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Appendix A: Chemical model

In this Appendix we describe the “minimal” chemical model introduced in Sect. 3 to compute the ionisation fraction. The charged species considered are H⁺, , molecular ions mH⁺ (e.g. HCO⁺), “metal” ions M⁺ (e.g. Mg⁺), electrons e, and negatively charged dust grains, whereas the neutral species are H₂, heavy molecules m (e.g. CO), metal atoms M, (e.g. Mg) and neutral grains. Charged and neutral grains are collectively indicated as g. We indicate with x(i) the abundance of each species i with respect to H₂. The abundance of the neutral species is fixed. In particular, we assume x(m) ≃ 6 × 10^-4 and x(M) ≃ 4 × 10^-8. All rate coefficients are estimated at T = 10 K.

Appendix A.1: Minimal chemical network

Protons are produced by CR ionisation of H₂ at a rate ϵζ^H₂, with ϵ ≃ 0.05 (Shah & Gilbody 1982). They are mainly destroyed by charge transfer (CT) with molecules (at a rate β ≃ 10^-9 cm³ s^-1) and by recombination on grains (at a rate α_gr, see Eq. (A.8)) (A.1)The formation of is driven by CR ionisation of H₂ at a rate (1 − ϵ)ζ^H₂, while destruction is due to CT with heavy molecules, dissociative recombination (DR, at a rate α_dr ≃ 10^-6 cm³ s^-1), and recombination on grains (A.2)The formation of molecular ions mH⁺ occurs by CT of and heavy molecules, while destruction occurs by DR and recombination on grains (A.3)Metal ions are formed by CT with and mH⁺, and destroyed by recombination with free electrons and on grains (A.4)Note that CT with metal atoms can be neglected with respect to DR if x(e) ≫ (β/α_dr)x(M) ≃ 10^-3x(M).

Dust grains are assumed to be negatively charged (charge −1) or neutral. The total number density of grains is obtained from the MRN size distribution (Mathis et al. 1977) (A.5)between a minimum (a_min) and a maximum (a_max) grain radius (see also Sect. 3.2). The normalisation constant C is obtained by imposing that the mass density of grains is equal to a fraction q = 0.01 of the gas density. We assume that grains are spherical and have density ρ_g = 2 g cm^-3 (Flower et al. 2005). For a_max ≫ a_min we obtain (A.6)Under these assumptions, the number density of grains is (A.7)and is strongly dependent on a_min.

The coefficient of recombination of positive ions on negatively charged grains was computed by Draine & Sutin (1987) assuming the MRN size distribution, (A.8)where ψ is a numerical coefficient equal to −2.5 for an e–H⁺ plasma and −3.8 for a heavy-ion plasma. For simplicity, we adopt the same value of α_gr for all positively charged ions, assuming a typical ion mass m_i = 25m_H. The actual value of α_gr is larger by a factor 5 and 3 for H⁺ and H, respectively.

Appendix A.2: Evaluation of the ionisation fractions

The equation of charge neutrality is (A.9)Equations (A.1)–(A.4) then become

where we have defined (A.14)Thus, the equation of charge neutrality (A.15)becomes (A.16)We solve Eq. (A.16) as a cubic equation for x(e) assuming that all grains have charge −1. When x(e) becomes negative, we set x(e) = 0 and we solve Eq. (A.16) for x(g).

Appendix B: Dependence of diffusion coefficients on grain radius

Fig. B.1 Ambipolar (upper panel), Hall (middle panel), and Ohmic (lower panel) diffusion coefficients as a function of the mean grain radius computed at different molecular hydrogen densities: 105 (thick solid line), 106 (thick dashed line), 107 (thick dash-dotted line), 108 (dotted line), 109 (thin solid line), 1010 (thin dashed line), 1011 cm-3 (thin dash-dotted line), and 1012 cm-3 (thin dotted line).

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In this Appendix we compute the resistivities using Eqs. (5)–(7) varying the minimum radius of the grain size distribution and fixing the H₂ density to evaluate the sensitivity of the diffusive terms to the grain size. For the magnetic field strength we assume | B | = (n/ cm^-3)^0.47μG (Crutcher 1999), in order to be independent on specific models. The MRN size distribution given by Eq. (A.5) implies that the mean value of the square of the grain radius weighted on the grain distribution (namely the quantity that enters the equation for the momentum transfer rate coefficients), defined by (B.1)

is close to . We vary the minimum grain radius between 10^-7 to 10^-5 cm, fixing a_max = 3 × 10⁵ cm. This corresponds to values of ranging from 2.2 × 10^-7 to 1.5 × 10^-5 cm. Figure B.1 shows that the coefficient of ambipolar diffusion is not monotonic with the grain radius but presents an absolute minimum at larger radii with increasing H₂ densities. On the other hand, the Hall term increases with the grain radius for any value of n(H₂), starting to decrease only at very large grain size (a_min ≳ 10^-5 cm). Finally, the Ohmic resistivity becomes important at high densities (n ≳ 10¹¹ cm^-3) and increases monotonically with grain size.

Appendix C: Contributions to the diffusion time of the magnetic field

	Fig. C.1 Ambipolar, Hall, and Ohmic contribution to the diffusion time (red dotted lines) compared with the dynamical time (black solid lines). Labels show log 10(t/ yr).
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For the sake of completeness, in Fig. C.1 we show separately the diffusion time for ambipolar, Hall, and Ohmic diffusion computed with ζ^H₂ from model L₂. As expected, at the high densities reached in the disc region, ambipolar diffusion time is more than one order of magnitude larger than the dynamical time scale. On the contrary, Hall and Ohmic diffusion times are comparable and lower than the dynamical time scale in a region whose radius decreases from about 50 AU to 25 AU with decreasing minimum grain size. This is another way to say that gas-magnetic field decoupling is due to Hall and Ohmic diffusion at densities higher than n ~ 10¹⁰ cm^-3.

© ESO, 2014