Effect of magnetic field on the rotating filamentary molecular clouds
aa r X i v : . [ a s t r o - ph . GA ] F e b Effect of magnetic field on the rotating filamentarymolecular clouds
P. Aghili • K. KokabiAbstract
Purpose of this work is study evolution ofmagnetized rotating filamentary molecular cloud’s. Wewill consider cylindrical symmetric filamentary molec-ular clouds at the early stage of evolution. For the firsttime we consider rotation of filamentary molecular inpresence of an axial and azimuthal magnetic field with-out any assumption of density and magnetic functions.We show that in addition to decreasing of radial col-lapse velocity, the rotational velocity also affected bymagnetic field. Existence of rotation yields to fragmentof filament. Moreover, we show that the magnetic fieldhave significant effect on the fragmentation of filamen-tary molecular clouds.
Keywords:
Filamentary molecular clouds; Magneticfield; Rotation.
Giant molecular clouds are one of the most impor-tant star-formation regions. Filamentary structures arepresent in many molecular clouds. They may be birth-place for stars or even many body systems. So, ana-lyzing their evolution and dynamics help us to betterunderstanding of star formation process. Physical pro-cesses like self-gravitation process, thermal process andmagnetic field play the main roll in the star-formationprocess. (Larson 1985, Nakamura1995, Nakajima 1996,Shadmehri 2005, Van Loo et al. 2014).Various magnetic field observed in the filamentarymolecular cloud (Heiles 2000; Goldsmith et al. 2008).There are various configuration of magnetic field with
P. AghiliK. KokabiSchool of Physics, Damghan University, Damghan, [email protected] respect to the filament such as parallel or orthogonal(Poidevin et al 2011; Vrba 1976). Value of magneticfield is also different at different place of order micro G(Opher et al. 2009; Ferriere 2009). Already, effect ofmagnetic field on the filamentary molecular cloud stud-ied, some of them used a function for magnetic field,and some of them assumed special magnetic field axis(Nagasava 1987). Now our main goal is to study effectof magnetic field on the rotational filamentary molec-ular cloud without any assumption about function ofmagnetic field. Already (Khesali et al. (2014)) we stud-ied effect of magnetic field on the filamentary molecularcloud without rotation and found no fragment. Now,we will show that effect of rotational motion is fragmentof filament which also affected by magnetic field.Considering various magnetic fields that observed inmolecular clouds, result gained for spherical symmet-ric collapse in the rotation by (Ulrich 1976). Withgeneralizing those results to disk structure by (Cassenand Moosman 1981) and to isothermal crash (collapse-Decay) by (Terbey, Shu and Cassen 1984) the resultswere analyzing. (Mendoza, Tejeda and Nagel 2009 hereafter MTN). Generalized this idea with creation of afluid continuous growing flow model for a definite gascloud by considering outer region observations of fil-amentary cloud. We will study the effect of magneticfield on the other filament parameters like density, rota-tional velocity and etc. We will study cooling functionchanges and rotation effect on the molecular clouds.
In this section, as regards that we have symmetry in ourproblem, we will consider the cloud as a long cylinderwith symmetrical z axis, and considering magnetic fieldas B = B ϕ ˆ ϕ + B z ˆ k ( B is function of radial distance and time, i.e. B ( r, t ) ) considering Ideal gas and energyequations, we have main equations as: ∂ρ∂t + ∇ · ( ρu ) = 0 , (1) ∂v r ∂t + v r ∂v r ∂r + 1 ρ ∂P∂r + ∂ψ∂r = v ϕ r − πρ (cid:20) B ϕ r ∂∂r ( rB ϕ ) + B z ∂B z ∂r (cid:21) , (2)1 r ∂∂r ( r ∂ψ∂r ) = 4 πGρ (3) ∂B∂t = ∇ × ( v × B ) (4)1 γ − (cid:20) ∂p∂t + v r ∂p∂r (cid:21) + γγ − pr ∂∂r ( rv r )+ A ν ρ T ν = 0 , (5) ∂∂t ( rv ϕ ) + v r ∂∂r ( rv ϕ ) = 0 , (6)where ρ , V r , V ϕ , ψ , P and B are: gas density, ra-dial velocity, azimuthal velocity, gravitational poten-tial, pressure and magnetic field. Also ν and A ν are con-stants. These constants can be determined by Spritzer(1978). All changes are depend on R (the distancefrom cylinder axis) and time. For simplification, wewill introduce dimensionless variables and then solvethe problem using the following translations ρ → ρ ˆ ρ, t → t ˆ t, v → v ˆ v, r → r ˆ r,p → p ˆ p, B → B ˆ B, ψ → ψ ˆ ψ, (7)whereˆ ρ = ρ , ˆ P = P , ˆ ψ = P ρ , ˆ r = s P πGρ , ˆ t = P A ν ρ T ν , ˆ v = ˆ r ˆ t , ˆ B = p πP . (8)We choose the physical scale for length and timewhich are 1PC and 10 yr, respectively. Thus, thegravitational constant is G = 4 . × − . In this man-ner, the temperature and density units are 130 K and6 . × − grCm , respectively. And thee unit of mag-netic field is chosen equal to 2.85 µG . By considering the modified equations (7), we will re-place dimensionless functions of ( r, t ) with the functionof ( η, t ) for self-similar variable η = rt n . We use thefollowing forms of the physical variables, ρ ( r, t ) = R ( η ) t ε ,v r ( r, t ) = v r ( η ) t ε ,ψ ( r, t ) = ψ ( η ) t ε ,B z ( r, t ) = B z ( η ) t ε ,B ϕ ( r, t ) = B ϕ ( η ) t ε ,P ( r, t ) = P ( η ) t ε ,v ϕ ( r, t ) = v ϕ ( η ) t ε , (9)By equilibrating the time powers, local equations willbe obtained as follows, ε R ( η ) − nη dR ( η ) dη + 1 η ddη ( ηR ( η ) v r ( η )) = 0 , (10) ε v r ( η ) − nη dv r ( η ) dη + v r ( η ) dv r ( η ) dη + 1 R ( η ) dP ( η ) dη + dψ ( η ) dη = − B z ( η ) R ( η ) dB z ( η ) dη − B ϕ ( η ) ηR ( η ) d ( ηB ϕ ( η )) dη + v ϕ η , (11)( v r ( η ) + ε η ) v ϕ ( η ) + ( v r ( η ) − nη ) η dv ϕ ( η ) dη = 0 , (12)1 η ddη ( η dψ ( η ) dη ) = R ( η ) , (13) ε B ϕ ( η ) − nη dB ϕ ( η ) dη + ddη ( v r ( η ) B ϕ ( η )) = 0 , (14) ε B z ( η ) − nη dB z ( η ) dη + 1 η ddη ( ηv r ( η ) B z ( η )) = 0 , (15)1 γ − ε P ( η ) − nη P ( η ) dη + v r ( η ) dP ( η ) dη )+ γγ − P ( η ) η ddη ( ηv r ( η )) + R ( η ) − ν P ( η ) ν = 0 , (16) where ε - ε , and n are obtained as follows, ε = − , ε = 2 ε = ε = 1 ν − ,n = 2 ν − ν − , ε = ε = ε = 3 − νν − . (17)Now the equations (10)-(16) can be solved by using thefourth-order Runge-kuttta method which is consideredin the next section. In this section, we will use the observed outer conditionsfor filamentary clouds to integrate the equations fromoutside to inside of the cloud. We will study behavior ofphysical quantities. According to the observation of fil-amentary clouds, density is higher in the inner regionsthan outer regions. The outer density of filamentarycloud’s is like ISM density. For the reason we will choose n = 10 cm − as typical density’s value. We expectthat this value increases in the center (Hanawa 1996).Density in the center is about n = 10 cm − as typ-ical density’s value (Li and Goldsmith 2012; Henshowet al 2013). According to the boundary conditions andfree parameter ν , we can solve a set of ordinary equa-tions. We can consider ν = 2 . n ≈ cm − in agreement with recent work of Ysardet al. (2013). Shadmehri (2005) and Chapman et al.(2011) show that density increased with increasing oftoroidal magnetic field component, in agreement withour result which illustrated by the Fig. 1. -3 -2 -1024 log R log Fig. 1
Density profile corresponding to γ = 1 .
66, anddifferent initial value for azimuthal component of magneticfield.
According to the observation, the magnetic field hasvarious shapes in the molecular cloud and makes differ-ent angels with filamentary cloud’s axis in outer region(Planck Collaboration 2016). Tilley and Pudritz (2003)supposed that magnetic field in outer region in onlyazimuthal. Shadmehri (2005) supposed that the az-imuthal component is dominate in the outer region, wesuppose (in agreement with Wareing et al. 2016) thatprimary axial component of the magnetic field is smallerthan azimuthal magnetic field for outer region. Also,magnetic field in molecular cloud is of order µG (Khe-sali et al 2014). Collapse velocity in outer region is lessthan 3 km/s. So we choose v ϕ = 0 .
001 and v r = − . γ = 1 . -3 -2 -1020406080100B z log B B B
Fig. 2
Axial magnetic field corresponding to γ = 1 .
66, anddifferent initial value for azimuthal component of magneticfield.
We show variation of axial and azimuthal magneticfields by Figs. 2 and 3, It is clear that value of mag-netic field increased at center in agreement with previ-ous works (Tilley and Pudritz 2003, Shadmehri 2005,Fiege and Pudritz 2000). Also we show that increasinginitial azimuthal magnetic field in outer edge of cloudcauses to increasing central magnetic field components.It is completely coincide with the Fig. 1 which showmagnetic field freezing of filament.Previous works has been shown that magnetic fieldhelp to avoid gravitational collapse (Shadmehri 2005,K¨ortgen and Banerjee 2015, Nakajima and Hanawa1996). In agreement with mentioned work we show inthe Fig. 4 that radial velocity decreased with magneticfield in the center which is indeed magnetic braking ef-fect.In the Fig. 5 we can see that azimuthal velocityincreased at center, but increasing of magnetic field re-duced radial velocity due to the magnetic braking.We can see that when more energy is released fromthe filament the in fall velocity is increased as expected -3.0 -2.5 -2.0 -1.5 -1.0 -0.5050100150200250 log Fig. 3
Azimuthal magnetic field corresponding to γ =1 .
66, and different initial value for azimuthal component ofmagnetic field. -3 -2 -1-3-2-10 log(-V r ) log Fig. 4 radial velocity corresponding to γ = 1 .
66, and dif-ferent initial value for azimuthal component of magneticfield. in the filament collapsing process. In that case in theFigs. 6 and 7 we see that increasing ν increases thevalue of radial and azimuthal velocities.The Figs. 8 and 9 show effect of increasing energyrelease from the system on the density and magneticfield in agreement with the previous work (Khesali etal 2014).In the Figs. 10 we draw density profile in termsof z -direction at the constant radius, and show thatdensity profile affected by variation of magnetic field. Itis illustrated that increasing of magnetic field decreaseschange of density, so it help to keep filament. Hence wecan see less fragmentation in presence of magnetic fieldat the early stage. In this paper, we studied rotational filamentary molec-ular cloud’s collapse in presence of magnetic field at the -3 -2 -10.000.010.020.03 V log Fig. 5
Azimuthal velocity corresponding to γ = 1 .
66, anddifferent initial value for azimuthal component of magneticfield. -3 -2 -1-3-2-10 log(-V r ) log Fig. 6 radial velocity corresponding to γ = 1 .
66, and dif-ferent initial value for ν . early stage. We studied filamentary cloud’s dynamic ofthe early stage with azimuthal and axial fields with-out introducing a function for magnetic field and den-sity functions. Previous studies and observational datasuch as Arzoumanian et al. (2011) indicated that thefilament density decreased with radius and our studyalso confirms these results as illustrated by the Fig. 1.Also, Kirk et al. (2015) has been studied the effectof magnetic fields in filamentary structure using thesimulation method. We know that magnetic field hasimportant role in the formation of the filament. Also,density increases with increasing of initial azimuthalmagnetic field (Fig. 1). Values of azimuthal and ax-ial magnetic fields is larger in center as illustrated byFig. 2 and Fig. 3, which show freezing of magneticfield. There are several models to formation of filamentlike Padoan et al. (2001) and Nakamura (2008) whereimportance of magnetic field have been studied. For ex-ample, K¨ortgen and Banerjee (2015) have been shownthat star formation due primary magnetic field heavilydelayed or suppressed. Therefore, magnetic field canreduces radial collapse velocity which verified with the -3 -2 -10.000.010.020.03 V log Fig. 7
Azimuthal velocity corresponding to γ = 1 .
66, anddifferent initial value for ν . -3 -2 -101234 log R log Fig. 8
Density profile corresponding to γ = 1 .
66, anddifferent initial value for ν . Figs. 4 of our work in agreement with other works likeShadmehri (2005) and Khesali et al. (2014). Differenceof our new results with Khesali et al. (2014) is presenceof filament fragment due t rotational motion. Behaviorof radial and azimuthal velocity plotted in Fig. 4 andFig. 5. In the Fig. 4 we can see that radial velocity issmaller at center to yields zero in agreement with obser-vation (K¨ortgen and Banerjee 2015). In the Fig. 5 wecan see that azimuthal velocity is larger at center inter-preted as magnetic braking. It means that bigger valueof magnetic field help to stability of cloud. We knowthat rotational velocity is small at the early stage. It isclear that rotational velocity affected by the magneticfield. We shown that presence of magnetic field reducesvalue of rotational parameter and yield to more stabil-ity and this confirmed by several observations. In Figs.6, 7, 8 and 9 effect of cooling function on the physicalsystem parameters have been shown. In the Figs. 6and 7 we can see that radial and azimuthal velocitiesincreased by exit of energy from the system. The Figs.8 and 9 increasing exit energy effect on the density andmagnetic field have been shown respectively. -3 -2 -104080120 log
Fig. 9
Azimuthal magnetic field corresponding to γ =1 .
66, and different initial value for ν . B =0.05 B =0.06 B =0.07R z
Fig. 10
Density profile corresponding to γ = 1 .
66 withconstant radius, and different initial value for azimuthalcomponent of magnetic field.
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