Gravitational equilibrium and the mass limit for dust clouds

We show the existence of a new class of astrophysical objects where the self-gravity of the dust is balanced by the force arising from shielded electric fields on the charged dust. The problem of equilibrium dust clouds is formulated in terms of an equation of hydrostatic force balance together with an equation of state. Because of the dust charge reduction at high dust density, the adiabatic index reduces from two to zero. This gives rise to a mass limit MAS for the maximum dust mass that can be supported against gravitational collapse by these fields. If the total mass MD of the dust in the interstellar cloud exceeds MAS, the dust collapses, while in the case MD < MAS, equilibrium may be achieved. The physics of the mass limit is similar to the Chandrasekhar's mass limit for compact objects, such as white dwarfs and neutron stars.


Introduction
The gravitational collapse of dust in interstellar clouds plays an important role in the formation of stars and other planetary bodies [1,2]. In the cloud the interstellar radiation fields and other mechanisms ionize the gas and hence the dust grains acquire a net non-negligible electric charge [3,4]. The gravitational collapse of the charged dust in the astrophysical context has been investigated by a number of authors [5]- [11]. One of the main conclusions from these studies is that the threshold for the Jeans instability is set up by the dust acoustic waves [12] rather than by the neutral sound waves. Recently, the importance of gravitational-like instabilities in dusty plasmas has been recognized in processes which are responsible for star and planet formation [13].
As is well known, one of the major problems encountered in the study of the gravitational collapse of matter is the formation of a legitimate equilibrium. Since the dust kinetic pressure is too weak, it cannot balance the self-gravity of the dust on astrophysically relevant scales. One way to resolve this problem is to ignore the equilibrium altogether and consider the local stability of perturbations much smaller than the global scale. This study is of limited scope, as it does not address the stability of global eigenmodes which are expected to play a major role in the collapse. The other alternative, frequently used hitherto, is to balance the selfgravity of the dust by charge separation electric fields in the background plasma. Since the dimensions of the cloud are typically much larger than the plasma Debye radius λ d , the plasmadust system will be quasi-neutral and hence the charge separation electric fields are expected to be weak.
In this paper, we discuss the possibility of a new way of balancing the self-gravity of the dust. Specifically, we show that the self-gravity of the dust can be balanced by the pressure P E arising from shielded electric fields in the background plasma. This pressure arises essentially due to an inhomogeneous distribution of the dust in the background plasma. Such inhomogeneities are bound to occur during the collapse of the dust in a cloud. The importance of this effect lies in the fact that the dust equilibrium constructed by balancing the self-gravity with P E are of astrophysically relevant length and mass scales. We present an equation of state for P E , which relates it to the dust number density n d , i.e. P E = P E (n d ). Equilibrium configurations are obtained by balancing P E with the self-gravity of the dust. Specifically, we construct one-parameter family of spherically symmetric and static dust configurations. Furthermore, we show that as the dust number density increases during the collapse, P E also increases. However, due to the dust charge reduction, which occurs at high dust density [14]- [17], the increase in the pressure is not sufficient to balance the self-gravity. This results in an upper limit M AS on the total mass M D of the dust in the cloud for the equilibrium. For M D > M AS , the dust-cloud collapses, while for M D < M AS an equilibrium may be obtained. The physics of this mass limit is very similar to the Chandrasekhar's mass limit for compact objects, such as white dwarfs and neutron stars [18]- [20]. In the dust equilibria near the mass limit, the dust is 'compactly' packed at high densities. These dust clouds, which are a new class of astrophysical objects where gravity is supported by shielded electric fields, may exist inside small (<0.1 pc) interstellar clouds.

The model
The simple model we study consists of a dust cloud of finite extent imbedded in background electron-ion plasma of much larger extent. Our model is governed by and where n e (n i ) are the electron (ion) densities, p e (p i ) are the electron (ion) pressure, n d (ρ d ) is the dust number (mass) density and ϕ( ) are the electro-static (gravitational) potential. On the slow timescale of the collapse, we neglect the inertia of electrons and ions in equations (1) and (2). We have also neglected the dust kinetic pressure in equation (3) and assumed quasi-neutrality in equation (5). This is appropriate for the dust cloud of dimensions much larger than the plasma Debye radius. The first term due to the electric field in the right-hand side of equation (3) is finite and arises due to an inhomogeneous distribution of charged dust grains. We will show shortly that this force can be expressed as the gradient of P E and can balance the self-gravity given by the second term in the right-hand side of equation (3). The dust charge q d in equation (3) is not constant but is a function of n d [14,15]. It decreases with the increase in n d . It is important to take into account this effect as the dust charge reduction at high dust density encountered during the collapse may be substantial. The equilibrium dust charge q d is given by where I e (I i ) are the electron (ion) thermal flux impinging on the dust grain surface, and I PH is the flux of photoelectrons [11]. These charging currents are given by where a is the grain radius, T is the background plasma temperature, m e (m i ) are the electron (ion) mass, ψ is the dust surface potential relative to the local plasma potential ϕ, and PH is the flux of photoelectrons per unit area from the dust surface. If the ions and electrons are thermalized within the interstellar cloud, which is a reasonable assumption [15], then n i and n e are given by the Boltzmann relation: n i = n 0 exp(−qϕ/T ) and n e = n 0 exp(qϕ/T ). The local plasma potential ϕ is taken to be zero at infinity, where n d = 0 and n i = n e = n 0 . However, within the cloud, ϕ may not be zero and an electric field arises due to an inhomogeneous dust distribution. As shown by Havnes et al [15], equations (7)-(9) together with (1), (2) and (5) can be combined to obtain two equations which can be solved to relate ϕ and ψ as functions of n d (for given n 0 , a and T ) i.e. ϕ = ϕ(n d ) and ψ = ψ(n d ). Thus, the Coulomb force on the dust behaves like pressure [16,17]. The dust number density n d or the dust mass density ρ d is parametrized by a dimensionalless number p = aTn d /q 2 n 0 . Due to lighter mass of electrons, the dust charge q d , ψ and the plasma potential ϕ are negative in the presence of thermal fluxes. By solving the aforementioned equations, Havnes et al [21] have obtained simple polynomial approximations to ϕ and ψ as a function of p, given by where constants a j , b j , c j and d j depend on the species of the interstellar gas and I PH . These constants are tabulated in [21]. In the isolated dust grain limit p → 0, ϕ → 0 and ψ → 2.5 for H ions, and I PH = 0. As p increases, ψ and q d approach zero, while ϕ increases monotonically and saturates, i.e. ϕ → c 2 /d 2 = 1.87. The dust charge reduction at high dust density is due to the mutual screening of the grains [22], and has been verified in laboratory experiments and numerical simulations [23,24].

Equilibrium configurations
We begin our calculations by defining the pressure P E due to the shielded electric fields. Since q d n d is a function of ϕ, we may rewrite (3) as where P E is an effective pressure defined by P E = q d n d dϕ. The latter, which expels charged dust particles from regions of high density, may balance the self-gravity of the dust in the cloud 5 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT to provide equilibrium, i.e.
This is the equation of force balance for the dust. In this paper, we construct equilibria where dust is static i.e. ν d = 0. Eliminating q d n d through the quasi-neutrality condition (5) and imposing the condition P E = 0 at ϕ = 0 (the pressure is purely due to the electrostatic potential), we obtain Here, P E is a function of n d via ϕ in equation (10) (for given n 0 , a and T ). Thus, equation (13) together with (10) constitutes an equation of state for our problem. The other equation between P E and n d is obtained by taking the divergence of equation (12) (after dividing by ρ d ) and using equation (4) to eliminate . For a particular species of an interstellar gas, these two equations can be solved with appropriate boundary conditions (mentioned later) to obtain the spatial profiles of n d and P E in the equilibrium configuration.

The mass limit
In this section, we show that there is an upper limit on the total mass of the dust in the cloud for the equilibrium and point out that the physics of this limit is similar to Chandrasekhar's mass limit of compact objects [18]- [20]. In order to see the mass limit we note from equations (10) and (13) that P E increases with n d . Now, in the low dust density limit p 1, we have ϕ → c 1 p 1, P E ≈ n 0 T ϕ 2 , and thus P E ∝ p 2 , i.e. P E increases with the increase in n d , with an adiabatic index = 2. On the other hand, in the high-density limit p → ∞, ϕ → c 2 /d 2 , P E saturates and the adiabatic index goes to zero. This reduction in is responsible for the limit on the total dust mass M D in the cloud for the equilibrium. Basically, if M D is large then the gravity compresses grains to high densities and de-charges (substantial charge reduction) them before the pressure P E has had a chance to stop the gravity. Subsequent collapse then takes place as if the dust is uncharged. On the other hand, if M D is small, the charge reduction is not substantial and P E may balance the gravity. Because of the reduction of the adiabatic index for the pressure from two to zero due to the charge reduction, the pressure is not able to resist the self-gravity as the density increases, resulting in the mass limit. This scenario is very similar to the physics of Chandrasekhar's mass limit [18]- [20]. In the latter, the adiabatic index for the hydrostatic pressure reduces from 5/3 to 4/3 due to relativistic effects, resulting in the mass limit for white dwarfs. In the next section, we construct equilibrium solutions and calculate the mass limit.

Solutions
The problem of the equilibrium of dust clouds may be formulated in terms of a fluid with the force balance ∇P E = −ρ d ∇ , together with an equation of state. We look for spherically symmetric and static (ν d = 0) solutions. Taking the divergence of equation (12) (after dividing by ρ d ) and using Poisson's equation (4), the force balance equation in spherical coordinates can be written as where we have normalized pressure P E by n 0 T , the dust density by p, and the distance r by the scale length L, given by This parameter gives a typical dimension of the dust cloud in terms of other parameters. As stated earlier, equation (14) is supplemented with the equation of state given by Equations (14) and (16) may be solved to obtain P(r) and p(r) for a given species of interstellar gas and a value of I PH . The boundary conditions at r = 0 are p = p c , dp dr = 0, at r = 0, where p c is the normalized central dust density in the cloud. Using these boundary conditions, we construct one-parameter family of spherically symmetric and static solutions characterized by p c . The normalized radius R of the configuration is given by the condition p(R) = 0 and the total mass M D of the dust cloud is given by and λ d = (T/4πn 0 q 2 ) 1/2 is the plasma Debye radius. The integral I, which is a dimensionless number of order unity, should be evaluated by solving (14) and (16). Thus, as in the theory of stellar configurations, there is a definite mass-radius relationship for these configurations [18]. We next discuss the low-density limit.

Low dust density limit
In this limit p 1, ϕ < 1 and P = (c 1 p) 2 with = 2. Thus, equation (14) is linear in p and is given by The solution of (19), with the boundary conditions given in equation (17), is p = p c sin r/r, I = p c π and R = π.
These tenuous dust equilibria are characterized by a radius, which is independent of p c while M D αρc (ρc is the central dust mass density). For arbitrary p c , equations (14) and (16) may be solved numerically to obtain the value of the integral I and the equilibrium solution.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

The mass limit
Next, we discuss the mass limit M AS . To evaluate this limit, we consider the case I PH = 0, where P E is strongest. In this case, all charges are of the same sign (which is negative) and maximum for given values of n 0 , a and T . From the tables given in [21],  (>0.01), we find that I saturates with a broad maximum I max = 0.85 at p c = 2. The value of R is constant and is equal to π for p c < 0.01. In the range 0.01 < p c < 2 it decreases from π to 2.27. After this it decreases very slowly for equilibria on the other side of the maximum. In dust clouds with p c 2, dust is compactly packed at high density, the dust charge is small and electrons are depleted; they are absorbed by the dust.
From the values of I and p c obtained above, we may plot a universal equilibrium curve in terms of the normalized variables as well as a specific curve in physical units of M D and ρ c . Substituting the maximum value of I, and expressing the constants L and λ d in terms G, n H , T and a, we obtain the mass limit M AS and L as which are the main results of our paper. The important scalings of M AS and L against different parameters are evident in equation (21). Specifically, we note that the strong dependence on a is important, as any error in a will severely affect the values of M AS and L.
To plot the equilibrium curve in terms of M D and ρ c , we choose the parameters which characterize a standard interstellar cloud in the HII region [4]: namely n 0 = n H = 10 7 m −3 and T = 5 × 10 3 K. For the dust size we take a = 3 × 10 −7 m. The equilibrium curve is plotted in figure 1. For these parameters, L = 5 × 10 9 m and in figure 1 there is broad maximum at M D ≈ 8 × 10 18 kg. Thus, the value of the mass limit for the parameters of HII cloud is M AS ≈ 8 × 10 18 kg. In the dust equilibria near the mass limit, which are roughly of the mass M D M AS ≈ 8 × 10 18 kg and sizes L = 5 × 10 9 m, the dust is compactly packed with a central density n dc ∼ 2 × 10 5 m −3 . This is about eleven orders of magnitude higher than the average interstellar grain density ∼10 −6 m −3 . At the centre of these clouds, p ≈ 1 and ϕ ≈ 0.7. Thus, in the vicinity of the centre of such clouds the ion density is roughly two times and the electron density is half of its value outside the cloud which is n 0 = 10 7 m −3 . These compact dust objects are roughly of the size of the Sun and contain dust mass equal to that of a small satellite. However, as it is seen from equation (21), these parameters are sensitive functions of the dust size a and the plasma temperature T. Any small error in these parameters will severely affect the values of M D , M AS and L. For instance, if we increase the dust size three times, i.e a = 10 −6 m, then the mass of the equilibrium increases by four orders of magnitude and M AS ≈ 10 22 kg. This is roughly equal to the mass of a small planet like Pluto.
The effect of Coulomb correlations defined by the parameter c = q 2 d /dT d (d is the mean inter-particle distance and T d is the kinetic temperature associated with the random motion of the dust grain) may become significant in the compact dust configurations. As a function of p, c has a maximum at p = 0.2. Depending on whether we take T d to be equal to the gas temperature or lower, max c ∼ 5 or higher. Thus, a significant portion of compact dust equilibrium (with p c in the range 0.1-2) may be in a strongly correlated Coulomb liquid or even a crystalline state.
The overall dimensions of the dust cloud ranges from 0.01 to 50 pc [4]. The dust in these clouds is typically silicate or polycyclic aromatic hydrocarbon (PAH) distributed inhomogenously. Now, if as a conservative estimate, we take the average interstellar dust mass density ∼3 × 10 −28 kg m −3 [4] then in a small-sized (∼0.1 pc) dust cloud the total dust mass is M D ∼ 3 × 10 19 kg, which is roughly equal to the mass limit M AS ≈ 8 × 10 18 kg. Hence, in small interstellar clouds with dimensions <0.1 pc the electric fields are sufficiently strong to hold the dust against gravity in a stable equilibrium, while in bigger clouds with dimensions >0.1 pc the dust will collapse. Of course these numbers and conclusions depend sensitively on the values of the dust radius and T , and the fact that the whole of the dust cloud is characterized by the HII region parameters given above. These numbers may also be affected by a finite I PH , which is a function of the interstellar UV radiation field and the work function of the grain material.
What are the astrophysical implications of these considerations? This requires further investigation. However, one possibility is that since the compact dust equilibria are fluffy aggregates of a 'Coulombic matter' that is held together by Coulomb correlations, these could be a precursor to a proto-planetary or proto-stellar core formation. For example, if the electric fields become weak over a period of time then these aggregates will slowly contract and become denser to give rise to van der Waals correlations and the formation of a more solid body.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT It should be noted that the steady-state dust cloud with a constant dust charge requires constant recycling of the background plasma at the grain surface. This in turn requires a balance between the absorption of plasma and re-ionization of subsequently produced neutral H by the UV radiation in the cloud. In the equilibria near the mass limit, the dense core is usually much smaller than the tenuous envelope. Hence, the optical depth in the core τ = nσz (where σ is the cross-section and z is roughly the radius of the core) is expected to be small so that UV radiation intensity is enough to ionize the neutral hydrogen. However, this point requires further careful consideration which will be taken in a subsequent publication.

Conclusions
We have obtained the equilibrium configurations of dust clouds imbedded within plasma background by balancing the pressure P E due to shielded electric fields with the self-gravity of the dust. The pressure P E arises due to an inhomogeneous distribution of charged dust grains within the cloud, and tries to expel dust from the regions of high dust density. Furthermore, we have derived an equation of state which relates P E and the dust number density n d . The problem of equilibrium can be posed in terms of an equation for the force balance together with an equation of state. Due to the charge reduction at high dust number density, the adiabatic index is shown to reduce from two to zero. This results in an upper limit for the total mass of the dust in the dust cloud equilibrium. The physics of this mass limit is similar to Chandrasekhar's mass limit. This mass limit is numerically evaluated for the HII region parameters. It is found to depend critically on the dust radius and the gas temperature. Based on this limit it is argued that in clouds smaller than 0.1 pc, dust may be held in equilibrium by shielded electric fields, while in bigger clouds it will collapse. An implication of this scenario to planet formation is briefly discussed. In our model, we have ignored a number of effects, e.g. dust rotations, ambient magnetic fields, dust size distribution, etc. These will be covered in a subsequent paper.