Response of a Multicomponent Interstellar Medium to External Gravitational Influences V
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Astronomy Reports, Vol. 46, No. 5, 2002, pp. 347–351. Translated from Astronomicheski˘ı Zhurnal, Vol. 79, No. 5, 2002, pp. 387–391. Original Russian Text Copyright c 2002 by Antonov, Baranov. Response of a Multicomponent Interstellar Medium to External Gravitational Influences V. A. Antonov and A. S. Baranov Pulkovo Astronomical Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140 Russia Received April 30, 2001; in final form October 19, 2001 Abstract—The contribution of cosmic rays as a collection of charged particles to the generation of currents in the interstellar medium and to the total viscosity of this medium is analyzed. Our estimates show that this contribution is negligible under the conditions commonly found in our Galaxy but can become important when an excess dust concentration above the mean level is combined with a local increase in the cosmic-ray density. c 2002 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION more detailed analysis of the joint response of various The response of the interstellar medium (ISM) to components to external influences brings to light cer- external gravitational influences, or, more precisely, tain features that are not evident in previous studies, the elasticity and rigidity of the medium, is not as which, as a rule, were restricted to the formal intro- straightforward a problem as is often believed [1, 2]. duction of a single sound speed [3]. Studies usually fail to adequately allow for the multi- component nature of the ISM. To perform a more de- tailed analysis, we consider here a model with an in- 2. PRINCIPAL EQUATIONS finite uniform medium of fairly complex composition, neglecting self-gravity or including it in the external Our adopted assumptions imply that electrons, gravitational field. We use a four-component model ions, and cosmic dust should individually satisfy the for the interstellar medium. Appropriate subscripts— one-dimensional hydrodynamical equation for a cold e, i, d,orc—are used to distinguish between param- medium and the continuity equation eters referring to these different components. ∂w ∂w e A We restrict our analysis to the case of a constant + w = F + E + , (2) gravitational perturbation in the form of a one- ∂t ∂z m mn ∂n ∂ dimensional running wave with the field strength + (nw)=0. given in the (complex) form ∂t ∂z − F = εe iωt+iκz, (1) In (2), w is the z velocity of the component consid- ered, e and m the charge and mass of the particle where ε is a small parameter used for subsequent lin- type considered, n the particle volume density, E the earization, t is time, and ω and κ are the frequency and intensity of the electric field along the z axis, and A wavenumber of the perturbation. The wave propa- the specific force per unit volume due to friction. To gates along the z axis; the two remaining coordinates simplify our analysis, we take e and m to represent do not enter into our computations. mean parameters, although their values can differ for We take the phase velocity v0 = ω/κ to be in the different ions or dust particles. Linearization of (2) interval V v0 c,whereV is the characteristic yields thermal velocity of particles of ordinary matter (not cosmic rays) and c is the speed of light. Accord- ∂w e A = F + E + , (3) ingly, the hydrodynamical equations below neglect ∂t m mn the pressure, whereas the cosmic-ray component, on ∂δn ∂w the contrary, is relativistic and therefore essentially + n =0, ∂t ∂z unresponsive to the gravitational field. The only role played by cosmic rays is as a static source of elec- where we have introduced the perturbation δn of the tromagnetic, or, more specifically, positive charge, particle-number density to denote its departure from due to the predominance of nuclei over electrons. A the initial density n.Wemustalsotakeintoaccount 1063-7729/02/4605-0347$22.00 c 2002 MAIK “Nauka/Interperiodica” 348 ANTONOV, BARANOV the Poisson equation for the electric field, which in the 3. COSMIC PLASMA WITHOUT DUST one-dimensional case reduces to In the absence of dust, the system of equations (8) ∂E =4πρ, (4) reduces to two equations: ∂z 4πee ωwe − (eenewe + einiwi) (9) where ρ is the total space density of emerging ωme charges; i.e., i − q (w − w )=iε, m n ei i e A = eα δnα. (5) e e α 4πei ωwi − (eenewe + einiwi) ωmi In (5) and the following equations, α can represent i any of the subscripts e, i, d,orc mentioned above. We − qei(we − wi)=iε. m n then isolate the arguments t and z in the usual way i i and go over, as in (1), to the complex form ∂/∂t = It is convenient to introduce the macroscopic −iω, ∂/∂z = iκ. In view of (4), Eqs. (3) acquire the parameters X = menewe + miniwi,Z = eenewe + form einiwi, which are related to the mass-flow rate and electric field in obvious ways. The velocities of the 4πieα Aα −iωwα = ε − ρ + , (6) components can be expressed in terms of these mακ mαnα quantities as follows: κnα δnα = wα. eiX − miZ ω we = , (10) ne(meei − miee) We now specify the frictional force, which is also meZ − eeX electromagnetic but is manifest irregularly via col- wi = . n (m e − m e ) lisions. Friction in plasma has been the subject of i e i i e numerous studies [4, 5]. Its analog in stellar dynamics We now multiply the first and second equations of is so-called dynamical friction [6–8]. (9) by mene and mini or by eene and eini, respec- Friction is the result of the relative motions of tively. After summation, we obtain in either case particles, and must therefore be included at this stage. 4π ωX + n e Z = iε(m n + m n ), (11) We assume that the systematic velocities wα (or ω c c e e i i rather, their differences) are small even compared to 4π e2n e2n the thermal velocities, so the frictional forces can also ωZ − e e + i i Z be linearized, yielding formulas of the form ω me mi iqei − Aα = qαβ(wβ − wα),qαβ ≥ 0, (7) + [(mene + mini)Z + ecncX]= iεecnc. memenine β When deriving (11), we took into account the initial where, by virtue of the equality of action and reaction, electrical-neutrality condition the coefficients qαβ should obey the symmetry rule eene + eini = −ecnc. (12) qαβ = qβα. The parameter β = α can be excluded from the summation, since a given component does This transformation gives rise to the small param- not act on itself. eter n /n ≈ n /n . Solution of system (11) yields c e c i Simple manipulations with (5)–(7)enableusto qei(mene + mini) derive the following algebraic relations for the induced Z = εecnc − i (13) ωm m n n velocities: e i e i 4π e2n e2n 4πeα × − e e i i ωw − e n w (8) ω + α ωw α α α ω me mi α α 2 −1 iqei 4π(ecnc) − i − + m n + m n − . qαβ(wβ wα)=iε. m m n n e e i i ω2 mαnα e i e i β The parameter X can then be derived from the first Recall that we assumed that wα =0for cosmic equation of (11). rays. We neglect the scattering of cosmic rays on ions Amorespecific analysis requires some numerical and electrons, and vice versa, due to the smallness of estimates. As noted above, we neglect the presence of these effects. heavy elements in cosmic rays and among ions, and ASTRONOMY REPORTS Vol. 46 No. 5 2002 RESPONSE OF A MULTICOMPONENT INTERSTELLAR MEDIUM 349 −7 can therefore use the parameters for protons: mi = 10 . It is clear that we have with a large margin −24 −10 2 1.67 × 10 , ei = ec = −ee =4.8 × 10 (here and Ω ω ω, ω˜ ω,andΩ ω ω˜ /ω, so both the below, we use CGS units). We adopt the volume numerator and denominator in (16) reduce to single −10 densities ne ≈ ni ≈ 0.08 and nc ≈ 0.8 × 10 [9]. terms: When finding q , we take into account the fact εe n ω ei Z = − c c . (17) that, after each free flight, an individual electron im- Ω2 parts an average momentum of me(we − wi) to the Here, an important point is the computation of the ion component, and the additional coefficient ne must be introduced to allow for the total number of elec- work done by the gravitational field on the moving trons per unit volume. This transfer of momentum plasma. The work per period per unit volume will be occurs during the free-flight time [4] 2π W = Re εX 2 3 ω ≈ meV τ 4 , (14) 16πeeneL or,inviewofthefirst equation in (11) and for- mula (17), where L ≈ 15 is the well-known Coulomb logarithm 2 2 and the velocity V refers to the electrons. Further- 8π (ecnc) ω 2 −28 W = 3 2 ε . more, me =9.1 × 10 . ω Ω It is known [9, 10] that the electron temperature We are interested in the ratio of this work per in the interstellar gas can vary substantially, and we ≈ × 6 period to the density of gravitational energy: adopt an average value of V 5.5 10 .Comparing − (14) and the accompanying arguments to the defini- ε2 1 64π3G(e n )2ω µ = W = c c ∼ 2 × 10−9 tion of qei in (7), we now find that 8πG ω3Ω2 4 2 16πeeneL (G is the gravitational constant). Therefore, the elec- qei = 3 . (15) meV trostatic effect of cosmic rays in the Galaxy is negli- −30 gible.