## Abstract

We present a multi-fluid theory for the Jeans instability accounting for an attractive force between two equally charged dust particles in a self-gravitating plasma. Our analyses which includes the electrostatic energy between two charged dust grains provides a possibility of resolving the ‘Jeans swindle’, in addition to obtaining a Jeans instability with a faster growth rate. The relevance of our investigation to the formation of planetesimals and collapse of interstellar clouds in star forming regions is discussed.

It is a well established that the Jeans instability (Jeans 1929) plays a very important role in the formation of planetesimals and stars in interstellar media (Eddington 1988). In astrophysical fluids, the collapse of an object is attributed to a self-gravitational force that is responsible for producing the Jeans instability. The hydrostatic gas pressure force sets a threshold (Chandrasekhar 1961) for the Jeans instability. The equilibrium of a homogeneous neutral fluid is, however, not defined, and this is thus referred to as the Jeans swindle. The latter has been resolved (Chandrasekhar 1961; Spitzer 1978; Binney & Tremaine 1988) by considering the neutral gas pressure inhomogeneity and the presence of an external magnetic field.

In dusty plasmas, which are composed of electrons, ions and charged dust grains, the neutral mass density is replaced by the dust mass density and the gas pressure by the sum of the electron and ion pressures. Thus, the Jeans instability threshold is set-up by the dust acoustic waves (DAWs; Rao *et al*. 1990; Shukla & Mamun 2002), instead of the usual sound waves. The importance of the gravitational-like instabilities in a dusty plasma has been recognized (Bingham & Tsytovich 2001; Bingham *et al*. 2002) in the context of processes which are responsible for the production of stars, planets and smaller bodies such as comets and asteroids. Specifically, Bingham & Tsytovich (2001) pointed out the possibility of dust agglomeration due to a shadowing dust attraction force.

In a recent paper, Delzanno & Lapenta (2005) generalized the Jeans instability analysis of Verheest *et al*. (1997), Pandey *et al*. (1994), Rao & Verheest (2000) and Verheest *et al*. (2003), taking into account a Lennard–Jones-like shielding potential for charged dust particles in a collisionless dusty plasma, ignoring dust charge fluctuations. The authors of Pandey *et al*. (1994) resolve the Jeans swindle by suggesting that in equilibrium the dc electric force acting on the dust grains is balanced by the self-gravitation force. On the other hand, the model of Delzanno & Lapenta (2005) did not resolve the Jeans swindle, but found a broader spectrum of unstable Jeans modes with faster growth rates. The results were applied to the dense cores of molecular clouds.

In this brief report, we present a multi-fluid theory for a modified Jeans instability in a partially ionized dusty plasma, taking into account the dust charge fluctuations as well as the electrostatic energy (which includes an attractive force) associated with overlapping Debye spheres (Resendes *et al*. 1998) around two isolated charged dust grains which have similar polarity. Consideration of the electrostatic energy between two isolated grains resolves the Jeans swindle ambiguity, as in equilibrium the self-gravitation force is balanced by the electrostatic interaction force between the two dust grains. Hence, the unperturbed gravitational potential is , where *Z*_{d0} is the unperturbed dust charge number, *e* is the magnitude of the electron charge, *m*_{d} is the dust mass, and *d* is the dust grain separation distance. Furthermore, our new dispersion relation involving the electrostatic energy reveals that the dust grain attraction can cause further destabilization of the Jeans mode, or even produce a novel instability in a non-self-gravitating dusty plasma. Physically, the instability is caused by an adverse phase lag between the DAW potential and the dust charge fluctuations.

The dynamics of dust acoustic perturbations (DAPs) in a self-gravitating dusty plasma is governed by Boltzmann distributed electrons and ions with the perturbed number densities(1)the dust continuity equation(2)and the dust momentum equation(3)where *n*_{j1} (≪*n*_{j0}) is a small density perturbation in the equilibrium value *n*_{j0} (*j* equals ‘e’ for the electrons, ‘i’ for the ions, and ‘d’ for the dust grains), , *T*_{e}(*T*_{i}) is the electron (ion) temperature, **v**_{d} is the dust fluid velocity, *ν*_{d} is the dust-neutral collision frequency, *φ* is the electrostatic potential of the DAPs, *m*_{d} is the dust mass, *V*_{Td} is the dust thermal speed, and *ϕ*_{g} is the gravitational potential determined from Poisson's equation(4)where *G* is the gravitational constant. The interaction potential energy perturbation associated with two dressed dust grains is (Resendes *et al*. 1998; Shukla & Mamun 2002)(5)where *Z*_{d1} (≪*Z*_{d0}) is a small perturbation in the equilibrium dust charge, is the effective dusty plasma Debye radius (Shukla & Mamun 2002), and *λ*_{De} and *λ*_{Di} are the electron and ion Debye radii, respectively. Equations (1)–(4) are closed by using the quasi-neutrality condition , which gives(6)where is associated with the dust charge perturbation (Shukla & Stenflo 2002)(7)which comes about when the dust charging time is much longer than the time-period of the DAPs. We have denoted the dust radius by *r*_{d}, and(8)and(9)where and *ω*_{pe}(*ω*_{pi}) is the electron (ion) plasma frequency.

We note that in the right-hand side of equation (3) appear various forces on the dust fluid. The first and second terms correspond to the electrostatic force and the dust pressure gradient, while the third and fourth terms are due to self-gravitational and attractive forces. The attractive force involves the sum of the electrostatic energies associated with the electron and ion Debye spheres and a grain Debye sphere that overlap (Resendes *et al*. 1998; Shukla & Mamun 2002).

By combining (2)–(7) and Fourier transforming, we obtain the new dispersion relation(10)where *ω* is the frequency, *k* is the wavenumber, , is the Jeans frequency, and is the dust plasma frequency. The solutions of (10) are(11)where and . The modified Jeans instability occurs if(12)

The maximum growth rate above threshold is(13)indicating that dust grain attraction enhances the growth rate of the Jeans instability. Remarkably, we see from (12) and (13) that there is also instability in a dusty non-self-gravitating plasma due to the attractive force involving the dust charge fluctuations in our dusty plasma. We find that for the dust grain attraction induced Jeans instability is dominant. For the interstellar dust clouds (Whittet 1992; Evans 1994), which are composed of a hydrogen plasma and charged dust grains, we typically have (Verheest & Cadez 2002) *n*_{d0}=10^{−7} cm^{−3}, *m*_{d}=4×10^{−12} g (for water ice, micron sized grains), and *Z*_{d0}=1500 electron charges per grain for plasma temperatures of *T*_{e}≈*T*_{i}=10^{4} K. At *n*_{e0}=4.85×10^{−3} and *n*_{i0}=5×10^{−3} cm^{−3}, the effective dusty plasma Debye radius *λ*_{D} is 70 m. The dust temperature is *T*_{d}=30 K. Such plasma parameters yield *Ω*_{J}=5.8×10^{−13} s^{−1}, *V*_{Td}=3.2×10^{−2} cm s^{−1} and *C*_{DP}=2.8 cm s^{−1}. The ratio *C*_{DP}/*Ω*_{J} turns out to be a third of an astronomical unit (AU). Thus, wavelengths larger than several tens of AU can be excited due to the dust grain attraction.

The effect of dust size distributions on the above analysis can easily be incorporated (Meuris 1997*a*,*b*) by assuming that the dust density is given by a power law distribution and that the dust mass is a function of the dust radius. We suppose that and , where *p* is a power law index, the normalization constant *C*_{p} is proportional to *n*_{i0}−*n*_{e0}, and the dust surface potential *ϕ*_{s} is the same for all the grains. For a particular grain size *r*_{d0}, we thus replace by and *ν*_{d} by , where *n*_{n}, *v*_{n} and *m*_{n} are the density, velocity and mass of the neutral atoms.

To summarize, we have investigated the properties of a modified Jeans instability in a self-gravitating dusty plasma, taking into account the dust charge fluctuation inherent in the electrostatic interaction energy of two isolated dust grains whose Debye spheres overlap. We have found that the latter allows us to overcome the Jeans swindle, in addition to obtaining a new dispersion relation which admits an instability with an enhanced growth rate. The underlying physics of the present instability differs significantly from that in Delzanno & Lapenta (2005), which investigated the gravitational instability by assuming that dust particles in a plasma can have a Lennard–Jones-like shielding potential instead of a Coulomb-like electrostatic interaction among the dust particles. It should be noted that Delzanno & Lapenta (2005) used a model force on charged dust grains which interact through a potential (eqn (2) in Delzanno & Lapenta 2005) containing a well. In view of the numerical examples given above, we conclude that the present instability involving charged dust grain attraction could be responsible for the collapse of astrophysical objects, giving rise to the birth of stars and planets in dusty plasmas that are ubiquitous in interstellar clouds.

## Acknowledgements

This research was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 591 entitled ‘Gleichgewichtsferner Plasmen’, as well as by the Swedish Research Council.

## Footnotes

- Received May 11, 2005.
- Accepted October 12, 2005.

- © 2005 The Royal Society