Appendix B: Dust implementation

Figure B.1: Dust sublimation surfaces geometry for the adopted radiation field.

As shown by Safier if there is dust in the disk, the wind is powerful enough to drag it along. Thus disk winds are dusty winds. Dust is important for the wind thermal structure mainly as an opacity source affecting the photoionisation heating at the wind base. To compute the dust opacity we need a description of its size distribution, its wavelength dependent absorption cross-section and the inner dust sublimation surface. In the inner flow zones and for high accretion rates the strong stellar and boundary layer flux will sublimate the dust, creating a dust free inner cavity (see Fig. B.1). Results on the evolution of dust in accretion disks by Schmitt et al. (1997) show that at the disk surface the initial dust distribution isn't much affected by coagulation and sedimentation effects. Thus we assume a MRN dust distribution (Mathis et al. 1977; Draine & Lee 1984):

$${\rm d}n_{\rm i}=n_{\rm H}\,A_i\,a^{-3.5}\,{\rm d}a$$ (B.1)

where ${\rm d}n_{\rm i}$ is the number of particles of type i ("astronomical silicate'' - Sil or graphite - C) with sizes in $[a,a+{\rm d}a]$, and $0.005~\mbox{$\mu$ m} \leq a \leq 0.25 ~\mbox{$\mu$ m}$, $A_{\rm Sil}=10^{-25.11}$ cm^2.5 H^-1 and $A_{\rm C}=10^{-25.16}$cm^2.5 H^-1. We then proceed by averaging all relevant grain quantities function of size and species (F_i(a)) by the size/species distribution,

 

$$\langle F_i(a)\langle_a = \int_{a_{\rm min}}^{a_{\rm max}} \sum_{i={\rm Sil,C}} F_i(a) \, \frac{{\rm d}n_{\rm i}}{N\rm _T}\cdot$$ (B.2)

In order to compute the sublimation radius, some description of the dust temperature must be made. For simplicity, we assume the dust to be in thermodynamic equilibrium with the radiation field, the dominant dust heating mechanism. In our case, the central source radiation field will dominate throughout the jet, except probably in the recollimation zone, where the strong gas emission overcomes the central diluted field. However in this region dust is no longer relevant for the gas thermodynamics and we will therefore only consider dust heating by the central source. The dust temperature $T_{\rm gr}$ for a grain of size a is obtained by equating the absorbed to the emitted radiation (e.g., Tielens & Hollenbach 1985),

 

$$4 \pi a^2 \langle Q_a\rangle(T_{\rm gr}) \sigma T_{\rm gr}^4 = \pi a^2 \,\int_0^\infty Q_a^{\rm abs}(\nu) 4\pi J_\nu {\rm d}\nu$$ (B.3)

where a is the grain size $\langle Q_a\rangle(T_{\rm gr})$ is the Planck-averaged emissivity (Draine & Lee 1984; Laor & Draine 1993; Draine & Malhotra 1993), $\sigma$ is the Stefan-Boltzmann constant, $Q_a^{\rm abs}(\nu)$ is the dust absorption efficiency and $4\pi J_\nu$ is the central source radiation flux at the grain position given by Eq. (15). Averaging out the previous equation by the size/species distribution (Eq. (B.2)) we obtain,

$$4 \langle Q_a^{\rm em}(T_{\rm gr})\rangle \sigma T_{\rm gr}^4 = \... ...angle Q_a^{\rm abs}\rangle(\nu) F(\nu) {\rm e}^{-\tau_\nu(r,\theta)} {\rm d}\nu$$ (B.4)

where we describe the central source flux by $F(\nu)$ which is attenuated only by the dust opacity $\tau$. For simplicity $F(\nu)$ is taken as exactly the same as in Safier, i.e. a classical boundary layer (Bertout et al. 1988). The sublimation radius is obtained from the previous expression by noting that at its position $\tau_\nu=0$,

$$\frac{\langle r_{\rm sub}(\theta)\rangle}{R_\ast}=\sqrt{\frac{g_\... ...ngle T_{\rm bl}^4 }{4 \langle Q_a^{\rm em}(T_{\rm sub})\rangle T_{\rm sub}^4 }}$$ (B.5)

where $T_{\rm bl}$ and $T_{\ast}$ are the boundary layer and star temperatures, $R_\ast$ the stellar radius and $g_{\rm bl}(\theta)$/ $g_\ast(\theta)$ are the $\theta$ dependent terms of the radiation field (given in Bertout et al. and Safier). We assume a dust sublimation temperature $T_{\rm sub}$ of 1500 K.

With the dust sublimation radius in hand we can now proceed to compute the dust optical depth defined as,

$$\tau_\nu(r,\theta) = \int_{r_{\rm in}(\theta)}^{r} \kappa_\n... ...}^{r} n_{\rm H} (r',\theta) \overline{\sigma(\nu)_a} {\rm d}r'$$ (B.6)

where $r_{\rm in}(\theta)$ is the radius inside which there is no dust. This radius is given by the inner flow line $r_{\varpi_{\rm i}}(\theta)$ and by the sublimation radius $\langle r_{\rm sub}(\theta)\rangle$ (see Fig. B.1) such that $r_{\rm in} (\theta) = {\rm max} (\langle r_{\rm sub}\rangle;r_{\varpi_{\rm i}})$. The dust absorption cross-section ( $\overline{\sigma(\nu)_a}$) is,

$$\overline{\sigma(\nu)_a}=\int_{a_{\rm min}}^{a_{\rm max}} \pi a^2... ...nu) A_{\rm Sil} + Q^{\rm abs}_{\rm C}(a,\nu) A_{\rm C} \big] a^{-3.5} {\rm d}a.$$ (B.7)

Using the self-similarity of $n_{\rm H} (r,\theta)$ we can integrate Eq. (B.6) to obtain,

$$\tau_\nu(r,\theta) =n_{\rm H}(r,\theta)\,\overline{\sigma(\nu)} \, 2r\Big(\sqrt{\frac{r}{\langle r_{\rm in}(\theta)\rangle}}-1\Big)$$ (B.8)

which was used in Eq. (15). We note that at large distances from the source, the optical depth converges to a finite value, proportional to $\dot{M}_{\rm acc}$ and whose $\theta$variation is function of the self similar wind solution and central source radiation field. Thus for high accretion rates, although the central source radiation hardens, the outer zones of the wind base are less photoionized than for smaller accretion rates.