2 Dynamical structure

2.1 General properties

A precise disk wind theory must explain how much matter is deviated from radial to vertical motion, as well as the amount of energy and angular momentum carried away. This implies a thorough treatment of both the disk interior and its matching with the jets, namely to consider magnetized accretion-ejection structures (hereafter MAES). The only way to solve such an entangled problem is to take into account all dynamical terms, a task that can be properly done within a self-similar framework.

In this paper, we use the models of Ferreira (1997) describing steady-state, axisymmetric MAES under the following assumptions: (i) a large scale magnetic field of bipolar topology is threading a geometrically thin disk; (ii) its ionization is such that MHD applies (neutrals are well-coupled to the magnetic field); (iii) some active turbulence inside the disk produces anomalous diffusion allowing matter to cross the field lines. Two extra simplifying assumptions were used: (iv) jets are assumed to be cold, i.e. powered by the magnetic Lorentz force only (the centrifugal force is due to the Lorentz azimuthal torque), with isothermal magnetic surfaces (the midplane temperature varying as $T_0 \propto r^{-1}$) and (v) jets carry away all disk angular momentum. This last assumption has been removed only recently by Casse & Ferreira (2000a).

All solutions obtained so far display the same asymptotic behavior. After an opening of the jet radius leading to a very efficient acceleration of the plasma, the jet undergoes a refocusing towards the axis (recollimation). All self-similar solutions are then terminated, most probably producing a shock (Gomez de Castro & Pudritz 1993; Ouyed & Pudritz 1993). This systematic behavior could well be imposed by the self-similar geometry itself and not be a general result (Ferreira 1997). Nevertheless, such a shock would occur in the asymptotic region, far away from the disk. Thus, we can confidently use these solutions in the acceleration zone, where forbidden emission lines are believed to be produced (Hartigan et al. 1995).

  
2.2 Model parameters

The isothermal self-similar MAES considered here are described with three free dimensionless local parameters (see Ferreira 1997, for more details) and four global quantities:
(1) the disk aspect ratio

$$\varepsilon= \frac{h(\varpi)}{\varpi}$$ (1)

where $h(\varpi)$ is the vertical scale height at the cylindrical radius $\varpi$;
(2) the MHD turbulence parameter

$$\alpha_{\rm m}= \frac{\nu_{\rm m}}{V_{\rm A}h}$$ (2)

where $\nu_{\rm m}$ is the required turbulent magnetic diffusivity and $V_{\rm A}$ the Alfvén speed at the disk midplane; this diffusivity allows matter to cross field lines and therefore to accrete towards the central star. It also controls the amplitude of the toroidal field at the disk surface.
(3) the ejection index

$$\xi= \frac{{\rm d} \ln \dot M_{{\rm acc}} (\varpi)}{{\rm d} \ln \varpi}$$ (3)

which measures locally the ejection efficiency ($\xi=0$ in a standard accretion disk), but also affects the jet opening (a higher $\xi$translates in a lower opening);
(4) M_* the mass of the central protostar;
(5) $\varpi_{\rm i}$ the inner edge of the MAES;
(6) $\varpi_{\rm e}$ the outer edge of the MAES, a standard accretion disk lying at greater radii. This outer radius is formally imposed by the amount of open, large scale magnetic flux threading the disk and producing jets;
(7) $\dot{M}_{\rm acc}$, the disk accretion rate fueling the MAES and measured at $\varpi_{\rm e}$.

For our present study, we keep only $\xi$ and $\dot{M}_{\rm acc}$ as free parameters and fix the values of the other five as follows: The disk aspect ratio was measured by Burrows et al. (1996) for HH 30 as $\sim$0.1 so we fix $\varepsilon =0.1$. The MHD turbulence parameter is taken $\alpha _{{\rm m}}=1$ in order to have powerful jets (Ferreira 1997). The stellar mass is fixed at $M_* =0.5 \ M_{\odot}$, typical for T Tauri stars, and the inner radius of the MAES is set to $\varpi_{\rm i} =$ 0.07 AU (typical disk corotation radius for a 10 days rotation period): inside this region the magnetic field topology could be significantly affected by the stellar magnetosphere-disk interaction. The outer radius is kept at $\varpi_{{\rm e}}= 1 \rm {AU}$ for consistency with the one fluid approximation (Appendix C) and the atomic gas description. Regarding atomic consistency, Safier (1993a) solved the flow evolution assuming inicially all H bound in H₂. He found H₂ to completely dissociate at the wind base, for small $\varpi _0$. However, after a critical flow line footpoint H₂ would not completely dissociate, therefore affecting the thermal solution. This critical footpoint was at 3 AU for his MHD solution nearer our parameter space.

We note that our two free parameters are still bounded by observational constraints: mass conservation relates the ejection index $\xi$ to the accretion/ejection rates ratios,

$$2 \dot{M}_{{\rm J}}\simeq \xi \dot{M}_{{\rm acc}} \ln\frac{\varpi_{\rm e}}{\varpi_{\rm i}}\cdot$$ (4)

The observational estimates for the ratio of mass outflow rate by mass accretion rate are $\dot{M}_{{\rm J}}/\dot{M}_{{\rm acc}}\simeq 0.01$ (Hartigan et al. 1995). The uncertainties affecting these estimates can be up to a factor of 10 (Gullbring et al. 1998; Lavalley-Fouquet et al. 2000). The range of ejection indexes considered here (0.005-0.01) is kept compatible with Hartigan's canonical value. The accretion rates $\dot{M}_{\rm acc}$ are also kept free but inside the observed range of $10^{-8}~M_\odot ~{\rm yr}^{-1}$ to $10^{-5}~M_\odot~{\rm yr}^{-1}$ in T Tauri stars (Hartigan et al. 1995).

Table 1 provides a list of some disk and jet parameters. These local parameters were constrained by steady-state requirements, namely the smooth crossing of MHD critical points. Disk parameters are useful to give us a view of the physical conditions inside the disk. Thus, the required magnetic field B₀ at the disk midplane and at a radial distance $\varpi _0$ is

$$B_0 = 0.3\ \zeta \left ( \frac{M_*}{M_{\odot}} \right)^\frac... ...1\,\rm {AU}} \right)^{\frac{\xi}{2} - \frac{5}{4}} \ \mbox{G}.$$ (5)

The global energy conservation of a cold MAES writes

$$P_{\rm acc} = 2P_{\rm jet}\ +\ 2P_{\rm rad}$$ (6)

where $P_{\rm acc}$ is the mechanical power liberated by the accretion flow, $P_{\rm jet}$ the total (kinetic, thermal and magnetic) power carried away by one jet and $P_{\rm rad}$ the luminosity radiated at one surface of the disk. For the solutions used, the accretion power is given by

$$P_{\rm acc} = \eta\ \frac{GM_* \dot M_{\rm acc}}{2\varpi_{\rm i}} 0.1\eta \left ( \frac{M_*}{M_{\odot}} \right) \left ( \frac{\dot M_{\rm acc}} {10^{-7}~M_{\odot}~\rm {yr}^{-1}} \right )$$ =

$$\times \left ( \frac{\varpi_{\rm i}}{0.07\,{\rm AU}} \right)^{-1}\ L_{\odot}$$ (7)

where $L_{\odot}$ is the solar luminosity and the efficiency factor $\eta= (\varpi_{\rm i}/\varpi_{\rm e})^\xi - (\varpi_{\rm i}/\varpi_{\rm e})$ depends on both the local ejection efficiency $\xi$ and the MAES radial extent. Typical values for our solutions are $\eta\simeq 0.9$. The ratio $P_{\rm jet}/P_{\rm rad}$ is given in Table 1. The jet parameters, mass load $\kappa$ and magnetic lever arm $\lambda$, have the same definition as in Blandford & Payne (1982). They are given here to allow a comparison with the solutions used in Safier's work.

 

 

Table 1: Isothermal MAES parameters. With $\varepsilon =0.1$ and $\alpha _{{\rm m}}=1$, the only parameter remaining free is $\xi$. Here, the magnetic lever arm $\lambda$, mass load $\kappa$ and initial jet opening angle $\theta _0$ are presented to ease comparison with Safier's models. However these parameters do not uniquely determine the MHD solution.

Solution	$\xi$	$\zeta$	$P_{\rm jet}/P_{\rm rad}$	$\kappa$	$\lambda$	$\theta_0 (\hbox{$^\circ$ })$
A	0.010	0.729	1.46	0.014	41.6	50.6
B	0.007	0.690	1.46	0.011	59.4	52.4
C	0.005	0.627	1.52	0.009	84.2	55.4

2.3 Physical quantities along streamlines

Figure 1: Several wind quantities along a streamline for model A (long-dashed line), B (solid) and C (dashed): jet nuclei density $\tilde n$, velocity, magnetic field, and Lorentz force. For the latter three vectors, poloidal components ( $v_{\rm p}$, $B_{\rm p}$, $(\vec J \times \vec B)_{\rm p}$) are plotted in black and toroidal components ($v_\phi$ $B_\phi$, $(\vec J \times \vec B)_\phi$) in red. The field line is anchored at $\varpi _0=0.1$ AU, around a $1~M_{\odot}$ protostar, with an accretion rate $\dot M_{\rm acc}= 10^{-6}~M_{\odot}~\rm{yr}^{-1}$.

In order to obtain a solution for the MAES, a variable separation method has been used allowing to transform the set of coupled partially differential equations into a set of coupled ordinary differential equations (ODEs). Hence, the solution in the $(\varpi,z)$space is obtained by solving for a flow line and then scaling this solution to all space. Once a solution is found (for a given set of parameters in Sect. 2.2), the evolution of all wind quantities Q along any flow line is given by:

$$Q(\varpi,z) Q_0(\varpi_0) Q_\chi(\chi)$$ (8)

where the self-similar variable $\chi =z/\varpi _0$ measures the position along a streamline flowing along a magnetic surface anchored in the disk at $\varpi _0$. In particular, the flow line shape equation is given by $\varpi(z)= \varpi_0 \Psi(\chi)$, where the function $\Psi(\chi)$ is provided by solving the full dynamical problem. In Fig. 1 we plot the values of the jet nuclei density $\tilde n$ and the poloidal and toroidal components of jet velocity, magnetic field and Lorentz force. This is done for our 3 models, along a streamline with $\varpi _0$ = 0.1 AU, $M_{*}= 1 \ M_{\odot}$ and $\dot M_{\rm acc}= 10^{-6}~M_{\odot}~\rm{yr}^{-1}$. Values for other $\varpi _0$ and $\dot{M}_{\rm acc}$ can easily be deduced from these plots using the following scalings

 

$$\tilde{n} \propto \dot M_{\rm acc} M_\ast^{-\frac{1}{2}} \varpi_0^{\xi - \frac{3}{2}}$$

$$\propto M_\ast^\frac{1}{2} \varpi_0^{-\frac{1}{2}}$$ v

$$\propto \dot M_{\rm acc}^{\frac{1}{2}} M_\ast^\frac{1}{4} \varpi_0^{\frac{\xi}{2} - \frac{5}{4}}$$ B (9)

$$\propto \dot M_{\rm acc}^{\frac{1}{2}} M_\ast^\frac{1}{4} \varpi_0^{\frac{\xi}{2} - \frac{9}{4}}.$$ J

The terminal poloidal velocity is $v_{\rm {p},\infty} \simeq \sqrt{G M_\ast/ \xi \varpi_0}$ so that solutions with smaller opening angles also reach smaller terminal velocities, with higher terminal densities. The point where $v_\phi$ reaches a minimum is also the point where the jet reaches its maximum width (we call it recollimation point), before the jet starts to bend towards the axis. The numerical solution becomes unreliable as we move away from this point. The MHD solution is stopped at the super-Alfvénic point, which is reached nearer for higher $\xi$. An illustration of the resulting ($\varpi, z$) distribution of density $\tilde n$ and total velocity modulus for model A can be found in Fig. 1 of Cabrit et al. (1999).