Pulsational instability of complex charge-fluctuating magnetized turbulent astroclouds

Abstract

We develop a theoretic model to study the linear stability behaviour of pulsational (gravito-electrostatic) mode in a self-gravitating, magnetized, collisional, turbulent and unbounded dust molecular cloud (DMC). The analytic model consists of lighter electrons and ions; and massive charged dust grains with partial ionization over the geometrically infinite extension. The semi-empirically obtained Larson logatropic equation of state, correlating all the thermo-turbo-magnetic pressures concurrently, is included afresh to model the constituent fluid turbulence pressures arising because of multiple randomized aperiodic flow scales of space and time. A linear normal mode analysis over the slightly perturbed composite cloud, relative to the defined homogeneous hydrostatic equilibrium, results in a unique mathematical construct of generalized polynomial (octic) dispersion relation with different coefficients sensitively dependent upon the diversified equilibrium cloud parameters. The main features of the modified pulsational mode dynamics are numerically explored over a commodious window of parametric values. It is shown and established that the grain mass introduces a dispersive stabilizing effect to the mode (with enhancement in phase speed), and vice-versa. A spatiotemporal illustrative tapestry is also portrayed for further confirmation of the dispersive mode with sporadic properties. The tentative application of our findings in different space and astrophysical circumstances is briefly outlined.

Appendices

Appendix A: Description of normalization procedure

We present the estimated values of the normalizing parameters in our defined normalizing scheme in Table 1. The different inputs employed in the estimation are validated for the realistic space, astrophysical and cosmic environments from different sources available in the literature (Shukla and Mamun 2002; Verheest 2002; Khare and Shukla 2006; Karmakar and Borah 2013; Karmakar and Haloi 2017; Karmakar and Haloi 2017; Karmakar and Das 2017). The dust mass used for the calculation lies in the range of $\approx 10^{ - 9}$–$10^{ - 21}$ kg (Verheest 2002). The cloud plasma temperature is $T_{p} =1.00$ eV (Verheest 2002; Karmakar and Borah 2013; Dutta et al. 2016)

Table 1 Adopted astrophysical normalizing scheme

Appendix B: Various dispersion coefficients

The derived polynomial dispersion relation (see (38)) has diversified dispersion coefficients, which are dependent on the equilibrium cloud plasma parameters, presented respectively as

$$\begin{aligned} A_{7} = \bigl[ a_{1}^{ - 1}a_{2} + a_{8}^{ - 1}a_{9} + a_{15}^{ - 1}a_{16} - a_{19}a_{21}^{ - 1} \bigr], \end{aligned}$$

with

$$\begin{aligned} &a_{1} = ( \beta_{1} F_{edc} ) K^{ - 1}, \\ & a_{2} = i ( 2 - \alpha_{1} )K + i \bigl( \beta_{1} F_{edc}^{2} - n_{e0}e\mu \bigr)K^{ - 1}, \\ & a_{8} = ( \beta_{2} F_{idc} ) K^{ - 1}, \\ & a_{9} = i ( 2 + \alpha_{2} )K + i \bigl( \beta_{2} F_{idc}^{2} - n_{i0}e\mu \bigr)K^{ - 1}, \\ & a_{15} = - K^{ - 1}, \\ & a_{16} = F_{cn} K^{ - 1}, \\ & a_{19} = - F_{nc} K^{ - 1}, \\ & a_{21} = - i K^{ - 1}; \end{aligned}$$

$$\begin{aligned} A_{6} &= \bigl[ a_{1}a_{3} + a_{8}^{ - 1} \bigl\{ a_{9} \bigl( a_{15}^{ - 1}a_{16} - a_{19}a_{21}^{ - 1} \bigr) \\ &\quad{}+ a_{10} + a_{1}^{ - 1} ( a_{2}a_{9} - a_{4}a_{7} ) - a_{21}^{ - 1}a_{22} \\ &\quad{}- a_{15}^{ - 1}a_{17} - a_{1}^{ - 1}a_{2} \bigl( a_{16}a_{15}^{ - 1} - a_{19}a_{21}^{ - 1} \bigr) \bigr\} \bigr], \end{aligned}$$

with

$$\begin{aligned} &a_{3} = ( \alpha_{1}F_{edc} )K + ( n_{e0}e\mu )F_{edc} K^{ - 1}, \\ & a_{4} = i\alpha_{1}K + i ( n_{i0}e\mu )K^{ - 1}, \\ & a_{7} = i\alpha_{1}K + i ( n_{e0}e\mu )K^{ - 1}, \\ & a_{10} = - ( \alpha_{2}F_{idc} ) K + ( n_{i0}e\mu F_{idc} ) K^{ - 1}, \\ & a_{17} = 2i \biggl( \frac{T_{d}}{T_{p}} \biggr)K + i \biggl\{ n_{dc0} \biggl( \frac{q_{d0}^{2}}{e} \biggr)\mu + m_{d}n_{dc0} \rho_{0}^{ - 1} \biggr\} K^{ - 1}, \\ & a_{22} = 2i \biggl( \frac{T_{d}}{T_{p}} \biggr)K - im_{d}n_{dn0}\rho_{0}^{ - 1}K^{ - 1}; \end{aligned}$$

$$\begin{aligned} A_{5} &= ( a_{8}a_{15}a_{21} )^{ - 1} \bigl[ a_{1}^{ - 1} \bigl\{ a_{2}a_{8} ( a_{15}a_{22} + a_{17}a_{21} ) \\ &\quad{} - ( a_{15}a_{19} - a_{16}a_{21} ) ( a_{2}a_{9} + a_{3}a_{8} - a _{4}a_{7}) \\ &\quad{}+ (a_{15}a_{21}) (a_{2}a_{10} + a_{3}a_{9} - a_{5}a_{7} ) \\ &\quad{} + a_{3}a_{4}a_{6}a_{11}a_{21} - a_{6}a_{8}a_{10}a_{21} \bigr\} \\ &\quad{} + a_{8} ( a_{16}a_{22} - a_{17}a_{19} - a_{18}a_{19} + a_{16}a_{20} ) \\ &\quad{}+ a_{9} ( a_{15}a_{22} + a_{17}a_{21} ) - a_{10} ( a_{15}a_{19} - a_{16}a_{21} ) \\ &\quad{}- a_{6}a_{13}a_{21} \bigr], \end{aligned}$$

with

$$\begin{aligned} &a_{5} = - ( \alpha_{1}F_{idc} ) K - ( n_{i0}e\mu F_{idc} ) K^{ - 1}, \\ & a_{6} = - i ( q_{d0}n_{dc0} \mu )K^{ - 1}, \\ & a_{11} = - i\alpha_{3}K - i ( n_{e0}q_{d0} \mu )K^{ - 1}, \\ & a_{13} = i\alpha_{3}K + i ( n_{i0}q_{d0} \mu )K^{ - 1}, \\ & a_{18} = - im_{d}n_{dn0}\rho_{0}^{ - 1}K^{ - 1}; \end{aligned}$$

$$\begin{aligned} A_{4} &= ( a_{8}a_{15}a_{21} )^{ - 1} \bigl[ a_{8} ( a_{17}a_{22} - a_{18}a_{20} ) \\ &\quad{}+ a_{9} ( a_{16}a_{22} - a_{17}a_{19} - a_{18}a_{19} + a_{16}a_{20} ) \\ &\quad{}+ a_{10} ( a_{15}a_{22} + a_{17}a_{21} ) + a_{6} ( a_{13}a_{19} - a_{14}a_{21} ) \\ &\quad{}+ a_{1}^{ - 1} \bigl[ a_{2}a_{8} ( a_{16}a_{22} - a_{17}a_{19} - a_{18}a_{19} + a_{16}a_{20} ) \\ &\quad{} + ( a_{15}a_{22} + a_{17}a_{21} ) ( a_{2}a_{9} + a_{3}a_{8} - a_{4}a_{7} ) \\ &\quad{}- ( a_{15}a_{19} - a_{16}a_{21} ) \bigl\{ a_{2}a_{10} + a_{3} ( a_{4} + a_{9} ) - a_{5}a_{7} \bigr\} \\ &\quad{}- a_{6}a_{21} ( a_{2}a_{13} - a_{7}a_{13} + a_{8}a_{12} ) \\ &\quad{}+ a_{3}a_{15}a_{21} ( a_{5} + a_{10} ) \\ &\quad{}+ a_{6}a_{11} \bigl\{ a_{21} ( a_{4} - a_{9} ) + a_{8}a_{19} \bigr\} \bigr] \bigr], \end{aligned}$$

with

$$\begin{aligned} &a_{12} = ( \alpha_{3}F_{edc} )K + ( n_{e0}q_{d0}\mu F_{ed} )K^{ - 1}, \\ & a_{20} = - i \bigl( m_{d}n_{dc0} \rho_{0}^{ - 1} \bigr) K^{ - 1}; \end{aligned}$$

$$\begin{aligned} A_{3} &= ( a_{8}a_{15}a_{21} )^{ - 1} \bigl[ a_{9} ( a_{17}a_{22} - a_{18}a_{20} ) \\ &\quad{}+ a_{10} ( a_{16}a_{22} - a_{17}a_{19} - a_{18}a_{19} + a_{16}a_{20} ) \\ &\quad{}- a_{6} ( a_{13}a_{22} - a_{14}a_{19} ) \\ &\quad{}+ a_{1}^{ - 1} \bigl\{ a_{2} a_{8} ( a_{17}a_{22} - a_{18}a_{20} ) \\ &\quad{} + ( a_{16}a_{22} - a_{17}a_{19} - a_{18}a_{19} + a_{16}a_{20} ) \\ &\quad{}\times( a_{2}a_{9} + a_{3}a_{8} - a_{4}a_{7} ) \\ &\quad{}+ ( a_{15}a_{22} + a_{17}a_{21} ) \bigl\{ a_{2}a_{10} + a_{3} ( a_{4} + a_{9} ) - a_{5}a_{7} \bigr\} \\ &\quad{}+ a_{6} \bigl\{ a_{2} ( a_{13}a_{19} - a_{14}a_{21} ) - 2a_{3}a_{13}a_{21} \\ &\quad{}- a_{4} ( a_{11}a_{19} - a_{12}a_{21} ) + a_{5}a_{11}a_{21} \\ &\quad{}- a_{7} ( a_{13}a_{19} - a_{14}a_{21} ) \\ &\quad{}- a_{11} ( a_{8}a_{22} - a_{9}a_{19} + a_{10}a_{21} ) \\ &\quad{}+ a_{12} ( a_{8}a_{9} - a_{9}a_{21} ) \bigr\} \bigr\} \bigr], \end{aligned}$$

with

$$\begin{aligned} a_{14} = - ( \alpha_{3}F_{idc} )K - ( n_{i0}q_{d0}\mu F_{id} )K^{ - 1}; \end{aligned}$$

$$\begin{aligned} A_{2} &= ( a_{8}a_{15}a_{21} )^{ - 1} \bigl[ a_{10} ( a_{17}a_{22} - a_{18}a_{20} ) - a_{6}a_{14}a_{22} \\ &\quad{}+ a_{1}^{ - 1} \bigl\{ ( a_{17}a_{22} - a_{18}a_{20} ) ( a_{2}a_{9} + a_{3}a_{8} - a_{4}a_{7} ) \\ &\quad{} + ( a_{16}a_{22} - a_{17}a_{19} - a_{18}a_{19} + a_{16}a_{20} ) \\ &\quad{}\times\bigl\{ a_{2}a_{9} + a_{3} ( a_{4} + a_{9} ) - a_{5}a_{7} \bigr\} \\ &\quad{}- a_{6} \bigl\{ a_{2} ( a_{13}a_{22} - a_{14}a_{19} ) - 2a_{3} ( a_{13}a_{19} - a_{14}a_{21} ) \\ &\quad{} - a_{4} ( a_{11}a_{22} - a_{12}a_{19} ) + a_{5} ( a_{11}a_{19} - a_{12}a_{21} ) \\ &\quad{}- a_{7} ( a_{13}a_{22} - a_{14}a_{19} ) + a_{11} ( a_{9}a_{22} -a_{10}a_{19}) \\ &\quad{}+a_{12}(a_{8}a_{22}-a_{9}a_{19}+a_{10}a_{21}\bigr\} \\ &\quad{}+a_{3}(a_{15}a_{22}+a_{17}a_{21})(a_{5}a_{10}) \bigr\} \bigr], \end{aligned}$$

$$\begin{aligned} A_{1} &= \bigl[ ( a_{1}a_{8}a_{15}a_{21} )^{ - 1} \\ &\quad{}\times\bigl[ ( a_{17}a_{22} - a_{18}a_{20} ) ( a_{2}a_{10} + a_{3}a_{9} - a_{4}a_{7} ) \\ &\quad{}- a_{6} \bigl[ a_{2}a_{14}a_{22} \\ &\quad{}+ a_{3} \bigl\{ a_{13} ( a_{22} - a_{19} ) - a_{14} ( a_{19} - a_{21} ) \bigr\} \\ &\quad{} - a_{4} ( a_{11}a_{22} - a_{12}a_{19} ) + a_{5} ( a_{11}a_{19} - a_{12}a_{21} ) \\ &\quad{}- a_{7} ( a_{13}a_{22} - a_{14}a_{19} ) + a_{11} ( a_{9}a_{22} - a_{10}a_{19} ) \\ &\quad{} + a_{12} ( a_{8}a_{22} - a_{9}a_{19} + a_{10}a_{21} ) \bigr] \\ &\quad{}+ \bigl\{ a_{16} ( a_{20} + a_{22} ) - a_{19} ( a_{17} + a_{18} ) \bigr\} \\ &\quad{}\times\bigl\{ a_{3} ( a_{4} + a_{10} ) - a_{5}a_{7} \bigr\} + a_{5} ( a_{15}a_{22} + a_{17}a_{21} ) \bigr] \bigr], \end{aligned}$$

$$\begin{aligned} A_{0} &= ( a_{1}a_{8}a_{15}a_{21} )^{ - 1} \bigl[ a_{3} ( a_{17}a_{22} - a_{18}a_{20} ) ( a_{5} + a_{10} ) \\ &\quad{}- a_{6} \bigl\{ a_{12} ( a_{8}a_{22} - a_{9}a_{19} + a_{10}a_{21} ) \\ &\quad{} + a_{22} ( 2a_{3}a_{14} - a_{5}a_{12} ) \bigr\} \bigr]. \end{aligned}$$