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The response of filamentary and spherical clouds to the turbulence and magnetic field
New Astronomy 61 (2018) 69–77
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The response of filamentary and spherical clouds to the turbulence and magnetic field
T
Mahmoud Gholipour Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) -Maragha, P. O. Box: 55134 - 441, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: ISM: structure Stars: formation
Recent observations have revealed that there is a power-law relation between magnetic field and density in molecular clouds. Furthermore, turbulence has been observed in some regions of molecular clouds and the velocity dispersion resulting from the turbulence is found to correlate with to the cloud density. Relating to these observations, in this study, we model filamentary and spherical clouds in magnetohydrostatic equilibrium in two quiescent and turbulent regions. The proposed equations are expected to represent the impact of magnetic field and turbulence on the cloud structure and the relation of cloud mass with shape. The Virial theorem is applied to consider the cloud evolution leading to important conditions for equilibrium of the cloud over its lifetime. The obtained results indicate that under the same conditions of the magnetic field and turbulence, each shape presents different responses. The possible ways for the formation of massive cores or coreless clouds in some regions as well as the formation of massive stars or low-mass stars can be discussed based on the results of this study. It should be mentioned that the shape of the clouds plays an important role in the formation of the protostellar clouds as well as their structure and evolution. This role is due to the effects of magnetic fields and turbulence.
1. Introduction For several decades, it has been widely argued that the process of star formation is primarily controlled by the interplay between the gravity and magnetostatic support, regulated by neutral-ion drift (Mestel and Spitzer, 1956; Mac Low and Klessen, 2004). However, both observational and numerical recent works suggest that supersonic interstellar turbulence rather than magnetic fields controls the star formation (Mac Low and Klessen, 2004; Klessen, 2004). Generally, in astrophysics, different turbulence, such as supersonic, compressible, magnetized, and incompressible turbulence are considered. Most studies on the turbulence deal with turbulence in an incompressible medium, which is characteristic of many terrestrial applications. Generally, the root-mean-square velocities are considered to be subsonic, and the density is taken almost constant. The energy dissipation occurs then totally on the scales of the smallest vortices, where the dynamical scale is shorter than the length on which viscosity acts (Mac Low and Klessen, 2004). Based on the molecular clouds and condensations, Larson (1981) showed that the internal velocity dispersion of any region is well correlated with its size and mass. He reported that these correlations have a power-law behavior. The dependence of the velocity dispersion on the region size is similar to the Kolmogoroff law for the subsonic turbulence, which suggests that the observed motions are part
of a common hierarchy of interstellar turbulent motions (Larson, 1981; Heyer and Dame, 2015). The studied regions are often gravitationally bound and approximately in Virial equilibrium. By the way, they cannot be created by a simple gravitational collapse, and it seems the molecular clouds are formed at least partly by supersonic hydrodynamics processes (Larson, 1981). On the other hand, the observations of the magnetic fields in molecular clouds can give us important information about the structure and evolution of the clouds (Crutcher, 2012). The techniques for observing and mapping magnetic fields in molecular clouds have been comprehensively reviewed by Crutcher (2012). The use of dust polarization, Faraday rotation, masers, Goldreich–Kylafis (GK) effect, and Zeeman effect are some of techniques for observing and mapping magnetic fields in molecular clouds (e.g. Gholipour, 2017). The only available technique for directly measuring magnetic field strengths in the ISM is the Zeeman effect (e.g., Crutcher et al. 1993; Crutcher, 2012). Magnetic field strengths from masers would be very important in confirming the relationship between magnetic field, B and gas density, ρ (i.e. B = αρn , where α and n are constant; see Crutcher, 2012 or Gholipour, 2017). In the recent two decades, due to advances in observational instruments, numerous observational works have been carried out to determine the direction of the magnetic fields in the molecular clouds.
E-mail address: [email protected]. https://doi.org/10.1016/j.newast.2017.12.003 Received 11 September 2017; Received in revised form 5 December 2017; Accepted 7 December 2017 Available online 12 December 2017 1384-1076/ © 2017 Elsevier B.V. All rights reserved.
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In this way, the toroidal magnetic fields have been targeted by many observational studies (e.g. André et al., 2014; Han, 2013; Verschuur, 2013; Contreras et al., 2013; Girart et al., 2013; Novak et al., 2003; Matthews et al., 2001). For instance, Girart et al. (2013) presented high angular resolution observations of a massive star forming core at 880 μ m using the Submillimeter Array. They found that the dust polarization shows a complex magnetic field, compatible with a toroidal configuration. As another example, we can mention to the work of Novak et al. (2003). They observed the linear polarization emission from the Galactic center, using the polarimetric detector system at the South Pole. The resulting polarization map covered 30 pc in Galactic latitude and 170 pc along the Galactic plane, and thus included a significant fraction of the central molecular zone. Their map presented that the magnetic fields were often toroidal in these regions. Recently, Gholipour (2017) studied the stability of the spherical cloud core with a toroidal magnetic field. He considered a power-law relation between the magnetic field and density in a quiescent region (i.e. without turbulence). He found that the presence of the magnetic field can significantly affect a density structure and the total mass of the cloud. In this paper, the response of filamentary and spherical clouds on the turbulence and toroidal magnetic fields is considered. Since observations of star-forming regions, i.e. molecular clouds, show a powerlaw relation between the magnetic field and density, we study both classes of clouds (i.e. filamentary and spherical clouds) with this mentioned relation, in magnetohydrostatic equilibrium and with turbulence. Then, the Virial theorem will be applied for discussing the time evolution of the clouds. In this way, in Section 2 we model the magnetic field in filamentary clouds. In Section 3 we consider spherical clouds. In both cases we consider quiescent and turbulent regions. The evolution of clouds is considered in Section 4 in terms of the Virial theorem. In Section 5, we discuss the obtained results and compare the results with the observations. Finally, we give our conclusions in Section 5.
result is 2
α 2ρ2n − 1 ⎛ (3n − 1) dρ 1 dP 1 d 2P 1 dρ ⎞ dP + − ⎜⎛ ⎟ + rρ dr ρ dr 2 ρ dr dρ 4π ⎜ rρ dr ⎝ ⎠ ⎝ 2
1 dρ ⎞ n d 2ρ ⎞ ⎟ + 2n (n − 1) ⎜⎛ + ρ dr ρ dr 2 ⎟ ⎝ ⎠ ⎠ + 4πGρ = 0.
Eq. (3) shows the density structure, if we know the equation of state (i.e. knowing P(ρ)). However, the equation of state is sensitive to turbulence (e.g. Gehman et al., 1996). Thus, we consider this equation in two important regions as follows: 1- quiescent region 2- turbulent region. 2.2. Quiescent clouds Here, we investigate the density structure of an isothermal quiescent cloud (i.e. P = cs2 ρ where cs is the isothermal sound speed). With defining β = α 2/4πcs2, Eq. (3) can be rewritten as follows
(1 + nβρ2n − 1)
+
4πG ρ = 0. cs2
−1 2
4πGρ r0 = ⎜⎛ 2 c ⎟⎞ ⎝ cs ⎠
(4)
,
(5)
and also,
σ = βρc2n − 1 .
(6)
Therefore, we can rewrite Eq. (4) as follows 2
d 2ϱ (1 + 2n (1 − n) σρ2n − 1) 1 ⎛ dϱ ⎞ (1 + (3n − 1) σρ2n − 1) 1 dϱ − + ⎜ ⎟ 1 + nσ ϱ2n − 1 ϱ ⎝ dξ ⎠ 1 + nσ ϱ2n − 1 dξ 2 ξ dξ ϱ2 + = 0. 1 + nσ ϱ2n − 1
(7) Furthermore, the boundary conditions are given by (e.g. Stahler and Palla, 2004)
2.1. Magnetic field in molecular clouds
ϱ (ξ = 0) = 1,
The scaling of magnetic field strength with density can be parameterized as B = αρn where α and n are the field strength and field gradient, respectively (e.g. Crutcher et al., 2010; Crutcher, 2012). In a cylindrical system of coordinates, we study the impact of a toroidal magnetic field on a self-gravitating cloud which is in an equilibrium state. The equation of magnetohydrostatic equilibrium is (e.g. Mestel, 2003; Gholipour, 2017)
dϱ dξ
= 0. (ξ = 0)
(8)
Eq. (7) describes the density structure of a quiescent cloud in the presence of the toroidal field configuration. The resulting density as a function of radius is depicted in Fig. 1, for different values of the parameters σ and n. 2.3. Turbulent clouds
(1)
Now, we consider turbulence effect by adding a term that attempts to model the turbulence observed in the molecular clouds (e.g. Gehman et al., 1996; Lizano and Shu, 1989; see also Mestel, 2003). The equation of state is then given by
where P is the gas pressure and Φ is the gravitational potential related to the gas density by Poisson’s equation
∇2 Φ = 4πGρ ,
1 ⎛ dρ ⎞2 ρ2 ⎝ dr ⎠
This equation shows an isothermal density structure of a cloud in the presence of the toroidal magnetic field in the cylindrical coordinate system. We recast Eq. (4) into a dimensionless form. We define a nondimensional radius as ξ = r / r0, and a non-dimensional density as ϱ = ρ / ρc where ρc is the central density and r0 is
In this section, we study the role of magnetic field and turbulence in filamentary clouds. In a cylindrical coordinate system (r, ϕ, z), we consider an axisymmetric and long filament along the z-axis. Hence, all variables are assumed to only depend on the radial distance r. The cloud is assumed to be exposed to a toroidal magnetic field (e.g. André et al., 2014; Girart et al., 2013; Han, 2013; Verschuur, 2013; Contreras et al., 2013). Many observational studies (e.g. Tritsis et al., 2015; Crutcher, 2012; 1999; Crutcher et al., 2010; Mouschovias and Tassis, 2010) and some analytical works (e.g. Li et al., 2015; Bodenheimer, 2011, Mouschovias, 1985; 1979; 1976a; 1976b) indicate this where there is a power-law relation between magnetic field strength and density in the molecular clouds.
1 (∇ × B) × B = 0, 4π
1 d 2ρ 1 dρ + (1 + (3n − 1) βρ2n − 1) ρ dr 2 rρ dr
− (1 + 2n (1 − n) βρ2n − 1)
2. Filamentary clouds
− ∇P − ρ ∇Φ +
(3)
(2)
ρ P = Pgas + Pturb = cs2 ρ + p0 log ⎛⎜ ⎞⎟, ρ ⎝ c⎠
where G is Newton’s constant. If one applies the cylindrical divergence on Eq. (1) and then substitutes the obtained result into Eq. (2), the final 70
(9)
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Fig. 1. The non-dimensional density against the non-dimensional radius for a quiescent filamentary cloud (a) n = 0.45 (b) n = 0.65 .
κ ⎞ dρ κ d 2ρ κ dρ 2 κ dρ 2 1 ⎛ 1 1 ⎜1 + ⎟ + ⎜⎛1 + ⎟⎞ 2 − 3 ⎛ ⎞ − 2 ⎜⎛1 + ⎟⎞ ⎛ ⎞ rρ ⎝ ρ ⎠ dr ρ⎝ ρ ⎠ dr ρ ⎝ dr ⎠ ρ ⎝ ρ ⎠ ⎝ dr ⎠
where p0 is a constant, which may be determined empirically. The motivation for the functional form adopted in Eq. (9) which is the empirical power-law correlation vturb ∼ ρ−1/2 is often called Larson’s law (e.g. Mestel, 2003). Thus, we can write 1 2
2
+
1 2
p dP vturb = ⎜⎛ turb ⎟⎞ = ⎜⎛ 0 ⎟⎞ . dρ ⎝ ⎠ ⎝ρ⎠
α 2ρ2n − 1 ⎛ (3n − 1) dρ 4πGρ n d 2ρ ⎞ 1 dρ ⎞ ⎟ + 2n (1 − n) ⎜⎛ + + rρ dr ρ dr ρ dr 2 ⎟ 4πcs2 ⎜⎝ cs2 ⎝ ⎠ ⎠ = 0.
(10)
(11)
Eq. (10) leads to the logarithm relationship between turbulent pressure and density (see Eq. (9)). According to Eqs. (3) and (9), and with defining κ = p0 / cs2, we can write
After transporting variables to the dimensionless forms, Eq. (11) leads to
71
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Fig. 2. The non-dimensional density against the non-dimensional radius for a turbulent filamentary cloud n = 0.5, (a) σ = 0.5 (b) σ = 1.5 .
(1 + γ ϱ−1 + (3n − 1) σρ2n − 1) 1 dϱ d 2ϱ + 1 + γ ϱ−1 + nσ ϱ2n − 1 dξ 2 ξ dξ
to the isothermal Lane–Emden (hereafter LE) equation for a non-magnetized quiescent cloud (i.e. σ = γ = 0 ). As Eq. (12) shows, the nondimensional density is related to three parameters as follows: 1) the field strength σ, 2) the field gradient n, 3) the turbulence parameter γ. Fig. 2 shows the impact of turbulence on a filamentary cloud.
2
(1 + γ ϱ−2 + 2n (1 − n) σρ2n − 1) 1 ⎛ dϱ ⎞ ⎜ ⎟ 1 + γ ϱ−1 + nσ ϱ2n − 1 ϱ ⎝ dξ ⎠ ϱ2 + = 0, 1 − 1 + γ ϱ + nσ ϱ2n − 1 −
(12) 3. Spherical clouds
where γ is the turbulence parameter (κ/ρc). In addition, the boundary conditions are given by
ϱ (ξ = 0) = 1,
dϱ dξ
Here, we consider the impact of toroidal magnetic field and turbulence on a spherical cloud. This process is a bit more complicated than the filamentary clouds. In spherical clouds, the magnetic field is often composed of two components as poloidal and toroidal (i.e. ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ B = Bθ + Bϕ ). If the poloidal component is sufficiently weak (i.e.
= 0. (ξ = 0)
(13)
Eq. (12) leads to Eq. (7) for a quiescent cloud (i.e. γ = 0 ), and also leads 72
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Bθ ≪ Bϕ), we can assume only the toroidal component is dominant. Moreover, a purely toroidal field makes it easier to compare the obtained results with the results of previous section.
defining κ = p0 / cs2, we can write
κ ⎞ dρ κ d 2ρ κ dρ 2 κ dρ 2 2 ⎛ 1 1 ⎜1 + ⎟ + ⎜⎛1 + ⎟⎞ 2 − 3 ⎛ ⎞ − 2 ⎜⎛1 + ⎟⎞ ⎛ ⎞ rρ ⎝ ρ ⎠ dr ρ⎝ ρ ⎠ dr ρ ⎝ dr ⎠ ρ ⎝ ρ ⎠ ⎝ dr ⎠
3.1. Magnetic field in quiescent clouds
2
We study the equilibrium structure of a self-gravitating cloud affected by the toroidal magnetic field. The equation of magnetohydrostatic equilibrium and the Poisson’s equation have been given by Eqs. (1) and (2), respectively. We take the divergence on Eq. (1), and then we substitute the obtained result into Eq. (2). Thereupon, the final result is
(20) As previous section, we can rewrite Eq. (20) to the non-dimensional form as follows
(2 + 2γ ϱ−1 + (4n − 1) σρ2n − 1) 1 dϱ d 2ϱ + 1 + γ ϱ−1 + nσ ϱ2n − 1 dξ 2 ξ dξ
α 2ρ2n − 1 ⎛ 1 (4n − 1) dρ 2 dP 1 d 2P 1 dP dρ + − 2 + + rρ dr ρ dr 2 ρ dr dr 4π ⎜ r 2 rρ dr ⎝
2
(1 + 2γ ϱ−2 + 2n (1 − n) σρ2n − 1) 1 ⎛ dϱ ⎞ ⎜ ⎟ 1 + γ ϱ−1 + nσ ϱ2n − 1 ϱ ⎝ dξ ⎠ ϱ2 σ ϱ2n 1 + + = 0, 1 2 1 2 1 − n − − 1 + γ ϱ + nσ ϱ2n − 1 1 + γ ϱ + nσ ϱ ξ
2
1 dρ ⎞ n d 2ρ ⎞ + 2n (n − 1) ⎛⎜ + ⎟ ρ dr 2 ⎟ ⎝ ρ dr ⎠ ⎠ + 4πGρ = 0.
−
(14)
The density structure of an isothermal quiescent cloud, with defining β = α 2/4πcs2, can be written as follows
(1 + nβρ2n − 1)
ϱ (ξ = 0) = 1,
2
− (1 + 2n (1 − +
1 ⎛ dρ ⎞ ρ2 ⎝ dr ⎠
βρ2n − 1 4πG + 2 ρ = 0. r2 cs
−1 2
(15)
,
(17)
Then, Eq. (15) will be
dξ 2
+
(2 + (4n − 1 + nσ ϱ2n − 1
n) σρ2n − 1)
(1 + 2n (1 − 1 dϱ − 1 + nσ ϱ2n − 1 ξ dξ
2
where K is the internal energy, Ψ is the gravitational energy and UB is the magnetic field energy. Note that we neglect the effect of external pressure at the surface in the Virial theorem (i.e. Ps = 0 ). Now, we can begin the study of time-evolution of a cloud. First of all, we consider the dimensional mass M for a spherical cloud. We have
1 ⎛ dϱ ⎞ ⎜ ⎟ ϱ ⎝ dξ ⎠
σ ϱ2n 1 1 + nσ ϱ2n − 1 ξ 2 ϱ2 + = 0. 1 + nσ ϱ2n − 1
s
ϱξ 2dξ ,
(24)
where ’c’ and ’s’ denote the center and surface of the cloud, respectively. Also, we introduce non-dimensional mass as follows
Also, the boundary conditions are given by (e.g. Gholipour and NejadAsghar, 2013)
M , M0
Mξ =
= 0. (ξ = 0)
∫ ρdV = 4πρc r03 ∫c
M= (18)
dϱ dξ
(23)
2K + Ψ + UB = 0, 1) σρ2n − 1)
+
ϱ (ξ = 0) = 1,
(22)
The time evolution of a cloud, which would be in an equilibrium state, will be considered over its lifetime in this section. The critical question, that emerges here, is: would the cloud remain in an equilibrium state (force balance) over its lifetime? The Virial theorem has a suitable response for this question. The Virial theorem should be satisfied, if the cloud wants to stay in force balance over its lifetime (e.g., Mestel, 1965a; 1965b)
and also,
d 2ϱ
= 0. (ξ = 0)
4. Virial theorem in molecular clouds
(16)
σ = βρc2n − 1 .
dϱ dξ
Indeed, Eq. (21) leads to Eq. (18) for a quiescent cloud (i.e. γ = 0 ), and leads to the isothermal LE equation for a non-magnetized quiescent cloud (i.e. σ = γ = 0 ). As Eq. (21) shows, the non-dimensional density structure depends on three parameters as follows: 1) the field strength σ, 2) the field gradient n, 3) the turbulent parameter γ. Fig. 3.b shows the role of turbulence and magnetic field on the density structure.
This equation is the isothermal density structure of a cloud core in the presence of the toroidal magnetic field in the spherical coordinate system. It is useful to recast Eq. (15) into dimensionless form. As in the previous section, we define a non-dimensional radius as ξ = r / r0 and a non-dimensional density as ϱ = ρ / ρc where ρc is the central density and r0 is
4πGρ r0 = ⎜⎛ 2 c ⎟⎞ ⎝ cs ⎠
(21)
where γ is the turbulent parameter as κ/ρc. The boundary conditions are given by
1 d 2ρ 1 dρ + (2 + (4n − 1) βρ2n − 1) ρ dr 2 rρ dr n) βρ2n − 1)
α 2ρ2n − 1 ⎛ 1 (4n − 1) dρ 1 dρ ⎞ n d 2ρ ⎞ ⎟ + + 2n (n − 1) ⎜⎛ + 2 ⎜ 2 rρ dr ρ dr 2 ⎟ 4πcs ⎝ r ⎝ ρ dr ⎠ ⎠ 4πGρ 0. + = cs2
+
(19)
M0 = 4πρc r03,
(25)
so that we have
Eq. (18) presents a modified form of the LE equation: in the absence of the toroidal magnetic field (i.e. α = 0 or σ = 0 ) one obtains the isothermal LE equation. Fig. 3.a presents the density as a function of radius for the spherical clouds, with different values of σ and γ.
Mξ =
∫c
s
ϱξdξ .
(26)
In the following, the internal energy is given by
K=
3.2. Turbulent clouds Regarding to Eqs. (9) and (14) in spherical coordinates, and with
3 2
∫ PdV = 6πcs2 ρc r03 ⎛ ∫c ⎜
s
ϱξ 2dξ + γ
⎝
Also, the gravitational energy is 73
∫c
s
log ϱ ξ 2dξ ⎞⎟. ⎠
(27)
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Ψ=
1 2
∫ ρΦdV = 2πcs2 ρc r03 ∫c
s
ϱΦϱ ξ 2dξ ,
on a spherical cloud. This matter can be discussed in the core-less clouds. The part (b) of this figure reveals the effect of turbulence versus the magnetic field. The turbulence in a spherical cloud can balance somewhat the magnetic field’s effect. Fig. 4 shows the comparison between the results for different cloud shapes. Under the same conditions, each shape gives different responses to the magnetic field. This difference is seen most significantly at small radii. In other words, the structure of spherical clouds is related to the interplay between magnetic field and turbulence. While, the structure of a filamentary cloud is the result of cooperation between magnetic field and turbulence. Thus, we expect that a quasi-spherical dense core will be seen more in turbulent-dominated regions than other regions. Although, the turbulence increases the density over the radius in each shape, but this increase is limited by the Virial theorem. According to this theorem, the maximum value of the turbulence parameter can be obtained from the equation of the density structure so that the cloud leaves equilibrium when we have γ > γmax. The relation for maximum value is same for two shapes.
(28)
where Φϱ = Φ/ cs2 is non-dimensional potential. The magnetic field energy can be written as
1 1 2 B2dV = α 8π 8π s 1 ϱ2nξ 2dξ . = α 2ρc2n r03 c 2
∫
UB =
∫ ρ2ndV ,
∫
(29)
It is better to work with the below dimensionless variables, as follows:
hξ =
∫c
s
ϱ2nξdξ ,
Ψξ = −
∫c
s
ξ ϱΨdξ ,
Γξ =
∫c
s
log ϱ ξdξ .
(30)
The Virial theorem then leads to
Ψξ − 3Mξ − 3γ Γξ ⎞ σ = ⎜⎛ ⎟, hξ ⎝ ⎠
(31)
or 1
α=
4π ⎛ Ψξ − 3Mξ − 3γ Γξ ⎞ 2 ⎜ ⎟ cs . hξ ρc2n − 1 ⎝ ⎠
5.2. Observations and simulations (32) Recent observations and simulations of spiral arm clouds show that the most filaments have a magnetic field, compatible with a toroidal configuration (Seifried and Walch, 2015; Pillai et al., 2015; Zhang et al., 2014; Tomisaka, 2014; Hennebelle, 2013; Palmeirim et al., 2013; Sugitani et al., 2011; Gaensler et al., 2011). Furthermore, the starforming cores in nearby spiral-arm clouds are often reported along dense filaments (Konyves et al., 2015; Polychroni et al., 2013). These observations and observations of the Galactic center suggest that filaments may be fundamental building blocks of the molecular clouds in the star formation processes, regardless of whether the clouds are located along spiral arms or near the Galactic Center (Federrath, 2016). For instance, Polychroni et al. (2013) presented Herschel survey maps of the molecular clouds in Orion A. They extracted both the filaments and dense cores in this region. They identified which of the dense sources are proto-stellar or pre-stellar. Then, their associations with the identified filaments were studied. They found that most of the prestellar sources (71%) are located in the filaments. As we know, there is a lot of massive stars in the Orion which may be discussed based on the results of this study. However, the Herschel data indicate the fact that core-less clouds and the low-mass clouds should be seen in an quasispherical shape (André et al., 2013). Recently, regarding many observations of massive turbulent filaments, Beuther et al. (2015) showed that turbulence plays an important role in supporting the filament against the collapse (see also Klassen et al., 2016). This matter fully agrees with our results. Apart from many observations, numerous simulations and analytical works emphasized the potential importance of filaments for star formation (André et al., 2014). Simulations of turbulence have consistently shown that the gas is quickly compressed into a hierarchy of sheets and filaments (e.g. Padoan et al., 2001; Vazquez-Semadeni, 1994). On the other hand, sheets have a natural tendency to fragment into filaments (e.g. Miyama et al., 1987; André, 2015). In this regard, many simulations indicate this fact, in which the star-forming cores should be located along the dense filaments (e.g., Krumholz et al., 2007; Tilley and Pudritz, 2004; Mac Low and Klessen, 2004; Bonnell et al., 2003; Ostriker et al., 1999). These issues are consistent with the results of this work.
Regarding to the fact that ‘α’ should be real, we can obtain an important condition as follows
Ψξ ≥ 3(Mξ + γ Γξ ).
(33)
According to this equation, the critical turbulent parameter can be written as
Ψξ − 3Mξ ⎞ γmax = ⎜⎛ . ⎟ 3Γξ ⎝ ⎠α = 0
(34)
Eq. (34) represents the upper limit of the turbulence parameter in a cloud when it wants to remain in force balance. For γ > γmax, the cloud goes to an unstable phase. One can show that Eq. (34) is satisfied for both shapes. 5. Results and observations 5.1. Results and discussion The results of this work are summarized in Figs. 1–4. Figs. 1 and 2 concern the filamentary clouds, while Fig. 3 is related to spherical clouds. Fig. 1 shows the non-dimensional density against the non-dimensional radius for a quiescent, filamentary cloud (i.e. γ = 0 ). We notice that σ is proportional to the magnetic fields strength. As this figure shows, an increase in field strength increases the density as a function of the radius. Also, the effect of the field gradient is significant through the clouds. It seems that a magnetic field can be effective in formation of a massive core in a quiescent, filamentary cloud. As we will see further, this subject will be vice versa in the spherical clouds. According to Eq. (1) and a toroidal configuration of the magnetic field, the Lorentz force has two terms: the first term supports cloud against the gravity and another term acts vice versa. In filamentary clouds, the first term (that supports the cloud against the gravity) is dominant. However, in spherically cloud it is different and the second term is dominant. Fig. 2 presents an overview of the turbulence effect on filamentary clouds. The comparison between parts (a) and (b) shows the role of turbulence is more important than the magnetic field’s role. According to Eqs. (1) and (9), the turbulence in both cases supports the cloud against the gravity. Fig. 3 shows the impact of a toroidal magnetic field and turbulence on a spherical cloud. Part (a) of this figure presents the role of magnetic field on the quiescent regions, while the part (b) shows the effect of turbulence and magnetic field together. As part (a) of Fig. 3 shows, the toroidal magnetic field has a destructive effect on a quiescent spherical clouds. It also shows that a weak magnetic field can produce this effect
6. Conclusion Observed molecular clouds are seen in two main shapes: as filamentary and spherical clouds. Based on the observational evidence, the filamentary clouds have a high potential for the birth of massive stars. On the other hand, it is widely accepted that a magnetic field and turbulence have particular impacts on the structure and evolution of 74
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Fig. 3. The non-dimensional density against the non-dimensional radius for a spherical cloud with n = 0.5 (a) γ = 0 (quiescent region) (b) γ = 0.5 (turbulent region).
small radii, while the impact of turbulence can be seen in large radii of the cloud. Note that there is a limitation the effect of turbulence for both shapes, if the cloud wants to remain in force balance over its lifetime. This limitation leads to a maximum of the turbulence parameter that can be directly determined from the density structure equations and the Virial theorem. The interesting result of this work is that the mass of each shape is related to the magnetic field and turbulence. Magnetic field results in the decrease and increase of the mass of spherical and filamentary clouds, respectively. It is widely believed that a magnetic field in a molecular cloud supports the cloud against the gravity (e.g. Stahler and Palla, 2004; Bodenheimer, 2011). In the standard model of star formation for low-mass stars (Shu, Adams, & Lizano, 1987), a criterion for collapse in the presence of a magnetic field is often stated as a critical
both shapes. The geometry of clouds appears to play a key role in the mentioned processes. In this work, we have considered the effect of a toroidal magnetic field on the structure and evolution of the molecular clouds in quiescent and turbulent regions. In each region, a power-law relation between the magnetic field and density is assumed for two shapes in magnetohydrostatic equilibrium. In this case, the density structure was obtained from the magnetohydrostatic equation and Poisson’s equation for a known equation of state. Figs. 1–4 show that the clouds (with different shapes) give different response to the magnetic field. In other words, the toroidal magnetic field plays a dual role in the molecular clouds related to their shapes. The response of two shapes is approximately the same to the turbulence so that the turbulence increases the density as a function of the radius for the two shapes. Furthermore, the magnetic field effectively acts at 75
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Fig. 4. The non-dimensional density against the non-dimensional radius for both shapes n = 0.5 (a) σ = 0.5 (b) σ = 1.
mass-to-flux ratio (i.e. (M /ΦB )crit = C0/ G where C0 is a constant related to the geometry of the cloud, M is the total mass and ΦB is the magnetic flux of the cloud) (Mouschovias & Spitzer, 1976). If the ratio of the actual M/ΦB to the critical value is greater than ‘1’, the cloud is in supercritical phase and can contract (Bodenheimer, 2011). In the standard model, the magnetic field does not have any role in total mass of the cloud. This lack comes from the fact that the relation between the magnetic field and density is not considered in this model. However, the results of this work showed that M and ΦB are related to the field strength and turbulence. According to the obtained results, the clouds related to massive stars, are expected to be seen in filament configurations. Whereas lowmass and core-less clouds are expected to be seen in spherical configurations. These results are fully consistent with the observations.
Because of the giant molecular clouds, the birth place of massive stars, are often organized in filamentary configurations in a spiral arm (e.g. Bally et al., 1987; Busquetetal, 2013; Ke et al., 2015). In this regard, observations indicate that massive stars in the Galaxy form from turbulent cores (e.g. McKee and Tan, 2003). Regarding to the obtained results of this work, we can classify the clouds as four major classes: 1) core-less clouds 2) low-mass cloud cores 3) moderate-mass cloud cores 4) high-mass cloud cores. Note that we limited ourselves to the regions such that the magnetic field is compatible with toroidal configuration. The first class of clouds can be seen in quasi-spherical clouds at quiescent regions. This is consistent with the observational evidence (Benson and Myers, 1989; Goodman et al., 1998; Barranco and Goodman, 1998). The second class of clouds belongs to the quasispherical clouds at turbulence-dominated regions. This class might be a 76
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site to the birth of very low-mass star such as M stars or brown-dwarf stars. The filamentary clouds in quiescent regions are in the third class. Solar-mass stars can be seen in this class (Kauffmann et al., 2010). The fourth class of clouds can be seen in filamentary clouds at turbulencedominated regions. This class has a suitable potential for the birth massive stars (OB stars). The third and fourth class can be seen in the galactic center and spiral arms of Galaxy.
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Contents lists available at ScienceDirect
New Astronomy journal homepage: www.elsevier.com/locate/newast
The response of filamentary and spherical clouds to the turbulence and magnetic field
T
Mahmoud Gholipour Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) -Maragha, P. O. Box: 55134 - 441, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: ISM: structure Stars: formation
Recent observations have revealed that there is a power-law relation between magnetic field and density in molecular clouds. Furthermore, turbulence has been observed in some regions of molecular clouds and the velocity dispersion resulting from the turbulence is found to correlate with to the cloud density. Relating to these observations, in this study, we model filamentary and spherical clouds in magnetohydrostatic equilibrium in two quiescent and turbulent regions. The proposed equations are expected to represent the impact of magnetic field and turbulence on the cloud structure and the relation of cloud mass with shape. The Virial theorem is applied to consider the cloud evolution leading to important conditions for equilibrium of the cloud over its lifetime. The obtained results indicate that under the same conditions of the magnetic field and turbulence, each shape presents different responses. The possible ways for the formation of massive cores or coreless clouds in some regions as well as the formation of massive stars or low-mass stars can be discussed based on the results of this study. It should be mentioned that the shape of the clouds plays an important role in the formation of the protostellar clouds as well as their structure and evolution. This role is due to the effects of magnetic fields and turbulence.
1. Introduction For several decades, it has been widely argued that the process of star formation is primarily controlled by the interplay between the gravity and magnetostatic support, regulated by neutral-ion drift (Mestel and Spitzer, 1956; Mac Low and Klessen, 2004). However, both observational and numerical recent works suggest that supersonic interstellar turbulence rather than magnetic fields controls the star formation (Mac Low and Klessen, 2004; Klessen, 2004). Generally, in astrophysics, different turbulence, such as supersonic, compressible, magnetized, and incompressible turbulence are considered. Most studies on the turbulence deal with turbulence in an incompressible medium, which is characteristic of many terrestrial applications. Generally, the root-mean-square velocities are considered to be subsonic, and the density is taken almost constant. The energy dissipation occurs then totally on the scales of the smallest vortices, where the dynamical scale is shorter than the length on which viscosity acts (Mac Low and Klessen, 2004). Based on the molecular clouds and condensations, Larson (1981) showed that the internal velocity dispersion of any region is well correlated with its size and mass. He reported that these correlations have a power-law behavior. The dependence of the velocity dispersion on the region size is similar to the Kolmogoroff law for the subsonic turbulence, which suggests that the observed motions are part
of a common hierarchy of interstellar turbulent motions (Larson, 1981; Heyer and Dame, 2015). The studied regions are often gravitationally bound and approximately in Virial equilibrium. By the way, they cannot be created by a simple gravitational collapse, and it seems the molecular clouds are formed at least partly by supersonic hydrodynamics processes (Larson, 1981). On the other hand, the observations of the magnetic fields in molecular clouds can give us important information about the structure and evolution of the clouds (Crutcher, 2012). The techniques for observing and mapping magnetic fields in molecular clouds have been comprehensively reviewed by Crutcher (2012). The use of dust polarization, Faraday rotation, masers, Goldreich–Kylafis (GK) effect, and Zeeman effect are some of techniques for observing and mapping magnetic fields in molecular clouds (e.g. Gholipour, 2017). The only available technique for directly measuring magnetic field strengths in the ISM is the Zeeman effect (e.g., Crutcher et al. 1993; Crutcher, 2012). Magnetic field strengths from masers would be very important in confirming the relationship between magnetic field, B and gas density, ρ (i.e. B = αρn , where α and n are constant; see Crutcher, 2012 or Gholipour, 2017). In the recent two decades, due to advances in observational instruments, numerous observational works have been carried out to determine the direction of the magnetic fields in the molecular clouds.
E-mail address: [email protected]. https://doi.org/10.1016/j.newast.2017.12.003 Received 11 September 2017; Received in revised form 5 December 2017; Accepted 7 December 2017 Available online 12 December 2017 1384-1076/ © 2017 Elsevier B.V. All rights reserved.
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In this way, the toroidal magnetic fields have been targeted by many observational studies (e.g. André et al., 2014; Han, 2013; Verschuur, 2013; Contreras et al., 2013; Girart et al., 2013; Novak et al., 2003; Matthews et al., 2001). For instance, Girart et al. (2013) presented high angular resolution observations of a massive star forming core at 880 μ m using the Submillimeter Array. They found that the dust polarization shows a complex magnetic field, compatible with a toroidal configuration. As another example, we can mention to the work of Novak et al. (2003). They observed the linear polarization emission from the Galactic center, using the polarimetric detector system at the South Pole. The resulting polarization map covered 30 pc in Galactic latitude and 170 pc along the Galactic plane, and thus included a significant fraction of the central molecular zone. Their map presented that the magnetic fields were often toroidal in these regions. Recently, Gholipour (2017) studied the stability of the spherical cloud core with a toroidal magnetic field. He considered a power-law relation between the magnetic field and density in a quiescent region (i.e. without turbulence). He found that the presence of the magnetic field can significantly affect a density structure and the total mass of the cloud. In this paper, the response of filamentary and spherical clouds on the turbulence and toroidal magnetic fields is considered. Since observations of star-forming regions, i.e. molecular clouds, show a powerlaw relation between the magnetic field and density, we study both classes of clouds (i.e. filamentary and spherical clouds) with this mentioned relation, in magnetohydrostatic equilibrium and with turbulence. Then, the Virial theorem will be applied for discussing the time evolution of the clouds. In this way, in Section 2 we model the magnetic field in filamentary clouds. In Section 3 we consider spherical clouds. In both cases we consider quiescent and turbulent regions. The evolution of clouds is considered in Section 4 in terms of the Virial theorem. In Section 5, we discuss the obtained results and compare the results with the observations. Finally, we give our conclusions in Section 5.
result is 2
α 2ρ2n − 1 ⎛ (3n − 1) dρ 1 dP 1 d 2P 1 dρ ⎞ dP + − ⎜⎛ ⎟ + rρ dr ρ dr 2 ρ dr dρ 4π ⎜ rρ dr ⎝ ⎠ ⎝ 2
1 dρ ⎞ n d 2ρ ⎞ ⎟ + 2n (n − 1) ⎜⎛ + ρ dr ρ dr 2 ⎟ ⎝ ⎠ ⎠ + 4πGρ = 0.
Eq. (3) shows the density structure, if we know the equation of state (i.e. knowing P(ρ)). However, the equation of state is sensitive to turbulence (e.g. Gehman et al., 1996). Thus, we consider this equation in two important regions as follows: 1- quiescent region 2- turbulent region. 2.2. Quiescent clouds Here, we investigate the density structure of an isothermal quiescent cloud (i.e. P = cs2 ρ where cs is the isothermal sound speed). With defining β = α 2/4πcs2, Eq. (3) can be rewritten as follows
(1 + nβρ2n − 1)
+
4πG ρ = 0. cs2
−1 2
4πGρ r0 = ⎜⎛ 2 c ⎟⎞ ⎝ cs ⎠
(4)
,
(5)
and also,
σ = βρc2n − 1 .
(6)
Therefore, we can rewrite Eq. (4) as follows 2
d 2ϱ (1 + 2n (1 − n) σρ2n − 1) 1 ⎛ dϱ ⎞ (1 + (3n − 1) σρ2n − 1) 1 dϱ − + ⎜ ⎟ 1 + nσ ϱ2n − 1 ϱ ⎝ dξ ⎠ 1 + nσ ϱ2n − 1 dξ 2 ξ dξ ϱ2 + = 0. 1 + nσ ϱ2n − 1
(7) Furthermore, the boundary conditions are given by (e.g. Stahler and Palla, 2004)
2.1. Magnetic field in molecular clouds
ϱ (ξ = 0) = 1,
The scaling of magnetic field strength with density can be parameterized as B = αρn where α and n are the field strength and field gradient, respectively (e.g. Crutcher et al., 2010; Crutcher, 2012). In a cylindrical system of coordinates, we study the impact of a toroidal magnetic field on a self-gravitating cloud which is in an equilibrium state. The equation of magnetohydrostatic equilibrium is (e.g. Mestel, 2003; Gholipour, 2017)
dϱ dξ
= 0. (ξ = 0)
(8)
Eq. (7) describes the density structure of a quiescent cloud in the presence of the toroidal field configuration. The resulting density as a function of radius is depicted in Fig. 1, for different values of the parameters σ and n. 2.3. Turbulent clouds
(1)
Now, we consider turbulence effect by adding a term that attempts to model the turbulence observed in the molecular clouds (e.g. Gehman et al., 1996; Lizano and Shu, 1989; see also Mestel, 2003). The equation of state is then given by
where P is the gas pressure and Φ is the gravitational potential related to the gas density by Poisson’s equation
∇2 Φ = 4πGρ ,
1 ⎛ dρ ⎞2 ρ2 ⎝ dr ⎠
This equation shows an isothermal density structure of a cloud in the presence of the toroidal magnetic field in the cylindrical coordinate system. We recast Eq. (4) into a dimensionless form. We define a nondimensional radius as ξ = r / r0, and a non-dimensional density as ϱ = ρ / ρc where ρc is the central density and r0 is
In this section, we study the role of magnetic field and turbulence in filamentary clouds. In a cylindrical coordinate system (r, ϕ, z), we consider an axisymmetric and long filament along the z-axis. Hence, all variables are assumed to only depend on the radial distance r. The cloud is assumed to be exposed to a toroidal magnetic field (e.g. André et al., 2014; Girart et al., 2013; Han, 2013; Verschuur, 2013; Contreras et al., 2013). Many observational studies (e.g. Tritsis et al., 2015; Crutcher, 2012; 1999; Crutcher et al., 2010; Mouschovias and Tassis, 2010) and some analytical works (e.g. Li et al., 2015; Bodenheimer, 2011, Mouschovias, 1985; 1979; 1976a; 1976b) indicate this where there is a power-law relation between magnetic field strength and density in the molecular clouds.
1 (∇ × B) × B = 0, 4π
1 d 2ρ 1 dρ + (1 + (3n − 1) βρ2n − 1) ρ dr 2 rρ dr
− (1 + 2n (1 − n) βρ2n − 1)
2. Filamentary clouds
− ∇P − ρ ∇Φ +
(3)
(2)
ρ P = Pgas + Pturb = cs2 ρ + p0 log ⎛⎜ ⎞⎟, ρ ⎝ c⎠
where G is Newton’s constant. If one applies the cylindrical divergence on Eq. (1) and then substitutes the obtained result into Eq. (2), the final 70
(9)
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Fig. 1. The non-dimensional density against the non-dimensional radius for a quiescent filamentary cloud (a) n = 0.45 (b) n = 0.65 .
κ ⎞ dρ κ d 2ρ κ dρ 2 κ dρ 2 1 ⎛ 1 1 ⎜1 + ⎟ + ⎜⎛1 + ⎟⎞ 2 − 3 ⎛ ⎞ − 2 ⎜⎛1 + ⎟⎞ ⎛ ⎞ rρ ⎝ ρ ⎠ dr ρ⎝ ρ ⎠ dr ρ ⎝ dr ⎠ ρ ⎝ ρ ⎠ ⎝ dr ⎠
where p0 is a constant, which may be determined empirically. The motivation for the functional form adopted in Eq. (9) which is the empirical power-law correlation vturb ∼ ρ−1/2 is often called Larson’s law (e.g. Mestel, 2003). Thus, we can write 1 2
2
+
1 2
p dP vturb = ⎜⎛ turb ⎟⎞ = ⎜⎛ 0 ⎟⎞ . dρ ⎝ ⎠ ⎝ρ⎠
α 2ρ2n − 1 ⎛ (3n − 1) dρ 4πGρ n d 2ρ ⎞ 1 dρ ⎞ ⎟ + 2n (1 − n) ⎜⎛ + + rρ dr ρ dr ρ dr 2 ⎟ 4πcs2 ⎜⎝ cs2 ⎝ ⎠ ⎠ = 0.
(10)
(11)
Eq. (10) leads to the logarithm relationship between turbulent pressure and density (see Eq. (9)). According to Eqs. (3) and (9), and with defining κ = p0 / cs2, we can write
After transporting variables to the dimensionless forms, Eq. (11) leads to
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Fig. 2. The non-dimensional density against the non-dimensional radius for a turbulent filamentary cloud n = 0.5, (a) σ = 0.5 (b) σ = 1.5 .
(1 + γ ϱ−1 + (3n − 1) σρ2n − 1) 1 dϱ d 2ϱ + 1 + γ ϱ−1 + nσ ϱ2n − 1 dξ 2 ξ dξ
to the isothermal Lane–Emden (hereafter LE) equation for a non-magnetized quiescent cloud (i.e. σ = γ = 0 ). As Eq. (12) shows, the nondimensional density is related to three parameters as follows: 1) the field strength σ, 2) the field gradient n, 3) the turbulence parameter γ. Fig. 2 shows the impact of turbulence on a filamentary cloud.
2
(1 + γ ϱ−2 + 2n (1 − n) σρ2n − 1) 1 ⎛ dϱ ⎞ ⎜ ⎟ 1 + γ ϱ−1 + nσ ϱ2n − 1 ϱ ⎝ dξ ⎠ ϱ2 + = 0, 1 − 1 + γ ϱ + nσ ϱ2n − 1 −
(12) 3. Spherical clouds
where γ is the turbulence parameter (κ/ρc). In addition, the boundary conditions are given by
ϱ (ξ = 0) = 1,
dϱ dξ
Here, we consider the impact of toroidal magnetic field and turbulence on a spherical cloud. This process is a bit more complicated than the filamentary clouds. In spherical clouds, the magnetic field is often composed of two components as poloidal and toroidal (i.e. ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ B = Bθ + Bϕ ). If the poloidal component is sufficiently weak (i.e.
= 0. (ξ = 0)
(13)
Eq. (12) leads to Eq. (7) for a quiescent cloud (i.e. γ = 0 ), and also leads 72
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Bθ ≪ Bϕ), we can assume only the toroidal component is dominant. Moreover, a purely toroidal field makes it easier to compare the obtained results with the results of previous section.
defining κ = p0 / cs2, we can write
κ ⎞ dρ κ d 2ρ κ dρ 2 κ dρ 2 2 ⎛ 1 1 ⎜1 + ⎟ + ⎜⎛1 + ⎟⎞ 2 − 3 ⎛ ⎞ − 2 ⎜⎛1 + ⎟⎞ ⎛ ⎞ rρ ⎝ ρ ⎠ dr ρ⎝ ρ ⎠ dr ρ ⎝ dr ⎠ ρ ⎝ ρ ⎠ ⎝ dr ⎠
3.1. Magnetic field in quiescent clouds
2
We study the equilibrium structure of a self-gravitating cloud affected by the toroidal magnetic field. The equation of magnetohydrostatic equilibrium and the Poisson’s equation have been given by Eqs. (1) and (2), respectively. We take the divergence on Eq. (1), and then we substitute the obtained result into Eq. (2). Thereupon, the final result is
(20) As previous section, we can rewrite Eq. (20) to the non-dimensional form as follows
(2 + 2γ ϱ−1 + (4n − 1) σρ2n − 1) 1 dϱ d 2ϱ + 1 + γ ϱ−1 + nσ ϱ2n − 1 dξ 2 ξ dξ
α 2ρ2n − 1 ⎛ 1 (4n − 1) dρ 2 dP 1 d 2P 1 dP dρ + − 2 + + rρ dr ρ dr 2 ρ dr dr 4π ⎜ r 2 rρ dr ⎝
2
(1 + 2γ ϱ−2 + 2n (1 − n) σρ2n − 1) 1 ⎛ dϱ ⎞ ⎜ ⎟ 1 + γ ϱ−1 + nσ ϱ2n − 1 ϱ ⎝ dξ ⎠ ϱ2 σ ϱ2n 1 + + = 0, 1 2 1 2 1 − n − − 1 + γ ϱ + nσ ϱ2n − 1 1 + γ ϱ + nσ ϱ ξ
2
1 dρ ⎞ n d 2ρ ⎞ + 2n (n − 1) ⎛⎜ + ⎟ ρ dr 2 ⎟ ⎝ ρ dr ⎠ ⎠ + 4πGρ = 0.
−
(14)
The density structure of an isothermal quiescent cloud, with defining β = α 2/4πcs2, can be written as follows
(1 + nβρ2n − 1)
ϱ (ξ = 0) = 1,
2
− (1 + 2n (1 − +
1 ⎛ dρ ⎞ ρ2 ⎝ dr ⎠
βρ2n − 1 4πG + 2 ρ = 0. r2 cs
−1 2
(15)
,
(17)
Then, Eq. (15) will be
dξ 2
+
(2 + (4n − 1 + nσ ϱ2n − 1
n) σρ2n − 1)
(1 + 2n (1 − 1 dϱ − 1 + nσ ϱ2n − 1 ξ dξ
2
where K is the internal energy, Ψ is the gravitational energy and UB is the magnetic field energy. Note that we neglect the effect of external pressure at the surface in the Virial theorem (i.e. Ps = 0 ). Now, we can begin the study of time-evolution of a cloud. First of all, we consider the dimensional mass M for a spherical cloud. We have
1 ⎛ dϱ ⎞ ⎜ ⎟ ϱ ⎝ dξ ⎠
σ ϱ2n 1 1 + nσ ϱ2n − 1 ξ 2 ϱ2 + = 0. 1 + nσ ϱ2n − 1
s
ϱξ 2dξ ,
(24)
where ’c’ and ’s’ denote the center and surface of the cloud, respectively. Also, we introduce non-dimensional mass as follows
Also, the boundary conditions are given by (e.g. Gholipour and NejadAsghar, 2013)
M , M0
Mξ =
= 0. (ξ = 0)
∫ ρdV = 4πρc r03 ∫c
M= (18)
dϱ dξ
(23)
2K + Ψ + UB = 0, 1) σρ2n − 1)
+
ϱ (ξ = 0) = 1,
(22)
The time evolution of a cloud, which would be in an equilibrium state, will be considered over its lifetime in this section. The critical question, that emerges here, is: would the cloud remain in an equilibrium state (force balance) over its lifetime? The Virial theorem has a suitable response for this question. The Virial theorem should be satisfied, if the cloud wants to stay in force balance over its lifetime (e.g., Mestel, 1965a; 1965b)
and also,
d 2ϱ
= 0. (ξ = 0)
4. Virial theorem in molecular clouds
(16)
σ = βρc2n − 1 .
dϱ dξ
Indeed, Eq. (21) leads to Eq. (18) for a quiescent cloud (i.e. γ = 0 ), and leads to the isothermal LE equation for a non-magnetized quiescent cloud (i.e. σ = γ = 0 ). As Eq. (21) shows, the non-dimensional density structure depends on three parameters as follows: 1) the field strength σ, 2) the field gradient n, 3) the turbulent parameter γ. Fig. 3.b shows the role of turbulence and magnetic field on the density structure.
This equation is the isothermal density structure of a cloud core in the presence of the toroidal magnetic field in the spherical coordinate system. It is useful to recast Eq. (15) into dimensionless form. As in the previous section, we define a non-dimensional radius as ξ = r / r0 and a non-dimensional density as ϱ = ρ / ρc where ρc is the central density and r0 is
4πGρ r0 = ⎜⎛ 2 c ⎟⎞ ⎝ cs ⎠
(21)
where γ is the turbulent parameter as κ/ρc. The boundary conditions are given by
1 d 2ρ 1 dρ + (2 + (4n − 1) βρ2n − 1) ρ dr 2 rρ dr n) βρ2n − 1)
α 2ρ2n − 1 ⎛ 1 (4n − 1) dρ 1 dρ ⎞ n d 2ρ ⎞ ⎟ + + 2n (n − 1) ⎜⎛ + 2 ⎜ 2 rρ dr ρ dr 2 ⎟ 4πcs ⎝ r ⎝ ρ dr ⎠ ⎠ 4πGρ 0. + = cs2
+
(19)
M0 = 4πρc r03,
(25)
so that we have
Eq. (18) presents a modified form of the LE equation: in the absence of the toroidal magnetic field (i.e. α = 0 or σ = 0 ) one obtains the isothermal LE equation. Fig. 3.a presents the density as a function of radius for the spherical clouds, with different values of σ and γ.
Mξ =
∫c
s
ϱξdξ .
(26)
In the following, the internal energy is given by
K=
3.2. Turbulent clouds Regarding to Eqs. (9) and (14) in spherical coordinates, and with
3 2
∫ PdV = 6πcs2 ρc r03 ⎛ ∫c ⎜
s
ϱξ 2dξ + γ
⎝
Also, the gravitational energy is 73
∫c
s
log ϱ ξ 2dξ ⎞⎟. ⎠
(27)
New Astronomy 61 (2018) 69–77
M. Gholipour
Ψ=
1 2
∫ ρΦdV = 2πcs2 ρc r03 ∫c
s
ϱΦϱ ξ 2dξ ,
on a spherical cloud. This matter can be discussed in the core-less clouds. The part (b) of this figure reveals the effect of turbulence versus the magnetic field. The turbulence in a spherical cloud can balance somewhat the magnetic field’s effect. Fig. 4 shows the comparison between the results for different cloud shapes. Under the same conditions, each shape gives different responses to the magnetic field. This difference is seen most significantly at small radii. In other words, the structure of spherical clouds is related to the interplay between magnetic field and turbulence. While, the structure of a filamentary cloud is the result of cooperation between magnetic field and turbulence. Thus, we expect that a quasi-spherical dense core will be seen more in turbulent-dominated regions than other regions. Although, the turbulence increases the density over the radius in each shape, but this increase is limited by the Virial theorem. According to this theorem, the maximum value of the turbulence parameter can be obtained from the equation of the density structure so that the cloud leaves equilibrium when we have γ > γmax. The relation for maximum value is same for two shapes.
(28)
where Φϱ = Φ/ cs2 is non-dimensional potential. The magnetic field energy can be written as
1 1 2 B2dV = α 8π 8π s 1 ϱ2nξ 2dξ . = α 2ρc2n r03 c 2
∫
UB =
∫ ρ2ndV ,
∫
(29)
It is better to work with the below dimensionless variables, as follows:
hξ =
∫c
s
ϱ2nξdξ ,
Ψξ = −
∫c
s
ξ ϱΨdξ ,
Γξ =
∫c
s
log ϱ ξdξ .
(30)
The Virial theorem then leads to
Ψξ − 3Mξ − 3γ Γξ ⎞ σ = ⎜⎛ ⎟, hξ ⎝ ⎠
(31)
or 1
α=
4π ⎛ Ψξ − 3Mξ − 3γ Γξ ⎞ 2 ⎜ ⎟ cs . hξ ρc2n − 1 ⎝ ⎠
5.2. Observations and simulations (32) Recent observations and simulations of spiral arm clouds show that the most filaments have a magnetic field, compatible with a toroidal configuration (Seifried and Walch, 2015; Pillai et al., 2015; Zhang et al., 2014; Tomisaka, 2014; Hennebelle, 2013; Palmeirim et al., 2013; Sugitani et al., 2011; Gaensler et al., 2011). Furthermore, the starforming cores in nearby spiral-arm clouds are often reported along dense filaments (Konyves et al., 2015; Polychroni et al., 2013). These observations and observations of the Galactic center suggest that filaments may be fundamental building blocks of the molecular clouds in the star formation processes, regardless of whether the clouds are located along spiral arms or near the Galactic Center (Federrath, 2016). For instance, Polychroni et al. (2013) presented Herschel survey maps of the molecular clouds in Orion A. They extracted both the filaments and dense cores in this region. They identified which of the dense sources are proto-stellar or pre-stellar. Then, their associations with the identified filaments were studied. They found that most of the prestellar sources (71%) are located in the filaments. As we know, there is a lot of massive stars in the Orion which may be discussed based on the results of this study. However, the Herschel data indicate the fact that core-less clouds and the low-mass clouds should be seen in an quasispherical shape (André et al., 2013). Recently, regarding many observations of massive turbulent filaments, Beuther et al. (2015) showed that turbulence plays an important role in supporting the filament against the collapse (see also Klassen et al., 2016). This matter fully agrees with our results. Apart from many observations, numerous simulations and analytical works emphasized the potential importance of filaments for star formation (André et al., 2014). Simulations of turbulence have consistently shown that the gas is quickly compressed into a hierarchy of sheets and filaments (e.g. Padoan et al., 2001; Vazquez-Semadeni, 1994). On the other hand, sheets have a natural tendency to fragment into filaments (e.g. Miyama et al., 1987; André, 2015). In this regard, many simulations indicate this fact, in which the star-forming cores should be located along the dense filaments (e.g., Krumholz et al., 2007; Tilley and Pudritz, 2004; Mac Low and Klessen, 2004; Bonnell et al., 2003; Ostriker et al., 1999). These issues are consistent with the results of this work.
Regarding to the fact that ‘α’ should be real, we can obtain an important condition as follows
Ψξ ≥ 3(Mξ + γ Γξ ).
(33)
According to this equation, the critical turbulent parameter can be written as
Ψξ − 3Mξ ⎞ γmax = ⎜⎛ . ⎟ 3Γξ ⎝ ⎠α = 0
(34)
Eq. (34) represents the upper limit of the turbulence parameter in a cloud when it wants to remain in force balance. For γ > γmax, the cloud goes to an unstable phase. One can show that Eq. (34) is satisfied for both shapes. 5. Results and observations 5.1. Results and discussion The results of this work are summarized in Figs. 1–4. Figs. 1 and 2 concern the filamentary clouds, while Fig. 3 is related to spherical clouds. Fig. 1 shows the non-dimensional density against the non-dimensional radius for a quiescent, filamentary cloud (i.e. γ = 0 ). We notice that σ is proportional to the magnetic fields strength. As this figure shows, an increase in field strength increases the density as a function of the radius. Also, the effect of the field gradient is significant through the clouds. It seems that a magnetic field can be effective in formation of a massive core in a quiescent, filamentary cloud. As we will see further, this subject will be vice versa in the spherical clouds. According to Eq. (1) and a toroidal configuration of the magnetic field, the Lorentz force has two terms: the first term supports cloud against the gravity and another term acts vice versa. In filamentary clouds, the first term (that supports the cloud against the gravity) is dominant. However, in spherically cloud it is different and the second term is dominant. Fig. 2 presents an overview of the turbulence effect on filamentary clouds. The comparison between parts (a) and (b) shows the role of turbulence is more important than the magnetic field’s role. According to Eqs. (1) and (9), the turbulence in both cases supports the cloud against the gravity. Fig. 3 shows the impact of a toroidal magnetic field and turbulence on a spherical cloud. Part (a) of this figure presents the role of magnetic field on the quiescent regions, while the part (b) shows the effect of turbulence and magnetic field together. As part (a) of Fig. 3 shows, the toroidal magnetic field has a destructive effect on a quiescent spherical clouds. It also shows that a weak magnetic field can produce this effect
6. Conclusion Observed molecular clouds are seen in two main shapes: as filamentary and spherical clouds. Based on the observational evidence, the filamentary clouds have a high potential for the birth of massive stars. On the other hand, it is widely accepted that a magnetic field and turbulence have particular impacts on the structure and evolution of 74
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Fig. 3. The non-dimensional density against the non-dimensional radius for a spherical cloud with n = 0.5 (a) γ = 0 (quiescent region) (b) γ = 0.5 (turbulent region).
small radii, while the impact of turbulence can be seen in large radii of the cloud. Note that there is a limitation the effect of turbulence for both shapes, if the cloud wants to remain in force balance over its lifetime. This limitation leads to a maximum of the turbulence parameter that can be directly determined from the density structure equations and the Virial theorem. The interesting result of this work is that the mass of each shape is related to the magnetic field and turbulence. Magnetic field results in the decrease and increase of the mass of spherical and filamentary clouds, respectively. It is widely believed that a magnetic field in a molecular cloud supports the cloud against the gravity (e.g. Stahler and Palla, 2004; Bodenheimer, 2011). In the standard model of star formation for low-mass stars (Shu, Adams, & Lizano, 1987), a criterion for collapse in the presence of a magnetic field is often stated as a critical
both shapes. The geometry of clouds appears to play a key role in the mentioned processes. In this work, we have considered the effect of a toroidal magnetic field on the structure and evolution of the molecular clouds in quiescent and turbulent regions. In each region, a power-law relation between the magnetic field and density is assumed for two shapes in magnetohydrostatic equilibrium. In this case, the density structure was obtained from the magnetohydrostatic equation and Poisson’s equation for a known equation of state. Figs. 1–4 show that the clouds (with different shapes) give different response to the magnetic field. In other words, the toroidal magnetic field plays a dual role in the molecular clouds related to their shapes. The response of two shapes is approximately the same to the turbulence so that the turbulence increases the density as a function of the radius for the two shapes. Furthermore, the magnetic field effectively acts at 75
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Fig. 4. The non-dimensional density against the non-dimensional radius for both shapes n = 0.5 (a) σ = 0.5 (b) σ = 1.
mass-to-flux ratio (i.e. (M /ΦB )crit = C0/ G where C0 is a constant related to the geometry of the cloud, M is the total mass and ΦB is the magnetic flux of the cloud) (Mouschovias & Spitzer, 1976). If the ratio of the actual M/ΦB to the critical value is greater than ‘1’, the cloud is in supercritical phase and can contract (Bodenheimer, 2011). In the standard model, the magnetic field does not have any role in total mass of the cloud. This lack comes from the fact that the relation between the magnetic field and density is not considered in this model. However, the results of this work showed that M and ΦB are related to the field strength and turbulence. According to the obtained results, the clouds related to massive stars, are expected to be seen in filament configurations. Whereas lowmass and core-less clouds are expected to be seen in spherical configurations. These results are fully consistent with the observations.
Because of the giant molecular clouds, the birth place of massive stars, are often organized in filamentary configurations in a spiral arm (e.g. Bally et al., 1987; Busquetetal, 2013; Ke et al., 2015). In this regard, observations indicate that massive stars in the Galaxy form from turbulent cores (e.g. McKee and Tan, 2003). Regarding to the obtained results of this work, we can classify the clouds as four major classes: 1) core-less clouds 2) low-mass cloud cores 3) moderate-mass cloud cores 4) high-mass cloud cores. Note that we limited ourselves to the regions such that the magnetic field is compatible with toroidal configuration. The first class of clouds can be seen in quasi-spherical clouds at quiescent regions. This is consistent with the observational evidence (Benson and Myers, 1989; Goodman et al., 1998; Barranco and Goodman, 1998). The second class of clouds belongs to the quasispherical clouds at turbulence-dominated regions. This class might be a 76
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site to the birth of very low-mass star such as M stars or brown-dwarf stars. The filamentary clouds in quiescent regions are in the third class. Solar-mass stars can be seen in this class (Kauffmann et al., 2010). The fourth class of clouds can be seen in filamentary clouds at turbulencedominated regions. This class has a suitable potential for the birth massive stars (OB stars). The third and fourth class can be seen in the galactic center and spiral arms of Galaxy.
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