In 1992, Allen et al [1] indicated that an electromagnetic wave may carry orbital angular momentum (OAM), and when propagating in a vacuum, it can transfer its OAM to particles if the wave is absorbed. Subsequently, growing interest has been given to the study of photon OAM states, which are generated in light waves [2–5]. It is widely known that the angular momentum of electromagnetic radiation has two distinct components. One is the intrinsic part associated with wave polarization or spin, and the other is the extrinsic part related to the orbital angular momentum that depends on the spatial radiation distribution [6, 7]. The light waves, which can be described using a Laguerre–Gaussian (LG) solution to the paraxial wave equation, often referred to as either helical waves or vortex waves somewhat interchangeably, have both spin and angular momentum [8]. In recent years, the studies of photon OAM states have been transposed to plasmas because the propagation of OAM optical beams in plasmas is associated with the excitation of plasma waves that may likewise carry OAM [9]. Based on the framework of plasma kinetic theory, Mendonça et al [10] have studied the dispersion relation and Landau damping of helical electron plasma waves with OAM in cylindrical geometry under the paraxial approximation. The results showed that the twisted plasmons can be excited by rotating electron beams. Since the twisted plasma wave carries orbital angular momentum in addition to the energy and momentum of the ordinary plasma wave, Blackman et al [11] further discussed the properties of twisted waves in plasmas with orbital angular momentum. It was shown that the existence of an orbital angular momentum and the corresponding radial structure of the plasma wave modified its dispersion and damping peculiarities. Following the work of Mendonça [10], Khan et al [12] revisited the helical structure of electrostatic plasma waves carrying OAM by introducing a variable transformation.
It is worth noting that the properties of gravitational systems are similar to the plasma system, i.e., their interaction forces are inversely proportional to the square of the distance. So, the methodology used to describe wave-wave and wave-particle interactions in plasma is also valid for self-gravitation systems [13–16]. That is, we can consider self-gravitation systems as a special kind of plasma system. The famous theory describing the instability of the self-gravitational system was the Jeans instability. In 1902, Jeans [17] introduced a new theory to explain the gravitational formation of structures and their evolution in the linear regime, and indicates that a self-gravitating infinite uniform gas at rest should be unstable against small perturbation in the forms of $\exp [{\rm{i}}({\boldsymbol{k}}\cdot {\boldsymbol{r}}-\omega t)]$. Later on, the Jeans theory has been widely used in the fields of astrophysics [18–20], plasma [21–23] and the complex fluid community [24, 25]. However, with the observation data involving the statistical description becoming increasingly more precise, the conventional Jeans instability struggles to explain some of the unique physical phenomena in star formation, such as the collapse time scale of some gigantic cold gas clouds [26, 27].
Based on the fact that optical beams with OAM are able to excite twisted density perturbations, Mendonça [28] et al theoretically explained the exchange of OAM between electromagnetic and electrostatic waves in a plasma. It was shown that the collimated beams described by LG functions lead to twisted acoustic and Langmuir modes in stimulated Raman and Brillouin backscattering, which implied that plasmon and phonon states carried OAM. The idea opened the door for various studies on OAM in plasma, which included new investigations on waves and instabilities [29]. The LG beams introduced other significant effects in the plasma due to intertwined helical wavefronts, for instance, the ring-shaped morphology of plasma turbulence in radio-pumping [30], twisted photon emission at single electron level [31], a twisted plasma accelerator [32], and so on. At present, the influence of OAM on fluctuations has been widely studied, but the research on the instability of the self-gravitation system needs to be deepened. So now, an open question, worthy of in-depth study, is whether the instability of the self-gravitation system is strongly influenced by the OAM. In the self-gravitation system of plasma, the OAM will provide a fundamentally new degree of freedom, which leads to novel vortex sources and the wave surfaces with constant phase are not plane surfaces but a complicated helical structure. It is, therefore, necessary to consider new cylindrical wave solutions to replace the usual plane wave solutions. Moreover, apart from the usual conservation laws for energy and momentum, the OAM modes also have to satisfy the conservation of angular momentum [33]. In what follows, considering the effects of OAM, we introduce a twisted Jeans model to briefly discuss the dynamics of self-gravitation systems under the perturbation with orbital angular momentum. The perturbed wave of the twisted Jeans model is similar to the twisted plasma wave, which is a helical structure along a symmetry axis z and spatially limited in the radial direction. A particularity of the twisted Jeans model is that the perturbation waves persist in their capacity to carry, in addition to the axial momentum, an orbital angular momentum, that is, the perturbation having an azimuthal component, which is a periodic function of the azimuthal angle θ in the cylindrical coordinate system. In other words, the usual plane wave solutions are replaced by OAM wave solutions described by LG functions in the twisted Jeans model. Based on the results of twisted plasma waves [11], OAM may act as a potential influencing factor to change the dispersion of perturbations of the self-gravitational system, thus further influencing the formation and evolution of the structure.
According to plasma kinetic theory, the dynamic behavior of a self-gravitating system can be described by the collisionless Boltzmann equation which can be written in the following way [34, 35]:
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial f\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)}{\partial t}+{\boldsymbol{v}}\cdot \displaystyle \frac{\partial f\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)}{\partial {\boldsymbol{r}}}\\ \quad +{\rm{\nabla }}\varphi \cdot \displaystyle \frac{\partial f\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)}{\partial {\boldsymbol{v}}}=0,\end{array}\end{eqnarray}$
where $f\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)$ is the matter-particle distribution function, and φ is the gravitational potential which satisfies the Poisson equation $\begin{eqnarray}{{\rm{\nabla }}}^{2}\varphi =4\pi G\int f\left({\boldsymbol{v}},{\boldsymbol{r}},t\right){\rm{d}}{\boldsymbol{v}}.\end{eqnarray}$
Along the standard lines, the distribution function can be divided into equilibrium and perturbation parts, i.e., $f\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)={f}_{0}\left({\boldsymbol{v}}\right)+{f}_{1}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)$, then one arrives at the linearized form of equations (1 ) and (2 ) which are given as follows4 ) can be reduced to
$\begin{eqnarray}\displaystyle \frac{\partial {f}_{1}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)}{\partial t}+v\cdot \displaystyle \frac{\partial {f}_{1}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)}{\partial {\boldsymbol{r}}}+{\rm{\nabla }}{\varphi }_{1}\cdot \displaystyle \frac{\partial {f}_{0}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)}{\partial {\boldsymbol{v}}}=0,\end{eqnarray}$
$\begin{eqnarray}{{\rm{\nabla }}}^{2}{\varphi }_{1}=4\pi G\int {f}_{1}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right){\rm{d}}{\boldsymbol{v}}.\end{eqnarray}$
Taking the OAM into account, we assume that the perturbed distribution function can be decomposed in LG modes, that is, the wave propagates along the z-axis as $\exp \left({\rm{i}}{kz}\right)$ with a slowly changing amplitude. In such a case, we have $\begin{eqnarray}{{\rm{\nabla }}}^{2}\varphi \simeq \left({{\rm{\nabla }}}_{\perp }^{2}-{k}^{2}+2{\rm{i}}k\partial /\partial z\ \right)\varphi .\end{eqnarray}$
Furthermore, the potential φ is assumed to be associated with waves with finite orbital angular momentum, such that they satisfy the paraxial equation $\left({{\rm{\nabla }}}_{\perp }^{2}+2{\rm{i}}k\partial /\partial z\ \right)\varphi =0$. This means that the Poisson's equation ( $\begin{eqnarray}{k}^{2}\varphi =4\pi \displaystyle \sum _{s}{q}_{s}\int {f}_{1}\left({\boldsymbol{v}}\right){\rm{d}}{\boldsymbol{v}}.\end{eqnarray}$
Then the solution can be expanded in a series of LG functions as follow [12]: $\begin{eqnarray}{\varphi }_{1}\left({\boldsymbol{r}},t\right)=\displaystyle \sum _{p,l}{\tilde{\varphi }}_{p,l}{F}_{p,l}\left(r,z\right)\exp \left({\rm{i}}l\theta \right)\exp \left({\rm{i}}{kz}-{\rm{i}}\omega t\right),\end{eqnarray}$
where ${\tilde{\varphi }}_{p,l}$, ${\tilde{f}}_{p,l}$ are the mode amplitudes and θ stands for the azimuthal angle. The radial wave number p ≥ 0 is an integer, which numerates radial mode; and the integer l is a positive or negative angular mode number, which denotes the OAM. The LG mode function is defined as $\begin{eqnarray}{F}_{p,l}\left(X\right)={A}_{p,l}{X}^{\left|l\right|}{L}_{p}^{\left|l\right|}\left(X\right)\exp \left(-X/2\ \right),\end{eqnarray}$
where $X={r}^{2}/{w}^{2}\left(z\right)$, $w\left(z\right)$ is the wave beam waist, and ${A}_{{pl}}=\tfrac{1}{2\sqrt{\pi }}\sqrt{\tfrac{\left(l+p\right)!}{p!}}$ is the normalization factor. The ${L}_{p}^{\left|l\right|}\left(X\right)$ is the Laguerre polynomial defined by $\begin{eqnarray}{L}_{p}^{\left|l\right|}\left(X\right)=\displaystyle \frac{{X}^{-\left|l\right|}{{\rm{e}}}^{X}}{p!}\displaystyle \frac{{{\rm{d}}}^{p}}{{\rm{d}}{X}^{p}}\left({X}^{\left|l\right|+p}{{\rm{e}}}^{-X}\right).\end{eqnarray}$
The set of functions Fp,l satisfy the orthogonality conditions $\begin{eqnarray}\begin{array}{l}\left\langle {F}_{{p}^{{\prime} },{l}^{{\prime} }}\left(X\right)\cdot | {F}_{p,l}\left(X\right)\right\rangle \equiv {\int }_{0}^{2\pi }{\rm{d}}\theta \\ \,\times {\int }_{0}^{\infty }r{\rm{d}}{{rF}}_{{p}^{{\prime} },{l}^{{\prime} }}\left(X\right){F}_{p,l}\left(X\right)\exp \left[{\rm{i}}\left(l-{l}^{{\prime} }\right)\theta \right]={\delta }_{{{pp}}^{{\prime} }}{\delta }_{{{ll}}^{{\prime} }},\end{array}\end{eqnarray}$
where ${\delta }_{{{pp}}^{{\prime} }}$ and ${\delta }_{{{ll}}^{{\prime} }}$ are the Kronecker symbols. To solve equations ( $\begin{eqnarray}{f}_{1}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)=\displaystyle \sum _{p,l}{\tilde{f}}_{p,l}{F}_{p,l}\left(r,z\right)\exp \left({\rm{i}}l\theta \right)\exp \left({\rm{i}}{kz}-{\rm{i}}\omega t\right),\end{eqnarray}$
where ${\tilde{f}}_{p,l}$ is the mode amplitudes. Considering the orthogonality condition of the LG modes, the algebraic expression between the perturbation distribution function ${\tilde{f}}_{p,l}$ and the potential amplitude ${\tilde{\varphi }}_{p,l}$ is obtained $\begin{eqnarray}{\tilde{f}}_{p,l}=4\pi G\displaystyle \frac{{{\boldsymbol{q}}}_{\mathrm{eff}}\cdot \partial {f}_{0}/\partial {\boldsymbol{v}}\ }{\omega -{{\boldsymbol{q}}}_{\mathrm{eff}}\cdot {\boldsymbol{v}}}{\tilde{\varphi }}_{p,l}.\end{eqnarray}$
Here, ${{\boldsymbol{q}}}_{\mathrm{eff}}=-{\rm{i}}{q}_{r}{\hat{{\boldsymbol{e}}}}_{r}+{{lq}}_{\theta }{\hat{{\boldsymbol{e}}}}_{\theta }+\left(k-{\rm{i}}{q}_{z}\right){\hat{{\boldsymbol{e}}}}_{z}$ is independent of r, ${q}_{\theta }={\int }_{0}^{\infty }{F}_{p,l}^{2}{\rm{d}}r$ and ${q}_{j}={\int }_{0}^{\infty }{F}_{p,l}\tfrac{{\rm{\partial }}{F}_{p,l}}{{\rm{\partial }}j}r{\rm{d}}r$ for j = r, z. Thus, the dielectric function for the gravitational response as $\begin{eqnarray}{\varepsilon }^{G}\left(\omega ,{{\boldsymbol{q}}}_{\mathrm{eff}}\right)=-1+\displaystyle \frac{4\pi G}{{k}^{2}}\int \displaystyle \frac{{{\boldsymbol{q}}}_{\mathrm{eff}}\cdot \partial {f}_{0}/\partial {\boldsymbol{v}}\ }{\omega -{{\boldsymbol{q}}}_{\mathrm{eff}}\cdot {\boldsymbol{v}}}{\rm{d}}{\boldsymbol{v}}=0,\end{eqnarray}$
which is similar to the dispersion relation of plane waves but is different in an effective wave vector. The frequency in the case of plane wave ω = kvz has shifted into a new form, i.e., $\omega ={{kv}}_{z}+{{lq}}_{\theta }{v}_{\theta }-{\rm{i}}\left({q}_{r}{v}_{r}+{q}_{z}{v}_{z}\right)$. It should be noticed that under the limit $\left|{q}_{r}\right|,\left|{q}_{z}\right|\ll \left|{{lq}}_{\theta }\right|$, the resonance frequency reduces to ω = kvz + lqθvθ, which leads to a poloidal contribution in the resonance point. Then, we obtain the dielectric function including the contribution from the azimuthal term as $\begin{eqnarray}\begin{array}{l}{\varepsilon }^{G}\left(\omega ,k,{{lq}}_{\theta }\right)=-1+\displaystyle \frac{4\pi G}{{k}^{2}}\\ \,\times \,\int \displaystyle \frac{k\partial {f}_{0}/\partial {v}_{z}+{{lq}}_{\theta }\partial {f}_{0}/\partial {v}_{\theta }\ }{\omega -{{kv}}_{z}-{{lq}}_{\theta }{v}_{\theta }}{\rm{d}}{\boldsymbol{v}}=0.\end{array}\end{eqnarray}$
For simplicity, at equilibrium, the velocity distribution function of the particle ${f}_{0}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)$ can be described by the Maxwellian distribution
$\begin{eqnarray}{f}_{0}\left({\boldsymbol{v}},{\boldsymbol{r}},t\right)=\displaystyle \frac{{\rho }_{0}}{{\left(2\pi {v}_{T}^{2}\right)}^{3/2\ }}\exp \left(-\displaystyle \frac{{{\boldsymbol{v}}}^{2}}{2{v}_{T}^{2}}\right),\end{eqnarray}$
where ρ0 and vT are the matter equilibrium density and dispersion velocity of the particles, respectively.To derive the dispersion properties of the self-gravitating system with OAM perturbation which obeys the Maxwellian distribution function (15 ), let us introduce the variable transformation [12],
$\begin{eqnarray}\begin{array}{l}F\left({v}_{z},{v}_{\theta }\right)=k\displaystyle \frac{\partial {f}_{e0}}{\partial {v}_{z}}+{{lq}}_{\theta }\displaystyle \frac{\partial {f}_{e0}}{\partial {v}_{\theta }},\\ \xi ={v}_{z}+\displaystyle \frac{{{lq}}_{\theta }}{k}{v}_{\theta },\eta ={v}_{\theta }-\displaystyle \frac{k}{{{lq}}_{\theta }}{v}_{z},\end{array}\end{eqnarray}$
and that leads to $\begin{eqnarray}{v}_{z}=\displaystyle \frac{\xi }{2}-\displaystyle \frac{\sigma \eta }{2},{v}_{\theta }=\displaystyle \frac{\eta }{2}+\displaystyle \frac{\xi }{2\sigma },J=\displaystyle \frac{\partial \left({v}_{z},{v}_{\theta }\right)}{\partial \left(\xi ,\eta \right)}=\displaystyle \frac{1}{2}.\end{eqnarray}$
In the above expression, $\sigma =\tfrac{{{lq}}_{\theta }}{k}$ is a dimensionless parameter characterizing the OAM, which is directly related to the helical phase structure of the plasma oscillation. After some simple algebraic operation, equation (14 ) can be rewritten as
$\begin{eqnarray}{\varepsilon }^{G}\left(\omega ,k,{{lq}}_{\theta }\right)=-1+\displaystyle \frac{4\pi G}{2{k}^{2}}\int \displaystyle \frac{F\left(\xi ,\eta \right)}{\omega -k\xi }{\rm{d}}\xi {\rm{d}}\eta {\rm{d}}{v}_{r},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}F\left(\xi ,\eta \right)=\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2\ }}\displaystyle \frac{k\xi {\rho }_{0}}{{v}_{T}^{3}}\\ \,\times \,\exp \left[-\displaystyle \frac{1}{2{v}_{T}^{2}}\left\{{\left(\displaystyle \frac{\xi }{2}-\displaystyle \frac{\sigma \eta }{2}\right)}^{2}+{\left(\displaystyle \frac{\eta }{2}+\displaystyle \frac{\xi }{2\sigma }\right)}^{2}+{v}_{r}^{2}\right\}\right].\end{array}\end{eqnarray}$
It should be noted that when the parameter σ goes to zero, some of the parameters or terms in equations (17 ) and (19 ) will become divergent mathematically, but the parameters vz and vθ will still remain convergent physically. As intermediate transition parameters, ξ and η are defined on the premise of the existence of OAM, so in this definition, σ is a non-zero value. In addition, ξ and η do not appear in the final result (see equation (24 )), which again shows that the definition of ξ and η does not affect the result. So, in this sense, the definitions for ξ and η are mathematical skills proposed by Khan [12]. Combining equations (18 ) and (19 ), one can get the dispersion equation20 ), then the critical wave number is written by
$\begin{eqnarray}{\varepsilon }^{G}\left(\omega ,k,{{lq}}_{\theta }\right)=-1+\displaystyle \frac{{k}_{J}^{2}}{{k}^{2}}\displaystyle \frac{1}{\sqrt{\pi }}Z\left(\zeta \right)=0,\end{eqnarray}$
where ${k}_{J}^{2}=\tfrac{4\pi G{\rho }_{0}}{{v}_{T}^{2}}$ denotes the classical Jeans wave number; the modified dispersion function $Z\left(\zeta \right)$ is expressed as $\begin{eqnarray}Z\left(\zeta \right)=\int \displaystyle \frac{X}{X-\zeta }\exp \left(-{X}^{2}\right){\rm{d}}X,\end{eqnarray}$
where $\zeta =\tfrac{\omega }{k\sqrt{2\left(1+{\sigma }^{2}\right)}{v}_{T}}$ and $X=\tfrac{\xi }{\sqrt{2\left(1+{\sigma }^{2}\right)}{v}_{T}}$. According to the paradigm of the Jeans instability [13, 21], the boundary between stable and unstable modes is obtained by setting ω = 0 (i.e., ζ = 0 ) in equation ( $\begin{eqnarray}{k}_{\sigma }^{2}={k}_{J}^{2},\end{eqnarray}$
which is identical to the results of classical Jeans critical values. That is, the critical wavenumber in a self-gravitational system with helical wave perturbation does not vary with the OAM parameter σ.Next, in order to derive the dispersion relation of unstable modes, we assume that the wavenumber k is real, and ω = iγ, where γ is real positive, then the dispersion relation equation (20 ) is transformed to23 ) is reduced to23 ) is numerically evaluated to give the growth rates of the Jeans modes. Figure 1 shows the normalized growth rate Ω ($\equiv \tfrac{\gamma }{\sqrt{4\pi G{\rho }_{0}}}$) versus the normalized wave number K ($\equiv \tfrac{k}{{k}_{J}}$) with different a parameter σ. It can be found that as σ increases, the growth rate for the instability increases, i.e., compared with the standard case, the Jeans instability is enhanced by the OAM in the self-gravitating matter systems. In light of Jeans theory, the Jeans instability arises when the thermal pressure is not strong enough to prevent the gravitational collapse of the matter system. The perturbation with OAM may exchange its energy and momentum with self-gravitating plasmas and lead to the excitation of oscillations and waves at the Jeans timescale. As the OAM parameter σ increases, then the energy increases that has been transferred to the matter particles and the related thermal pressure decrease, thus the growth rate of the Jeans instability increases.
$\begin{eqnarray}\displaystyle \frac{{k}^{2}}{{k}_{J}^{2}}=\displaystyle \frac{1}{\sqrt{\pi }}Z\left(\beta \right),\end{eqnarray}$
and the dispersion function $Z\left(\beta \right)$ is defined as $\begin{eqnarray}Z\left(\beta \right)=\int \displaystyle \frac{{X}^{2}}{{X}^{2}+{\beta }^{2}}\exp \left(-{X}^{2}\right){\rm{d}}X,\end{eqnarray}$
where $\beta =\tfrac{\gamma }{k\sqrt{2\left(1+{\sigma }^{2}\right)}{v}_{T}}$. Here, it is obvious that the dispersion relation of the self-gravitating system with OAM perturbation relies on the OAM parameter σ. According to the following integral formula [36] $\begin{eqnarray}{\int }_{0}^{\infty }\displaystyle \frac{{x}^{2}{{\rm{e}}}^{-{\mu }^{2}{x}^{2}}}{{x}^{2}+{\beta }^{2}}{\rm{d}}x=\displaystyle \frac{\sqrt{\pi }}{2\mu }-\displaystyle \frac{\pi \beta }{2}{{\rm{e}}}^{{\beta }^{2}}\mathrm{erfc}\left(\beta \mu \right),\end{eqnarray}$
the form of expression equation ( $\begin{eqnarray}\displaystyle \frac{{k}^{2}}{{k}_{J}^{2}}=1-\sqrt{\pi }\beta {{\rm{e}}}^{{\beta }^{2}}\mathrm{erfc}\left(\beta \right),\end{eqnarray}$
which will transform into the standard dispersion relation for stellar systems [13, 37] under the condition σ = 0, i.e., $\beta =\tfrac{\gamma }{k\sqrt{2}{v}_{T}}$. To intuitively study the effect of the OAM parameter, equation (
Figure 1. The normalized growth rate $\left({\rm{\Omega }}\equiv \tfrac{\gamma }{\sqrt{4\pi G{\rho }_{0}}}\right)$ versus the normalized wave number $\left(K\equiv \tfrac{k}{{k}_{J}}\right)$ with different σ. |
To summarize, the instabilities under the perturbation with OAM are investigated in a self-gravitating plasma based on kinetic theory. In the twisted mode, the wave fronts with constant phase are not plane surfaces but have a complicated helical structure. The kinetic dispersion relation in the present model is formally similar to that of the usual plane wave, but the wavenumber is replaced by an effective wave vector qeff in which such helical structure is embedded. The result shows that the Jeans instability is favored by the OAM effect in the self-gravitating matter systems, and therefore, structures are more easily formed in the twisted Jeans model. As we all know, a typical interstellar gas cloud, which the physical parameters are size ∼2pc, density ∼10−19kg · m−3 and Jeans length λJ ≈ 30pc, is stable against gravitational instability, and the collapse time scale can be calculated by the formula ${\tau }_{\mathrm{ff}}=\sqrt{{r}^{3}/{GM}\ }$ [38]. However, in light of current observation results, it has been found that the collapse times of some gigantic cold gas clouds deviate significantly from the typical Jeans time [26, 27]. The discrepancy between observed and theoretical values allows the evolution of interstellar gas clouds to become more enigmatic. Combining the results in our work, the perturbation with OAM will interact with the particles to promote the gravitational collapse of the gigantic molecular clouds, as well as lead to the collapse time being shorter than its theoretical Jeans time. We can, thus, conclude that consideration of OAM in self-gravitational systems may provide a promising way to explain some discrepancies in the evolution and dynamics of interstellar gas clouds.