Ionneutral friction and accretiondriven turbulence in selfgravitating filaments
^{1} Laboratoire AIM, ParisSaclay, CEA/IRFU/SAp − CNRS − Université Paris Diderot, 91191 GifsurYvette Cedex, France
email: patrick.hennebelle@lra.ens.fr
^{2} LERMA (UMR CNRS 8112), École Normale Supérieure, 75231 Paris Cedex, France
Received: 23 April 2013
Accepted: 8 October 2013
Recent Herschel observations have confirmed that filaments are ubiquitous in molecular clouds and suggest, that irrespective of the column density, there is a characteristic width of about 0.1 pc whose physical origin remains unclear. We develop an analytical model that can be applied to selfgravitating accreting filaments. It is based on the one hand on the virial equilibrium of the central part of the filament and on the other hand on the energy balance between the turbulence driven by accretion onto the filament and dissipation. We consider two dissipation mechanisms, the turbulent cascade and the ionneutral friction. Our model predicts that the width of the inner part of the filament is almost independent of the column density and leads to values comparable to what is inferred observationally if dissipation is due to ionneutral friction. On the contrary, turbulent dissipation leads to a structure that is bigger and depends significantly on the column density. Our model provides a reasonable physical explanation which could explain the observed filament width when they are selfgravitating. It predicts the correct order or magnitude though hampered by some uncertainties.
Key words: turbulence / stars: formation / ISM: structure / magnetic fields
© ESO, 2013
1. Introduction
With the recent observations performed with Herschel (e.g. André et al. 2010; Molinari et al. 2010), it has become clearer that filaments are ubiquitous in molecular clouds and that they likely play a central role in star formation. While the exact influence filaments may have on the star formation process remains to be clarified, it is important to understand the properties of these filaments since they are a direct consequence of the physics at play within molecular clouds. In this respect, a particularly intriguing observational result was found by Arzoumanian et al. (2011) who showed that the central widths of the interstellar filaments have a narrow distribution that is peaked around a value of 0.1 pc. Moreover, this characteristic width does not depend on the column density within the filament. Since it is believed that turbulence is important in molecular clouds and largely triggers their evolution, this result is counterintuitive at first sight since turbulence is responsible in a great variety of contexts for producing scalefree powerlaw distributions. This suggests that there is probably a physical process involved in setting this distribution, which unlike turbulence, presents a characteristic scale.
A recent proposal made by Fischera & Martins (2012, see also Heitsch 2013; Gomez & VázquezSemadeni 2013) is that it may result from selfgravitating equilibrium. By solving the hydrostatic equilibrium in an isothermal filament, Fischera & Martins (2012) show that the filament width does not vary significantly and remains on a scale close to the observed 0.1 pc. While this explanation is appealing, a few questions arise. First, it assumes the existence of some confining pressure outside the filament whose nature remains to be specified. Second, it fails to explain why the gravitationally unstable filaments that are collapsing also present this typical width.
In this paper, we explore the idea that the typical width of a selfgravitating filament is due to the combination of accretiondriven turbulence onto the filament, as suggested by the recent velocity dispersion measurements of Arzoumanian et al. (2013), and to the dissipation of this turbulence by ionneutral friction which does have a characteristic scale. We stress that although ambipolar diffusion is considered here, the underlying idea is totally different from the classical magnetically regulated star formation (e.g. Shu et al. 1987) in which the magnetic field is envisaged as the dominant support and ambipolar diffusion as the process through which the support can be circumvented. In the present picture clouds are typically supercritical. We also note that we do not address the reason why filaments form. As discussed in Hennebelle (2013) this may be due to the shear of the turbulence with a possible further amplification of the anisotropy by gravity for the most massive of them.
The plan of the paper is as follows. In Sect. 2, we present the various assumptions and physical processes used in our simple model. Section 3 describes the results and Sect. 4 concludes the paper.
2. Model and assumptions
2.1. Characteristics of the filament
Herschel observations (e.g. Palmeirim et al. 2013) suggest that the typical average structure of selfgravitating filaments is constituted by i) a central cylinder of nearly uniform density ρ_{f} and radius r_{f}; and ii) an envelope whose density profile is ∝r^{2}. This radial structure is reminiscent of many selfgravitating objects such as BonnorEbert spheres. More precisely, filaments that are collapsing in a selfsimilar manner are expected to present an envelope with a profile ∝r^{− 2/(2 − γ)} where γ is the adiabatic index of the gas (Kawachi & Hanawa 1998). As it is likely that selfgravitating filaments are collapsing in a way not too different from, although not identical to, a selfsimilar collapse, assuming an r^{2} profile is thus a wellmotivated assumption, both from observations and theory. One important difference with such selfsimilar solutions, however, is that the central density plateau does not seem to be shrinking with time.
If L is the length of the filament, the mass of the central part is obviously , where M_{f} is the mass and m_{f} the mass per unit length. The total mass of the central part plus the surrounding envelope is (1)where μ(x) = 1 + 2ln(x). It has been assumed that the density outside the filament is ∝1/r^{2} and that the filament stops at some radius r_{ext}. Below we assume r_{ext} ≃ L/2.
2.2. Gravitational potential within the filament
The gravitational potential, in the radial direction is obtained by the Gauss’s theorem (2)where .
2.3. Magnetic field
As the ionneutral friction dissipation depends on the magnetic field, it is necessary to know its dependence.
We proceed in two steps. First, we discuss the expected value of the magnetic field in the parent clump B_{c}. Second, we infer the value of the magnetic field in the filament B_{f} from the value of B_{c}.
2.3.1. Magnetic field in the parent clump
The magnetic field in the clump is assumed to be proportional to the square root of the density as has been observed (e.g. Crutcher 1999). Typical values are B_{0} ≃ 25 μG and n_{0} ≃ 10^{3} cm^{3}, where n_{0} = ρ_{0}/m_{p}. In the following we will use the km s^{1} as a fiducial value.
We note that the magnetic field dependence is still under debate and that there are alternative choices. First, as suggested by Basu (2000), the magnetic field could scale as , where σ is the velocity dispersion, rather than just as . Second, Crutcher et al. (2010) now favor B ∝ ρ^{2/3}. These relations do not represent large variations and would therefore not affect our results very significantly.
2.3.2. Magnetic field in the filament
To link the magnetic field in the filament to the magnetic field in the parent clump, we proceed as follows. First we assume that the magnetic field is perpendicular to the filament. This configuration is well supported by observations in massive filaments such as Taurus (Heyer et al. 2008; Palmeirin et al. 2013) or DR21 (e.g. Kirby 2009), and is also natural on physical grounds as the gas is expected to accumulate preferentially along the field lines. This implies that at least a fraction of the gas accreted by the filament, is not impeded by the magnetic Lorentz force. There may also be gas accreted perpendicularly to the field lines, which is therefore probably slowed down by magnetic pressure. However, if the field is strong, then most of the material is presumably channeled along the field lines, while if the field is weak it also has a weak influence.
Second, we assume flux freezing which at these scales is a reasonable assumption. This implies that the magnetic field in the filament is simply the magnetic field in the clump compressed along the direction perpendicular to the field and the filament axis. Thus B_{f} ≃ B_{c} × ηL/r_{f} since the matter that is inside the radius r_{f} comes from a distance comparable to the clump’s size, ηL, where η typically varies with time between 0 and 1/2. A similar reasoning can be applied to get a relation between ρ_{c} and ρ_{f} since . We thus obtain ρ_{f} = ρ_{c}(ηL/r_{f})^{2}. Combining the expression for B_{c} obtained above with the last expression, we get that is to say, the magnetic field in the filament is also expected to be nearly proportional to implying that the Alfvén velocity should remain nearly constant.
This relation is valid as long as flux freezing can be assumed. While this is a reasonable assumption in the collapsing envelope, which is not magnetically supported, it is not the case in the central part, which is presumably close to equilibrium. The typical ambipolar diffusion timescale is given by Eq. (13) below. For densities on the order of 10^{4 − 7} cm^{3}, the field is diffused in about 0.1−1 Myr, which is shorter than or comparable to, the accretion time of the filaments. Moreover, as emphasized by SantosLima et al. (2012) and Joos et al. (2013), turbulence also tends to diffuse out the field. However since selfgravitating filaments are probably accreting, the magnetic flux cannot leak out far away since it is confined by the infalling gas. Therefore, while the mean magnetic intensity within the central part of these filaments is probably on the order of , it is likely that the magnetic field gradient is greatly reduced with respect to the ideal MHD case.
2.4. Virial equilibrium
The virial theorem is applied to the filament inner part of radius r_{f}. The expression for a filament is (e.g. Fiege & Pudritz 2000) (3)where W_{grav} is the gravitational term, P and P_{ext} are the internal and external pressure, and V the volume; is the kinetic energy in the direction perpendicular to the filament axis and σ_{1D} is the nonthermal onedimensional velocity dispersion. This expression is, however, not strictly valid since our model filament is accreting. Additional terms should be taken into account (see Goldbaum et al. 2011 and Hennebelle 2012) which corresponds to the surface terms that do not cancel out as it is usually the case. When the surface terms are taken into account, the virial expression becomes (4)At this stage we do not consider the influence that the magnetic field may have on the equilibrium because it is not expected to change our conclusions qualitatively. In particular, as discussed in the previous section, the magnetic gradient within the central part of the filament is probably smoothed because of ambipolar diffusion. Another complication arises because the anisotropy introduced by the magnetic field is perpendicular to the filament axis, which would require a bidimensional analysis.
Using the different expressions obtained above we get (5)where P_{ram} is the ram pressure exerted by the incoming flow and where the pressure of the external medium has not been taken into account. The ram pressure will be estimated below. For the sake of simplicity, we will also use the simplified form of the virial equilibrium (6)
2.5. Mechanical energy balance
The mechanical energy balance within the cylinder of radius r_{f} leads to (7)Obviously, the left handside is the dissipation which can be due either to the turbulence cascade or to the friction between ions and neutrals as described below. We note that it is assumed that the turbulence is isotropic which is why σ_{3D} is used. The righthand side describes the source of turbulence which is due to the accretion onto the central part of the filament (Klessen & Hennebelle 2010). The efficiency ϵ_{eff} is not well known. Klessen & Hennebelle (2010) proposed that it can be related to the density contrast between the density of the incoming flow and the density of the actual gas in which energy is injected. In the present case, the accretion shock may not be clearly defined because of the turbulent nature of the flow. In any case, the present calculation remains largely indicative at this stage. Below, the value ϵ_{eff} = 0.5 is used because it leads to good agreement with the data. We stress that since our model remains indicative, ϵ_{eff} could also take into account various other uncertainties. For the sake of simplicity, we have also ignored terms associated to the volume variation and the external pressure (e.g. Goldbaum et al. 2011) as they do not modify the results substantially, but make the mechanical energy balance much more complex.
2.6. Accretion rate
The accretion rate remains uncertain since it is difficult to infer observationally (see however Palmeirim et al. 2013 for an estimate in the case of the Taurus B211/3 filament). Here, we consider two different possibilities. This will allow us to test the robustness of our conclusion.
2.6.1. Gravitational accretion rate
To estimate the accretion rate, we assume that it can be computed from the density within the parent clump and the infall velocity. Assuming that the parent clump has a cylindrical radius of about L/2 and a length equal to L, we get Ṁ = πL^{2}ρ_{c}V_{inf}.
The infall velocity is due to the gravitational field of the filament which is given by Eq. (2). We assume that the material which enters the clump at radius r ≃ L/2 has no initial velocity and we estimate the infall velocity at r = L/4 leading for V_{inf} to (8)We note that this is again a rough estimate, but since the gravitational potential varies logarithmically with r, this estimate does not depend severely on these assumptions.
The density within the clump is also needed to get the accretion rate and we assume that ρ_{c} = m_{p}n_{c} follows the Larson relations (Larson 1981; Falgarone et al. 2009; Hennebelle & Falgarone 2012), m_{p} being the mass per particle (9)where n_{c} is the clump gas density and σ_{3D} the internal rms velocity. The exact values of the various coefficients remain somewhat uncertain. Originally, Larson (1981) estimated η_{d} ≃ 1.1 and η ≃ 0.38, but more recent estimates (Falgarone et al. 2009) using larger sets of data suggest that η_{d} ≃ 0.7 and η ≃ 0.45 − 0.5. For the sake of simplicity, we use η_{d} = 1 and take n_{0} = 1000 cm^{3}. (10)The typical accretion timescale is simply given by τ_{accret} = M/Ṁ. With Eq. (10), it is easy to show that .
2.6.2. Turbulent accretion rate
As it is not clear what controls the accretion rates onto interstellar filaments, we also consider the turbulent accretion rate constructed from the Larson relations described in Hennebelle (2012), (11)where η_{acc} ≃ 0.7 − 0.8 depending on the exact choice of the parameter η and η_{d} that is retained. We will adopt η_{acc} ≃ 0.75 as a fiducial value. We typically have Ṁ_{0} = 10^{3} M_{⊙} yr^{1} for M_{0} = 10^{4} M_{⊙}.
2.6.3. Ram pressure
The ram pressure which appears in Eq. (5) can be estimated as follows. It is equal to the product of V_{inf}(r_{f})^{2} and ρ_{in} = Ṁ/(2πr_{f}LV_{inf}(r_{f})), where ρ_{in} is the density that is obtained assuming a constant accretion rate. We note that ρ_{in} < ρ_{f}, which implies that an accretion shock is connecting the infalling envelope and the central part of the filament. Since the above expression assumes that the flow is isotropic, and since we are assuming the structure of the flow rather than inferring an exact solution, this value remains hampered by large uncertainties and is certainly valid within a factor of a few. We have tested the influence of vaying the ram pressure by a factor of a few and found that it has a limited influence on the solution at low density, while it has no influence at high density.
2.7. Dissipation timescales
The dissipation timescale to be used in Eq. (7) is a crucial issue. Here we emphasize two dissipation mechanisms, the turbulent cascade time and the ambipolar diffusion time. Quantitative estimates of these two timescales have been recently estimated by Li et al. (2012) in turbulent two fluid MHD simulations. They found that under typical molecular cloud conditions both contribute, but the second dominates over the first.
2.7.1. Dissipation by turbulent cascade
First, we consider the standard turbulent cascade timescale, which is the crossing time of the system (12)The energy cascades to smaller and smaller scales until the size of the eddies reaches the viscous scale.
2.7.2. Dissipation by ionneutral friction
Second, we investigate the dissipation induced by the ionneutral friction. Its expression was first inferred by Kulsrud & Pearce (1969, see also Lequeux 2005) and is given by (13)where v_{A} is the Alfvén speed, λ is the wavelength assumed to be equal to r_{f} and ν_{ni} is the ionneutral coupling coefficient. The reason we choose λ ≃ r_{f} is that if most of the energy is dissipated at a scale much smaller than r_{f}, then the relevant time would be the crossing or cascading time that would be necessary for the energy to cascade from r_{f}. In this case, the timescale would thus be similar to the turbulent cascade timescale discussed above. The coefficient γ_{damp} = 3.5 × 10^{13} cm^{3} g^{1} s^{1} is the damping rate. The ion density, ρ_{i} is assumed to be where C = 3 × 10^{16} cm^{−3/2} g^{1/2}. For wavelengths λ < λ_{c} = πv_{A}/(γ_{damp}ρ_{i}), the critical wavelength, the Alfvén waves do not even propagate except at very small wavelengths when the ions and the neutrals are entirely decoupled. With n ≃ 10^{4} cm^{3} and v_{A} ≃ 1 km s^{1}, λ_{c} ≃ 5 × 10^{2} pc.
We note that the expression stated in Eq. (13) is strictly valid only for Alfvén waves. The corresponding expression for the compressible modes has been inferred by Ferrière et al. (1988). They are slightly more complex as it entails the angle between the field and the direction of propagation, but the order of magnitude is not different.
3. Results
To infer the radius of the filament as a function of the central density ρ_{f} we have to combine Eqs. (2), (6), and (7) together with the accretion rate given by Eq. (10) or Eq. (11). With the gravitational accretion rate (Eq. (10)), we obtain (14)where . With the turbulent accretion rate (Eq. (11)), we get the relation (15)where .
3.1. Dynamical equilibrium with turbulent dissipation
Combining Eq. (12) with Eq. (6), we get which in turn together with Eq. (15) leads to the expression (16)For the canonical value η_{acc} = 0.75, we find that r_{f} ∝ 1/ρ_{f}. That is to say, the typical filament radius decreases with the central density as displayed in Fig. 1 (see the line labeled turbulence). The value ϵ_{eff} = 0.5 has been used in this calculation. For larger values of η_{acc} we still get significant variations of r_{f} with ρ_{f}. For η_{acc} = 1 we would even predict that the central density is independent of r_{f}. A filament radius independent of the central density is obtained only for η_{acc} ≃ 3/2. The gravitational accretion rate expression leads to a very similar expression with r_{f} ∝ 1/ρ_{f} and the corresponding expression is not given here for conciseness as the obvious conclusion is that this behaviour is incompatible with a nearly constant filament width.
Fig. 1 Filament radius as a function of filament density for four models. The curve labeled turbulence shows the result for a turbulent crossing time (Eq. (16)); the three curves labeled friction show results when ionneutral friction time is assumed to be the energy dissipation time (Eqs. (17)−(18)). Frictions A and C used a gravitational accretion rate and Friction B a turbulent rate. 

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3.2. Dynamical equilibirum with ionneutral friction
Combining Eqs. (13) and (6), with (14), we infer (17)where we see that r_{f} does not depend on ρ_{f}. That is to say, the width of the filament does not change with its column density as suggested from the results of Arzoumanian et al. (2011). To test the robustness of this result, it is worth investigating what the turbulent accretion rate stated by Eq. (11) is predicting. The corresponding expression is (18)As can be seen for an accretion rate exponent η_{acc} of the order of 0.75, we find that , which implies a very weak dependence on the filament radius r_{f}. For a value of η_{acc} = 1, we have which is still a shallow dependence as shown in Fig. 1 where the two expressions stated by Eqs. (17) and (18) are displayed (labeled as friction A and B, respectively).
Fig. 2 Amplitude of the various terms which appear in the virial equilibrium (Eq. (5)). While at high density the filament equilibrium is due to the balance between gravity and velocity dispersion, it is due to the balance between ram and thermal pressure at low density. 

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3.3. A more accurate model
Finally, we investigate the solutions where the thermal support and ram pressure are considered as stated in Eq. (5). The corresponding curve is labeled as friction C. Equation (5) is an ordinary differential equation in r_{f}, which can be solved using a standard RungeKutta method. Since it is necessary to specify a radius and a density to start the integration, we have explored various cases. We found that for a large range of radii (r_{start} ≃ 0.01 − 0.1 pc) at low density, the solutions quickly converge towards the one that is presented here and for which the radius at n = 10^{3} cm^{3} is equal to about 0.05 pc.
As can be seen, more variability is introduced, particularly at low density, where we see that the filament radius decreases at low density. In order to better understand the physical meaning of this solution we plot in Fig. 2, the values of the various virial terms as a function of density.
While the equilibrium between gravity and turbulent support at high density is robust and independent of the choice of the boundary condition, r_{start}, the behaviour at low density is less robust and varies with it.
It is important to stress a few points. First, the ram pressure term which causes most of the variation remains uncertain since our model is not fully selfconsistent in the sense that the density and velocity fields, although reasonable, are not proper solutions of the problem. Second, in the low density regime, the filament is not gravitationally accreting and it is likely that the validity of the model is questionable.
3.4. Comparison of the two dissipation timescales
It is enlighting to compare the values of the dissipation timescales as a function of density for the filament radius corresponding to model B. As expected, the turbulent dissipation timescale is much longer than the ionneutral friction timescale for densities lower than 10^{6} cm^{3}. It also increases with density while the turbulent timescale decreases with density. This behaviour is the very reason which explains the nearly constant width of the filaments in our model because as can be seen in Fig. 3, the accretion time presents the same dependence.
Fig. 3 Comparison between the turbulent dissipation and ionneutral friction times as a function of density. Also shown is the accretion timescale. It scales exactly as the ionneutral friction time does. 

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To summarize, assuming that the relevant timescale for energy dissipation within the central part of the filament is the turbulent crossing time, we find under reasonable assumptions for the accretion rate that the width changes significantly with density. This is because τ_{diss,c} ∝ 1/. On the other hand, when we assume that the relevant timescale for energy dissipation is the ionneutral friction time, we find that the width varies much less with the filament density. This is because, . Since the relevant timescale is the shortest one, which corresponds to the smallest value of r_{f}, one expects ionneutral friction to be the dominant mechanism for energy dissipation up to densities of a few 10^{5} − 10^{6} cm^{3} (see Fig. 1).
Fig. 4 Comparison between the models and the filaments width distribution (adapted from Arzoumanian et al. 2011). 

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3.5. Comparison with observations
Finally, we confront the present models with the Herschel observational result of Arzoumanian et al. (2011). Figure 4 shows filament width as a function of filament column density. As can be seen, the model based on ionneutral friction and gravitational accretion (friction B and C) work very well. We note that the model based on turbulent dissipation predicts a constant column density and a variable radius.
4. Conclusion
We have presented a simple model to describe the evolution of accreting selfgravitating filaments within molecular clouds. It assumes virial equilibrium between gravity and turbulence and mechanical energy balance between accretion which drives turbulence in the filament and the dissipation of this energy. We show that while dissipation based on turbulent cascade fails to reproduce the narrow range of radius inferred from Herschel observations, dissipation based on ionneutral friction leads to a filament width that depends only weakly on the filament density and is very close to the ≃0.1 pc value, although our analytical approach is hampered by significant uncertainties. We conclude that the combination of accretiondriven turbulence and ionneutral friction is a promising mechanism to explain the structure of selfgravitating filaments and deserves further investigation.
Acknowledgments
We thank the anonymous referee for a constructive and helpful report. We thank Doris Arzoumanian and Evangelia Ntormousi for discussions on this topic. P.H. acknowledge the financial support of the Agence National pour la Recherche through the COSMIS project. This research has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007−2013 Grant Agreements No. 306483 and No. 291294).
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All Figures
Fig. 1 Filament radius as a function of filament density for four models. The curve labeled turbulence shows the result for a turbulent crossing time (Eq. (16)); the three curves labeled friction show results when ionneutral friction time is assumed to be the energy dissipation time (Eqs. (17)−(18)). Frictions A and C used a gravitational accretion rate and Friction B a turbulent rate. 

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In the text 
Fig. 2 Amplitude of the various terms which appear in the virial equilibrium (Eq. (5)). While at high density the filament equilibrium is due to the balance between gravity and velocity dispersion, it is due to the balance between ram and thermal pressure at low density. 

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In the text 
Fig. 3 Comparison between the turbulent dissipation and ionneutral friction times as a function of density. Also shown is the accretion timescale. It scales exactly as the ionneutral friction time does. 

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In the text 
Fig. 4 Comparison between the models and the filaments width distribution (adapted from Arzoumanian et al. 2011). 

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In the text 