New self-gravito-acoustic mode in degenerate quantum plasmas

TL;DR

This paper predicts a new self-gravito-acoustic mode in cold self-gravitating degenerate quantum plasmas, which depends on the interplay of gravitational and degenerate pressures, with implications for white dwarf physics.

Contribution

It introduces and theoretically describes a novel perturbation mode in degenerate quantum plasmas, absent when degenerate pressure of light particles is neglected.

Findings

Prediction of a new SGAM in SGDQPs

SGAM depends on degenerate pressure of light particles

Application to white dwarf plasma

Abstract

The existence of a new perturbation mode [`self-gravito-acoustic mode' (SGAM)] in cold self-gravitating degenerate quantum plasmas (SGDQPs) is theoretically predicted. This new SGAM is developed in the perturbed SGDQPs, in which the compression is mainly provided by the self-gravitational pressure of the heavy particle species, and the rarefaction is mainly provided by the degenerate pressure of the light particle species. The SGAM is a new perturbation mode since it completely disappears if the degenerate pressure of the light particle species is neglected. The prediction of this new SGAM is applied in a white dwarf SGDQP.

Equations

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

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

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

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

$$1+\sum _j\frac{{\mu }_j}{{\omega }^2-{\beta }_j{k}^2}-\sum _s\frac{{\delta }_s}{{\alpha }_s{k}^2}=0.$$

$${\omega }^2=k^2\frac{{\sum }_j{\mu }_j}{{\sum }_s\frac{{\delta }_s}{{\alpha }_s}-k^2}.$$

$${\omega }^2=k^2\left(\sum _j{\mu }_j\right)\left(\sum _s\frac{{\alpha }_s}{{\delta }_s}\right).$$

Full text

The self-gravitating degenerate quantum plasma medium Shapiro2004 ; Manfredi2005 ; Shukla2011 ; Haas2011 , which is common in many astrophysical compact objects (ACOs) like white dwarfs, neutron stars and black holes Shapiro2004 , significantly differs from other plasma media because of its extremely low temperate and extra-ordinarily high density. The ingredients of this self-gravitating degenerate quantum plasma (SGDQP) medium varies not only from one ACO to other, but also from one part to other parts of the same ACO. Thus, depending on ACO or its regions, the SGDQP medium can be assumed to contain non-inertial degenerate light particle (viz. electron or/and positron or/and (non-zero mass quark) species, and inertial degenerate heavy (compared to electron mass) particle (viz. proton or/and neutron or/and $~{}^{4}_{2}$He or/and $~{}^{12}_{~{}6}$C or/and $~{}^{16}_{~{}8}$O), $~{}^{56}_{26}$Fe, etc.) species. The degeneracy of these particles arises due to Heisenberg’s uncertainty principle. This is due to the fact that in a medium of extremely low temperature and extra-ordinarily high density, the particles are located in an extra-ordinarily confined space, and that the momenta of highly compressed particles are extremely uncertain. Thus, ts particles must move very fast on average, and gives rise to a very high pressure. This pressure is known as ‘degenerate pressure’, which depends on the degenerate particle number density, but not on thermal temperature. This generate pressure is given by Shapiro2004 $P=3\hbar^{2}n^{5/3}/5m$ for non-relativistic limit. We note that the subscript $s$ and $j$ on $P$, $n$, and $m$ are used for the species $s$ and $j$, respectively. This outward degenerate pressure is counter balanced by the inward self-gravitational pressure to form ACOs like white dwarfs, neutron stars, black holes, etc. according to Chandrasekhar limit Shapiro2004 .

The SGDQPM medium in ACOs (particularly, in white dwarfs and neutron stars) cannot be at equilibrium in general. This can be perturbed by many reasons (e. g. merging of two small ACOs Abbott2016 , splitting up a large ACO according to the Chandrasekhar limit Shapiro2004 , gravitational interaction among nearby ACOs Shapiro2004 , etc.). Once the SGDQP medium in any ACO is perturbed, a perturbation mode [‘self-gravito-acoustic mode’ (SGAM)] developed by the compression (due to self-gravitational pressure on heavy particle species) and rarefaction (due to degenerate pressure on light particle species), and propagates through the medium. The dynamics of this SGAM in such a SGDQP medium is described by

[TABLE]

where $n_{s}$ ($n_{j}$) is the number density of the degenerate non-inertial light (inertial heavy) particle species $s$ ($j$), and is normalized by $n_{s0}$ ($n_{j0}$) in which $n_{s0}$ ($n_{j0}$) 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 $C_{q}~{}(=\sqrt{\pi}\hbar n_{e0}^{1/3}/m_{q}$, in which $q$ represents one of the species $j$ which have the maximum mass density (for example in white dwarfs it is $~{}^{12}_{~{}6}$C Koester1990 ; $\psi$ is self-gravitational potential normalized by $C_{q}^{2}$; $\alpha_{s}=(m_{q}/m_{s})^{2}(n_{s0}/n_{e0})^{2/3}$ (which represents the degeneracy effect of the non-inertial light particle species $s$) $\beta_{j}=(m_{q}/m_{j})^{2}(n_{j0}/n_{e0})^{2/3}$ (which represents the degeneracy effect of the inertial heavy particle species $j$), $m_{s}$ ($m_{j}$) 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 the time scale $\tau$, which is the inverse of the Jeans frequency, $\omega_{Jq}=(4\pi Gm_{q}n_{q0})^{1/2}$, in which $G$ is the universal gravitational constant); $x$ is the space variable normalized by the scale length $L_{q}$ (defined as $L_{q}=C_{q}/\omega_{Jq}$); $\delta_{s}=m_{s}n_{s0}/m_{q}n_{q0}$, and $\mu_{j}=\omega_{Jj}/\omega_{Jq}$. It should be noted here that the counter balance between the outward degenerate pressure and the inward self-gravitational pressure of non-inertial particle species $s$ gives rise to (1), which is however valid for the perturbation mode whose phase speed is much smaller than $C_{e}=\sqrt{\pi}\hbar n_{e0}^{1/3}/m_{e}$.

We first linearize (1)$-$(4) by assuming that $n_{s,j}=1+\tilde{n}_{s,j}$, $u_{j}=0+\tilde{u}_{j}$, and $\psi=0+\tilde{\psi}$, where $\tilde{n}_{s,j}$, $\tilde{u}_{j}$, and $\tilde{\psi}$ represent the perturbed part of $n_{s,j}$, $u_{j}$, and $\psi$. We then assume that all of these perturbed quantities are directly proportional to $\exp(-i\omega{t}+ikx)$, where $\omega$ is the angular frequency of the SGAM, and is normalized by $\omega_{Jh}$, and $k$ is the propagation constant, and is normalized by $L_{q}^{-1}$. This assumption allows us to express the linear dispersion relation for the SGAM as

[TABLE]

This is the general dispersion relation for the SGPM propagating in any SGDQP medium containing arbitrary numbers of light non-inertial degenerate particle species $s$, and of inertial heavy (compared to electron mass) degenerate particle species $j$ . Thus, this general dispersion relation is valid for any SGDQP system. However, to analyze this dispersion relation analytically, we first consider that all the heavy particle species $j$ are non-degenerate (i.e. $\beta_{j}=0$). This limiting case leads (5) to

[TABLE]

It is obvious from (6) that when the non-inertial inertial particle species $s$ are assumed to be non-degenerate (i.e. $\alpha_{s}=0$), (6) reduces $\Omega=i\sqrt{\sum_{j}\omega_{Jj}^{2}}$, where $\Omega$ is the dimensional form of $\omega$. The latter means that for non-inertial light light particle species (non-degenerate plasma medium), (6) reduces to a well-known purely growing unstable Jeans mode with the growth rate $\sqrt{\sum_{j}\omega_{Jj}^{2}}$. However, for a degenerate plasma system (e.g. ACOs like white dwarfs and neutron stars), one cannot neglect $\alpha_{s}$ any way Shapiro2004 ; Horn1991 ; Koester1990 ; Chandrasekhar1931 . It is obvious from (6) that $\omega\rightarrow\infty$ at $k\rightarrow k_{c}$ (where $k_{c}=\sqrt{\sum_{s}\delta_{s}/\alpha_{s}}$), and that for $k\ll k_{c}$, (6) yields

[TABLE]

This represents the linear part of the dispersion relation for the predicted new SGAM in a degenerate plasma containing arbitrary number of degenerate light particle species $s$, and of non-degenerate inertial particle species $j$. This is, in deed, represents a stable new SGAM since the latter disappears when the effect of degeneracy all light particle species $s$ is neglected. This new stable mode exists for the wavelength satisfying the condition $k<k_{c}$. However, to examine the dispersion properties of this new mode for whole wavelength range, we apply (6) in a white dwarf SGDQP containing non-inertial degenerate electron species, and inertial non-degenerate nuclear ($~{}^{12}_{~{}6}$C) species (which has maximum mass density Koester1990 ), and is denoted by $q$ ), and inertial non-degenerate heavy nuclear ($~{}^{56}_{26}$Fe) species (which is the heaviest particle species, and is denoted by $h$).

To compare among the effects of degeneracy in non-inertial electron species , and inertial $~{}^{12}_{~{}6}$C and $~{}^{56}_{26}$Fe species, we have as estimated that $\alpha_{e}=4.85\times 10^{8}$, $\beta_{q}=0.30$, and $\beta_{h}=2.43\times 10^{-4}$ for white dwarf SGDQP parameters (viz. $n_{e0}=10^{29}~{}{\rm cm^{-3}}$, $n_{q0}=0.99n_{e0}/Z_{q}$ and $n_{h0}=(n_{e0}-Z_{q}n_{q0})/Z_{h}$, in which $Z_{q}~{}(Z_{h})$ is the number of protons in $q$ ($h$) species. Thus the assumption ($\beta_{j}=0$) used in (6 is quite valid for a white dwarf SGDQP. To see the nature of the dispersion curve in between $k\gg k_{c}$ and $k\ll k_{c}$, we now numerically solve (6) for typical parameters corresponding to a white dwarf SGDQP Shapiro2004 ; Koester1990 ; Vanderburg2015 , where $m_{q}=12m_{p}$ and $m_{h}=56m_{p}$ (in which $m_{p}$ is the proton mass); $n_{e0}=10^{29}~{}{\rm cm^{-3}}$, $n_{q0}=0.995n_{e0}/Zq$ ($n_{q0}=n_{e0}/Zq$ means that no other nuclei of heavy elements are present). The numerical results are shown in figure 1.

To conclude, a new SGAM in a cold SGDQP is identified for the first time. The physics of this new SGAM is that if any SGDQP system is perturbed from its equilibrium state by its compression, the degenerate pressure brings it back to its equilibrium state, but during this action it is expanded more than its equilibrium state according to Newtons first law of motion, and again the self-gravitational pressure brings the system back to its equilibrium state, but again during this action, it is compressed more than it equilibrium state. These compression and rarefaction of the medium continue, and consequently, a low-frequency, long-wavelength new SGAM propagates through the medium. Though the result is applied in a particular SGDQP system like white dwarf, it can be applied to any SGDQP system since a SGDQP system containing an arbitrary number of degenerate, non-inertial, light particle species $s$, and an arbitrary number of degenerate, inertial, heavy particle species is considered. Recent discovery Abbott2016 of the gravitational waves Abbott2016 produced by merging of two black holes leads us to expect that in near future the signatures of similar or different kind of new wave/mode like SGAM in other astrophysical compact objects like white dwarfs and neutron stars Shapiro2004 should be detected.