# AMBIPOLAR DIFFUSION REVISITED

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3 AMBIPOLAR DIFFUSION 75 29: Instituto de Astronomía, UNAM - Magnetic Fields in the Universe II: From Laboratory and Stars to the Primordial Universe (AS27). The quantities γ and /C are, respectively, the usual drag coefficient between ions and neutrals and the height-averaged reciprocal coefficient for the ion mass-abundance (see Shu 992). An attractive feature of this present approach is that we can delay specifying the numerical value of the product γc until it comes to specifying dimensional scalings appropriate to specific astronomical objects, as long as the combination of parameters given by equation (4) below is a small number compared to unity. We define the dimensionless ratio λ of mass per unit area to flux per unit area according to λ = 2πG/2 Σ B z. (8) Next we need expressions for Θ and z in terms of λ for a magnetized singular isothermal disk, the form that our inner core approaches asymptotically at the moment of gravomagneto catastrophe. These relations are derived in AS27. The resulting relationships have the elegance of simplicity, and we assume that the following expressions hold for all times: Θ = 2 + ( ) λ2 λ 2 a 2 + λ 2 and z = + λ 2 πgσ. (9) Combined with equations (), (2), (6), the relationships (8) and (9) give us a closed set of equations to solve for Σ, u, B z, λ, Θ, and z. With the form of the magnetic induction equation specified, we now construct a similarity transformation that changes the original description in terms of the variables (ϖ, t) to a description of (x, t); specifically, we use the relations x = ϖ a, Σ(ϖ, t) = σ(x, t), () a t 2πG t u(ϖ, t) = aṽ(x, t), B z (ϖ, t) = a G /2 t β(x, t). With this formulation, the equation of continuity is given by t σ ( t + + x ) σ + (xṽ σ) =, () x x x and the force equation becomes t ṽ t + (x + ṽ) ṽ x + σ (Θ σ) = (2) x ydy ( y ) x 2 K σ(y, t) [f(x, t)f(y, t) ]. x Fig.. The dimensionless fluid fields from the self-similar collapse calculation. The top panel shows the reduced density field σ(ξ), as a function of the similarity variable ξ, for varying choices of the flux ratio f. The bottom panel shows the analogous plot for the reduced velocity field v(ξ). The induction equation can then be written [ σ 2 t f ] [ ( )] + (x + ṽ) f = ɛ f 2 x σ b, t x x x + f 2 (3) where f /λ and where we have defined the dimensionless diffusion coefficient 8πG ɛ. (4) γc Notice that ɛ is a small dimensionless parameter in this problem, and is essentially the ratio of dynamical time to the diffusion time (see also Galli & Shu 993). In addition, the reduced radial magnetic field b is defined in terms of the integral b(x, t) = x 2 K ( y x ) σ(y, t)f(y, t) ydy. (5) Next, we can eliminate the time dependence from the equations of motion to obtain reduced equations in terms of the variable ξ only (where ξ is a scaled version of x). To carry out this reduction, we must solve for the solution at a given value of the flux ratio f = f and then iterate (note that we are omitting several pages of mathematics in this present description see AS27 for the full treatment). Figure shows the resulting solutions for the reduced

6 78 ADAMS 29: Instituto de Astronomía, UNAM - Magnetic Fields in the Universe II: From Laboratory and Stars to the Primordial Universe Fig. 5. Distribution of time scales for core formation through ambipolar diffusion in the presence of turbulent fluctuations. The histogram shows the results of, numerical simulations of the ambipolar diffusion process with different realizations of the fluctuations. The smooth dashed curve shows the analytically predicted distribution for the given level of turbulence; the dotted curve shows the wider distribution that would result with a longer time scale for producing independent realizations of the turbulence. The vertical line on the right hand side of the plot shows the (single) value of the time scale for no turbulent fluctuations. of the chaotic nature of the fluctuations, the ambipolar diffusion time scale will take on a full distribution of values for effectively the same initial states. Fluctuations in both the magnetic field strength and the density field are expected to be present in essentially all star forming regions. Molecular clouds are observed to have substantial non-thermal contributions to the observed molecular line-widths; these non-thermal motions are generally interpreted as arising from MHD turbulence. Indeed, the size of these non-thermal motions, as indicated by the observed line-widths, are consistent with the magnitude of the Alfvén speed. As a result, the fluctuations are often comparable in magnitude to the mean values of the fields (in other words, the relative amplitudes of the fluctuations are of order unity). Background fluctuations can lead to a net change in the diffusion rate because magnetic diffusion is a nonlinear process. As many authors have derived previously (e.g., Shu 992), the dimensionless diffusion equation takes the schematic form b τ = ( µ b 2 b µ ), (7) where b is the magnetic field strength, µ is the Lagrangian mass coordinate, and we have ignored density variations. If the magnetic field fluctuates about its mean value on a time scale that is short compared to the diffusion time, then we let b b( + η), where η is the relative fluctuation amplitude. In the simplest case where the fluctuations are spatially independent, the right hand side of equation is multiplied by a cubic factor ( + η) 3. Although a linear correction would average out over time, this nonlinear term averages to a value greater than unity, and the corresponding diffusion time scale grows shorter. We have derived a more rigorous treatment of this effect by including both magnetic field and density fluctuations (Fatuzzo & Adams 22) into the standard one-dimensional ambipolar diffusion problem (Shu 983, 992). For the case of long wavelength fluctuations, the rate of ambipolar diffusion increases by a significant factor Λ. The corresponding decrease in the magnetic diffusion time helps make ambipolar diffusion more consistent with observations. In addition, the stochastic nature of the process makes the ambipolar diffusion time take on a distribution of different values. Figure 5 shows the resulting distribution of ambipolar diffusion times for a cloud layer with uniform fluctuations of amplitude A =.886; this level of fluctuations is consistent with the non-thermal line widths observed in star forming regions. The solid curve (histogram) shows the result of, numerical simulations with different realizations of the fluctuations. The dashed curve shows the analytic prediction for the time scale distribution a gaussian with a peak value given by the expectation value and with a width predicted by application of the central limit theorem. The dotted curve depicts a wider gaussian distribution that applies for longer fluctuation time scales (resulting in fewer realizations of the turbulent fluctuations). In the absence of fluctuations, the cloud maintains a single value for its ambipolar diffusion time, as shown by the delta-function spike at τ e GENERALIZED SELF-SIMILAR COLLAPSE Motivated by recent observations that show starless molecular cloud cores exhibit subsonic inward velocities, we revisit the collapse problem for polytropic gaseous spheres. In particular, we provide a generalized treatment of protostellar collapse in the spherical limit (Fatuzzo et al. 24) and find semi-analytic (self-similar) solutions, corresponding numerical solutions, and purely analytic calculations of the mass infall rates (the three approaches are in

7 AMBIPOLAR DIFFUSION : Instituto de Astronomía, UNAM - Magnetic Fields in the Universe II: From Laboratory and Stars to the Primordial Universe Fig. 6. Comparison of self-similar, analytic, and numerical determinations of m for varying initial inward velocities u in and for isothermal initial conditions. The m values calculated from the self-similar equations of motion (solid curve), numerically solving the partial differential equations (dashed curve), and an analytic estimate (dotted curve). All models use initial states with isothermal static Γ = and dynamic γ =. good agreement). This study focuses on collapse solutions that exhibit nonzero inward velocities at large radii, as observed in molecular cloud cores, and extends previous work in four ways: () The initial conditions allow nonzero initial inward velocity. (2) The starting states can exceed the density of hydrostatic equilibrium so that the collapse itself can provide the observed inward motions. (3) We consider different equations of state, especially those that are softer than isothermal. (4) We consider dynamic equations of state that are different from the effective equation of state that produces the initial density distribution. This work determines the infall rates over a wide range of parameter space, as characterized by four variables: the initial inward velocity u in, the overdensity Λ of the initial state, the index Γ of the static equation of state, and the index γ of the dynamic equation of state. For the range of parameter space applicable to observed cores, the resulting infall rate is about a factor of two larger than found in previous theoretical studies (those with hydrostatic initial conditions and u in = ). This work allows for a greater understanding of the infall rate Ṁ, which is perhaps the most important variable in the star formation problem. For Fig. 7. Mass infall rates as a function of time for flattopped cores. The initial (asymptotic) inward velocities are taken to be u in = (solid curve),.5 a S (dotted curve), and. a S (dashed curve). isothermal initial conditions, the mass infall rate is given by Ṁ = m a 3 s/g so that the dimensionless parameter m specifies the infall rate. For example, Figure 6 shows the values of m calculated using three different approaches: the standard selfsimilar formulation (as in Shu 977), an analytic estimate, and a fully numerical determinations. The resulting values of m are shown as a function of the initial inward velocity. All of these cases use initial states with isothermal static Γ = and dynamic γ =. Notice that the three approaches are in good agreement, and that the infall rate is a smoothly increasing function of the initial velocity. For observed values of u in, the infall rate is roughly twice the canonical value (for hydrostatic initial conditions). Figure 7 shows the mass infall rates as a function of time for flat-topped cores with varying initial infall speeds. This set of numerical collapse calculations begins with typical parameters observed in flattopped cores with sound speed a S =.2 km/s, and assumes a central region of nearly constant density spanning r C 6 cm. The initial states are taken to be overdense at the percent level. The solid curve shows the resulting mass infall rate (expressed in M yr ) for the case with no initial velocity. The dotted (dashed) curve shows the corresponding mass infall rate for starting inward speeds of u in =.5 a S (u in = a S ). When u in, the mass infall rates reach their peak values more quickly, the peak values are smaller, and the infall rates more rapidly approach their asymptotic values (as predicted by

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