Appendix C: Consistency checks

  
C.1 Dynamical assumptions

Figure C.1: Top left: we plot the relevant drift speeds normalized to the fluid poloidal velocity. The worst case of one fluid approximation violation is obtained for model C, $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=1$ AU. Top right: we plot the $\tau _{\alpha \beta }$ versus dynamical $\tau _{\rm dyn}$time-scales (s) versus $\chi$, normalized to the Keplerian period at the line footpoint (for a 1 $M_\odot$ star). We plot only the longer time-scales ( $\tau _{\rm in} = \tau _{\rm ni}/f_i$ and $\tau _{\rm en} = \tau _{\rm ne}/f_{\rm e}$). The worst case is obtained again for model C, $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=1$ AU. Bottom left: ideal MHD tests for the worst situation (model C, $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=1$ AU). As expected from our heated winds, the ambipolar diffusion term is the dominant one. Bottom right: ratios of the thermal pressure gradient to the Lorentz force versus $\chi$, along ( $\beta _\parallel$, solid) and across ( $\beta _\perp$, dash) a magnetic surface anchored at $\varpi _0$. The worst case for $\beta _\parallel$ is obtained for model A, $\varpi _0=1$ AU, $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$, the best for model A, $\varpi _0=0.1$ AU and $\dot{M}_{\rm acc}=10^{-5}~M_\odot~{\rm yr}^{-1}$. With respect to $\beta _\perp$, the worst case is for model C, $\varpi _0=1$ AU and $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$, while the best is for model A, $\varpi _0=0.1$ AU and $\dot{M}_{\rm acc}=10^{-5}~M_\odot~{\rm yr}^{-1}$. Although definitely not negligible in some models, those compatible with observations do not show important deviations from the "cold'' jet approximation.

First, local charge neutrality is always achieved. For example, we achieve a maximum Debye length of $r_{{\rm D}}\sim10^5 {\rm cm}$ at the outer radius of the recollimation zone (model C, lowest $\dot{M}_{\rm acc}$).

Second, single fluid approximation requires that relative velocity drifts of all species ($\alpha=$ ions, electrons, neutrals) $\Vert\vec{v} - \vec{v}_{\alpha}\Vert / \Vert\vec{v}\Vert$ are smaller than unity. These drifts are higher for lower accretion rates and at the outer wind base (due to the decrease in density and velocity, see Eq. (9)). In Fig. C.1 we present the worst case for the drift velocities, showing that our jets can be indeed approximated by single fluid calculations.

We assumed gas thermalization, which is achieved only if collisional time-scales between species $\tau_{\alpha\beta} = 1/\nu_{\alpha \beta}$ are much smaller than the dynamical time-scale $\tau_{{\rm dyn}} = \varpi_0 ({\rm d} v_z/{\rm d} \chi)^{-1}$. In the collision network considered here, the longer time-scales involve collisions with neutrals. However, even in the worst situation (see Fig. C.1), after the wind base they remain comfortably below the above dynamical time-scale. Our dynamical jet solutions were derived within the ideal MHD framework. This assumption requires that all terms in the right hand side of the generalized Ohm's law (Eq. (A.11)) are negligible when compared to the electromotive field $\vec{v} \times \vec{B}/c$. We consider Ohm's term $\Vert\eta \vec{J} \Vert$, Hall's effect $\Vert\vec{J}\times\vec{B}\Vert/c\,e n_{\rm e}$ and the ambipolar diffusion term $(\frac{\overline{\rho_{\rm n}}}{\rho})^2 \Vert(\vec{J} \times\vec{B}) \times\vec{B}\Vert/c^2 \overline{m_{\rm in} n_{\rm i} \nu_{\rm in}}$ (effects due to the electronic pressure gradient are small compared to the Lorentz force - Hall's term -). In Fig. C.1 we present the worst case for our ideal MHD checks. We find that deviations from ideal MHD remain negligible, despite the presence of ambipolar diffusion. As expected, this is the dominant diffusion process in our (non turbulent) MHD jets. Ambipolar diffusion is larger for low accretion rates and at the outer wind base (because the ratio of the ambipolar to the electromotive term scales as $(\dot{M}_{{\rm acc}}\,f_i)^{-1}$).

The worst case for the previous three tests is, as expected, for the model that attains the lowest density: Model C, with the lowest accretion rate ( $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$) and at the outer edge footpoint ( $\varpi_0=1~{\rm AU}$).

The dynamical jet evolution was calculated under the additional assumption of negligible thermal pressure gradient (cold jets). Since it is the gradient that provides a force, one should not just measure (along one field line) the relative importance of the gas pressure to the magnetic pressure (usual $\beta= P/(B^2/8\pi)$parameter). Instead, we compare the thermal pressure gradient to the Lorentz force, along ( $\beta _\parallel$) and perpendicular ( $\beta _\perp$) to the flow, namely

$$\beta_\parallel \frac{\nabla_\parallel P}{F_\parallel}= c \frac{\vec{v}_{\rm p} \cdot \nabla P}{ \vec{v}_{\rm p} \cdot (\vec J \times \vec B)}$$ = (C.1)

$$\beta_\perp \frac{\nabla_\perp P}{F_\perp} = c \frac{\nabla a \cdot \nabla P}{ \nabla a \cdot (\vec J \times \vec B)}\cdot$$ = (C.2)

Here $a(\varpi, z)$ is the poloidal magnetic flux function, hence $\nabla a$ is perpendicular to a magnetic surface. High values of $\beta _\parallel$ imply that the thermal pressure gradient plays a role in gas acceleration, whereas high values of $\beta _\perp$show that it affects the gas collimation.

In Fig. C.1 we plot the worst case of cold fluid violation and best case of cold fluid validity. Again the worst case appears at lower accretion rates and in outer wind zones. It can be seen that high values of $\beta _\perp$ and $\beta _\parallel$ can be attained, hinting at the importance of gas heating on jet dynamics (providing both enthalpy at the base of the jet and/or pressure support against recollimation further out). We underline that models inconsistent with the cold fluid approximation are those found to have the largest difficulty in meeting the observations (Paper II). Conversely, models that better reproduce observations also fulfill the cold fluid approximation. For those models, the thermal pressure gradient appears to be fairly negligible with respect to the Lorentz force.

   
C.2 Thermal assumptions

Figure C.2: Ignored heating/cooling terms (in erg s-1 cm-3). Left: we compare the cooling term $- \frac{3}{2}k \tilde{n} T Df_{\rm e}/Dt$ (dashed) with adiabatic cooling $\Lambda _{\rm adia}$(solid) for the worst case (model C, $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=0.1$ AU). Right: grain heating/cooling rate $\vert\Gamma _{\rm gr}\vert$ (dashed) compared with ambipolar diffusion heating $\Gamma _{\rm drag}$ (solid) for the worst case (model A, $\dot{M}_{\rm acc}=10^{-5}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=1$ AU). $\Gamma _{\rm gr}$ is only significant at the very base of the wind, and will not affect the thermal state further out in the jet.

Finally, we check that all ignored heating/cooling processes are not relevant when compared to adiabatic cooling and ambipolar diffusion heating.

The first ignored process is the term $- \frac{3}{2}k \tilde{n} T {\rm D}f_{\rm e}/{\rm D}t$. This term decreases for increasing accretion rate and $\varpi _0$ due to the lower ionizations found in these regions. It is plotted in Fig. C.2 for the worst case (model C, $\dot{M}_{\rm acc}=10^{-8}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=0.1$ AU). There, it reaches at most $\sim$13% of $\Lambda _{\rm adia}$. Typical values for higher accretion rates are only $\simeq$0.1% of $\Lambda _{\rm adia}$.

Next we consider heating/cooling of the gas by collision with dust grains, given by Hollenbach & McKee (1979):

$$\Gamma_{{\rm gr}} = n \langle n_{{\rm gr}} \sigma_{{\rm gr}} \ran... ...k_{{\rm B}} T }{\pi m_{{\rm H}} } } f(2k_{{\rm B}} T_{\rm gr} - 2k_{{\rm B}} T)$$ (C.3)

where $\langle n_{{\rm gr}} \sigma_{{\rm gr}}\rangle$ is computed from the adopted MRN distribution, and f=0.16 is the sticking parameter that takes into account charge and accommodation effects for a warm gas (Hollenbach & McKee 1979). With these values the grain heating/cooling becomes,

$$\Gamma_{{\rm gr}} = 4.78 \times 10^{-34} n^2 (T_{{\rm gr}}-T) \hspace{.5cm} \mbox{erg s$^{-1}$\space cm$^{-3}$ }.$$ (C.4)

This term increases with increasing $\varpi _0$ and accretion rate. In Fig. C.2 we compare it (in absolute value) with $\Gamma _{\rm drag}$ in the case where its contribution is the most important (model A, $\dot{M}_{\rm acc}=10^{-5}~M_\odot~\mbox{yr}^{-1}$ and $\varpi _0=1$ AU). $\Gamma _{\rm gr}$ is initially positive (dust hotter than the gas), but changes sign at $\chi \simeq 0.4$, where the gas becomes hotter than the dust, becoming an effective cooling term. It is only significant at the very base of the wind, where it exceeds the ambipolar diffusion heating term by a factor $\simeq$3.5. However, this effect will not have important consequences in terms of observational predictions: we will show in next section that the thermal state in the hot plateau (where forbidden line emission is excited) is not sensitive to the initial temperature. Furthermore, the outer streamlines at $\varpi_0 \simeq 1$ AU contribute much less to the line emission than the inner ones. At lower accretion rates ${\le} 10^{-6}~M_\odot~\mbox{yr}^{-1}$, $\Gamma _{\rm gr}$ is always $\le$10% of $\Gamma _{\rm drag}$.

Heating due to cosmic rays, which could be important in the outer tenuous zones of the wind is (Spitzer & Tomasko 1968),

$$\Gamma_{\rm cr} = n_{\rm H} \zeta \Delta E = 1.9\times 10^{-28} \, n_{\rm H} \mbox{ erg s$^{-1}$\space cm$^{-3}$ }$$ (C.5)

where $\zeta$ is the ionization rate which we took as $\zeta=3.5\times 10^{-17}$ s^-1 (Webber 1998) and $\Delta E = 3.4$ eV is the average thermal energy transmitted to the gas by each ionization (Spitzer & Tomasko 1968). This effect is at most ${\sim} 3.6\times 10^{-6}$ times $\Gamma _{\rm drag}$ for model A, $\dot{M}_{\rm acc}=10^{-5}~M_\odot~{\rm yr}^{-1}$ and $\varpi _0=0.1$ AU. Finally the thermal conductivity along magnetic field lines was computed with the Spitzer conductivity for a fully ionized gas (Lang 1999) and is irrelevant (at most ${\sim}10^{-6}$ of the adiabatic cooling term) the maximum being achieved at the recollimation zone where the physical validity of our MHD solutions ends. It should be pointed out that Nowak & Ulmschneider (1977) compute the thermal conductivity for a partially ionized mixture in ionization equilibrium and found that for low temperatures $T\sim10^{3-4}$K the Spitzer expression underestimates the conductivity by a factor 10². However this is still too small to be important.

   
C.3 Dependence on initial conditions

Figure C.3: Effect of initial temperature on the thermal evolution for model B, $\dot M_{\rm acc}= 10^{-6}~M_{\odot}~\rm{yr}^{-1}$. The several initial temperatures are in dashed 50 K, 100 K, 500 K, 1000 K, 2000 K and 3000 K. In solid we plot the solution obtained by our standard initial conditions. Note that the almost vertical evolution of the temperature for very low initial temperatures is not an artifact. The field line anchored at $\varpi _0=0.1$ AU crosses the sublimation surface at $\chi \simeq 0.5$.

Formally, our temperature integration is an initial value problem. In the absence of a self-consistent description of the disc thermodynamics, there is some freedom in the initial temperature determination. It is therefore crucial to check that the subsequent thermal evolution of the wind does not depend critically on the adopted initial value.

Safier obtained the initial temperature by assuming the poloidal velocity at the slow magnetosonic point ( $v_{\rm p,s}$) to be the sound speed for adiabatic perturbations $T_{\rm s} = \mu m_{\rm H} v_{\rm p,s}^2/ \gamma k_{{\rm B}}$. Here, we have chosen to compute the initial temperature assuming local thermal equilibrium ${\rm D}T/{\rm D}t=0$. Our method produces lower initial temperatures than Safier due to adiabatic cooling.

For high accretion rates $\dot{M}_{\rm acc}\,\geq 10^{-6}~M_\odot~{\rm yr}^{-1}$ our initial temperature versus $\varpi _0$ has a minimum at the beginning of the dusty zone: inside the sublimation cavity, the thermal equilibrium is between photoionization heating and adiabatic cooling. Just beyond the dust sublimation radius, photoionization heating is strongly reduced, but the ionization fraction is still too high for efficient drag heating, resulting in a low initial equilibrium temperature.

The initial ionization fraction is similarly determined by assuming local ionization equilibrium ${\rm D}f_A^{\rm i}/{\rm D}t = 0$ for all elements. It decreases with $\varpi _0$. For $\dot{M}_{\rm acc}=10^{-5}~M_\odot~{\rm yr}^{-1}$ and $\varpi_0 \geq 0.8$ AU, the initial ionization fraction is set to a minimum value by assuming that Na is fully ionized, which is somewhat arbitrary. However, as gas is lifted up above the disk plane, the dust opacity decreases and the gas heats up, so that ionization becomes dominated by other photoionized heavy species and by protons, all computed self consistently.

In order to check that our results do not depend on the initial temperature, we have run model B for a broad range of initial temperatures. As shown in Fig. C.3), we find that the thermal and ionization evolution quickly becomes insensitive to the initial temperature. If we start with a temperature lower than the local isothermal condition, the dominant adiabatic cooling is strongly reduced, and the gas strongly heats up, quickly converging to our nominal curve. If we start with a higher initial temperature, adiabatic cooling is stronger, and we have the characteristic dip in the temperature found by Safier. Our choice of initial temperature has the advantage of reducing this dip, which is somewhat artificial (see Fig. C.3). In either case, we conclude that our results are robust with respect to the choice of initial temperature. In particular, the distance at which the hot plateau is reached, which has a crucial effect on line profile predictions, is unaffected.