Self-gravitational solitary waves in astrophysical compact objects

TL;DR

This paper theoretically investigates the existence and properties of self-gravitational solitary waves in white dwarfs and neutron stars using a pseudo-potential approach, revealing their negative potential and estimating their amplitude and width.

Contribution

It introduces the first theoretical analysis of SGSWs in astrophysical compact objects, detailing their conditions, polarity, and characteristics in white dwarfs and neutron stars.

Findings

SGSWs exist with negative potential in white dwarfs and neutron stars.

Estimated amplitude and width of SGSWs in white dwarfs: ~-1.5 ergs/gm and 50 cm.

Estimated amplitude and width of SGSWs in neutron stars: ~-7.76×10^4 Joules/kg and 5.5 km.

Abstract

The condition for the existence of self-gravitational solitary waves (SGSWs), and their polarity in any astrophysical compact object are theoretically found for the first time. The pseudo-potential approach, which is valid for arbitrary amplitude SGSWs, is emplyed. The general analytical results are applied in white dwarfs and neutron stars to identify the basic features (polarity, amplitude, and width) of the SGSWs formed in them. It found for the first time that the SGSWs exist with negative self-gravitational potential in perturbed states of white dwarfs and neutron stars. It is also estimated that for their typical degenerate plasma parmeters, the amplitude and the width of the SGSWs (moving with the speed 2 cm/s) in white dwarfs are $\sim -1.5$ ergs/gm and $\sim 50$ cm, respectively, and those of the SGSWs (moving with the speed $\sim 27$ m/s) in neutron stars are $\sim -7.76\times 10^4$ Joules/kg, and $5.5$ km, respectively.

Tables

Table 1. Table 1: Typical approximate values of the physical quantities, m q subscript 𝑚 𝑞 m_{q} , m h subscript 𝑚 ℎ m_{h} , ρ e ​ 0 subscript 𝜌 𝑒 0 \rho_{e0} , ρ q ​ 0 subscript 𝜌 𝑞 0 \rho_{q0} , ρ n ​ 0 subscript 𝜌 𝑛 0 \rho_{n0} , C q subscript 𝐶 𝑞 C_{q} , L q subscript 𝐿 𝑞 L_{q} , and ω J ​ q subscript 𝜔 𝐽 𝑞 \omega_{Jq} corresponding to white dwarfs Shapiro2004 ; Horn1991 ; Koester2002 ; Vanderburg2015 and neutron stars Shapiro2004 ; Horn1991 ; Witze2014 .

Physical quantity	White dwarfs	Neutron stars
$m_q$ (gm)	$12m_p$111 where $m_p\left(=1.67\times {10}^{-24}\mathrm{gm}\right)$ is the proton mass.	$4m_p$
$m_h$ (gm)	$56m_p$	$85m_p$
${\rho }_{e0}\left(\mathrm{gm}{\mathrm{cm}}^{-3}\right)$	$1.82\times {10}^3$	$9.10\times {10}^{11}$
${\rho }_{p0}\left(\mathrm{gm}{\mathrm{cm}}^{-3}\right)$	$0$	$1.67\times {10}^{13}$
${\rho }_{n0}\left(\mathrm{gm}{\mathrm{cm}}^{-3}\right)$	$0$	$1.53\times {10}^{14}$
${\rho }_{g0}\left(\mathrm{gm}{\mathrm{cm}}^{-3}\right)$	$6.35\times {10}^6$	$3.17\times {10}^{15}$
${\rho }_{h0}\left(\mathrm{gm}{\mathrm{cm}}^{-3}\right)$	$3.60\times {10}^5$	$2.50\times {10}^{13}$
$C_q\left(\mathrm{cm}{s}^{-1}\right)$	$1.17\times {10}^6$	$2.78\times {10}^9$
${\omega }_{Jq}\left(s^{-1}\right)$	$2.31\times {10}^{-2}$	$5.15\times {10}^2$
$L_q\left(\mathrm{cm}\right)$	$5.07\times {10}^7$	$5.40\times {10}^6$

Equations

$$\frac{\partial \psi }{\partial x}=-\frac{3}{2}{\alpha }_s^{\prime }\frac{\partial {\rho }_s^{\frac{2}{3}}}{\partial x},$$

$$\frac{\partial {\rho }_j}{\partial t}+\frac{\partial }{\partial x}({\rho }_j{u}_j)=0,$$

$$\frac{\partial u_j}{\partial t}+u_j\frac{\partial u_j}{\partial x}=-\frac{\partial \psi }{\partial x}-\frac{3}{2}{\beta }_j\frac{\partial {\rho }_j^{\frac{2}{3}}}{\partial x},$$

$$\frac{{\partial }^2\psi }{\partial x^2}=\sum _s{\sigma }_s({\rho }_s-1)+\sum _j{\mu }_j({\rho }_j-1),$$

$${\rho }_s={\left(1-\frac{2}{3}{\alpha }_s\psi \right)}^{\frac{3}{2}},$$

$${\rho }_j={\mu }_j{\left[1-b_j+b_j\sqrt{1-\frac{2\psi }{{\mathcal{M}}^2{b}_j^2}}\right]}^{-1},$$

$$\frac{{\partial }^2\psi }{\partial {\xi }^2}=\sum _s{\sigma }_s({\rho }_s-1)+\sum _j{\mu }_j({\rho }_j-1),$$

$$\displaystyle\frac{1}{2}\biggr{(}\frac{d\psi}{d\xi}\biggr{)}^{2}+V(\psi)=0.$$

$$V(\psi )=-\sum _s{\delta }_s\left[\frac{3}{5{\alpha }_s}-\psi -\frac{3}{5{\alpha }_s}{\left(1-\frac{2}{3}{\alpha }_s\psi \right)}^{\frac{5}{2}}\right]$$

$$-\sum _j{\mu }_j\left[b_j{\mathcal{M}}^2-\psi -b_j{\mathcal{M}}^2{Y}_j+(1-b_j){\mathcal{M}}^2{Z}_j\right],$$

$$V(\psi )=\frac{1}{2}{C}_2{\psi }^2+\frac{1}{2}{C}_3{\psi }^3+\cdot \cdot \cdot ,$$

$$C_2=\sum _s{\delta }_s{\alpha }_s-\sum _j\frac{{\mu }_j\left(1+\frac{5}{3}{\beta }_j\right)}{({\mathcal{M}}^2-{\beta }_j)},$$

$$C_3=-\frac{1}{9}\sum _s{\delta }_s{\alpha }_s^2$$

$$-\frac{1}{3}\sum _j\frac{{\mu }_j{\left(1+\frac{5}{3}{\beta }_j\right)}^2[{\mathcal{M}}^2\left(3+\frac{5}{3}{\beta }_j\right)-2{\beta }_j]}{({\mathcal{M}}^2-{\beta }_j{)}^3}.$$

$${\mathcal{M}}^2<{\mathcal{M}}_c^2=\frac{1}{2{\alpha }_e{\delta }_e}\left[{\beta }_1+\sqrt{{\beta }_1^2-4{\alpha }_e{\beta }_q{\delta }_e{\mu }_h}\right],$$

$$C_3{|}_{\mathcal{M}={\mathcal{M}}_c}=C_3^c=-\frac{1}{9}{\delta }_e{\alpha }_e^2-\frac{{\mu }_h}{{\mathcal{M}}_c^4}$$

$$-\frac{1}{3}\frac{{\left(1+\frac{5}{3}{\beta }_1\right)}^2[{\mathcal{M}}_c^2\left(3+\frac{5}{3}{\beta }_q\right)-2{\beta }_l]}{({\mathcal{M}}_c^2-{\beta }_q{)}^3},$$

$$\frac{d\psi }{d\xi }=\psi \sqrt{-C_2-C_3\psi }.$$

$$\psi =-\left(\frac{C_2}{C_3}\right)\mathrm{Sec}{\mathrm{h}}^2(\sqrt{-C_2}\xi ).$$

Full text

The astrophysical compact objects (ACOs) are completely different from other terrestrial bodies because of their (ACOs’) extremely low temperate and extremely high density Chandrasekhar1931 ; Shapiro2004 ; Horn1991 . They are, in fact, degenerate quantum plasma Manfredi2005 ; Shukla2006 ; Markcloud2007 ; Shukla2011a ; Haas2011 ; Shukla2011b ; MAS2016 ; Brodin2017 ; MAS2017 systems containing degeneare light particle species (viz. electron species with/without positron or/and quark (non-zero mass Fraga2005 ) species depending on the class of ACOs under consideration), degnerate heavy particle (compared to electron mass) species Chandrasekhar1931 ; Koester1990 ; Koester2002 ; Shapiro2004 (viz. $~{}^{4}_{2}$He or/and $~{}^{12}_{~{}6}$C or/and $~{}^{16}_{~{}8}$O species with/without proton or/and neutron species depending on the class of ACOs under consideration), and non-degenerate species of low densed heavy elements Shapiro2004 ; Witze2014 ; Vanderburg2015 (viz. $~{}^{56}_{26}$Fe or/and $~{}^{85}_{37}$Rd or/and $~{}^{96}_{42}$Mo depending again on the class of ACOs under consideration).

The degeneracy of the light and heavy particle species (viz. electron or proton or neutron or $~{}^{4}_{2}$He or $~{}^{12}_{~{}6}$C or $~{}^{16}_{~{}8}$O species) arises due to Heisenberg’s uncertainty principle, $\Delta p\Delta x\geq\hbar/2$ (where $\hbar$ is the reduced Planck constant, $\Delta p$ is the uncertainty in momentum of the species, and $\Delta x$ is the uncertainty in position of the species). This indicates that the momenta of highly compressed particles are extremely uncertain, since the particles are located in a very confined space. Therefore, even though the degenerate plasma is very cold, the plasma particles must move very fast on average, and give rise to a very high pressure (known as degenerate pressure), which does not depend on thermal temperature, but on degenerate particle number density. This leads us to a conclusion that in order to compress degenerate particles to an object into a very small space, a tremendous force (which is the self-gravitational force in ACOs like white dwarfs, neutron stars, black holes, etc. Shapiro2004 ) is required to control its particles’ momenta. Thus, if any ACO is disturbed/perturbed due any reason (viz. merging Abbott2016 of two small ACOs, fragmentation Shapiro2004 of a large ACO, gravitational interaction Shapiro2004 among neighboring ACOs, etc.), the self-gravitational perturbation mode [in which the compression (rarefaction) is provided by the self-gravitational attraction pressure (degenerate pressure)] is developed. The propagation of this self-gravitational perturbation (SGPM) cannot be usually explained by linear analysis. Therefore, for the first time a theoretical investigation is made on the nonlinear propagation of this SGPM by employing the pseudo-potential approach, which is valid for arbitrary amplitude SGPM. It is first time shown here that the SGPM nonlinearly propagate as self-gravitational solitonic potential signals (SGSWs) with negative potential in the ACOs like white dwarfs and neutron stars.

We consider a general degenerate quantum plasma system like an ACO containing arbitrary number of degenerate species $s$ of non-inertial light particles Chandrasekhar1931 ; Shapiro2004 ; Horn1991 (viz. electrons or/and positrons or/and non-zero mass quarks Fraga2005 , etc.), degenerate species $j$ of inertial heavy (compared to electron mass) particles Shapiro2004 ; Horn1991 ; Koester1990 ; Koester2002 (viz. $~{}^{4}_{2}$He or $~{}^{12}_{~{}6}$C or $~{}^{12}_{~{}6}$O or/and protons or/and neutrons, etc.), and non-degenerate species $h$ of inertial heavy elements Horn1991 ; Vanderburg2015 ; Witze2014 (viz. $~{}^{56}_{26}$Fe or/and $~{}^{85}_{37}$Rd or/and $~{}^{96}_{42}$Mo, etc.). The heavy elements, viz. $~{}^{56}_{26}$Fe, $~{}^{85}_{37}$Rd, $~{}^{96}_{42}$Mo, etc., are assumed to be non-degenerate just because of their low density and high mass.

The nonlinear dynamics of the SGPM, whose phase speed is much smaller than $C_{e}$ (where $C_{e}=\sqrt{\pi}\hbar n_{e0}^{1/3}/m_{e}$ in which $m_{e}$ is the electron mass, and $n_{e0}$ is the equilibrium degenerate electron number density) is described by

[TABLE]

where $\rho_{s}$ ($\rho_{j}$) is the number density of the degenerate light (heavy) particle species $s$ ($j$), and is normalized by $\rho_{s0}$ ($\rho_{j0}$) in which $\rho_{s0}$ ($\rho_{j0}$) is the equilibrium mass density of the degenerate particle species $s$ ($j$); $u_{j}$ is the degenerate fluid speed of the species $j$, and is normalized by $C_{q}$ in which $C_{q}=(\sqrt{\pi}\hbar n_{e0}^{1/3}/m_{q})$, and $q$ represents the heaviest particle species among all the degenerate particle species (e. g. $m_{q}$ is the mass of $~{}^{12}_{~{}6}$C for white dwarfs and of $~{}^{4}_{2}$He for neutron stars); $\psi$ is the self-gravitational potential, and is normalized by $C_{q}^{2}$; $\alpha_{s}^{\prime}=(m_{q}/m_{s})^{2}(n_{s0}/n_{e0})^{2/3}$, $\beta_{j}=(m_{q}/m_{j})^{2}(n_{j0}/n_{e0})^{2/3}$, in which $m_{s}$ ($m_{j}$) is the mass of the degenerate particle species $s$ ($j$), and $n_{s0}$ ($n_{j0}$) is the equilibrium number density of the degenerate particle species $s$ ($j$); $t$ is the time variable normalized by $\tau_{q}=(4\pi G\rho_{q0})^{-1/2}$; $x$ is the space variable normalized by $L_{q}=C_{q}\tau_{q}$; $\sigma_{s}=\rho_{s0}/\rho_{q0}$, and $\mu_{j}=\rho_{j0}/\rho_{q0}$ in which $\rho_{q0}$ is the mass density of the degenerate species $q$ (e. g. $~{}^{12}_{~{}6}$C for white dwarfs and $~{}^{4}_{2}$He for neutron stars). We note that $j=h$ when $\beta_{j}=0$. The typical approximate values of the physical quantities, viz. $m_{q}$, $m_{h}$, $\rho_{e0}$ $\rho_{q0}$, $\rho_{n0}$, $C_{q}$, $\omega_{Jq}$, and $L_{q}$, corresponding to white dwarfs and neutron stars are tabulated in Table I.

We note that $\rho_{h0}$ in both white dwarfs and neutron stars are determined by the charge neutrality condition, and that $m_{n}\simeq m_{p}$.

We now study the possibility for the formation of the SGSSs, and their basic features by the pseudo-potential approach Bernstein1957 ; Cairns1995 , which is valid for arbitrary amplitude SGSWs. To do so, we first assume that all dependent variables in (1)-(4) depend only on a single variable $\xi=x-{\cal M}t$, where ${\cal M}$ is the speed of the the SGSWs, and is normalized by $C_{q}$, and $\xi$ is normalized by $L_{q}$. On using (1)-(4), this assumption leads us to express $\rho_{s}$ and $\rho_{j}$ (under the steady state condition) as

[TABLE]

where $\alpha_{s}=1/\alpha_{s}^{\prime}$, and $b_{j}=(1-\beta_{j}/{\cal M}^{2})/(1+5\beta_{j}/3)$. We note that the species $j$ includes the species $q$ and $h$, and that $\rho_{j}=\rho_{q}$ when $j$ represents the heaviest degenerate species (e. g. $~{}^{12}_{~{}6}$C for white dwarfs Koester1990 ; Koester2002 , and $~{}^{4}_{2}$He for neutron stars Shapiro2004 ; Horn1991 ), and $\rho_{j}=\rho_{h}$ when $j$ represent non-degenerate species ($\beta_{j}=0$) of heavy elements (e. g. $~{}^{56}_{26}$Fe in white dwarfs Horn1991 ; Vanderburg2015 and $~{}^{85}_{37}$Rd in neutron stars Horn1991 ; Witze2014 ).

We first substitute (5) and (6) into (7), and then integrate the resulting equation over $\xi$ to obtain an energy integral in the form Bernstein1957 ; Cairns1995

[TABLE]

This energy integral represents the motion of a pseudo particle of unit mass moving with pseudo-speed $d\psi/d\xi$ in a pseudo-potential $V(\psi)$, where $\psi$ is the pseudo-position and $\xi$ is the pseudo-time Bernstein1957 . The pseudo-potential $V(\psi)$ appeared in (8) is given by

[TABLE]

where $b_{j}=(1-\beta_{j}/{\cal M}^{2})/(1+5\beta_{j}/3)$, $Y_{j}=[1-2\psi/(b_{j}^{2}{\cal M}^{2})]^{1/2}$, $Z_{j}=\ln(1-b_{j}+b_{j}Y_{j})$, and the integration constant is chosen in such a way that $V(\psi)=0$ at $\psi=0$. We note that the energy integral defined by (8) with the pseudo-potential $V(\psi)$ defined by (9) is valid for any ACO containing arbitrary number of degenerate species $s$ of light particles (viz. electrons, positrons, etc.), arbitrary number of degenerate species $j$ of heavy particles (viz. protons, neutrons, $~{}^{4}_{2}$He nuclei, $~{}^{12}_{~{}6}$C nuclei, etc.), and arbitrary number of non-degenerate species $h$ ($\beta_{h}=0$) of heavy elements (viz. $~{}^{56}_{26}$Fe, $~{}^{85}_{37}$Rd, $~{}^{96}_{42}$Mo, etc.). The Taylor series expansion of $V(\psi)$ around $\psi=0$ is

[TABLE]

where

[TABLE]

It is obvious from (10) that $V(0)=0$ and $(dV/d\psi)|_{\psi=0}=0$. This means that (i) the energy integral defined by (8) with the pseudo-potential $V(\psi)$ defined by (9) gives rise to the existence of SGSWs if and only if ($d^{2}V/d\psi^{2})|_{\psi=0}<0$, i. e. $C_{2}<0$, and thus $C_{2}|_{{\cal M}={\cal M}_{c}}=0$ yields ${\cal M}_{c}$ (the critical value of ${\cal M}$ bellow which the SGSWs exist); (ii) the SGSWs exist with $\psi>0$ if $C_{3}|_{{\cal M}={\cal M}_{c}}>0$, and with $\psi<0$ if $C_{3}|_{{\cal M}={\cal M}_{c}}<0$.

To find the expression for ${\cal M}_{c}$ and to examine whether $C_{3}|_{{\cal M}={\cal M}_{c}}>0$ or $C_{3}|_{{\cal M}={\cal M}_{c}}<0$, for an example an example, we first consider white dwarfs, which can be assumed to consist of degenerate species $s$ of light particles like Shapiro2004 electrons, degenerate species $q$ of heavy (compared to electron mass) particles like Koester2002 ; Shapiro2004 $~{}^{12}_{~{}6}$C, and non-degenerate species $h$ of heavy element like Vanderburg2015 $~{}^{56}_{26}$Fe). Thus, using (11) and (12), the conditions for the existence of SGSWs in white dwarfs become

[TABLE]

where $\beta_{1}=1+\mu_{h}+\beta_{q}(\alpha_{e}\delta_{e}+5/3)$. It is obvious from (14) that we cannot analytically predict $C_{3}^{c}<0$ for any possible value of $\beta_{q}$, which is $0.29$ for $~{}^{12}_{~{}6}$C. The numerical variation of $C_{3}^{c}$ with $\beta_{q}$ (for its range from [math] to $1$) implies that $C_{3}^{c}<0$ for $\beta_{q}=0-1$. Similarly, we can also use (11) and (12) to verify that the the conditions (viz. ${\cal M}<{\cal M}_{c}$ and $C_{3}^{c}<0$) for the existence of the SGSWs in neutron stars are satisfied. However, the expressions of ${\cal M}_{c}$ and $C_{3}^{c}$ for neutron stars are more complex than those for white dwarfs. This is because the neutron stars can be assumed to consist of a degenerate species of light particlesShapiro2004 (viz. electrons), three degenerate species of heavy (compared to electron mass) particles Shapiro2004 (viz. protons, neutrons, light nuclei like $~{}^{4}_{2}$He), and a non-degenerate ($\beta_{h}=0$) species of heavy element like Witze2014 $~{}^{85}_{37}$Rd. We note that in neutron stars $j=q$ for $~{}^{4}_{2}$He species, and that $\beta_{j}$ is $0.74$, $0.93$, and $0.97$ for $~{}^{4}_{2}$He, proton, and neutron species, respectively.

To examine the basic features (amplitude and width) of these SGSWs, we limit $V(\psi)$ up to the terms containing $\psi^{3}$ (since $\psi\ll 1$ always valid because of its normalization by $C_{q}^{2}$, which is $1.36\times 10^{12}{\rm cm^{2}~{}s^{-2}}$ for white dwarfs, and $7.73\times 10^{18}{\rm cm^{2}~{}s^{-2}}$ for neutron stars). This limit allows us to express (8) as

[TABLE]

We now look for the solitonic signal solution of this first order ordinary nonlinear differential equation (15), and under a localized boundary condition, viz. $\psi\rightarrow 0$ at $\xi\rightarrow\pm\infty$, its stationary solitonic signal solution can be written as

[TABLE]

This represents a solitonic signal solution of (15), and means that the SGSWs with $\psi<0$ exist in the degenerate plasma system under consideration since $C_{2}<0$ and $C_{3}<0$. We have graphically shown the SGSWs that can be formed for the parameters corresponding to white dwarfs and neutron stars. These are displayed in figures 1 and 2,

which clearly indicates that the SGSWs exist with $\psi<0$) in ACOs (like white dwarfs and neutron stars) and that the magnitude of the amplitude and the width of the SGSWs increase with the rise of ${\cal M}$ within the range of ${\cal M}_{c}<{\cal M}<0$. We have also numerically solved (8) [with the pseudo-potential $V(\psi)$ defined by (9)] for the parameters given in Table I, and used in figures 1 and 2, and have found the existence of the SGSWs, which are completely identical to those shown in figures 1 and 2.

To summarize, we have considered a general self-gravitating degenerate quantum plasma system [containing an arbitrary number of degenerate species of non-inertial light particles, (viz. electrons, positrons, non-zero mass quarks, etc.), arbitrary number of degenerate species of inertial heavy (compared to electron mass) particles (viz. protons, neutrons, nuclei of light elements like $~{}^{4}_{~{}2}$He, $~{}^{12}_{~{}6}$C, $~{}^{16}_{~{}8}$O, etc.), and an arbitrary number of non-degenerate species of inertial heavy elements (viz. $~{}^{56}_{26}$Fe, $~{}^{85}_{37}$Rd, $~{}^{96}_{42}$Mo, etc.)], and have found the conditions for the existence of the SGSWs in ACOs. The pseudo-potential approach, which is valid for the arbitrary amplitude SGSWs, is used. The general analytical results have been applied in white dwarfs (containing a degenerate electron species Shapiro2004 , a degenerate $~{}^{12}_{~{}6}$C species Koester2002 ; Shapiro2004 , and a non-degenerate $~{}^{56}_{26}$Fe species Vanderburg2015 ) and neutron stars (containing degenerate electron, proton, neutron, and $~{}^{4}_{2}$He species Shapiro2004 , and a non-degenerate $~{}^{85}_{37}$Rd species Witze2014 ) to identify the basic features (viz. polarity, amplitude, width, etc.) of the SGSWs, which are found to be formed in them. It is shown that the SGSWs exist with negative self-gravitational potential in both white dwarfs and neutron stars for ${\cal M}_{c}<{\cal M}<0$. It is observed that the speed ($V_{0}$), magnitude of the amplitude ($|\psi_{m}|$) and the width ($\delta$) of the SGSWs (in both white dwarfs and neutron stars) increase with the rise of their normalized speed (${\cal M}$) within the range $0<{\cal M}<{\cal M}_{c}$. It has been carefully checked that the analysis for arbitrary amplitude SGSWs gives the same results as that for small amplitude SGSWs does since $|\psi|\ll 1$ always valid because of its normalization by $C_{q}^{2}$, which is $1.36\times 10^{12}{\rm cm^{2}~{}s^{-2}}$ for white dwarfs, and $7.73\times 10^{18}{\rm cm^{2}~{}s^{-2}}$ for neutron stars. It is also observed that the dimensional (non-normalized) values of $V_{0}$, $\psi_{m}$, and $\delta$ in neutron stars are much higher than those in white dwarfs. It has been estimated that for ${\cal M}=10^{-6}$ and for the parameters given in Table I, $V_{0}$, $|\psi_{m}|$, and $\delta$ are $\sim 2$ cm/s, $\sim 1.5$ ergs/gm, and $\sim 50$ cm, respectively, in white dwarfs, and are $\sim 27$ m/s, $\sim 7.76\times 10^{4}$ Joules/kg, and $5.5$ km, respectively, in neutron stars.

Recent discovery Abbott2016 of the gravitational waves Abbott2016 ; Kurkela2016 ; Ho2016 produced by merging of two black holes, and the results of the present investigation indicate that the signatures of the nonlinear structures like SGSWs exist in ACOs like white dwarfs and neutron stars. The SGSWs are associated with the perturbation mode generated in ACOs (viz. white dwarfs and neutron stars) due to the departure from their equilibrium states by any event (viz. merging of two small ACOs Abbott2016 ; Kurkela2016 ; Ho2016 , splitting up a large ACO according to the Chandrasekhar limit Chandrasekhar1931 ; Shapiro2004 , gravitational interaction among nearby ACOs Shapiro2004 , etc.) or by any other reasons.

The degenerate quantum plasma system considered here is generalized to an arbitrary number of species of degenerate light non-inertial particles, degenerate heavy (compared to electron mass) inertial particles, and non-degenerate inertial heavy nuclear species of heavy elements. The degenerate quantum plasma system is also very general from the view of arbitrary mass densities of all the species comprising the system. Therefore, the investigation presented here can be applied for any ACO, and can be very useful in understanding the basic features of nonlinear localized structures (viz. SGSWs) in any ACO (viz. white dwarfs, neutron stars, and black holes Shapiro2004 ).