Gravitational instability of non-isothermal filamentary molecular clouds, in presence of external pressure
Mohammad Mahdi Motiei, Mohammad Hosseinirad, Shahram Abbassi
MMNRAS , 1–14 (2021) Preprint 4 February 2021 Compiled using MNRAS L A TEX style file v3.0
Gravitational instability of non-isothermal filamentary molecularclouds, in presence of external pressure
Mohammad Mahdi Motiei, (cid:63) Mohammad Hosseinirad , and Shahram Abbassi Department of Physics, School of Sciences, Ferdowsi University of Mashhad, Mashhad, PO Box 91775-1436, Iran School of Astronomy, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5531, Tehran, Iran
Accepted . Received ; in original form
ABSTRACT
Filamentary molecular clouds are omnipresent in the cold interstellar medium. Ob-servational evidences show that the non-isothermal equations of state describe thefilaments properties better than the isothermal one. In this paper we use the loga-tropic and the polytropic equations of state to study the gravitational instability ofthe pressure-confined filaments in presence of a uniform axial magnetic field. To fullyexplore the parameter space we carry out very large surveys of stability analysis thatcover filaments with different radii in various magnetic fields. Our results show thatfor all the equations of state the instability of thinner filaments is more sensitive to themagnetic field variations than the thicker ones. Moreover, for all the equations of state,an intermediate magnetic field can entirely stabilize the thinner filaments. Albeit forthe thicker ones this effect is suppressed for the magnetic field stronger than B (cid:39) µ G. Key words: instabilities – MHD – ISM: clouds – methods: numerical.
The cold interstellar molecular gas in the Galaxy has beenrevealed to have filamentary structures of parsec-scale (0.5 -100 pc) (e.g. Schneider & Elmegreen 1979; Bally et al. 1987;Goldsmith et al. 2008; Andr´e 2017), regularly harbouringclumps and dense cores (e.g. Bergin & Tafalla 2007; Miet-tinen & Harju 2010; Jackson et al. 2010; Wang et al. 2011;Miettinen 2012; Wang et al. 2014; Contreras et al. 2016; Hen-shaw et al. 2016; Wang et al. 2016; Feng et al. 2016; Kainu-lainen et al. 2017). The filamentary molecular clouds (MCs)unveiled by unprecedented images of
Herschel Space Obser-vatory (Pilbratt et al. 2010), represent a common width of ∼ . pc (Arzoumanian et al. (2019), see also Panopoulouet al. 2017; Andr´e 2017; Roy et al. 2019, for a recent de-bate on the existence of such a universal width) at leastin the nearby Gould belt, while they extend over a widerange in column density. The filamentary MCs, are iden-tified both in non-star-forming (Men’shchikov et al. 2010;Miville-Deschˆenes et al. 2010; Ward-Thompson et al. 2010)and star-forming (K¨onyves et al. 2010; Bontemps et al. 2010)regions which emphasizes their importance to better under-stand the theory of star formation (Andr´e et al. 2014).In the filamentary picture of formation of stars, thelarge-scale turbulent flows are assembled into a network of (cid:63) E-mail: [email protected] (MMM); [email protected] (MH)[email protected] (SA) filaments due to the supersonic shocks (e.g. Klessen et al.1998; McKee & Ostriker 2007; Dib et al. 2007; Padoan et al.2014; Pudritz & Kevlahan 2013) or combination with themagnetic field which is most probably perpendicular to thefilaments (e.g. Nakamura & Li 2008; Chen & Ostriker 2014;Inutsuka et al. 2015; Federrath 2016; Klassen et al. 2017; Li& Klein 2019). It is also possible that the global collapse ofthe parent cloud under its self-gravity, governs the formationprocess (e.g. Burkert & Hartmann 2004; Hartmann & Burk-ert 2007; V´azquez-Semadeni et al. 2007; G´omez & V´azquez-Semadeni 2014; Wareing et al. 2016; Camacho et al. 2016).If the filaments are gravitationally unstable, they will frag-ment onto the cores and finally form clusters of stars (Lada& Lada 2003) provided that the conditions for the subse-quent fragmentation are met.Filaments have been subject to many investigationssince almost the mid-point of the twentieth century, whenthe groundbreaking work by Chandrasekhar & Fermi (1953)showed that a poloidal magnetic field is able to completelystabilize a very long uniform incompressible cylinder of gas.Ten years later, Stod´olkiewicz (1963) derived the magneto-static equilibrium of an isothermal gas cylinder threaded bya longitudinal magnetic field proportional to the square rootof its initial density. Physical explanation of the filamentaryclouds was interesting enough to encourage other authorsfor more detailed theoretical investigations (see e.g. Larson1985; Nagasawa 1987; Inutsuka & Miyama 1992; Nakamuraet al. 1993; Matsumoto et al. 1994; Gehman et al. 1996a,b; © a r X i v : . [ a s t r o - ph . GA ] F e b Motiei et al.
Inutsuka & Miyama 1997; Fischera & Martin 2012a; Fre-undlich et al. 2014; Hanawa & Tomisaka 2015; Sadhukhanet al. 2016; Hosseinirad et al. 2017; Hanawa et al. 2017;Hanawa et al. 2019). In addition, various simulations of thecylindrical geometry have been performed for more realisticstudies in non-linear regime (see e.g. Steinacker et al. 2016;Gritschneder et al. 2016; Heigl et al. 2016, 2018, 2020; Ntor-mousi & Hennebelle 2019; Clarke et al. 2017; Clarke et al.2020).Filaments are seldom found in isolation but under theexternal pressure of the ambient medium (Fischera & Mar-tin 2012a; Fischera & Martin 2012b). Nagasawa (1987) per-formed global linear stability analysis for an infinitely longisothermal magnetized filament. Nagasawa (1987) showedthat both a non-confined and a pressure-confined filamentare gravitationally unstable for a specific range of wave-lengths. More specifically, Nagasawa (1987) found that apoloidal magnetic field can increase the stability of a fila-ment and interestingly entirely stabilize it if the filament isthin enough. The linear stability analysis of self-gravitatingobjects under the effect of external pressure in other environ-ments, have been also the matter of many studies (see e.g.Miyama et al. 1987; Nagai et al. 1998; Durrive & Langer2019, for sheet-like gas layers and also Chou et al. 2000;Lee & Hong 2007; Kim et al. 2012, for gas disks). Recently,Anathpindika & Di Francesco (2020) have reported the ex-ternal pressure could affect the peak central density, the col-umn density, the morphology and the star formation of thefilaments.Based on the column density maps extracted from
Her-schel images (Arzoumanian et al. 2011; Juvela et al. 2012;Palmeirim et al. 2013), the radial density profiles of thefilaments in Gould belt, could not be properly describedby a simple isothermal model (Stod´olkiewicz 1963; Ostriker1964), but instead they are best fitted by the softer (i.e. pro-files which are shallower at distances away from the centre)polytropic models with indices γ p < (Palmeirim et al. 2013;Toci & Galli 2015). Moreover, for the filaments in the IC5146region, a modification to the simple isothermal model thatsupposes a very long subcritical pressure-confined cylinderwith different masses per unit length, can account for thisproblem (Fischera & Martin 2012a; see also Heitsch 2013for a similar but accreting model), which is also the casefor a near-critical cylinder wrapped by a helical magneticfield (Fiege & Pudritz 2000b). In addition, polytropic fil-aments with indices less than but near the unity that areundergoing gravitational collapse, have also shallow densityprofiles (Kawachi & Hanawa 1998; Nakamura & Umemura1999; Shadmehri 2005).In a recent paper, Hosseinirad et al. (2018, hereafterH18) carried out a similar analysis to the work that hadbeen done by Nagasawa (1987), but for the aforementionedpolytropic equation of state (PEOS) as well as the logatropicequation of state (LEOS) (Lizano & Shu 1989). They usedthe non-ideal magnetohydrodynamic (MHD) framework fora filament threaded by a poloidal magnetic field in the ab-sence of the external pressure. They found that without theeffect of magnetic field, filaments with these two softer typesof equations of state (EOSs) are more susceptible to thegravitational instability than a filament with the isothermalEOS (IEOS). More specifically, they realized that while thegravitational instability in a moderate magnetized filament is generally sensitive to the type of EOS, the instability issuppressed in the strongly magnetized one, regardless of itsEOS type. Here, we aim to elucidate how a pressure-confinedfilament with the LEOS or the PEOS responds to the lin-ear perturbations, therefore combining the study by Naga-sawa (1987) and H18, albeit in the ideal MHD for simplicity.We will investigate this problem in the non-ideal MHD in aforthcoming paper.The outline of this paper is as follows. In Section 2.1,we explain the ideal MHD equations considering self-gravity.Section 2.2 introduces the non-isothermal EOSs we use inthis paper. The equilibrium state and perturbations are de-scribed in sections 2.3 and 2.4, respectively. Section 2.5 dealswith the boundary conditions. Computation method is givenin section 2.6. Section 3 and Section 4 contain the results andconclusions of this investigation. We consider an infinitely long cylinder of gas with finiteradius as the filament. The filament is threaded by a uni-form magnetic field parallel to its long axis so B = (0 , , B z ) which does not affect the unperturbed structure. Our set ofequations include the equation of motion (1), the inductionequation (2), the continuity equation (3) and the Poisson’sequation (4) as ρ ∂ u ∂ t + ρ ( u · ∇ ) u + ∇ p + ρ ∇ ψ − π ( ∇ × B ) × B = , (1) ∂ B ∂ t + ∇ × ( B × u ) = , (2) ∂ ρ∂ t + ∇ · ( ρ u ) = , (3) ∇ ψ = π G ρ. (4)In equations (1) to (4), ρ , u , p , B and ψ indicate the neu-tral gas density, the velocity of the fluid, the gas pressure,the magnetic field strength and the gravitational potential,respectively. Equations (1) to (4) must be accompanied by a relation be-tween the pressure and the density in order to be complete.There are many studies that use the IEOS (see e.g. Naga-sawa 1987; Inutsuka & Miyama 1992; Fischera & Martin2012b; Heigl et al. 2016; Hosseinirad et al. 2017). We con-sider three different types of non-isothermal EOSs. In thefollowing we describe them briefly.
By applying a non-isothermal barotropic equation of state,Gehman et al. (1996a,b) studied the observed turbulenceeffect in MCs. They added a term to the IEOS in order tomodel this effect. This EOS is softer than the IEOS and there
MNRAS000
MNRAS000 , 1–14 (2021) ravitational instability of molecular clouds is theoretical and empirical support for using that (Gehmanet al. 1996a,b, and Lizano & Shu 1989). They proposed theGEOS form as p = c ρ + P ln( ρ/ρ c ) . (5)In this equation c s and ρ c are the isothermal sound speedand the density at the filament axis, respectively. P is anempirical constant which its value changes between 10 and70 picodynes cm − (Gehman et al. 1996b). McLaughlin & Pudritz (1996) considered a pure logarithmicEOS as p = p c [1 + A ln( ρ/ρ c )] , (6)where p c and ρ c are the pressure and the density along the fil-ament axis, respectively and A is an empirical constant about0.2 for molecular cloud cores. They claimed that this EOSis the simplest and the most successful model that containsthe important properties of the giant MCs and their internalstructures such as cores. Also Fiege & Pudritz (2000a) foundthat this logatropic model is in agreement with the existingdata and although it was based on the core data, they usedthe same value of A for the filamentary clouds. This EOS isthe softest one among the EOSs we use in this text. Palmeirim et al. (2013) argued that the structure of B211filament in the Taurus MC is well described by a poly-tropic cylindrical filament with an EOS as p ∝ ρ γ where γ = . ± . . Toci & Galli (2015) analysed the observa-tional properties of the filamentary clouds in the cylindri-cal symmetry with the PEOS and the polytropic exponent / (cid:62) γ p (cid:62) / (the polytropic indices − (cid:62) n (cid:62) − . ) where γ p = + / n . In a more general way −∞ < n < − (Viala &Horedt 1974; Maloney 1988). In this paper we use the PEOSwith negative index as p = p c ( ρ/ρ c ) + / n , (7)where p c and ρ c are the same as in equation (6). The di-mensionless forms of equations (5) to (7) are available inappendix A. We use the cylindrical coordinates ( r , φ, z ) by assuming thefilament centre at the origin. The filament is very long andits radius is confined. The initial magnetic field B = B ˆ z isuniform and has not any effect on the equilibrium of the fil-ament. Solving a combination of equations (1) and (4) givesus the density profile at the equilibrium state. For a filamentwith the IEOS we have ρ ( r ) = ρ c (1 + r H ) − , (8)where H is the radial scale length (Stod´olkiewicz 1963; Os-triker 1964) as H = c s (cid:112) π G ρ c (9) where G is the gravitational constant. For a typical MCwith the central density of 4 × − g cm − and the ther-mal sound speed of 0.2 km s − , H ≈ . pc. The magneticfield strength B is in the unit of (4 πρ c ) / c s and consideringthe mentioned values for ρ c and c s , B = is equivalent with B (cid:39) . µ G. See Appendix A for more details. Since thereare no analytical solutions for the non-isothermal EOSs pre-sented in § κ = . , . , . , in the GEOS where κ = P / ( c ρ c ) , the dimensionless pa-rameter A = . in the MPEOS and the polytropic indices n = − . , − , − , − in the PEOS. By applying a small perturbation of δ r to the surface of thefilament (Gehman et al. 1996b), the perturbed version ofequations (1) to (4) in dimensionless form to the first ordergives us ρ ∂ u ∂ t + ∇ p + ρ ∇ ψ + ρ ∇ ψ − ( ∇ × B ) × B = , (10) ∂ B ∂ t + ∇ × ( B × u ) = , (11) ∂ ρ ∂ t + ∇ ρ · u + ρ ∇· u = , (12) ∇ ψ = ρ . (13)In these equations, the subscript 0 indicates the unperturbedparameters and the subscript 1 shows the perturbed quan-tities. Since our EOSs are barotropic, we can linearize themas p = dpd ρ ( ρ ) ρ ≡ p (cid:48) ( ρ ) ρ . (14)On the other hand the density, the velocity, the magneticfield and the gravitational potential are as ρ ( x , t ) = R (cid:2) f ( r ) exp ( ikz − i ω t ) (cid:3) , (15) u ( x , t ) = R (cid:2) v ( r ) exp ( ikz − i ω t ) (cid:3) , (16) B ( x , t ) = R (cid:2) b ( r ) exp ( ikz − i ω t ) (cid:3) , (17) ψ ( x , t ) = R (cid:2) φ ( r ) exp ( ikz − i ω t ) (cid:3) , (18)where R refers to the real part, k is the wave number (alongthe filament axis) and ω denotes to the angular frequency. f ( r ) , v ( r ) , b ( r ) and φ ( r ) are the amplitudes of the perturba-tions. We use these forms and equations (10) to (12) in orderto get the linearised dimensionless forms by considering w = i ω v r in the following equations ρ ( ddr + r ) w + ( ω − k P (cid:48) ) f − k ρ φ + w d ρ dr = , (19) (cid:20) P (cid:48) + B ρ (cid:18) − k ω P (cid:48) (cid:19)(cid:21) d fdr + A f + (cid:18) ρ − k ω B (cid:19) d φ dr + A φ + A w = , (20) MNRAS , 1–14 (2021)
Motiei et al. ( ddr + r ) d φ dr − k φ − f = , (21)where A = − k B ω ρ (cid:18) P (cid:48)(cid:48) − P (cid:48) ρ (cid:19) d ρ dr + (cid:18) P (cid:48)(cid:48) − B ρ (cid:19) d ρ dr + d ψ dr , (22) A = k B ω ρ d ρ dr , (23) A = B ω (cid:20) k + ρ d ρ dr − (cid:18) ρ d ρ dr (cid:19) − r ρ d ρ dr (cid:21) − ρ . (24) In this section we closely follow Nagasawa (1987) approachfor obtaining boundary conditions. The external pressureconfines the filament to the finite radius R . The perturbedsurface of this filament will have the radius r = R + δ r exp( ikz − i ω t ) . (25)On the deformed surface of the filament, the r -componentof the velocity is then defined as v r ( R ) = − i ωδ r . (26)On the other hand the pressure on the boundary must beequal to the external pressure. This leads to p ( R ) = ρ ( R ) + κ ln[ ρ ( R )] (GEOS) , + A ln[ ρ ( R )] (MPEOS) ,ρ ( R ) + / n (PEOS) , (27)in the dimensionless form and to the first order of the per-turbation we will have dp dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R δ r + p ( R ) + B B z ( R ) = B B ext z ( R ) , (28)It is necessary for the gravitational potential and its radialderivative to be continuous on the border. So ψ ( R ) = ψ ext ( R ) , (29) d ψ dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + ρ ( R ) δ r = d ψ ext dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R . (30)We consider a very hot and low density environment outof the filament ( r > R ) . We solve Laplace’s equation for ex-terior gravitational potential in cylindrical coordinates andwrite the solution with the modified Bessel function of thesecond type and order m ( K m ). Because we only investigateaxisymmetric and unstable modes, we use m = order inthe modified Bessel function, so we restrict ourselves to theaxisymmetric mode ( m = ). This is because it was shownby (Nagasawa 1987) that non-axisymmetric modes ( m (cid:62) )are stable against perturbation. This will let us to recastequation (30) into d ψ dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + ρ ( R ) δ r = − k K ( kR ) K ( kR ) ψ ( R ) . (31)Furthermore, we consider that there is no electric currentoutside the filament. So, B z should be continuous on the Table 1.
Units of the filament radius ( R ) , the magnetic fieldstrength ( B ) , the fastest perturbation growth rate ( ω fast ) andthe critical wave number ( k critic ) for a typical MC with thecentral density of 4 × − g cm − and the thermal sound speedof 0.2 km s − .Parameter Unit Parameter = 1 is equivalent with R H
B c s (cid:112) πρ c µ G ω fast (cid:112) π G ρ c . Myr − k critic / H − boundary. Considering equations (27) and (28) we will havefor the GEOS (cid:18) + κρ (cid:19)(cid:18) ρ + d ρ dr δ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + B B z ( R ) = − B iK ( kR ) K ( kR ) B r ( R ) , (32)for the MPEOS (cid:18) A ρ (cid:19)(cid:18) ρ + d ρ dr δ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + B B z ( R ) = − B iK ( kR ) K ( kR ) B r ( R ) , (33)and for the PEOS (cid:18) + n (cid:19) ρ / n (cid:18) ρ + d ρ dr δ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R + B B z ( R ) = − B iK ( kR ) K ( kR ) B r ( R ) . (34)We set the boundary conditions along the filament axis ( r = as f = , d φ dr = , w = . (35)Eventually, equation (31), one of the equations (32) to (34)and equation (35) are our boundary conditions. Applying the boundary conditions at the centre and the sur-face of the filament, one can solve equations (19) to (21)which demonstrate a disguised eigenvalue problem. This canbe done by different methods. Here, we finite difference ourequations over a 2000 point equally spaced mesh grid. Thisgives rise to a system of algebraic block-tridiagonal matrixequations which could be solved with any standard matrixsolver. We take advantage of a flexible relaxation techniquebased on the Newton-Raphson-Kantorovich (NRK) stan-dard algorithm (Garaud 2001). This algorithm needs an ini-tial guess to advance. For non-magnetic calculations, thealgorithm converges rapidly after a few iterations using areasonable guess for each of the dependent variables. Forcases with the magnetic field, we start with the previousnon-magnetic results as the initial guess. To calculate dis-persion relation, we fix ω and consider the eigenvalue k asa dependent variable. The algorithm successively adjusts k along with the other dependent variables until it converges. By considering the effect of external pressure of the en-vironment which confines the filament boundary, we try
MNRAS000
MNRAS000 , 1–14 (2021) ravitational instability of molecular clouds .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k − . − . − . − . − . . ω R = 1 , Isothermal .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k R = 2 , Isothermal .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k R = 3 , Isothermal .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k R = 4 , Isothermal B = 5 B = 0.5 B = 0.2 B = 00 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / − . − . − . − . − . . ω R = 1 , κ = 0.1 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 2 , κ = 0.1 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 3 , κ = 0.1 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 4 , κ = 0.1 B = 5 B = 0.5 B = 0.2 B = 00 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / − . − . − . − . − . . ω R = 1 , κ = 0.2 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 2 , κ = 0.2 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 3 , κ = 0.2 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 4 , κ = 0.2 B = 5 B = 0.5 B = 0.2 B = 00 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / − . − . − . − . − . . ω R = 1 , κ = 0.5 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 2 , κ = 0.5 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 3 , κ = 0.5 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 . k (1 + κ ) / R = 4 , κ = 0.5 B = 5 B = 0.5 B = 0.2 B = 0 Figure 1.
The GEOS dispersion relations for κ = (isothermal), . , . , . , different filament radii ( R ). R = is equivalent with R (cid:39) . pc and B = is equivalent with B (cid:39) µ G. In each panel the vertical axis is ω in the unit of π G ρ c and the horizontal axes are the wavenumbers ( k ) which is multiplied by (1 + κ ) / to account for using the thermal sound speed as the velocity unit. Units are given in Table1. For R = , B = and κ (cid:62) . , | ω | is too small to be recognized. to study the stability of the filament under influence ofvarious magnetic field strengths and different EOSs. Foreach EOS dispersion relation, we select the minimum of ω or equivalently the maximum of | ω | and substitute it in ω fast = (cid:112) | ω | to find the fastest perturbation growth rate.A system with larger ω fast is more prone to the instabil-ity and vice versa. So ω fast is a good indicator in order tostudy the instability of filaments. Also we define k critic asthe non-zero wave number corresponding to ω = , herewhich is also the largest unstable one. The wave numberswith ω < and k < k critic are unstable. The unit of the fila-ment radius ( R ) , the magnetic field strength ( B ) , the fastestperturbation growth rate ( ω fast ) and the critical wave num- ber ( k critic ) and their equivalent values when they are equalto 1 are summarized in Table 1. We adopt different val-ues for the filament radius, the magnetic field strengths, theturbulence parameters ( κ in the GEOS) and the polytropicindices ( n in the PEOS) to get as high as possible resolu-tion for dispersion relations in the different EOSs. For ex-ample, we set R = . , . , . , , . , ..., . , , . , B = , . , . , . , . , . , . , , , , and κ = , . , . , . , for the GEOS to provide the dispersion relations and investi-gate the effect of R , B and κ on the instability of this type offilaments. Albeit settings κ to the values larger than 1 leadsthe minimum value of | ω | to become about zero for the radiiless than 1 and for all the magnetic field strengths. It is worth MNRAS , 1–14 (2021)
Motiei et al. noting that R = is equivalent with R = H (cid:39) . pc andconsidering the density of 4 × − g cm − and the thermalsound speed of 0.2 km s − , B = is equivalent with B (cid:39) . µ G. Similarly we set R = . , . , . , , , , , , and A = . for the MPEOS with the same magnetic fieldsto achieve dispersion relations. For the PEOS filaments,we use the same R and B values as for the MPEOS and n = − . , − , − , − .Fig. 1 shows the dispersion relations for the IEOS (aGEOS with κ = ) and the GEOS. Although we calculatethe dispersion relations using so many values for the fila-ment radii, the magnetic field strengths and the turbulenceparameters ( κ ) , here, we draw the dispersion relations foronly R = , , , and B = , . , . , in order to observetheir impact on the stability of the filament. Panels in eachcolumn of Fig. 1 show specific radii and ones in each rowrepresent filaments with a specific turbulence parameter. AsNagasawa (1987), Gehman et al. (1996b) and H18 reported,because of the magnetic field disability to prevent fluid con-traction along the field direction, dispersion relation curvesare nearly overlapped for the magnetic fields larger than B (cid:39) or (cid:39) µ G. We compare the dispersion relations ofour models with the work by Nagasawa (1987) and find thatthe two studies reproduce fairly similar results. In all thepanels of Fig. 1 it is obvious that increasing the magneticfield leads to more stability by decreasing ω fast . This alsocauses to the reduction of k critic value. We can see for each κ and in a specific magnetic field, when the radius of a filamentincreases, ω fast increases as well and so the filament becomesmore unstable. With a focus on the radius of filaments, theinstability of filaments with smaller radii are more sensitiveto the magnetic field strength than the larger ones. On theother hand for a filament with a specific radius and in a fixedmagnetic field, when κ increases, ω fast decreases and so thefilament becomes more stable. Interestingly, this is in con-trast to the behaviour of infinite filaments (here by infinitewe mean large in radius) (H18). In addition, clearly none ofthe models represent the radial instability (RI) which occursat a non-zero ω corresponding to k = , in any panel thatis again in contrast to the infinite filaments (H18).To understand the problem with more details, we per-formed very large surveys of stability analysis that includethe effect of various magnetic field strengths, different radiiand several values of κ . Fig. 2 represents the result of foursurveys for the GEOSs with κ = . , . , . and 1. In eachpanel, the colour bar shows ω fast . The darker colour the larger ω fast and the more unstable filament. Areas with lightercolour, have lower ω fast and so represent more stable models.It is clear that the lower right region of each panel whichrepresents the larger radii i.e. the thicker filaments in thelower magnetic field strength, is the most unstable area. Incontrast, the upper left region in each panel which indicatesthe thinner filaments in the stronger magnetic fields are themost stable ones. In particular, if we compare panels witheach other, we find that when κ increases, the colour of aspecific area becomes lighter that means the correspondingfilament is more stable. It is also of the note that there aresome areas with different radii and magnetic fields but withthe same colour i.e. the same ω fast . This means that just byknowing the ω fast , it is not possible to find the magnetic fieldin a filament or its radius exclusively.In Fig. 3 we can see the effect of filament radius and the magnetic field on k critic in the GEOS dispersion relation forthe same κ values as the Fig. 2. This figure shows that k critic has very weak dependency on κ on the whole. Furthermore, ifwe compare this figure with Fig. 2, we find that models thatare in the upper left corner of the panels i.e. the thinnerfilaments in the stronger magnetic fields, which have thesmallest ω fast , have also the smallest k critic . On the other hand,there is an area with larger k critic that contains the thinnerfilaments in the weaker magnetic field strengths (the lowerleft region in all the panels of Fig. 3). It is also observed thatin all the panels, by increasing the filament radii in magneticfields B (cid:38) . ( log B (cid:38) − . , indicated by horizontal dashedline), k critic also increases. However in B < . , an inversetrend is observed. B = . is equivalent with B (cid:39) . µ G.Notwithstanding this different treatment in the stronger andthe weaker magnetic field regimes, k critic is converged to about0.5 for R (cid:38) . in all the panels.Fig. 4, demonstrates the dispersion relations for theMPEOS. Similar to the GEOS, the dispersion relations wereplotted for several values of the filament radius, the samemagnetic field strengths and A = . . Because for R (cid:38) ,there are no noticeable changes in the dispersion relations,in this figure, we select only R = . , . , . , , , , in or-der to see ω fast and k critic variations clearly. Each panel showsthe dispersion relation for the aforementioned filament radii.Like the GEOS dispersion relations (see Fig. 1), for all theradii, increasing the magnetic field, increases the stability ofthe filament. Yet also the stabilization effect due to the mag-netic field is saturated for B > (Nagasawa 1987; Gehmanet al. 1996b, H18). In small radii, the effect of magnetic fieldon the dispersion relation is stronger. As an example a fila-ment with the radius of R = . is almost stable when B = .This is also the case for a smaller radius of R = . , but foran order of magnitude weaker magnetic field. This filamentis completely stable when B = . It should be also noted thatfor the radius of R = in the absence of magnetic field andfor the radius of R = also in presence of a relatively weakmagnetic field of B = . the filament is radially unstable. Inaddition, it is noticeable that RI disappears entirely when B (cid:62) . .Much the same as the GEOS, the stability analysis iselaborated in the left-hand panel of Fig. 5 which shows in-stability of the MPEOS filaments with different radii in vari-ous magnetic fields with more details. Comparing this panelwith those of Fig. 2, one can see that the general treatmentis the same but the MPEOS is a little more unstable thanthe GEOSs. By incrementally increasing the magnetic field,same as the GEOS, all the filaments become more stablegradually. Moreover, the upper left region has the lightestcolour and manifestly shows that the most stable filamentsare the thinnest ones in the strongest magnetic field regime.Furthermore, it should be noted that similar to the Fig. 2for the GEOS, some models have the same ω fast in spite ofhaving different radii because of the effect of magnetic field.As it could be observed in the panel, like the GEOS, in-creasing the radius leads to more instability regardless ofthe magnetic field strength. For R > , increasing the radiushas not noticeable effect on the instability for all the mag-netic field strength. The right-hand panel of Fig. 5 showsthe effect of filament radius and the magnetic field on k critic .Here, the overall picture is very similar to the GEOSs, spe-cially one with κ = . . The only difference is that the very MNRAS000
Motiei et al. noting that R = is equivalent with R = H (cid:39) . pc andconsidering the density of 4 × − g cm − and the thermalsound speed of 0.2 km s − , B = is equivalent with B (cid:39) . µ G. Similarly we set R = . , . , . , , , , , , and A = . for the MPEOS with the same magnetic fieldsto achieve dispersion relations. For the PEOS filaments,we use the same R and B values as for the MPEOS and n = − . , − , − , − .Fig. 1 shows the dispersion relations for the IEOS (aGEOS with κ = ) and the GEOS. Although we calculatethe dispersion relations using so many values for the fila-ment radii, the magnetic field strengths and the turbulenceparameters ( κ ) , here, we draw the dispersion relations foronly R = , , , and B = , . , . , in order to observetheir impact on the stability of the filament. Panels in eachcolumn of Fig. 1 show specific radii and ones in each rowrepresent filaments with a specific turbulence parameter. AsNagasawa (1987), Gehman et al. (1996b) and H18 reported,because of the magnetic field disability to prevent fluid con-traction along the field direction, dispersion relation curvesare nearly overlapped for the magnetic fields larger than B (cid:39) or (cid:39) µ G. We compare the dispersion relations ofour models with the work by Nagasawa (1987) and find thatthe two studies reproduce fairly similar results. In all thepanels of Fig. 1 it is obvious that increasing the magneticfield leads to more stability by decreasing ω fast . This alsocauses to the reduction of k critic value. We can see for each κ and in a specific magnetic field, when the radius of a filamentincreases, ω fast increases as well and so the filament becomesmore unstable. With a focus on the radius of filaments, theinstability of filaments with smaller radii are more sensitiveto the magnetic field strength than the larger ones. On theother hand for a filament with a specific radius and in a fixedmagnetic field, when κ increases, ω fast decreases and so thefilament becomes more stable. Interestingly, this is in con-trast to the behaviour of infinite filaments (here by infinitewe mean large in radius) (H18). In addition, clearly none ofthe models represent the radial instability (RI) which occursat a non-zero ω corresponding to k = , in any panel thatis again in contrast to the infinite filaments (H18).To understand the problem with more details, we per-formed very large surveys of stability analysis that includethe effect of various magnetic field strengths, different radiiand several values of κ . Fig. 2 represents the result of foursurveys for the GEOSs with κ = . , . , . and 1. In eachpanel, the colour bar shows ω fast . The darker colour the larger ω fast and the more unstable filament. Areas with lightercolour, have lower ω fast and so represent more stable models.It is clear that the lower right region of each panel whichrepresents the larger radii i.e. the thicker filaments in thelower magnetic field strength, is the most unstable area. Incontrast, the upper left region in each panel which indicatesthe thinner filaments in the stronger magnetic fields are themost stable ones. In particular, if we compare panels witheach other, we find that when κ increases, the colour of aspecific area becomes lighter that means the correspondingfilament is more stable. It is also of the note that there aresome areas with different radii and magnetic fields but withthe same colour i.e. the same ω fast . This means that just byknowing the ω fast , it is not possible to find the magnetic fieldin a filament or its radius exclusively.In Fig. 3 we can see the effect of filament radius and the magnetic field on k critic in the GEOS dispersion relation forthe same κ values as the Fig. 2. This figure shows that k critic has very weak dependency on κ on the whole. Furthermore, ifwe compare this figure with Fig. 2, we find that models thatare in the upper left corner of the panels i.e. the thinnerfilaments in the stronger magnetic fields, which have thesmallest ω fast , have also the smallest k critic . On the other hand,there is an area with larger k critic that contains the thinnerfilaments in the weaker magnetic field strengths (the lowerleft region in all the panels of Fig. 3). It is also observed thatin all the panels, by increasing the filament radii in magneticfields B (cid:38) . ( log B (cid:38) − . , indicated by horizontal dashedline), k critic also increases. However in B < . , an inversetrend is observed. B = . is equivalent with B (cid:39) . µ G.Notwithstanding this different treatment in the stronger andthe weaker magnetic field regimes, k critic is converged to about0.5 for R (cid:38) . in all the panels.Fig. 4, demonstrates the dispersion relations for theMPEOS. Similar to the GEOS, the dispersion relations wereplotted for several values of the filament radius, the samemagnetic field strengths and A = . . Because for R (cid:38) ,there are no noticeable changes in the dispersion relations,in this figure, we select only R = . , . , . , , , , in or-der to see ω fast and k critic variations clearly. Each panel showsthe dispersion relation for the aforementioned filament radii.Like the GEOS dispersion relations (see Fig. 1), for all theradii, increasing the magnetic field, increases the stability ofthe filament. Yet also the stabilization effect due to the mag-netic field is saturated for B > (Nagasawa 1987; Gehmanet al. 1996b, H18). In small radii, the effect of magnetic fieldon the dispersion relation is stronger. As an example a fila-ment with the radius of R = . is almost stable when B = .This is also the case for a smaller radius of R = . , but foran order of magnitude weaker magnetic field. This filamentis completely stable when B = . It should be also noted thatfor the radius of R = in the absence of magnetic field andfor the radius of R = also in presence of a relatively weakmagnetic field of B = . the filament is radially unstable. Inaddition, it is noticeable that RI disappears entirely when B (cid:62) . .Much the same as the GEOS, the stability analysis iselaborated in the left-hand panel of Fig. 5 which shows in-stability of the MPEOS filaments with different radii in vari-ous magnetic fields with more details. Comparing this panelwith those of Fig. 2, one can see that the general treatmentis the same but the MPEOS is a little more unstable thanthe GEOSs. By incrementally increasing the magnetic field,same as the GEOS, all the filaments become more stablegradually. Moreover, the upper left region has the lightestcolour and manifestly shows that the most stable filamentsare the thinnest ones in the strongest magnetic field regime.Furthermore, it should be noted that similar to the Fig. 2for the GEOS, some models have the same ω fast in spite ofhaving different radii because of the effect of magnetic field.As it could be observed in the panel, like the GEOS, in-creasing the radius leads to more instability regardless ofthe magnetic field strength. For R > , increasing the radiushas not noticeable effect on the instability for all the mag-netic field strength. The right-hand panel of Fig. 5 showsthe effect of filament radius and the magnetic field on k critic .Here, the overall picture is very similar to the GEOSs, spe-cially one with κ = . . The only difference is that the very MNRAS000 , 1–14 (2021) ravitational instability of molecular clouds R − . − . − . − . . . . l og B κ = 0.1 . . . . . . ω f a s t R − . − . − . − . . . . l og B κ = 0.2 . . . . . . ω f a s t R − . − . − . − . . . . l og B κ = 0.5 . . . . . . ω f a s t R − . − . − . − . . . . l og B κ = 1 . . . . . . ω f a s t Figure 2.
Effect of the filament radius ( R ) and the magnetic field ( B ) on the GEOS filament instability with κ = . (upper left panel), κ = . (upper right panel), κ = . (lower left panel) and κ = (lower right panel). In each panel, the horizontal axis shows the radius offilaments, the vertical axis shows the logarithm of magnetic field strength and the colour bar represents ω fast . Units are given in Table 1.The darker shaded areas are more unstable. thin MPEOS filaments in the absence or presence of veryweak magnetic fields, have k critic almost twice larger thantheir GEOS counterparts.Fig. 6 represents the dispersion relations for the PEOS.As already mentioned in § n = − . , − and − as the polytropic indices and use R and B values same asin the MPEOS. We also compute the dispersion relation for n = − in order to examine the effect of smaller n on the in-stability. In this figure, panels in each row indicate filamentswith a specific polytropic index, while those in each columnshow specific radii. The filaments radii R > , do not shownoticeable changes in their dispersion relations, so we selectonly R = . , . , . , , , and in order to see theireffect on the filaments instability. Nevertheless, to see theimpact of these parameters on the shape of dispersion rela-tions more clearly, we draw the plots only for R = , , and and B = , . , . and . Here, once more the magneticfield strength B > causes the dispersion relation to nearlyoverlap similar to the GEOS and the MPEOS (Nagasawa1987; Gehman et al. 1996b, H18). In all the panels of Fig. 6it is obvious that increasing the magnetic field leads to the more stability by decreasing ω fast and for R (cid:46) , reductionof k critic value. H18 showed that the infinite filaments withthe PEOS are prone to the radial instability and a stronglyenough magnetic field could suppress the instability. Focus-ing on the R = and R = panels, we see that this is alsothe case for the pressure confined filaments with PEOSs andrelatively large radii. It should be noted that the smaller ra-dius is radially unstable just for n = − . while the larger oneis unstable for all the polytropic indices. The occurrence ofRI could be probably due to this fact that the effective soundspeed decreases when the density of the filament increases.For R < there is no sign of RI in the dispersion relations.In the following, the effect of R , B and n on the instabil-ity of filaments with the PEOS is studied by leveraging theresult of four large surveys in more details. Fig. 7 illustratesthe outcome for ω fast . The upper left stable region is a com-mon clear feature of all the panels which states that similarto the previous GEOS and MPEOS, the thinner filaments inthe stronger magnetic fields are the most stable ones. Look-ing at the panels, it is clear that the stability patterns aremore or less similar to the GEOS and MPEOS. All the four MNRAS , 1–14 (2021)
Motiei et al. R − . − . − . − . . . . l og B κ = 0.1 k c r i t i c R − . − . − . − . . . . l og B κ = 0.2 k c r i t i c R − . − . − . − . . . . l og B κ = 0.5 k c r i t i c R − . − . − . − . . . . l og B κ = 1 k c r i t i c Figure 3.
Effect of the filament radius ( R ) and the magnetic field ( B ) on the critical wave number ( k critic ) in the GEOS dispersion relationswith κ = . (upper left panel), κ = . (upper right panel), κ = . (lower left panel) and κ = (lower right panel). The horizontal andvertical axis in each panel shows the radius of filaments and the logarithm of magnetic field strength, respectively and the colour barrepresents k critic . Units are given in Table 1. The horizontal dashed line separates models with the magnetic fields larger than B (cid:39) . or log B (cid:39) − . from others in each panel. panels have a distinctive more stable region at the upperleft and an unstable region at the lower right corner. By de-creasing n , the former becomes a little larger while the latterfades out. This could indicate that the softer PEOSs (oneswith larger n ) are more unstable, possibly because these fil-aments have more mass per unit length. Moreover, like theGEOS and MPEOS by increasing the radius of a filament ina constant magnetic field, the stability decreases until R = or (cid:39) . pc where after this radius, the stability does notchange noticeably. Regarding the stabilizing effect of mag-netic field, the stability of filaments with smaller radii aremore sensitive to the magnetic field strength than the largerones. It is also worth noting that similar to Fig. 2 and 5,there is a degeneracy in determining B and R from a specific ω fast .Fig. 8 exhibits how the filament radius and the mag-netic field could affect k critic in the PEOSs with different n .The results are very similar to the GEOS and MPEOS, how-ever, one can see that for the thinner filaments in the low magnetic field regime, the GEOS has the greatest k critic whilethe MPEOS has the smallest one. In addition, by decreas-ing n , one can see that for the thinner filaments in the lowmagnetic field regime, k critic is a little increases. For all themodels, k critic is converged to (cid:39) . after R > .To further study the relationship between the criticalwavelength of the fragmentation ( λ critic = π/ k critic ) and theEOS, we calculate λ critic for the selected R , B , κ , n and A (Table 2). In this table because k critic is about 0 for some radii,magnetic fields and EOSs, we can not calculate λ critic forthose models (indicated as N/A in the table). As expected, λ critic is more sensitive to the magnetic field and the typeof EOS for thinner filaments. Also difference between themaximum and the minimum values of λ critic in the variousEOSs is greater for the filaments with smaller radii. MNRAS000
Effect of the filament radius ( R ) and the magnetic field ( B ) on the critical wave number ( k critic ) in the GEOS dispersion relationswith κ = . (upper left panel), κ = . (upper right panel), κ = . (lower left panel) and κ = (lower right panel). The horizontal andvertical axis in each panel shows the radius of filaments and the logarithm of magnetic field strength, respectively and the colour barrepresents k critic . Units are given in Table 1. The horizontal dashed line separates models with the magnetic fields larger than B (cid:39) . or log B (cid:39) − . from others in each panel. panels have a distinctive more stable region at the upperleft and an unstable region at the lower right corner. By de-creasing n , the former becomes a little larger while the latterfades out. This could indicate that the softer PEOSs (oneswith larger n ) are more unstable, possibly because these fil-aments have more mass per unit length. Moreover, like theGEOS and MPEOS by increasing the radius of a filament ina constant magnetic field, the stability decreases until R = or (cid:39) . pc where after this radius, the stability does notchange noticeably. Regarding the stabilizing effect of mag-netic field, the stability of filaments with smaller radii aremore sensitive to the magnetic field strength than the largerones. It is also worth noting that similar to Fig. 2 and 5,there is a degeneracy in determining B and R from a specific ω fast .Fig. 8 exhibits how the filament radius and the mag-netic field could affect k critic in the PEOSs with different n .The results are very similar to the GEOS and MPEOS, how-ever, one can see that for the thinner filaments in the low magnetic field regime, the GEOS has the greatest k critic whilethe MPEOS has the smallest one. In addition, by decreas-ing n , one can see that for the thinner filaments in the lowmagnetic field regime, k critic is a little increases. For all themodels, k critic is converged to (cid:39) . after R > .To further study the relationship between the criticalwavelength of the fragmentation ( λ critic = π/ k critic ) and theEOS, we calculate λ critic for the selected R , B , κ , n and A (Table 2). In this table because k critic is about 0 for some radii,magnetic fields and EOSs, we can not calculate λ critic forthose models (indicated as N/A in the table). As expected, λ critic is more sensitive to the magnetic field and the typeof EOS for thinner filaments. Also difference between themaximum and the minimum values of λ critic in the variousEOSs is greater for the filaments with smaller radii. MNRAS000 , 1–14 (2021) ravitational instability of molecular clouds . . . . . kA / − . − . − . − . − . . ω R = 0.25 , A = 0.2 . . . . . kA / R = 0.5 , A = 0.2 . . . . . kA / R = 0.75 , A = 0.2 . . . . . kA / R = 1 , A = 0.2 B = 5 B = 0.5 B = 0.2 B = 00 . . . . . . . kA / − . − . − . − . − . − . − . − . . ω R = 1 , A = 0.2 . . . . . . . kA / R = 2 , A = 0.2 . . . . . . . kA / R = 5 , A = 0.2 . . . . . . . kA / R = 10 , A = 0.2 B = 5 B = 0.5 B = 0.2 B = 0 Figure 4.
The MPEOS dispersion relations for different filament radii ( R ) and magnetic field strengths ( B ). The vertical axes are as Fig.1. The horizontal axes are the wave numbers ( k ) multiplied by A / to account for using the thermal sound speed as the velocity unit.Units are given in Table 1. For R = . and B > . , | ω | is too small to be recognized. Note to the different scale of the horizontal axesin the first and the second rows. The R = panel is repeated in the second row in order to better compare with the larger radii. − . . . . . R − . − . − . − . . . . l og B . . . . . . . . . ω f a s t − . . . . . R − . − . − . − . . . . l og B . . . . . . . . . k c r i t i c Figure 5.
Left-hand panel: Effect of the filament radius ( R ) and the magnetic field ( B ) on the instability of the MPEOS filaments. In eachpanel the horizontal axis shows the logarithm of filament radius, the vertical axis shows the logarithm of magnetic field strength andthe colour bar represents ω fast . The units of R , B and ω fast are as Fig. 2. The darker shaded areas are more unstable. Right-hand panel:Effect of the filament radius ( R ) and the magnetic field ( B ) on the critical wave number ( k critic ) in the MPEOS dispersion relation. Thehorizontal and vertical axes are as the left-hand panel and the unit of k critic are as Fig. 3. The horizontal dashed line separates modelswith the magnetic fields larger than B (cid:39) . or log B (cid:39) − . from others in each panel. Filamentary structures seem to be a natural early stage information of stars and clusters of stars. This has stimu-lated many investigations regarding the properties and evo-lution of these structures. In a pioneering work, Nagasawa(1987) showed that the pressure-confined filaments are grav-itationally unstable for a specific range of wavelengths and a poloidal magnetic field can increase their stability and in-terestingly entirely stabilize them if the filaments are thinenough.Recent observations show that the IEOS is not al-ways the best EOS for interpreting the filaments properties.Building on the work by Nagasawa (1987), H18 studied theinstability of filamentary MCs without the effect of exter-
MNRAS , 1–14 (2021) Motiei et al. . . . . . . k (1 + 1 /n ) / − . − . − . − . − . − . . ω R = 1 , n = -1.5 . . . . . . k (1 + 1 /n ) / R = 2 , n = -1.5 . . . . . . k (1 + 1 /n ) / R = 5 , n = -1.5 . . . . . . k (1 + 1 /n ) / R = 10 , n = -1.5 B = 5 B = 0.5 B = 0.2 B = 00 . . . . . . k (1 + 1 /n ) / − . − . − . − . − . − . . ω R = 1 , n = -2 . . . . . . k (1 + 1 /n ) / R = 2 , n = -2 . . . . . . k (1 + 1 /n ) / R = 5 , n = -2 . . . . . . k (1 + 1 /n ) / R = 10 , n = -2 B = 5 B = 0.5 B = 0.2 B = 00 . . . . . . k (1 + 1 /n ) / − . − . − . − . − . − . . ω R = 1 , n = -3 . . . . . . k (1 + 1 /n ) / R = 2 , n = -3 . . . . . . k (1 + 1 /n ) / R = 5 , n = -3 . . . . . . k (1 + 1 /n ) / R = 10 , n = -3 B = 5 B = 0.5 B = 0.2 B = 00 . . . . . . k (1 + 1 /n ) / − . − . − . − . − . − . . ω R = 1 , n = -4 . . . . . . k (1 + 1 /n ) / R = 2 , n = -4 . . . . . . k (1 + 1 /n ) / R = 5 , n = -4 . . . . . . k (1 + 1 /n ) / R = 10 , n = -4 B = 5 B = 0.5 B = 0.2 B = 0 Figure 6.
The PEOS dispersion relations for polytropic indices ( n ) of − . , − , − and − , different filament radii ( R ) and magnetic fieldstrengths ( B ). The vertical axes are as Fig. 1. The horizontal axes are the wave numbers ( k ) multiplied by (1 + / n ) / to account for usingthe thermal sound speed as the velocity unit. Units are given in Table 1. nal pressure, with the previously proposed non-isothermalEOSs, namely the GEOS, the MPEOS and the PEOS.In this paper, in a continuation of the previous work byH18, we have added the effect of external pressure. To thisaim, we use these three non-isothermal EOSs (described in § § R ) and mag-netic field ( B ). Moreover, by exploiting the growth rate ofthe fastest growing mode ( ω fast ) as a gravitational instabil-ity indicator, we are able to investigate the effect of filamentradius, magnetic field and type of EOS on the instability ofthe filaments. In summary, the results show that: (i) Similar to the infinite filaments, for all the EOSs, in-creasing the magnetic field strength, makes the pressure-confined filaments more stable.(ii) The instability in the thinner filaments is more sensi-tive to the magnetic field strength than the thicker ones.(iii) Unlike the infinite filaments, for the GEOS pressure-confined models considered in this study which have R (cid:46) . pc, for all the radii (specially larger ones), in a fixed magneticfield, when κ increases, the filaments become more stable.(iv) For all the EOSs, the thinner filaments are totallystabilized in an even intermediate magnetic field strength(e.g. models with R (cid:46) . pc in B (cid:38) µ G), while for thethicker ones this effect is suppressed for the magnetic fieldstronger than B (cid:39) µ G. MNRAS000
The PEOS dispersion relations for polytropic indices ( n ) of − . , − , − and − , different filament radii ( R ) and magnetic fieldstrengths ( B ). The vertical axes are as Fig. 1. The horizontal axes are the wave numbers ( k ) multiplied by (1 + / n ) / to account for usingthe thermal sound speed as the velocity unit. Units are given in Table 1. nal pressure, with the previously proposed non-isothermalEOSs, namely the GEOS, the MPEOS and the PEOS.In this paper, in a continuation of the previous work byH18, we have added the effect of external pressure. To thisaim, we use these three non-isothermal EOSs (described in § § R ) and mag-netic field ( B ). Moreover, by exploiting the growth rate ofthe fastest growing mode ( ω fast ) as a gravitational instabil-ity indicator, we are able to investigate the effect of filamentradius, magnetic field and type of EOS on the instability ofthe filaments. In summary, the results show that: (i) Similar to the infinite filaments, for all the EOSs, in-creasing the magnetic field strength, makes the pressure-confined filaments more stable.(ii) The instability in the thinner filaments is more sensi-tive to the magnetic field strength than the thicker ones.(iii) Unlike the infinite filaments, for the GEOS pressure-confined models considered in this study which have R (cid:46) . pc, for all the radii (specially larger ones), in a fixed magneticfield, when κ increases, the filaments become more stable.(iv) For all the EOSs, the thinner filaments are totallystabilized in an even intermediate magnetic field strength(e.g. models with R (cid:46) . pc in B (cid:38) µ G), while for thethicker ones this effect is suppressed for the magnetic fieldstronger than B (cid:39) µ G. MNRAS000 , 1–14 (2021) ravitational instability of molecular clouds − . . . . . R − . − . − . − . . . . l og B n = -1.5 . . . . . . . . ω f a s t − . . . . . R − . − . − . − . . . . l og B n = -2 . . . . . . . . ω f a s t − . . . . . R − . − . − . − . . . . l og B n = -3 . . . . . . . . ω f a s t − . . . . . R − . − . − . − . . . . l og B n = -4 . . . . . . . . ω f a s t Figure 7.
Effect of the filament radius ( R ) and the magnetic field ( B ) on the PEOS filaments instability with n = − . (upper left panel), n = − (upper right panel), n = − (lower left panel) and n = − (lower right panel). In each panel the horizontal axis shows the logarithmof the filament radius, the vertical axis shows the logarithm of magnetic field strength and the colour bar represents ω fast . The units of R , B and ω fast are as Fig. 2. The darker shaded areas are more unstable. Table 2. λ critic values in parsecs for the sample filaments radii( R ), the magnetic field strengths ( B ), the turbulence parame-ters ( κ ) in the GEOS, the polytropic indices ( n ) in the PEOSand the empirical constant ( A ) in the MPEOS. κ n AR (pc) B ( µ G) 0.0 1.0 -1.5 -4.0 0.20.018 00.0 0.102 0.073 0.171 0.117 0.2140.018 07.1 N/A a N/A 0.771 0.801 0.7130.018 70.9 N/A N/A 2.697 N/A 1.2500.035 00.0 0.193 0.142 0.301 0.219 0.3580.035 07.1 0.392 0.314 0.478 0.422 0.4910.035 70.9 1.324 6.505 0.599 0.970 0.5360.100 00.0 0.369 0.330 0.446 0.394 0.4800.100 07.1 0.395 0.381 0.452 0.416 0.4800.100 70.9 0.408 0.520 0.453 0.425 0.4800.175 00.0 0.391 0.419 0.452 0.414 0.4820.175 07.1 0.392 0.433 0.452 0.415 0.4820.175 70.9 0.393 0.451 0.452 0.416 0.482 a Not available data for this model (see the text). (v) There is no RI in the GEOS pressure-confined fila-ments. This is in contrast to the infinite GEOS filaments.(vi) In the absence of magnetic field, the MPEOS and thePEOS with n = − . and R (cid:38) . pc are radially unstable.The twice broader filaments of these two EOSs can also beradially unstable in presence of a weak magnetic field B (cid:39) µ G. The RI in the other less softer PEOSs ( n = − , − and − ) with the radius R = . pc is still suppressed by a weakmagnetic fields of B (cid:39) µ G for the first and B (cid:39) µ G for thenext two ones.(vii) In the PEOS, decreasing n has the same effect onthe filament instability as the increasing κ in the GEOS.(viii) Comparing the filaments with the same radius andin the same magnetic field, the MPEOS filaments are themost unstable ones, because of their softer EOS.(ix) The minimum spacing distance between clumps infilamentary MCs often is compared with λ critic (e.g. Hacar &Tafalla 2011; Contreras et al. 2016; Zhang et al. 2020) anddemonstrates diverse ranges of length (see Table 2). The pre-dicted λ critic is clearly dependent on the filament radii, the MNRAS , 1–14 (2021) Motiei et al. − . . . . . R − . − . − . − . . . . l og B n = -1.5 . . . . . . . . k c r i t i c − . . . . . R − . − . − . − . . . . l og B n = -2 . . . . . . . . k c r i t i c − . . . . . R − . − . − . − . . . . l og B n = -3 . . . . . . . . k c r i t i c − . . . . . R − . − . − . − . . . . l og B n = -4 . . . . . . . . k c r i t i c Figure 8.
Effect of the filament radius ( R ) and the magnetic field ( B ) on the critical wave number ( k critic ) in the PEOS dispersion relationswith n = − . (upper left panel), n = − (upper right panel), n = − (lower left panel) and n = − (lower right panel). In each panel thehorizontal axis shows the logarithm of filament radius, the vertical axis shows the logarithm of magnetic field strength and the colourbar represents k critic . The units of R , B and k critic are as Fig. 3. The horizontal dashed line in each panel separates the magnetic fieldslarger than or smaller than B (cid:39) . or log B (cid:39) − . . EOS and the magnetic field strength. This dependency ismore pronounced for the thinner filaments and is completelystrong for the thinnest ones. Caution is needed in interpret-ing this length scale. More specifically, it is interesting toinvestigate the fragmentation space within the thinner fila-ments (e.g. S´anchez-Monge et al. 2014).(x) In all the models it is observed that by decreasing thefilament radius (which means the higher external pressure), ω fast decreases or equivalently the minimum time needed forthe fragmentation ( τ min = /ω fast ) increases. This could corre-spond to a longer time needed for a clump to become unsta-ble and finally form protostars . Interestingly, this has beenalso reported by Anathpindika & Di Francesco (2020) re-cently. They have performed hydrodynamical simulation ofaccreting filaments in a medium with different external pres-sure and have shown that a higher external pressure leadsto a lower star formation rate.(xi) By comparing ω fast for the PEOSs, one can see thatthe softer ones have smaller τ min . Remarkably, in agreement with this result, in a hydrodynamical simulation of an ini-tially uniform polytropic gas within a periodic box and driv-ing turbulence, Federrath & Banerjee (2015) derived thatthe star formation rate increases for the softer PEOSs. DATA AVAILABILITY
No new data were generated or analysed in support of thisresearch.
ACKNOWLEDGEMENTS
Mohammad Mahdi Motiei and Mohammad Hosseiniradthank Mahmood Roshan and Najme Mohammad-Salehi foruseful discussions. The authors thank the anonymous refer-ees for the careful reading of the manuscript and their in-sightful and constructive comments. This research made useof
Scipy (Jones et al. 2001),
Jupyter (Kluyver et al. 2016)
MNRAS000
MNRAS000 , 1–14 (2021) ravitational instability of molecular clouds and Numpy (Walt et al. 2011). All figures were generated us-ing
Matplotlib (Hunter 2007). Also we have made extensiveuse of the NASA Astrophysical Data System Abstract Ser-vice. This work was supported by the Ferdowsi Universityof Mashhad under grant no. 50729 (1398/06/26).
REFERENCES
Anathpindika S., Di Francesco J., 2020, arXiv e-prints, p.arXiv:2012.01794Andr´e P., 2017, Comptes Rendus Geoscience, 349, 187Andr´e P., Di Francesco J., Ward-Thompson D., Inutsuka S.-I.,Pudritz R. E., Pineda J., 2014, in , Protostars and Plan-ets VI. University of Arizona Press ( arXiv:1312.6232 ),doi:10.2458/azu uapress 9780816531240-ch002, http://muse.jhu.edu/books/9780816598762/9780816598762-8.pdf
Arzoumanian D., et al., 2011, A&A, 529, L6Arzoumanian D., et al., 2019, A&A, 621, A42Bally J., Stark A. A., Wilson R. W., Langer W. D., 1987, ApJ,312, L45Bergin E. A., Tafalla M., 2007, ARA&A, 45, 339Bontemps S., et al., 2010, A&A, 518, L85Burkert A., Hartmann L., 2004, ApJ, 616, 288Camacho V., V´azquez-Semadeni E., Ballesteros-Paredes J.,G´omez G. C., Fall S. M., Mata-Ch´avez M. D., 2016, ApJ,833, 113Chandrasekhar S., Fermi E., 1953, ApJ, 118, 116Chen C.-Y., Ostriker E. C., 2014, ApJ, 785, 69Chou W., Matsumoto R., Tajima T., Umekawa M., Shibata K.,2000, ApJ, 538, 710Clarke S. D., Whitworth A. P., Duarte-Cabral A., Hubber D. A.,2017, MNRAS, 468, 2489Clarke S. D., Williams G. M., Walch S., 2020, arXiv e-prints, p.arXiv:2007.15358Contreras Y., Garay G., Rathborne J. M., Sanhueza P., 2016,MNRAS, 456, 2041Dib S., Kim J., V´azquez-Semadeni E., Burkert A., Shadmehri M.,2007, ApJ, 661, 262Durrive J.-B., Langer M., 2019, Journal of Fluid Mechanics, 859,362Federrath C., 2016, MNRAS, 457, 375Federrath C., Banerjee S., 2015, Monthly Notices of the RoyalAstronomical Society, 448, 3297Feng S., Beuther H., Zhang Q., Henning T., Linz H., Ragan S.,Smith R., 2016, A&A, 592, A21Fiege J. D., Pudritz R. E., 2000a, MNRAS, 311, 85Fiege J. D., Pudritz R. E., 2000b, MNRAS, 311, 105Fischera J., Martin P. G., 2012a, A&A, 542, A77Fischera J., Martin P. G., 2012b, A&A, 547, A86Freundlich J., Jog C. J., Combes F., 2014, A&A, 564, A7Garaud P., 2001, PhD thesis, https://users.soe.ucsc.edu/~pgaraud/Work/thesis.pdf
Gehman C. S., Adams F. C., Fatuzzo M., Watkins R., 1996a,ApJ, 457, 718Gehman C. S., Adams F. C., Watkins R., 1996b, ApJ, 472, 673Goldsmith P. F., Heyer M., Narayanan G., Snell R., Li D., BruntC., 2008, ApJ, 680, 428G´omez G. C., V´azquez-Semadeni E., 2014, ApJ, 791, 124Gritschneder M., Heigl S., Burkert A., 2016, ApJ, 834, 202Hacar A., Tafalla M., 2011, A&A, 533, A34Hanawa T., Tomisaka K., 2015, ApJ, 801, 11Hanawa T., Kudoh T., Tomisaka K., 2017, ApJ, 848, 2Hanawa T., Kudoh T., Tomisaka K., 2019, ApJ, 881, 97Hartmann L., Burkert A., 2007, ApJ, 654, 988Heigl S., Burkert A., Hacar A., 2016, MNRAS, 463, 4301Heigl S., Gritschneder M., Burkert A., 2018, MNRAS, 481, L1 Heigl S., Gritschneder M., Burkert A., 2020, 15, 1Heitsch F., 2013, ApJ, 776, 62Henshaw J. D., et al., 2016, MNRAS, 463, 146Hosseinirad M., Naficy K., Abbassi S., Roshan M., 2017, MNRAS,465, 1645Hosseinirad M., Naficy K., Abbassi S., Roshan M., 2018, MNRAS,465, 1645Hunter J. D., 2007, Computing In Science & Engineering, 9, 90Inutsuka S.-I., Miyama S. M., 1992, ApJ, 388, 392Inutsuka S., Miyama S. M., 1997, ApJ, 480, 681Inutsuka S.-i., Inoue T., Iwasaki K., Hosokawa T., 2015, A&A,580, A49Jackson J. M., Finn S. C., Chambers E. T., Rathborne J. M.,Simon R., 2010, ApJ, 719, L185Jones E., Oliphant T., Peterson P., et al., 2001, SciPy: Opensource scientific tools for Python,
Juvela M., et al., 2012, A&A, 541, A12Kainulainen J., Stutz A. M., Stanke T., Abreu-Vicente J., BeutherH., Henning T., Johnston K. G., Megeath S. T., 2017, A&A,600, A141Kawachi T., Hanawa T., 1998, Publications of the AstronomicalSociety of Japan, 50, 577Kim J.-g., Kim W.-t., Seo Y. M., Hong S. S., 2012, ApJ, 761, 131Klassen M., Pudritz R. E., Kirk H., 2017, MNRAS, 465, 2254Klessen R. S., Burkert A., Bate M. R., 1998, ApJ, 501, L205Kluyver T., et al., 2016, in ELPUB. pp 87–90K¨onyves V., et al., 2010, A&A, 518, L106Lada C. J., Lada E. A., 2003, /araa, 41, 57Larson R. B., 1985, MNRAS, 214, 379Lee S. M., Hong S. S., 2007, ApJS, 169, 269Li P. S., Klein R. I., 2019, MNRAS, 485, 4509Lizano S., Shu F. H., 1989, ApJ, 342, 834Maloney P., 1988, ApJ, 334, 761Matsumoto T., Nakamura F., Hanawa T., 1994, PASJ, 46, 243McKee C. F., Ostriker E. C., 2007, ARA&A, 45, 565McLaughlin D. E., Pudritz R. E., 1996, ApJ, 469, 194Men’shchikov A., et al., 2010, A&A, 518, L103Miettinen O., 2012, A&A, 542, A101Miettinen O., Harju J., 2010, A&A, 520, A102Miville-Deschˆenes M.-A., et al., 2010, A&A, 518, L104Miyama S. M., Narita S., Hayashi C., 1987, Progress of Theoret-ical Physics, 78, 1051Nagai T., Inutsuka S.-i., Miyama S. M., 1998, ApJ, 506, 306Nagasawa M., 1987, Progress of Theoretical Physics, 77, 635Nakamura F., Li Z.-Y., 2008, ApJ, 687, 354Nakamura F., Umemura M., 1999, ApJ, 515, 239Nakamura F., Hanawa T., Nakano T., 1993, PASJ, 45, 551Ntormousi E., Hennebelle P., 2019, A&A, 625, A82Ostriker J., 1964, ApJ, 140, 1056Padoan P., Haugbølle T., Nordlund ˚A., 2014, ApJ, 797, 32Palmeirim P., et al., 2013, A&A, 550, A38Panopoulou G. V., Psaradaki I., Skalidis R., Tassis K., AndrewsJ. J., 2017, MNRAS, 466, 2529Pilbratt G. L., et al., 2010, A&A, 518, L1Pudritz R. E., Kevlahan N. K.-R., 2013, Philosophical Transac-tions of the Royal Society of London Series A, 371, 20120248Roy A., et al., 2019, A&A, 626, A76Sadhukhan S., Mondal S., Chakraborty S., 2016, MNRAS, 459,3059S´anchez-Monge ´A., et al., 2014, A&A, 569, A11Schneider S., Elmegreen B. G., 1979, ApJS, 41, 87Shadmehri M., 2005, MNRAS, 356, 1429Steinacker J., Bacmann A., Henning T., Heigl S., 2016, A&A,593, A6Stod´olkiewicz J. S., 1963, Acta Astron., 13, 30Toci C., Galli D., 2015, MNRAS, 446, 2110V´azquez-Semadeni E., G´omez G. C., Jappsen A. K., Ballesteros-MNRAS , 1–14 (2021) Motiei et al.
Paredes J., Gonz´alez R. F., Klessen R. S., 2007, ApJ, 657,870Viala Y., Horedt G. P., 1974, A&AS, 16, 173Walt S. v. d., Colbert S. C., Varoquaux G., 2011, Computing inScience & Engineering, 13, 22Wang K., Zhang Q., Wu Y., Zhang H., 2011, ApJ, 735, 64Wang K., et al., 2014, MNRAS, 439, 3275Wang K., Testi L., Burkert A., Walmsley C. M., Beuther H.,Henning T., 2016, ApJS, 226, 9Ward-Thompson D., et al., 2010, A&A, 518, L92Wareing C. J., Pittard J. M., Falle S. A. E. G., Van Loo S., 2016,MNRAS, 459, 1803Zhang S., et al., 2020, arXiv e-prints, p. arXiv:2012.07738
APPENDIX A: DIMENSIONLESS FORM OF THEEQUATIONS OF STATE
We convert quantities and equations to the dimensionlessones as H18. The units are [ ρ ] = ρ c , (A1) [ t ] = (cid:112) π G [ ρ ] − , (A2) [ p ] = p c , (A3) [ u ] = (cid:115) [ p ][ ρ ] , (A4) [ r ] = [ t ][ u ] , (A5) [ ψ ] = [ u ] . (A6)The velocity unit is equal to the isothermal sound speed c s for the IEOS and the GEOS. For the MPEOS and the PEOS,it is assumed to be c s . The magnetic field unit is defined as [ B ] = (cid:112) π [ p ] , (A7) B = is equivalent with B (cid:39) . µ G. Using these units,we can rewrite the analytical solution of the density andgravitational potential of the isothermal filament as ρ ( r ) = (cid:18) + r (cid:19) − , (A8)and ψ ( r ) = (cid:18) + r (cid:19) . (A9)Moreover we achieve the dimensionless form of EOSs for theGEOS as p = ρ + κ ln ( ρ ) , (A10)where κ = P c ρ c and κ = gives the IEOS. The MPEOSdimensionless form is p = + A ln ( ρ ) , (A11)and finally the dimensionless form of the PEOS is p = ρ + / n . (A12) This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000