We apply the Hermite-Bieler theorem in the analysis of the effect of cosmic rays on the thermal stability of an initially uniform, static background. The cosmic rays were treated in a fluid approximation and the diffusion coefficient was assumed to be constant in time and space. The inclusion of cosmic rays does not alter the criterion for the thermal stability of a medium subjected to isobaric perturbations. It does alter the criteria for the stability of a medium perturbed by small amplitude sound waves. In the limit of a high background cosmic ray pressure to thermal pressure ratio, the instability in response to high frequency sound waves is suppressed

A. Y. Wagner

Boulares

Falle

Ferrière

Field

J. M. Pittard

McKenzie

S. A. E. G. Falle

T. W. Hartquist

Tytarenko

Webber

English

Crossref

Cosmic ray moderation of the thermal instability

CiteSeerX

We apply the Hermite-Bieler theorem in the analysis of the effect of cosmic rays on the thermal stability of an initially uniform, static background. The cosmic rays were treated in a fluid approximation and the diffusion coefficient was assumed to be constant in time and space. The inclusion of cosmic rays does not alter the criterion for the thermal stability of a medium subjected to isobaric perturbations. It does alter the criteria for the stability of a medium perturbed by small amplitude sound waves. In the limit of a high background cosmic ray pressure to thermal pressure ratio, the instability in response to high frequency sound waves is suppressed.

EDP Sciences OAI-PMH repository (1.2.0)

Wagner, A.Y.

Falle, S.A.E.G.

Hartquist, T.W.

Pittard, J.M.

White Rose Research Online

This is a repository copy of Cosmic ray moderation of the thermal instability.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/1250/Article:Wagner, A.Y., Falle, S.A.E.G., Hartquist, T.W. et al. (1 more author) (2005) Cosmic ray moderation of the thermal instability. Astronomy and Astrophysics, 430 (2). pp. 567-569. ISSN 0004-6361 https://doi.org/10.1051/0004-6361:20042138eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. A&A 430, 567–569 (2005)DOI: 10.1051/0004-6361:20042138c© ESO 2005Astronomy&AstrophysicsResearch noteCosmic ray moderation of the thermal instabilityA. Y. Wagner1, S. A. E. G. Falle2, T. W. Hartquist1, and J. M. Pittard11 School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UKe-mail: ayw@ast.leeds.ac.uk2 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UKReceived 7 October 2004 / Accepted 6 November 2004Abstract. We apply the Hermite-Bieler theorem in the analysis of the effect of cosmic rays on the thermal stability of aninitially uniform, static background. The cosmic rays were treated in a fluid approximation and the diffusion coefficient wasassumed to be constant in time and space. The inclusion of cosmic rays does not alter the criterion for the thermal stability of amedium subjected to isobaric perturbations. It does alter the criteria for the stability of a medium perturbed by small amplitudesound waves. In the limit of a high background cosmic ray pressure to thermal pressure ratio, the instability in response to highfrequency sound waves is suppressed.Key words. ISM: kinematics and dynamics – ISM: cosmic rays – hydrodynamics – instabilities1. IntroductionField (1965) deduced the criteria for the thermal stability ofa uniform, non-magnetic, single-fluid medium. The interstel-lar medium’s pressure has nonthermal contributions, includingthat due to cosmic rays, which modify the criteria for stability.Here, we investigate the effects of cosmic rays on the thermalstability of a uniform medium.In Sect. 2, we give the basic one-dimensional equationsgoverning the dynamics of a medium consisting of a thermalfluid and a cosmic ray fluid. Section 3 contains a descriptionof the method based on the Hermite-Bieler theorem and thestability criteria derived for general conditions. Section 4 pro-vides results for special limiting cases and Sect. 5 concludesthe paper.2. A one dimensional two-fluid descriptionTo simplify the treatment of the cosmic rays, they are some-times treated as a fluid (e.g. McKenzie & Völk 1982). The sec-ond velocity moment of the diffusion convection equation forcosmic rays gives∂Pc∂t+ u∂Pc∂x+ γcPc∂u∂x− χ∂2Pc∂x2= 0 (1)(e.g. Drury & Falle 1986). x is the spatial coordinate, u is themean speed of thermal particles, γc is an adiabatic constantequal to 4/3 for ultrarelativistic cosmic rays, χ is a spatial diffu-sion coefficient, and Pc is the cosmic ray pressure. The thermalfluid obeys∂ρ∂t+∂(ρu)∂x= 0 (2)ρ∂u∂t+ ρu∂u∂x+∂Pg∂x+∂Pc∂x= 0 (3)∂∂t(ρu22+Pgγg − 1)+∂∂x[(ρu22+Pgγg − 1)u]+∂(uPg)∂x+ u∂Pc∂x+ρL(ρ, T ) = 0 (4)Pg −ρRTµ= 0. (5)ρ, Pg, T, γg, µ, R and ρL are the mass density, pressure, temper-ature, ratio of specific heats, mean mass per particle, gas con-stant and net energy loss per unit volume per unit time, respec-tively of the thermal fluid. Equations (1) to (5) and an equationgiving L as a function of ρ and T constitute a closed set.3. The stability analysisWe assume that γc, χ, γg and µ are constant and that the back-ground values of Pc, ρ, u, Pg and T are constant and are denotedby the subscript 0. u0 = 0 and L(ρ0, T0) = 0. Pc1, ρ1, u1, Pg1and T1 are small perturbations of the state variables and areassumed to vary as exp(i(ωt + kx)).568 A. Y. Wagner et al.: Cosmic ray moderation of the thermal instabilitySubstitution of the expressions for the state variables, ap-propriate for the assumptions stated in the preceding paragraph,into Eqs. (1) to (5) yields the dispersion relationG(z) ≡ z4 − iz3(kkc+kTk)− z2(kTkc+ φ + 1)+iz(kTkφ +kkc+kT − kργgk)+kT − kργgkc= 0. (6)Herez ≡ ωak(7)a ≡(γgPg0ρ0)12(8)kT ≡µ(γg − 1)Ra∂L∂T∣∣∣∣∣T=T0(9)kρ ≡µ(γg − 1)ρ0RaT0∂L∂ρ∣∣∣∣∣ρ=ρ0(10)kc ≡aχ(11)φ ≡ γcPc0γgPg0· (12)Like Tytarenko et al. (2002) we make use of the Hermite-Bielertheorem (e.g. Levin 1964, Chapter VII) in the analysis of therelevant dispersion relation. We write G(z) = Gr(z) + iGi(z)where Gr and Gi are both real functions. The theorem impliesthat the physical system is stable if Gr and Gi have only simplereal roots, between two successive roots of one of these twopolynomials there lies exactly one root of the other, and at somepoint z = z′ on the real axisdGidz∣∣∣∣∣z=z′Gr(z′) −Gi(z′)dGrdz∣∣∣∣∣z=z′< 0. (13)The Hermite-Bieler theorem applied to the polynomialE(z) = z4 − iaz3 − bz2 + icz + d = 0 (14)gives the following stability conditionsd > 0 (15)ca>b2−√(b2)2− d (16)ca<b2+√(b2)2− d. (17)Direct comparison of Eq. (14) and the dispersion relationEq. (6) leads us to conclude that three conditions must be sat-isfied for the system to be stable with respect to the types ofperturbations we have specified. Those conditions arekT − kργgkc> 0 (18)kTφ +kT−kργg+ k2kckT +k2kc>12(φ +kTkc+ 1)−√14(φ +kTkc+ 1)2−kT − kργgkc(19)kTφ +kT−kργg+ k2kckT +k2kc<12(φ +kTkc+ 1)+√14(φ +kTkc+ 1)2−kT − kργgkc· (20)Condition (18) is closely related to the condition for the sta-bility of a medium containing no cosmic rays to be thermallystable with respect to isobaric perturbations; that condition forthe one fluid medium iskT − kρ > 0 (21)(Field 1965).4. Limiting casesEquations (19) and (20) are complicated for the general casebut simplify in various limits. We first define∆ ≡kT − kργg· (22)For the case in which φ ≫ 1,∣∣∣kTkc∣∣∣,∣∣∣kρkc∣∣∣,∣∣∣kkc∣∣∣ conditions (19)and (20) give0 <∼kTkT +k2kc<∼ 1 (23)for stability. As kc is always positive, this implies that whenthe cosmic ray pressure is very high and the perturbation is ona small scale relative to√|kckT|, sound waves will not growirrespective of the nature of L.In the case that 1 ≫ φ,∣∣∣kTkc∣∣∣,∣∣∣kρkc∣∣∣,∣∣∣kkc∣∣∣ conditions (19) and (20)imply that sound waves are stable if− ∆kTkckT − ∆<∼ k2 <∼ kTkc when kT − ∆ > 0 (24)kTkc <∼ k2 <∼ −∆kTkckT − ∆when kT − ∆ < 0. (25)We note that kT − ∆ > 0 is the condition for the stability ofsound waves and ∆ > 0 is the condition for the stability ofisobaric perturbations in a non-magnetic thermal fluid with nocosmic rays (Field 1965). When kT > 0 and ∆ > 0 condi-tion (24) is always satisfied and condition (25) is never satis-fied. Hence, kT − ∆ < 0 is a sufficient condition for instabilitywhen no cosmic rays are present and when they are but havea low pressure relative to the thermal pressure and the length-scale associated with cosmic ray diffusion is small compared tothe lengthscale associated with cooling and the wavelength ofthe sound wave.A. Y. Wagner et al.: Cosmic ray moderation of the thermal instability 569In the case that∣∣∣kTkc∣∣∣,∣∣∣kρkc∣∣∣,∣∣∣kkc∣∣∣ ≫ 1, φ condition (19) is sat-isfied ifk2 >∼ kTkckT(1 − φ) − ∆kT − ∆when kT − ∆ > 0 (26)k2 <∼ kTkckT(1 − φ) − ∆kT − ∆when kT − ∆ < 0 (27)and condition (20) is satisfied ifk2 >∼ k2c(φ − 1 + ∆kT)when kT > 0 (28)k2 <∼ k2c(φ − 1 + ∆kT)when kT < 0. (29)Thus, when the lengthscale associated with the cosmic ray dif-fusion is long compared to the lengthscale associated with thecooling and the wavelength of the perturbation, the criteria forthe stability of sound waves are somewhat more complicatedthan in the simplest limiting cases.5. ConclusionsThermal instability may play a role in the formation of con-densations in many diffuse astrophysical media, a fact that mo-tivated Field’s (1965) classic study. These media include theinterstellar medium, where the cosmic ray pressure in manyregions is comparable to the thermal pressure (e.g. Boulares &Cox 1990; Webber 1987; Ferrière 1998) and might be expectedto play a role in structure formation due to thermal instability.We have adopted a simplified treatment of the cosmicrays by using a fluid description and assuming a diffusioncoefficient that is constant in time and space. Given the uncer-tainties and difficulties in the calculation of the diffusion coef-ficient from first principles, a simple treatment of the problemof cosmic ray moderation of the thermal stability is justifiedas a first step. We have found that the criterion for the growthof isobaric perturbations is unaffected by the inclusion of cos-mic rays, irrespective of the diffusion coefficient and the ratioof cosmic ray to thermal pressure. The criteria for a medium tobe stable to perturbations by small ampitude sound waves arealtered by the inclusion of cosmic rays. In general these crite-ria are rather complicated. In the simple limit of a very highcosmic ray to thermal pressure, a medium is stable to all smallamplitude sound wave perturbations on scales small comparedto the geometric mean of the scale on which cosmic ray dif-fusion occurs and the scale associated with cooling. Thus, notsurprisingly, in cases in which the cosmic ray pressure is highcompared to the thermal pressure the instability of high fre-quency sound waves is suppressed.ReferencesBoulares, A., & Cox, D. P. 1990, ApJ, 365, 544Drury, L. O’C., & Falle, S. A. E. G. 1986, MNRAS, 223, 353Ferrière, K. 1998, ApJ, 497, 759Field, G. B. 1965, ApJ, 142, 531Levin, B. Ja. 1964, Trans. Math. Mono. 5. Am. Math. Soc.,Providence RIMcKenzie, J. F., & Völk, H. J. 1982, A&A, 116, 191Tytarenko, P. V., Williams, R. J. R., & Falle, S. A. E. G. 2002,MNRAS, 337, 117Webber, W. R. 1987, A&A, 179, 277

http://www.aanda.org/articles/aa/pdf/2005/05/aa2138.pdf

Cosmic ray moderation of the thermal instability

Abstract

Similar works

Full text

Available Versions

Crossref

CiteSeerX

EDP Sciences OAI-PMH repository (1.2.0)

White Rose Research Online