Damping of sound waves by bulk viscosity in reacting gases
DDamping of sound waves by bulk viscosity in reacting gases
Miguel H. Ib´a˜nez S. Pedro L. Contreras E.Centro de F´ısica Fundamental, Universidad de los Andes, M´erida 5101, Venezuela. [email protected]
Received ; accepted Now at Valle de Villa de Leyva, Colombia. a r X i v : . [ phy s i c s . p l a s m - ph ] D ec ABSTRACT
The very long standing problem of sound waves propagation in fluids is reex-amined. In particular, from the analysis of the wave damping in reacting gasesfollowing the work of Einsten (20), it is found that the damping due to thechemical reactions occurs nonetheless the second (bulk) viscosity introduced byLandau & Lifshitz (1987) is zero. The simple but important case of a recombiningHydrogen plasma is examined.
Subject headings:
Viscosity, sound waves, damping, Hydrogen plasma, reacting gas.
1. Introduction
Propagation of disturbances, in particular sound waves in hypothetical equilibriumfluids has been researched since the pioneer works (25; 20; 17; 20; 19) and their maincharacteristics have been well established, i.e. waves propagate with certain velocity, andare damped by the irreversible processes say viscosities, thermal conduction and chemicalreactions. Landau & Lifshitz (1987) introduce a bulk (second) viscosity coefficient ζ inthe equation of motion for accounting the dissipation of energy due to compression orexpansion through transferring kinetic energy into internal degrees of freedom (such aschemical reactions, excitation of atomic/molecular levels, etc.). However, in the case ofchemical reactions such approximation only holds if one neglects any others effects exceptthe density change δρ due to the chemical reaction.Henceforth, as it will be shown at the present note, the Landau approximation israther restrictive. In fact, if ξ is a parameter characterizing the degree of advance ofchemical reaction in the fluid (say, the concentration of one chemical component) and ξ itsrespective value at chemical equilibrium, which generally is a function of the equilibriumdensity and temperature, say ξ ( ρ , T ) (35); henceforth, as it can be realized, in the Landauapproximation (1987; 14) the second viscosity coefficient is ∼ ( ∂ξ /∂ρ ) T , therefore, when( ∂ξ /∂ρ ) T = 0 the acoustic wave damping is also zero. However, when ( ∂ξ /∂T ) ρ (cid:54) = 0 thesound waves could be damped nonetheless the Landau bulk viscosity coefficient is zero, asit will be shown below.The present analysis on the bulk viscosity is made for any reacting gas where thechemical reactions can be reduced to a net reaction that can be described by one parametermeasuring the advance of the reaction (37; 12). However, for context, the results are appliedto a Hydrogen plasma where the simple reaction H + + e − (cid:29) H + ( χ ) proceeds ( χ beingthe ionization potential). The knowledge of the above plasma is of particular importance in 4 –Astrophysics, say, the solar atmosphere (33; 34; 30; 2; 22), the interstellar gas (30; 31; 32)and more recently in the Intracluster gas (6; 26; 7; 8), in particular due to the fact thatwave dissipation have been invoked as one of the mechanisms of heat input. However, adetailed study of the thermal behavior of the above plasmas is out the scope of the presentstudy, which is particulary restricted to find an expression of the bulk viscosity coefficientin chemically active plasmas.
2. Basic Equations
In general, for a 1-D plane wave the wave number k and the frequency ω are related by k = ωc . (1)The parameter c is defined by the relation c = ± (cid:115) ∂p∂ρ , (2)where ∂p∂ρ = 1 ρ (cid:20) p − (cid:18) ∂ ( pV ) ∂V (cid:19)(cid:21) , (3)with V = 1 /ρ and the equilibrium values denoted with the subindex . The relation (1)formally obtained by non-dispersive media also holds for dispersive media for which c is a complex quantity (as well as k ) (1987) and only for disturbances propagating in anon-reacting ideal fluid becomes the adiabatic sound speed c = c s = (cid:115)(cid:18) ∂p∂ρ (cid:19) s . (4)Strictly speaking, the basic gas dynamic equations admit solutions in the form ∼ exp( βt + i k . r ), where β = σ − iω and k = k r + i k i , where σ and ω are real quantitiesand k r and k i are real vectors. Therefore, one may write the sound disturbance as 5 – ∼ exp( σt − k i x ) exp [ i ( k r x − ωt )] for the one-dimensional problem. The above can beinterpreted as a wave of frequency ω , wavelength λ = 2 π/k r , traveling along the x-axis with a phase velocity v = ω/k r and the amplitude ∼ exp( σt ) exp( − k i x ), the first factormeasures the attenuation (or growth if σ >
0) in time, and the second factor measures thespatial absorption (or amplification if k i <
0) in ordinary progressive wave propagationstudies (21; 18; 13). The present analysis is restricted to the spatial absorption of linearwave propagation in a chemically active fluids from where the bulk viscosity coefficient iscalculated.For reacting gases if the set of ”chemical reactions” which are in progress can bereduced to a single reaction (cid:80) j ν j A j = 0 where A j are the chemical symbols of the reagentsand the coefficients ν j are positive or negative integers, there is at least on component j for which the concentration ξ j = n j /n goes to zero when the reaction proceeds to a senseindefinitely, here n denotes the total number density of atoms and n j is the number densityfor gas particles of the j − th component. So, one may introduce the parameters ξ , and a ,such that ξ j = n j n = a ξ, ≤ ξ ≤
1; (5)where ξ denotes the degree of advance of the reaction and a the maximum number ofabundance ratio of the j − th component to the total number of nuclei.From the equation of continuity for the different components and the definition (5) onemay obtain the rate equation (35; 37; 12) dξdt + X ( ρ, T, ξ ( ρ, T )) = 0 , (6)where X ( ρ, T, ξ ( ρ, T )) is the net rate which at equilibrium X ( ρ , T , ξ ( ρ , T )) = 0. 6 –Additionally, an ideal-like state equation will be assumed, i.e. p = R ρ Tµ ( ξ ) , (7)where R is the gas the gas constant constant and µ ( ξ ) is the mean molecular weight, µ − = (cid:80) j ξ j .On the other hand the internal energy per unit mass becomes u = A ( ξ ) RT + χN aξ, (8)where χ and N denote the dissociation energy and the Avogadro’s number and A ( ξ ) = (cid:88) j ξ j γ j − , (9) γ j being the specific heat-ratio for the j − th component.For an adiabatic change, the energy equation can be written as RA ( ξ ) δTδt − pρ δρδt + RT B ( ξ, T ) δξδt = 0 , (10)where B ( ξ, T ) = dAdξ + aχk B T , (11)being k B the Boltzmann constant. For linear disturbances close to the equilibrium RA δT − p ρ δρ + RB T δξ = 0 , (12)where A = A ( ξ ) and B = B ( ξ , T ) are the equilibrium values of the functions A ( ξ ) and B ( ξ, T ).For fluctuations ∼ exp( − iωt ) from Eq.(6) follows that the disturbances δξ , δρ , and δT are related by the equation δξ = ξ ∗ ρ − iωτ δρ + ξ ∗ T − iωτ δT, (13) 7 –where τ = ( ∂X/∂ξ ) − is the relaxation time which is a positive quantity for chemicallystable gases; and where ξ ∗ ρ = ( ∂ξ /∂ρ ) T , and ξ ∗ T = ( ∂ξ /∂T ) ρ are the derivatives atequilibrium (37; 15; 12).Additionally, from Eqs.(7), (6),and (12) the Eq.(3) becomes ∂p∂ρ = p ρ [1 + Q ] , (14)the Q factor is given by Q = (1 − iωτ ) − ( µ B + µ ξ A ) ρξ ∗ ρ − T ξ ∗ T µ ξ /µµ [ A (1 − iωτ ) + T ξ ∗ T B ] , (15) µ ξ being the derivative of the molecular weight with respect to the chemical parameter.It is important to mention that the above relation (14) for a particular simply chemicalreaction was obtained in an early paper by (20).In the limiting when ωτ → ∞ (frozen chemistry), Q → /Aµ and in the oppositelimiting ω τ → Q = 1 − ( µ B + µ ξ A ) ρξ ∗ ρ − T ξ ∗ T µ ξ /µµ [ A + T ξ ∗ T B ] . (16)In the limiting case when the fluctuation δξ is only due to the change of density ξ ∗ T = 0,the Eq.(15) reduces to Q = (1 − iωτ ) − ( µ B + µ ξ A ) ρξ ∗ ρ µ A (1 − iωτ ) . (17)On the opposite limit when ξ ∗ ρ = 0, Q = (1 − iωτ ) − T ξ ∗ T µ ξ /µµ [ A (1 − iωτ ) + T ξ ∗ T B ] . (18)If in the limiting ξ ∗ T = 0 additionally ξ ∗ ρ = 0, henceforth Q = 1 /µ A = γ − (cid:112) ∂p/∂ρ = (cid:112) γp /ρ (being γ the specific heat ratio) becomes theisentropic sound speed c s in a non-reacting ideal gas, as it should be. 8 –It is interesting to point out that in the Landau approximation Landau & Lifshitz(1987) ( pp. − δξ is assumed to occur at constant entropy S , i.e. the change of pressure p is due only to the change of density δρ produced by thefluctuation in the chemical parameter δξ , ∂p∂ρ = 11 − iωτ (cid:2) c − iωτ c ∞ (cid:3) , (19)and c is given by c = (cid:18) ∂p∂ρ (cid:19) eq = (cid:18) ∂p∂ρ (cid:19) ξ + (cid:18) ∂p∂ξ (cid:19) ρ (cid:18) ∂ξ ∂ρ (cid:19) , c ∞ = (cid:18) ∂p∂ρ (cid:19) ξ . (20)From Eq. 15 one obtain the corresponding parameter Q L in the Landau approximation,i.e. Q L = − − iωτ µ ξ µ ρξ ∗ ρ . (21)Finally, in the limiting case when ξ ∗ ρ = 0, it follows that (cid:112) ∂p/∂ρ = (cid:112) p /ρ , i.e.the sound propagation would occur with the isothermal sound speed as it is expected.Additionally, at the Landau’s approximation the effects of the chemical reaction may beaccounted for introducing a second viscosity coefficient in the motion equation given by thefollowing expression ζ = ρ τ − iωτ (cid:2) c ∞ − c (cid:3) = ρ τ − iωτ p µ ξ µ ξ ∗ ρ , (22)i.e. the Landau bulk viscosity coefficient (in g cm − s − ), as it can be readily verified fromEq.(20).
3. Collisionally ionized Hydrogen plasma
For context, at the present section the above results will be applied to the simple butimportant examples of an ionized Hydrogen gas when it is collisionally ionized. As it will 9 –shown the damping of sound waves becoms zero at the Landau approximation, but differentfrom zero at Einstein approximation.A collisionally ionized Hydrogen plasma can be considered as a reacting plasma wherethe reaction H + + e − (cid:28) H + χ, (23)proceeds with the following expressions A = 1( γ − µ , B = 1 γ − χk B T , µ = 11 + ξ , (24) ξ being the degree of the ionization, χ the Hydrogen ionization potential and k B theBoltzmann constant, the sub-index 0 indicating equilibrium values has been omitted.Additionally, the generalized ionization recombination rate function (37; 12) becomes equalto X = N ρα ( T ) ξ − N ρq ( T ) ξ (1 − ξ ) = 0 , (25)therefore at equilibrium ξ ∗ ( T ) = q ( T ) α ( T ) + q ( T ) , (26)the total recombination coefficient α ( T ) is given by α ( T ) = 2 . × − T (cid:18) . . / − . (cid:19) cm s , (27)and the collisional ionization rate follows the expression q ( T ) = 5 . × − √ T exp( − Θ) , cm s (28)where Θ = 1 . × /T (27; 10; 11). The above approximation holds (24; 5) in the rangeof 3 . × (cid:46) T ( K ) (cid:46) . × .For a collisionally ionized Hydrogen plasma from Eq.(26) follows that ∂ξ ∗ /∂ρ = 0, andtherefore the second viscosity in the Landau approximation, Eq.(22), is also zero. However 10 –from Eq.(14) the speed of sound c becomes c = (cid:114) pρ (1 + Q ) , (29)with Q = ( γ −
1) [1 − µT ξ T / (1 − iωτ )][1 + ( γ − BµT ξ T / (1 − iωτ )] , (30)i.e. damping effect occurs due to the irreversible process inherent to the chemical reaction,as it follows from the fact than c becomes a complex quantity as well as the wave number k Eq.(1), and which can be written as k = k r + i k i , where k r , and k i are real quantities, k i being the damping coefficient which has to be a positive quantity (15).On the other hand, from Eqs. (25)-(28) the relaxation time becomes τ = 1 N ρ q ( T ) . (31)i.e. the relaxation time τ > πk i /k r and the phase velocity v ph /c T normalizedto the isothermal sound speed c T = (cid:112) p /ρ have been plotted in Figs. (1a) and (1b)respectively, as functions of temperature for three different values of ωτ (10 − dash line, 1thick line, and 10 point line). Regardless the value of ωτ the damping shows maxima, andthe phase speed shows minima at a temperature close to log T = 4 .
16, temperature at whichthe function ξ ∗ T becomes a maximum and the effect of the recombination-ionization processbecome important. At very low (neutral Hydrogen) as well as at very high temperatures(ionized Hydrogen), the damping tends to be zero Fig.(1a), and the sound velocity tends tobe the isentropic sound speed Fig.(1b), as it is expected from simple physical considerations.In Figs. (1c) and (1d) the damping per unit wave length (2 πk i /k r ) and the normalizedphase velocity ( v ph /c T ) are respectively plotted but as functions of ωτ for temperaturesslight lower (log T = 4 .
04, (dash line) and higher log T = 4 .
28, (point line) than log T = 4 .
16 11 –(thick line). The damping per unit wave length becomes a maximum very close to the ωτ value Fig. (1c) where the inflexion point of v ph /c T occurs as in Fig. (1d). Regardless ofthe temperature value, waves with ωτ (cid:28) ωτ (cid:29) In this section the results of Section 1 will be applied to a photo-ionized Hydrogenplasma model i.e. an optically thin Hydrogen plasma ionized by a background radiationfield of averaged photon energy E and photo-ionization rate ς . The net rate function X ( ρ, T, ξ ) is given by (5) as X ( ρ, T, ξ ) = N ρ [ ξ α − (1 − ξ ) ξq ] − (1 − ξ )(1 + φ ) ς, (32) α is the total recombination coefficient ( cm s − ) which is given be Eq.(27), q is thecollisional ionization rate ( cm s − ) according to (1), φ ( E, ξ ) is the number of secondaryelectrons which in general is a function of the energy mean photon energy E and theionization ξ , (28), and ς is the photo-ionization rate in s − (1). The last term of the righthand side of (32) just accoaunted for this effect.Therefore, the corresponding terms in theenergy equation Eq. (10), have to be be added for consistency. For accounted the heatinput and output of energy by radiation. So, instead of Equation (10) one obtains RA ( ξ ) δT − pρ δρ + RB ( ξ, T ) T δξ + δL ( ρ, T, ξ ) = 0 , (33)Where the net heat/cooling function becomes L ( ρ, T, ξ ) = N { ρ [(1 − ξ ) ξ Λ eH ] + ξ Λ eH + } − N { (1 − ξ )[ E h + (1 + φ ) χ ] }
12 –where Λ eH and Λ eH + are the cooling losses by e − H and e − H + collisions (15) neglectingsecondary electrons φ = 0According to (28), 0 . (cid:46) φ (cid:46) . . (cid:38) ξ (cid:38) − the exact value dependingon E (which depends on the particular optical depth in the gas) strictly speaking aself-consistent radiative transfer problem should be worked out, and which is out the scopeof the present paper, whose aim is restricted to obtain an indicative value of the bulkviscosity for a photo-ionized Hydrogen plasma. Therefore, if in a first approximation theproduction of secondary electrons is neglected ( φ = 0), from Eq. (32) an explicit form theionization ξ ∗ ( ρ, T ) at equilibrium can be obtained, i.e. ξ ∗ ( ρ, T ) = N ρq − ς + (cid:112) B p N ρ ( α + q ) , (34)with B p = ( N ρq + ς ) + 4 N ρας, (35)otherwise the solution for ξ at equilibrium becomes an implicit function of T and ρ , and forits calculation one must proceed numerically. The correction introduced by the secondaryelectrons is equivalent to an increase of the value of the photo-ionization rate, as it can beverified from Eq. (32)Therefore, from Eq.(34) one obtains ξ ∗ ρ = ρB pρ + 2 (cid:112) B p ς − B p (cid:112) B p N ρ ( α + q ) , (36)and ξ ∗ T = B pT ( α + q ) − B p ( α T + q T )4 (cid:112) B p N ρ ( α + q ) + 2 N ρ (cid:112) B p [ αq T − qα T + ¯ ς ( α T + q T )]4 (cid:112) B p N ρ ( α + q ) , (37)here B pT = ∂B p /∂T , B pρ = ∂B p /∂ρ , α T = ∂α/∂T and q T = ∂q/∂T . Similarly to theprevious sub-section, from Eqs. (1), (14), and (15) one may calculate both, the real andimaginary parts of c and k . However, for this particular plasma ξ ∗ is a function of both ρ ,and T instead of T only as given by Eq. (26). 13 –Fig.(2a) is a 3 D plot of the ionization rate ξ ∗ as function of T ( K ) and density n ( cm − )(in Fig. (2a) the red color refers to temperatures close to 5.000 K ; on the other hand, themagenta color gives the highest temperatures, which are of the order of 30.000 K , for a fixedvalue of the photo-ionization ς = 5 × − s − ). Fig.(2b) shows a 3 D plot of the ionization ξ ∗ as a function of the density n ( cm − ) and the photo-ionization ς ( s − ), for a fixed valueof temperature ( logT = 4 . K ), spanning in the range of values for the galactic interstellarmedium (16). In Fig. (2b) the color indicates the values of the density, red color refers todensities n near zero values, and magenta color indicates values of the density n close to100 ( cm − ). From both Figs. (2a), and (2b) follows that the effect of the ionizing radiationis to increase the ionization at any temperature, respect to that resulting by collisionsonly, however the strong ionization occurring at temperature ≈ × K is determined bycollisions, for galactic values of the photo-ionization rate ς ( s − ).As it can be verify the presence of the ionizing radiation field shifts the value of ξ ∗ T towards higher temperatures (log T = 4 .
21, for ς = 5 × − s − ) and smooth the change ofthe damping per unit wave length with the temperature for any wave frequency, as it canbe shown comparing Fig. (1a) with Fig. (3a) in which the damping 2 π k i /k r is plotted as afunction of T , for ς = 5 × − s − , and the same three values of ωτ shown in Fig. (1a) butfor a rate given by (32) instead of (25). The change of value of the maxima of the dampingper unit wave depends of the value of ωτ , in particular increases for ω τ = 1, additionallythey are shifted towards higher values of T following the shift of the maximum of ξ ∗ T asfollows from physical considerations.Accordingly, the change of the phase velocity produced (taking into account thephoto-ionization) can be seen comparing Fig. 3b with Fig. 1b. The minimum is shiftedtowards higher temperatures but they are smoothed at high frequencies ( ωτ ) as shown bycomparing the point lines ( ωτ = 10) in the the above two figures. 14 –At a particular temperature, the changes of the damping per unit wave length and thecorresponding to the phase velocity are small (for galactic vales of the photo-ionization ς )as it can be shown comparing Figs. (3c) with (1c) and Figs. (3d) with (1d), respectively.Generally, the qualitative and quantitative effects of the photo-ionization are small respectto those produced by collisions only in an atomic Hydrogen gas, as far as sound wavepropagation is concerned, and in the range of values of the parameters above considered. The aim of the present section is to compare the value of the three absorptioncoefficients corresponding to: (1) the bulk viscosity ˜ k b = c T k i /ω , (2) the dynamical viscosity˜ k ν , and (3) the thermal conduction ˜ k κ which are given by (18; 1987), i.e.˜ k ν = 2 ω ν c T γ / , (38)where ν is the kinematic viscosity, and˜ k κ = ω ( γ − χ c T γ / , (39)in which χ corresponds to the thermometric conductivity, (24; 29; 3; 18). The problemof sound wave propagation in a self-consistent model of the atomic gas in the galaxy andother plasmas of interest in Astrophysics, for which H e and ions of H e , and ions of heavyelements included will be published elsewhere.Incidentally, another irreversible process in plasmas, is due to the frictional forcebetween ions of mass m i (and velocity v i ), and neutral particles of mass m n (and velocity v n ) (3). The time scale for equalizing the velocities can be easily calculated from therespective Braginskii relations , from which one obtains the equation τ ni ≈ ( m i + m n ) (cid:104) σv (cid:105) ( ρ i + ρ n ) , (40) 15 –where (cid:104) σv (cid:105) is a mean value of the product of the cross-section and the relative velocityaveraged over all velocities. As it can be easily verified, generally τ ni << τ , additionally thefrictional damping becomes independent on the wave-length λ , and it is only important foroscillations with very high frequencies, and in plasmas with very low ionization (3; 23; 36).Such effect will not be considered at the present discussion.Figs. (4a), (4b), and (4c), are plots of the absorption coefficients k b (thin line), k ν (dashline), k κ (point line), and the total absorption k tot = k b + k ν + k κ (thick line) in unitsof cm − , as functions of the temperature T for n = 1 ( cm − ), a photo-ionization ratevalue of ς = 5 × − s − , and three different values of the frequency ω τ = 10 − , 1 and10, respectively. The relaxation time is plotted in Fig. (4d) for n = 1 and three valuesof the photo-ionization rate ς = 5 × − (dash line), 5 × − (thick line) and 10 − (point line), s − . Due to the fact that the effect of damping of sound waves is linear, it isworthy to calculate the total absorption coefficients due to the above three effects. Theabsorption by bulk viscosity becomes the dominant one in the range of temperatures whererecombination-ionization takes place 4 . × (cid:46) T (cid:46) T M ( ωτ ), where T M ( ωτ ) is a functionof the wave frequency, increasing when ωτ decreases as can be seen in the above Figs. (4a),(4b), and (4c). At high temperatures ( T > T M ) and high ionization, the thermal conduction(by electrons) dominates, instead, at low temperatures T (cid:46) . × K , the thermalconduction by neutral atoms becomes the dominating one. At frequencies ωτ (cid:38) . × (cid:46) T (cid:46) T M ( ωτ ). Fig.(4d) is a plot of therelaxation time τ (= | X ξ | − s ) as a function of temperature for three different values of thephoto-ionization rate ς = 5 × − (dash line), ς = 5 × − (thick line), ς = 10 − (point 16 –line). As it is expected the relaxation time τ sharply decreases at T ∼ ∼ × K andits value is close to τ ∼ y . Therefore, for a typical number density n ∼ ∼ . pc at high temperatures ( ∼ × K ),and ∼ pc at low temperatures ( ∼ K ), for frequencies ω ∼ − y − .In Summary, following the Einstein (1920) (20) work based on propagation of soundwaves in reacting gases, the bulk viscosity coefficient introduced by Landau & Lifshitz(1987) (Eq .22) has been generalized to chemically active gases. The bulk viscositycoefficient ∼ k i becomes the imaginary part of the wave vector k calculated from Eqs.(1,29 and 30). In particular, for a collisionally ionized Hydrogen gas, the bulk viscosityin the Landau approximation becomes zero, but it is different from zero at the presentapproximation, see results in section 3. For context, additionally the bulk viscosity is alsocalculated for a photo-ionized Hydrogen gas for values of parameters characteristic of thehigh latitude atomic gas in the Galaxy. 17 – REFERENCES
Black, J., H., 1981, MNRAS, 197, 555Bohm-Vitense, E., 1987, ApJ, 317, 750Braginskii, S., I., 1965, Rev. Plasma Phys., 1, 2015Bird, G. A., 1964 , ApJ, 139, 684Corbelli, E. & Ferrara, A., 1995, ApJ, 447, 720Einstein, A., 1929, Preussischem Akad. Wiss. 24, 38Fabian, A., C., Sanders, J.S., Allen, S.W., Crawford, C.S., Iwasawa, K. , Johnstone, R.M. ,Schmidt, R.W., & Taylor, G.B, 2003, MNRAS, 344L, 43FFabian, A., C., Reynolds, C.S., Taylor, G.B. & Dunn, R.J., 2005, MNRAS, 363, 891Ferland, G., J., Fabian, A.C., Hatch, N.A., Johnstone, R.M., Porter, R.L., Van Hoof, P. A.,& Williams, R.J., 2009, MNRAS, 392, 1475Fukue, T. & Kamaya, H., 2007, ApJ, 669, 363Hummer, D. G. & Seaton, M. J. 1963, MNRAS, 125, 437Hummer, D. G. 1963, MNRAS, 125, 461Ib´a˜nez, S. M. H. & and Parravano A., 1983, ApJ, 275, 181Ib´a˜nez, S. M. H. & Mendoza, B. C . A. 1987, Ap&SS, 137, 1Ib´a˜nez, S. M. H, 2009, ApJ, 695, 479Ib´a˜nez, S. M. H, 2004, Physics of Plasmas, 11, 5194 18 –Klessen, R. & Glover, S.C.O. 2014, arXiv:1412.5182Lamb, H., 1932,
Hydrodynamics
Dover, New YorkLifshitz, E. M. & Pitaevskii L. P., 1981,
Physical Kinetics , Pergamon Press, OxfordLandau, L. D. & Lifshitz, E. M., 1987,
Fluid Mechanics
Pergamon Press, LondonLighthill, J., 1978
Waves in Fluids
Cambrige University Press, CambridgeLinsay, R., 1951,
Physical Acoustics
Dowden, Hutchinson & Ross, Inc., StroudsburgMarkham, J., Beyer, R., & Lindsay, R., 1952, Reviews of Modern Physics, 23, 353Narain U. & Ulmschneider, P. 1990, Space Sci. Rev., 54, 377Nomura, et al., 1999, PASJ, 51, 337Parker, E., 1957, ApJ, 117, 431Lord Rayleigh, 1964,
Book Review: Scientific papers. LORD RAYLEIGH
Dover, New YorkRuszkowski M., Br¨uggen M, & Begelman, M., 2004, ApJ, 611, 158Seaton, M., J., 1959, MNRAS, 119, 81Shull, J., M. & Van Steemberg, M., E., 1985, ApJ, 298, 286Spitzer, L., 1962,
Physics of Fully Ionized Gases , Wiley, New YorkSpitzer, L., 1978,
Physical Processes in the Interestellar Medium , New YorkSpitzer, L., 1982, ApJ, 262, 315Spitzer, L., 1990, ARA&A, 28, 71Stein, R. F. & Schwartz, R. A., 1972, ApJ, 177, 807 19 –Stein, R. F. & Leibacher, J., 1974, ARA&A, 12, 407Vicenti, W. & Kruger, Ch., 1975
Introduction to Physical Gas Dynamics , Wiley, New YorkWatson, C. et al., 2004, ApJ, 608, 274Yoneyama, T., 1973, PASJ, 25, 349This manuscript was prepared with the AAS L A TEX macros v5.2. 20 –Fig. 1.—
The damping per unit wave length 2 πk i /k r as a function of temperaturefor three different values of the dimensionless frequency ωτ = 10 − (dash line), 1 (thickline ) and 10 (point line). The phase velocity v ph /c T normalized to the isothermalsound speed c T (= (cid:112) p /ρ ) as a function of temperature for three different values of thedimensionless frequency ωτ = 10 − (dash line), 1 (thick line ) and 10 (point line). Thedamping per unit wave length 2 πk i /k r as a function of the dimensionless frequency for threedifferent values of the temperature log T = 4 .
04 (dash line), log T = 4 .
16 (thick line) andlog T = 4 .
28, point lines).
The phase velocity v ph /c T normalized to the isothermalsound speed as a function of the dimensionless frequency for three different values of thetemperature log T = 4 .
04 (dash line), log T = 4 .
16 (thick line) and log T = 4 .
28, (pointlines). 21 –Fig. 2.—
The equilibrium ionization ξ ∗ as a function of temperature T ( K ) and density n (cm − ) for a photo-ionization rate ς = 5 × − s − (In Fig. , the red color refers totemperatures close to 5.000 K , and the magenta color refers to the highest temperatures ofthe order of the 30.000 K ). The equilibrium ionization ξ ∗ in a 3 D plot as a function ofthe density n ( cm − ) and the ionization rate ς ( s − ) for a fixed value of temperature ( log T =4 . K ), and where the color palette indicates the values of the density, i.e. red color refersto n around zero, and magenta color indicates values of the density n close to 100 ( cm − ). 22 –Fig. 3.— As Fig. (1a) for a photo-ionized gas with a rate given by the expression (32)and ς = 5 × − s − . As Fig. (1b) for a photo-ionized gas with a rate given by theexpression (32) and ς = 5 × − s − . As Fig. (1c) for ς = 5 × − s − . As Fig. (1d) for ς = 5 × − s − . 23 –Fig. 4.— The absorption coefficients k b (thin line), ν (dash line), k κ (point line) andthe total absorption (thick line) k tot = k b + k ν + k κ as functions of temperature T for n = 1cm − , a photo-ionization rate ς = 5 × − s − and a dimensionless frequency ωτ = 10 − . As Fig. (4a) for ωτ = 1. as Fig. (4a) for ωτ = 10. is a plot of the relaxationtime τ (= | X ξ | − ( s )) as a function of temperature T for three different values of the photo-ionization rate ς = 5 × − (dash line), ς = 5 × − (thick line), and ς = 10 −12