Wei-Heng Yang(杨伟恒)1,2, Hui Chen(陈辉),1,2, San-Qiu Liu(刘三秋)1,21Department of Physics, Nanchang University, Nanchang 330031, China 2Jiangxi Province Key Laboratory of Fusion and Information Control, Nanchang 330031, China
Abstract The effect of ion drag on the pulsational mode of gravitational collapse (PMGC) is investigated within the partially charged dusty plasma model by fluid dynamics. It is found that the ion drag force significantly enhances the instability of the PMGC. In addition, it is shown that the instability of the PMGC is influenced by the ratio of the abundances of charged to neutral grains. These results can be relevant for the planetesimal formation in dark interstellar clouds. Keywords：ion drag;Jeans instability;PMGC;dusty plasma
PDF (411KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Wei-Heng Yang(杨伟恒), Hui Chen(陈辉), San-Qiu Liu(刘三秋). Effect of ion drag on a pulsational mode of gravitational collapse. Communications in Theoretical Physics, 2020, 72(7): 075504- doi:10.1088/1572-9494/ab8a1e
1. Introduction
The dark interstellar clouds, which consist of ${{\rm{H}}}_{2}$, CO and the dusty grains of silicates, iron, etc, are viewed as ideal sites for star formation [1]. It is well known that these clouds are usually sub-divided into clusters of size10 pc and mass 103 to 104${\rm{M}}\odot $, which form low-mass stars due to gravitational collapse [2]. The mass of a star formed in a cloud core has been discussed by Nakano et al [3]. In particular, the HII regions in the dark interstellar clouds may exist within the dusty clouds and hydrogens in a low density diffused plasma state because it is relatively hotter, with the temperature ranging from 5000 K–10 000 K [4]. Star formation is a very hot topic in both plasma [5, 6] and astrophysics [7] owing to its important role in exploration of the Universe. As we all know, the conditions in which a cloud can successfully collapse into a protostar depend on whether the gravitational energy exceeds the sum of the thermal, rotational and magnetic energies. The dust in the HII regions is always negatively charged, depending on the following two mechanisms [7, 8]: (a) as it is immersed in such a low density and diffused plasma state, the dust can acquire a net electric charge; (b) due to the interstellar radiation fields, some of the dust can easily absorb charges from the plasma environment so that they will carry a net electric charge, whereas the other part will remain electrically neutral. Therefore, it is pertinent to study the gravitational collapse of a partially charged dusty plasma within the background of interstellar clouds.
On the other hand, dusty plasma is viewed as ‘the most abundant form of ordinary matter in the Universe’, and its linear and nonlinear physical phenomena have been widely discussed [9–14]. For example, a mathematical method used to solve some nonlinear physical phenomena in dusty plasma has been provided by Gao [15, 16]. The Jeans instability of dusty plasma was first researched by Pandy in 1994 [5]. After this, a series of works about the Jeans instabilities in dusty plasma have been widely investigated. The effect of electrostatic pressure on the Jeans instability of self-gravitating dusty cloud has been researched by Bezbaruah [17]. The Jeans instabilities in quantum dusty plasma on the basis of quantum effects has also been examined by Shukla et al [18–21]. The effect of secondary electron emission on Jeans instability has been discussed by Sarkar et al [22–24]. It has been found that secondary electron emission has a significant effect on the Jeans instability of the system. Meanwhile, they also further researched the effect of secondary electron emission on the Jeans instability in a complex plasma in the presence of nonthermal ions [8]. Similarly, Dwivedi, Sen and Bujarbarua studied the Jeans instability in dusty plasma in the background of a partially charged dusty plasma, and a meaningful phenomenon has been found in that a new mode of Jeans condensation, namely the pulsational mode of gravitational collapse (PMGC), may occur if a part of the dusty particles is ionized in the plasma environment [7]. However, they only considered the effect of the collision between uncharged dust and charged dust on the Jeans instability of dusty plasma, but ignored the effect of ion collision. As proposed by Gaurav and Avinash [6], the ion drag force significantly enhances the Jeans instability and the gravitational collapse of the dusty cloud. Therefore, in this paper, the effect of ion drag force aroused from ion–dust collision on the Jeans instability of partially charged dusty plasma in dark interstellar clouds has been examined in detail.
The remainder of this paper is organized as follows. Section 2 outlines the related work of the basic theories and the equations of our model. Based on the second part, using numerical methods and charts, these results are analyzed in section 3. A conclusion is given in section 4.
2. Partially charged dust model
A finite-sized charged dusty cloud embedded in a larger hydrogen plasma background composed of electrons and ions is considered. This larger-scale hydrogen plasma has a certain temperature, such as the HII region mentioned above, with 5000 K–10000 K. So far, some authors have shown that there are dust clouds in the $\mathrm{HII}$ region, from which dusty particles were charged in the context of such warm hydrogen plasma [3, 25, 26]. In the interstellar environment, there are various radiation mechanisms, secondary electron emission and other factors which will have a certain impact on the charge carried by the dusty surface. When very small dusty particles in cosmic plasma are exposed to ultraviolet radiation, electrons with high enough energy impact a single dusty particle and ionize dusty material, resulting in the generation of secondary electrons and the generation of a secondary electron current. Such a mechanism is equivalent to the flow of positive current to the dusty surface, which is a significant phenomenon in astrophysical plasmas [27–32]. However, a dusty charging mechanism different from other models is discussed here, which is decided on for the following reasons: a) because the dust and plasma in these clouds may be not homogeneously mixed, a part of the dust is closer to some ionizing sources; b) some parts of the cloud may be more ionized, where a relatively large number of charged dusty particles are produced; c) there may be local (small scale) instabilities and fluctuations of the plasma. Then there will be some parts of the dust that are charged in the cloud [33]. In fact, besides considering the important role of the charged-dust to neutral-dust ratio $\eta (\equiv {n}_{\mathrm{dc}0}/{n}_{\mathrm{dn}0})$ in the formation of stars, the impact of ion resistance on the formation of stars should also be taken into consideration in this article. Two assumptions about this model are presented as follows. a) We assumed that all the dusty particles in the model are the same size. Under these conditions, it has the same charge on the charged dusty particles for a given plasma environment. b) For any physical phenomena on a dusty inertial timescale, the electrons could be assumed to follow the Boltzmann distribution:$ \begin{eqnarray}{n}_{{\rm{e}}}\approx {n}_{{\rm{e}}0}\exp \left(e\phi /{T}_{{\rm{e}}}\right).\end{eqnarray}$In our model, the effect of electrostatic force between dust and other particles cannot be ignored. For the electrostatic force to play an important role, it should be of the order of gravitational force(${{Gm}}_{{\rm{d}}}/{q}_{{\rm{d}}}^{2}\sim O(1)$) [5]. By simulating the above molecular cloud model and deducing the dispersion relation of the Jeans instability in dusty plasma, the effect of ion drag on the above PMGC mode is studied. The friction coupling between neutral dusty particles and charged dusty particles is taken into consideration, as well as the friction coupling between ion resistance and charged dust. To further investigate the influence of these two factors on the Jeans instability in dusty plasma, the friction term in neutral dust dynamics is ignored. Considering four groups of components (ions, electrons, neutral dust and charged dust), our model is constructed by the following equations:$ \begin{eqnarray}\displaystyle \frac{\partial {n}_{\mathrm{dn}}}{\partial t}+{\rm{\nabla }}\cdot \left({n}_{\mathrm{dn}}{{\boldsymbol{v}}}_{\mathrm{dn}}\right)=0,\end{eqnarray}$$ \begin{eqnarray}\displaystyle \frac{\partial {n}_{\mathrm{dc}}}{\partial t}+{\rm{\nabla }}\cdot \left({n}_{\mathrm{dc}}{{\boldsymbol{v}}}_{\mathrm{dc}}\right)=0,\end{eqnarray}$$ \begin{eqnarray}\displaystyle \frac{\partial {n}_{{\rm{i}}}}{\partial t}+{\rm{\nabla }}\cdot \left({n}_{{\rm{i}}}{{\boldsymbol{v}}}_{{\rm{i}}}\right)=0,\end{eqnarray}$$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial {{\boldsymbol{v}}}_{\mathrm{dc}}}{\partial t}+({{\boldsymbol{v}}}_{\mathrm{dc}}\cdot {\rm{\nabla }}){{\boldsymbol{v}}}_{\mathrm{dc}}\\ =\,-\displaystyle \frac{{q}_{{\rm{d}}}}{{m}_{{\rm{d}}}}{\rm{\nabla }}\phi -{\rm{\nabla }}\varphi -{\nu }_{{\rm{i}}{\rm{d}}}\left({{\boldsymbol{v}}}_{\mathrm{dc}}-{{\boldsymbol{v}}}_{{\rm{i}}}\right)-{\nu }_{\mathrm{cn}}\left({{\boldsymbol{v}}}_{\mathrm{dc}}-{{\boldsymbol{v}}}_{\mathrm{dn}}\right),\end{array}\end{eqnarray}$$ \begin{eqnarray}\displaystyle \frac{\partial {{\boldsymbol{v}}}_{\mathrm{dn}}}{\partial t}+({{\boldsymbol{v}}}_{\mathrm{dn}}\cdot {\rm{\nabla }}){{\boldsymbol{v}}}_{\mathrm{dn}}=-{\rm{\nabla }}\varphi ,\end{eqnarray}$$ \begin{eqnarray}{{\rm{\nabla }}}^{2}\varphi =4\pi {{Gm}}_{{\rm{d}}}({n}_{\mathrm{dc}}+{n}_{\mathrm{dn}}),\end{eqnarray}$$ \begin{eqnarray}{{\rm{\nabla }}}^{2}\phi =4\pi e\left({n}_{{\rm{e}}}-{n}_{{\rm{i}}}-\displaystyle \frac{{q}_{{\rm{d}}}}{e}{n}_{\mathrm{dc}}\right),\end{eqnarray}$where ndn, ndc and ni represent number densities of neutral dust, charged dust and ions; ${v}_{{\rm{dn}}}$, ${v}_{{\rm{dc}}}$ and ${v}_{{\rm{i}}}$ are the velocities of neutral dust, charged dust and ions; md is the mass of dusty grain, G is the universal gravitational constant, φ is the gravitational potential, φ is the electrostatic potential, ${\nu }_{\mathrm{id}}$ is the collision frequency between ions and charged dust, and νcn is the binary collisional rate of momentum transfer from charged grains to neutral grains, respectively. The thermal term has been ignored here because we believed that the dust is cold.
Strictly speaking such dusty clouds are not homogeneous, which means this equilibrium cannot be regarded as homogeneous. However, to solve the actual problems, invoking the Jeans swindle, the zero order gravitational field is ignored and then the equilibrium can be viewed as homogeneous. Mathematically, the Poisson’s equation involved in the Jeans swindle is written as,$ \begin{eqnarray}{{\rm{\nabla }}}^{2}\varphi =4\pi {{Gm}}_{{\rm{d}}}\left({n}_{\mathrm{dc}}+{n}_{\mathrm{dn}}-{n}_{{\rm{d}}0}\right).\end{eqnarray}$In this equation, ${n}_{{\rm{d}}0}={n}_{\mathrm{dc}0}+{n}_{\mathrm{dn}0}$ models the Jeans swindle of the equilibrium gravitational force field [5]. In such uniform equilibrium, the velocities of three species and the electric field E are viewed as zero; meanwhile, the three species are distributed uniformly with constant densities. Along the standard procedure, the relevant physical variables can be expressed in the form of ${X}_{\alpha }={X}_{\alpha 0}+{X}_{\alpha 1}$, where ${X}_{\alpha 0}$ denotes its equilibrium status, and ${X}_{\alpha 1}$ denotes its perturbation (α=d, e, i for dust, electrons, ions, respectively). Substituting it in equations (2)–(9), and after Fourier analysis with the plane wave ansatz, i.e. ${X}_{\alpha 1}\propto \exp (-{\rm{i}}\omega t+{\rm{i}}{\boldsymbol{k}}\cdot {\boldsymbol{x}})$, we have:$ \begin{eqnarray}-{\rm{i}}\omega {n}_{\mathrm{dn}1}+{n}_{\mathrm{dn}0}{\rm{i}}{{kv}}_{\mathrm{dn}1}=0,\end{eqnarray}$$ \begin{eqnarray}-{\rm{i}}\omega {n}_{\mathrm{dc}1}+{n}_{\mathrm{dc}0}{\rm{i}}{{kv}}_{\mathrm{dc}1}=0,\end{eqnarray}$$ \begin{eqnarray}-{\rm{i}}\omega {n}_{{\rm{i}}1}+{n}_{{\rm{i}}0}{\rm{i}}{{kv}}_{{\rm{i}}1}=0,\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}-{\rm{i}}\omega {v}_{\mathrm{dc}1} & = & -\displaystyle \frac{{q}_{{\rm{d}}}}{{m}_{{\rm{d}}}}{\rm{i}}k{\phi }_{1}-{\rm{i}}k{\varphi }_{1}-{\nu }_{\mathrm{id}}\left({v}_{\mathrm{dc}1}-{v}_{{\rm{i}}1}\right)\\ & & -{\nu }_{\mathrm{cn}}\left({v}_{\mathrm{dc}1}-{v}_{\mathrm{dn}1}\right),\end{array}\end{eqnarray}$$ \begin{eqnarray}-{\rm{i}}\omega {v}_{\mathrm{dn}1}=-{\rm{i}}k{\varphi }_{1},\end{eqnarray}$$ \begin{eqnarray}-{k}^{2}{\varphi }_{1}=4\pi {{Gm}}_{{\rm{d}}}\left({n}_{\mathrm{dc}1}+{n}_{\mathrm{dn}1}\right),\end{eqnarray}$$ \begin{eqnarray}-{k}^{2}{\phi }_{1}=4\pi e\left({n}_{{\rm{e}}1}-{n}_{{\rm{i}}1}-\displaystyle \frac{{q}_{{\rm{d}}}}{e}{n}_{\mathrm{dc}1}\right).\end{eqnarray}$By a complicated but simple mathematical derivation, the dispersion relation for the dusty Jeans instability can be described in the following equation:$ \begin{eqnarray}\begin{array}{l}\left(1+{\rm{i}}\displaystyle \frac{{\nu }_{\mathrm{cn}}}{\omega }\right)\displaystyle \frac{{\omega }_{\mathrm{Jd}}^{2}}{{\omega }^{2}+{\omega }_{{\rm{J}}}^{2}-{\omega }_{\mathrm{Jd}}^{2}}+\left(1+{\rm{i}}\displaystyle \frac{{\nu }_{\mathrm{cn}}+{\nu }_{\mathrm{id}}}{\omega }\right)\\ \quad -\,\displaystyle \frac{{k}^{2}{C}_{\mathrm{scam}}^{2}}{{\omega }^{2}}-{\rm{i}}\displaystyle \frac{{{Zn}}_{\mathrm{dc}0}{\nu }_{\mathrm{id}}}{\omega {n}_{{\rm{i}}0}}=0,\end{array}\end{eqnarray}$where ${\omega }_{{\rm{J}}}=\sqrt{4\pi {{Gm}}_{{\rm{d}}}\left({n}_{\mathrm{dn}0}+{n}_{\mathrm{dc}0}\right)}$ and ${\omega }_{\mathrm{Jd}}=\sqrt{4\pi {{Gm}}_{{\rm{d}}}{n}_{\mathrm{dc}0}}$ are the corresponding Jeans frequencies, and ${C}_{\mathrm{scam}}^{2}=\tfrac{{q}_{{\rm{d}}}^{2}{n}_{\mathrm{dc}0}{T}_{{\rm{i}}}}{{e}^{2}{n}_{{\rm{i}}0}{m}_{{\rm{d}}}}$ is the dusty acoustic phase speed, $Z={q}_{{\rm{d}}}/e$. Introducing $K=k/{k}_{{\rm{J}}}$, ${\rm{\Omega }}=\omega /{\omega }_{{\rm{J}}}$, ${\eta }_{1}={n}_{\mathrm{dc}0}/{n}_{{\rm{i}}0}$, $\eta ={n}_{\mathrm{dc}0}/{n}_{\mathrm{dn}0}$, then equation (17) can be written in the following form$ \begin{eqnarray}\begin{array}{l}{{\rm{\Omega }}}^{4}+{\rm{i}}\left(\displaystyle \frac{{\nu }_{\mathrm{cn}}}{{\omega }_{{\rm{J}}}}+\displaystyle \frac{\left(1-Z{\eta }_{1}\right){\nu }_{\mathrm{id}}}{{\omega }_{{\rm{J}}}}\right){{\rm{\Omega }}}^{3}+\left(1-{K}^{2}\right){{\rm{\Omega }}}^{2}\\ +\,{\rm{i}}\left(\displaystyle \frac{{\nu }_{\mathrm{cn}}}{{\omega }_{{\rm{J}}}}+\displaystyle \frac{{\nu }_{\mathrm{id}}\left(1-Z{\eta }_{1}\right)}{{\omega }_{{\rm{J}}}}\displaystyle \frac{1}{1+\eta }\right){\rm{\Omega }}+{K}^{2}\left(\displaystyle \frac{1}{1+\eta }\right)=0,\end{array}\end{eqnarray}$where ${k}_{{\rm{J}}}\equiv {\omega }_{{\rm{J}}}/{C}_{\mathrm{scam}}$ is defined as the Jeans wave-number. It is obvious to see that when we ignored the terms ${n}_{\mathrm{dn}0}$ and ${\nu }_{\mathrm{id}}$, the above dispersion relationship is simplified to the case of the usual Jeans model ${\omega }^{2}={k}^{2}{C}_{{\rm{s}}}^{2}-{\omega }_{{\rm{J}}}^{2}$ [5]. On the other hand, when ${n}_{\mathrm{dc}0}\to 0$, one can obtain the usual Jeans mode proposed by Kolb [34]. In particular, the above dispersion relation reduces to the PMGC discussed by Dwivedi in the case of ${\nu }_{\mathrm{id}}\to 0$ [7]. In addition, since $Z{\eta }_{1}=\tfrac{{{Zn}}_{\mathrm{dc}0}}{{n}_{{\rm{i}}0}}\ll 1$, equation (18) can be reduced to the following form:$ \begin{eqnarray}\begin{array}{c}\begin{array}{l}{{\rm{\Omega }}}^{4}+{\rm{i}}\left(\displaystyle \frac{{\nu }_{{\rm{cn}}}+{\nu }_{{\rm{id}}}}{{\omega }_{{\rm{J}}}}\right){{\rm{\Omega }}}^{3}+\left(1-{\bar{K}}^{2}\right){{\rm{\Omega }}}^{2}\\ +\,{\rm{i}}\left(\displaystyle \frac{{\nu }_{{\rm{cn}}}}{{\omega }_{{\rm{J}}}}+\displaystyle \frac{{\nu }_{{\rm{id}}}}{{\omega }_{{\rm{J}}}}\displaystyle \frac{1}{1+\eta }\right){\rm{\Omega }}+{\bar{K}}^{2}\left(\displaystyle \frac{1}{1+\eta }\right)=0.\end{array}\end{array}\end{eqnarray}$
Since the aim of the present model is to discuss the effect of ion drag on the PMGC, the term νid shall be paid attention to in the following discussion. As shown in the following sections, the limit forms of the dispersion relation, which corresponds to two cases of the stability behavior of the above dusty mass distribution, are numerically calculated.
3. Numerical result analysis
For numerical calculation, we adopted some parameters given in the research of Avinash and Harpaz [35, 36]: $T=5000\,{\rm{K}}$, ${n}_{0}={n}_{{\rm{H}}}={10}^{-3}\,{{\rm{cm}}}^{-3}$, ${r}_{{\rm{d}}}\sim 3\times {10}^{-5}\,{\rm{cm}}$, with average dusty grain mass ${m}_{{\rm{d}}}\approx {10}^{-13}{\rm{g}}$. And the average dusty number density nd in these clouds is close to ${10}^{-6}\,{{\rm{cm}}}^{-3}$ [37, 38], ${q}_{{\rm{d}}}\approx 100e$, ${\omega }_{\mathrm{Jd}}\approx 3\times {10}^{-13}\,{{\rm{s}}}^{-1}$ and ${C}_{\mathrm{scam}}\approx 10\,{\rm{cm}}\,{{\rm{s}}}^{-1}$. As discussed by Nishi [39], the parameters $\eta ={n}_{\mathrm{dc}0}/{n}_{\mathrm{dn}0}$ can vary between ${10}^{-2}$ and 10−4. Thus we adopted it near 10−2 reasonably. The ion neutral collision frequency ${\nu }_{\mathrm{id}}\approx {10}^{-5}\,{{\rm{s}}}^{-1}$ [6] and the binary collisional rate of momentum transfer from charged grains to neutral grains can be estimated as ${\nu }_{\mathrm{cn}}\sim \pi {a}^{2}{n}_{\mathrm{dn}0}{v}_{\mathrm{td}}$ [7], where ${v}_{\mathrm{td}}$ ($=\sqrt{2{T}_{{\rm{d}}}/{m}_{{\rm{d}}}}$) represents the dusty thermal velocity. In fact, for this parameter νcn, two extreme limits in the background of cosmic radiation have been examined in this model. In the case of ${T}_{{\rm{d}}}\to 0$, using the above formula we can obtain ${\nu }_{\mathrm{cn}}\to 0$, which means the dusty mass distribution is frictionless. But in the other case of ${T}_{{\rm{d}}}\ne 0$, the charged dusty collision frequency ${\nu }_{\mathrm{cn}}\to \infty $, which means that the dust mass distribution is frictional.
Figure 1 shows the numerical results of equation (19), when ${\nu }_{\mathrm{cn}}\to 0$, ${\nu }_{\mathrm{id}}\to 0$. It can be found that as the ratio η increases, Ωr will decrease. Thus, the ratio of charged dusty particles to neutral dusty particles is important for the gravitational model. Physically, the gradual increase in η means an increase in the number of charged dusty particles, which leads to the exclusion of lighter electrons and the absorption of more heavy ions. Therefore, the heavy ions will absorb more electrons, which pronounces charge condensation and hence helps in gravitational collapse. As shown in the left part of figure 2, with the increase in η, the imaginary frequency Ωi gradually enhances, which means that dusty collapse will be more easily driven because the ratio η enhances the instability.
Figure 1.
New window|Download| PPT slide Figure 1.The normalized real part of frequency ${{\rm{\Omega }}}_{{\rm{r}}}\equiv {\omega }_{{\rm{r}}}/{\omega }_{{\rm{J}}}$ versus the normalized wave-number $K\equiv k/{k}_{{\rm{J}}}$ with different η in the case of ${\nu }_{\mathrm{cn}}\to 0$, ${\nu }_{\mathrm{id}}\to 0$, where η=0.01, 0.02, 0.03 for the blue dashed line, green solid line and yellow dotted line, respectively.
Figure 2.
New window|Download| PPT slide Figure 2.The normalized imaginary part of frequency ${{\rm{\Omega }}}_{{\rm{r}}}\equiv {\omega }_{{\rm{r}}}/{\omega }_{{\rm{J}}}$ versus the normalized wave-number $K\equiv k/{k}_{{\rm{J}}}$ with different η in the case of ${\nu }_{\mathrm{cn}}\to 0$, ${\nu }_{\mathrm{id}}\to 0$, where η=0.01, 0.02, 0.03 for the blue dashed line, green solid line and yellow dotted line, respectively.
In the case of ${\nu }_{\mathrm{cn}}\to \infty $, the effect of ion drag on the PMGC has been discussed in detail by the dispersion relation equation (17). As shown in figure 3, the black solid line represents the case of ${\nu }_{\mathrm{id}}=0$, which is consistent with the growth rate proposed by Dwivedi when ${\nu }_{\mathrm{cn}}\to \infty $. And it is also shown that with the increase in ${\nu }_{\mathrm{id}}$, the imaginary part of the frequency gradually rises, which means the ion drag force significantly enhances the instability. In other words, the ion drag enhances the instability of the pulsational mode, and it is conducive to the small amount of dust condensation in the dark interstellar clouds. Physically, it may have arisen as a result of mode competition. Contrary to both the thermal pressure and electromagnetic force, the condensation of dust is driven by the ion drag force and the self-gravity. The larger the ion drag force is, the more conducive to the condensation of small-scale dust, which is reflected by the collision frequency νid. Even when the ion drag force is strong enough to oppose other resistances, the arbitrary small amount of condensation of dust may occur. Therefore, a new mode of Jeans condensation, namely the PMGC, may exist in the dark interstellar clouds region and the ion drag force will act on this mode to help the small amount of condensation of dust, which may be one of the physical mechanisms for star formation.
Figure 3.
New window|Download| PPT slide Figure 3.The normalized imaginary part of frequency ${{\rm{\Omega }}}_{{\rm{i}}}\equiv {\omega }_{{\rm{i}}}/{\omega }_{{\rm{J}}}$ versus the normalized wave-number $K\equiv k/{k}_{{\rm{J}}}$ with different ${\nu }_{\mathrm{id}}$ in the case of ${\nu }_{\mathrm{cn}}\to \infty $, where ${\nu }_{\mathrm{id}}=0,1.0000{{\rm{e}}}^{-5},1.0001{{\rm{e}}}^{-5},1.0002{{\rm{e}}}^{-5}$ for the black thick line, yellow dashed line, red dot-draw line and blue dotted line, respectively. The left ordinate corresponds to a special case of ${\nu }_{\mathrm{id}}=0$, while the other cases are mated with the right ordinate.
4. Conclusions
The effect of ion drag on the PMGC existing in a self-gravitating system which is composed of charged dust, neutral dust, ions and electrons has been analyzed in the present work. Adopting fluid dynamics, with continuity equations, momentum equations, Poisson’s equation and the Boltzmann distribution of ions and electrons, the dispersion relations for the PMGC are found and then by carrying out numeral calculations, with the dispersion diagrams, the following conclusions have been obtained:
(a) A new mode of Jeans condensation, namely the PMGC, exists in the dark interstellar clouds, as shown in figure 1 (the real part of the dispersion relation) and figure 2 (the imaginary part of the dispersion relation).
(b) With the increase in η, the imaginary frequency Ωi also gradually enhances, which indicates the ratio of η is conducive to enhancing the instability of the PMGC.
(c) As shown in figure 3, the effect of ion drag on the PMGC is significant. With the increasing collision frequency ${\nu }_{\mathrm{id}}$, the imaginary part of the frequency is gradually raised. This phenomena occurs because the ion drag force drives the condensation of dust, which may contribute to star formation.
To sum up, the effect of ion drag on the PMGC is investigated and we found a significant result in that the ion drag enhances the instability of the PMGC, which means it aids the small amount of dust condensation existing in the dark interstellar clouds. Therefore, the result of the present work provides a plausible authority for the star forming and growing mechanism. As mentioned earlier, nonlinear physical phenomena have always been a hot topic in plasma physics, astrophysics and even photoelectric communication [40]. Inspired by the nonlinear Schrödinger equation [41–46], we will further research the nonlinear effects in dusty plasma systems based on those works.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 11 763 006, 11847023 and 11 863 004).
,1,2, San-Qiu Liu(刘三秋)1,21Department of Physics, 
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