ in the sink. Averaged over times $t=150 \dots 280$when the overall magnetic activity in the disc is relatively low.

Figure 24: As in Fig. 23, but averaged over a period of enhanced magnetic activity in the disc, $t=300 \dots 320$. A conical shell develops which crosses $z=\pm 1$ at $\varpi \approx 0.6$.

The value of $\beta = 0.005$ results in a disc temperature of $3\times10^3$ K in the outer parts and $1.5\times10^4~$K in the inner parts. As in Sect. 4.2, the disc terminates at an inner radius of $\varpi =0.25$. At $\varpi \approx 0.5$, the density in dimensional units is about 10^-9 g cm^-3, which is also the order of magnitude found for protostellar discs. The resulting magnetic field and outflow geometries are shown in Figs. 23 and 24.

A characteristic feature of models with a cooler disc is a more vigorous temporal behaviour with prolonged episodes of reduced overall magnetic activity in the disc during which the Alfvénic surface is closer to the disc surface, and phases of enhanced magnetic activity where the Alfvénic surface has moved further away. Figures 23 and 24 are representative of these two states. It is notable that the structured outflow of the type seen in the reference model occurs in states with strong magnetic field and disappears during periods with weak magnetic field.

Another interesting property of the cooler discs is that now a smaller fraction of matter goes into the wind (10-20%), and 80-90% is accreted by the central object, in a better agreement with the estimates of Pelletier & Pudritz (1992).

We note in passing that channel flow solutions typical of two-dimensional simulations of the magneto-rotational instability (Hawley & Balbus 1991) are generally absent in the simulations presented here. This is because the magnetic field saturates at a level close to equipartition between magnetic and thermal energies. The vertical wavelength of the instability can then exceed the half-thickness of the disc. In some of our simulations, indications of channel flow behaviour still can be seen. An example is Fig. 23 where the magnetic energy is weak enough so that the magneto-rotational instability is not yet suppressed.

According to the Shakura-Sunyaev prescription, turbulent viscosity and magnetic diffusivity are reduced in a cooler disc because of the smaller sound speed (cf. Eq. (10)). Since in the corona the dominant contributions to the artificial advection viscosity $\nu_{\rm adv}$ come from the poloidal velocity and poloidal Alfvén speed (and not from the sound speed), $c_\nu^{\rm adv}$ has to be reduced. Here we choose $\alpha _{\rm SS}=0.004$ and reduce $c_\nu^{\rm adv}$ by a factor of 10 to $c_\nu ^{\rm adv}=0.002$. Since we do not explicitly parameterize the turbulent magnetic diffusivity $\eta_{\rm t}$ with the sound speed (cf. Eq. (15)), also $\eta_{{\rm t}0}$ needs to be decreased, together with the background diffusivity $\eta_0$. We choose here values that are 25 times smaller compared to the previous runs, i.e. $\eta _{{\rm t}0}=4\times 10^{-5}$ and $\eta _0=2\times 10^{-5}$, which corresponds to $\alpha_{\rm SS}^{(\eta)}$ranging between 0.001 and 0.007. With this choice of coefficients, the total viscosity and magnetic diffusivity in the disc have comparable values of a few times 10^-5.

The effect of reduced viscosity and magnetic diffusivity is shown in Fig. 25. Features characteristic of the channel flow solution are now present, because the vertical wavelength of the magneto-rotational instability is here less than the half-thickness of the disc.

Figure 25: As in Figs. 23 and 24, but with $\eta _0=2\times 10^{-5}$, $\eta _{{\rm t}0}=4\times 10^{-5}$, $c_\nu ^{\rm adv}=0.002$, and $\alpha _0=-0.15$. Averaged over times $t=230 \dots 236$ (where the magnetic energy is enhanced), $\alpha _{\rm SS}=0.004$.