Effect of a tide on the Parker Jeans instability

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1 MNRAS 45, doi:1.193/mnras/stv611 Effect of a tide on the Parker Jeans instability Surajit Mondal 1 Sagar Chakraborty 1,2 1 Department of Physics, Indian Institute of Technology Kanpur, UP-2816, India 2 Mechanics, Applied Mathematics Group, Indian Institute of Technology Kanpur, UP-2816, India Accepted 215 March 17. Received 215 March 1; in original form 215 February 3 1 INTRODUCTION In this paper, we seek a modification of the Parker Jeans instability criterion in the presence of tides. Our anticipation is that a compressive tidal force can augment the instability to an extent such that the formation of stars, galaxies, etc. will be practically affected. While very recently Jog 213 has shown how tides can modify the Jeans instability, herein we turn our attention towards the Parker Jeans instability. Though several authors like Chou et al. 2, Kuwabara & Ko 26, etc. have worked on the Parker Jeans instability, the present paper is the first to discuss a dispersion relation for the Parker Jeans instability that includes tidal effects. Apart from the fact that the astrophysics community has tremendous current research interest in the gas collapse mechanism, the topic of this paper is of prime importance as both instabilities environmental effects play a major role in interstellar cloud formation star formation Mouschovias 1974; Mouschovias, Shu & Woodward 1974; Blit & Shu 198; Elmegreen 1982; Bekki et al. 22; Bekki & Couch 23; Renaud et al. 29; Sadavoy, Francesco & Johnstone 21; Pasetto et al Although the formation of clouds should be discussed taking into account many other instabilities, turbulence thermal pressure McKee & Ostriker 27; ennebelle & Falgarone 212, in this paper we exclusively consider the Parker Jeans instability to clearly underst the defining physics involved in the interplay between the tides the instability. It is widely accepted that the Jeans instability is the basic mechanism of star formation when the only acting inward force is gravity. It is sufficient to overcome the internal pressure of the interstellar gas clouds. Parker 1966 studied the role of magnetic fields in the formation of molecular clouds. e found that a system with a transverse magnetic field gravitational field is unstable to per- surajit@iitk.ac.in ABSTRACT By performing a linear stability analysis in the presence of an external tidal field, we have obtained a generalied dispersion relation for the Parker Jeans instability. We have found that a compressive tidal field helps in initiating the collapse of the interstellar cloud towards star formation. This effect is present even when the Jeans mass is more than the mass of the system under consideration. In such cases, of course, it is the magnetic field that makes the compressive tidal force effective via the Parker instability. We argue that this effect can be of importance in overcoming the disrupting effect of a rom magnetic field. Key words: hydrodynamics instabilities ISM: kinematics dynamics galaxies: interactions galaxies: magnetic fields galaxies: star formation. turbations in the plane of the fields for some wavenumbers. Later he went on to investigate the full three-dimensional stability of the system. In this paper we refer to the two-dimensional instability as the Parker instability. The Parker instability is seen in a perfectly conducting compressible plasma in a transverse gravitational magnetic field. Due to perturbations in the magnetic field, crests troughs are observed in the field lines. As the fluid flows towards a trough, the fluid pressure in the corresponding crest of the flux tube decreases. Then the buoyancy force pushes the flux tube up. But as the tube rises up, more fluid flows out of the crest. ence, the density in the trough will continue to increase. Elmegreen 1982 obtained a dispersion relation for the Parker Jeans instability showed that the self-gravitational forces can play an important role from the very onset of the Parker instability if the interstellar medium somehow gets slightly compressed during cloud formation. The compression can be due to spiral density shock waves Mouschovias 1974; Wada & Koda 24; Dobbs & Bonnell 26, cooling or tidal effects Bekki et al. 22. Tidal interactions have been used to explain the structure inside clouds Bekki 212 starbursts Bekki & Couch 23; Renaud et al ence, it is very imperative that one studies the stability of clouds in the presence of tides. In this paper we take up this issue technically by looking at the effect of tides on the dispersion relation of the Parker Jeans instability. 2 TE MODEL SYSTEM We consider an exponential non-rotating gas layer that has a constant gravitational acceleration g perpendicular to the unperturbed magnetic field. Without any loss of generality, we orient the y-axis parallel to the magnetic field the -axis along the direction of g, which is the self-gravity of the cloud. In realistic clouds, the gravitational attraction varies as tanh /, but a constant C 215 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

g brings out the essential features of the problem while retaining computational simplicity. g can be chosen to be 3.

2 g brings out the essential features of the problem while retaining computational simplicity. g can be chosen to be cm s 2 in the solar neighbourhood Oort 1965 at unit scale height = 16 pc Falgarone & Lequeux ere note that g is the self-gravitational attraction of the cloud. The magnetic field is denoted by B, the pressure due to cosmic rays by P CR,thefluid pressure by P the density by. The equilibrium configuration of the gas layer is given by = P = P CR = B2 P P CR B 2 = e /, 1 the quantities with subscript are for the mid-plane values. is the scale height, which is given by = P + B 2/8π + P CR. 2 g + T It can be easily seen why the scale height gets modified if we solve the equilibrium equation: P + P CR + B2 = g + T. 3 8π Let us denote the one-dimensional rms velocity of the interstellar medium by u. AlsoletP = u 2, α = B 2/8πP β = P CR /P. From the definitions of α β we get: = u2 1 + α + β. 4 g + T Thus, g + T has different values for ><, because the self-gravity changes sign when we cross the -axis while the external field T does not. In our work, we decide to take positive T for the compressive tidal force. The directions of g T are showninfig.1. Considering the directions of g T we get: = u α + β g + T u α + β g T if if >, <. We have taken α, β u to be constant throughout the layer. We have also assumed that the equilibrium configuration of the cloud is isothermal, but the perturbations are adiabatic with adiabatic index γ. Figure 1. We consider an exponential gas having a rectangular geometry. The pillbox shown in the figure extends to infinity in all three directions. The density is uniform along the x- y-axes but varies exponentially along the -axis. 5 Effect of a tide on Parker Jeans instability 1875 The equations of motion of a gas layer with gravitational acceleration g, magnetic field B 1, fluid pressure P 1, cosmic ray pressure P CR, density 1 tidal acceleration T are v t = P + B2 8π + P CR + 1 B B + g + T, 6 4π = v 7 t B = v B. 8 t When the cosmic rays move along the field line they do not feel any force Shu This implies that the field lines must be isobaric surfaces, that is, B P CR =. 9 Equations 6 9 define the dynamics in the system under consideration in this paper. 3 LINEAR STABILITY ANALYSIS The equilibrium solution of the immediately preceding set of equations are given by equations 1 4. To study the stability of this system, we perturb it slightly study the time evolution of the perturbation. 3.1 Linearied equations For the linear stability analysis, we perturb the equilibrium solution infinitesimally, i.e. we write B = B eq + δb, 1 P = P eq + δp, 11 = eq + δ 12 P CR = P CReq + δp CR. 13 Also for notational simplicity, we have taken v to be a perturbation: v = δv v eq =. Under the adiabatic approximation, δp = γp δ, 14 giving t δp = γv P γu2 v. 15 This equation follows straightforwardly from equation 7 after we put in the perturbations use equation 14. As the conductivity of the molecular cloud is very high, we can assume the flux freeing condition given by δb = v B. 16 t Also we can write δb = δa, 17 MNRAS 45,

3 1876 S. Mondal S. Chakraborty δa is the magnetic vector potential. We assume that δb lies in the y direction. From equations one can easily show that t δa = Bv. 18 From Parker 1966, one can see that equation 9 implies: δp CR = δa B P CR = δa P CR B. 19 We write the acceleration due to gravity as the sum of acceleration due to stars unperturbed gas due to perturbations as g = g unperturbedgas ˆk + T + δg gas = u2 1 + α + β ˆk + δg gas. 2 Now we write the perturbation equations, keeping only the linear terms as follows: t v y = B y δp + δp CR + y δa 8π +δg gas,y, 21 t v = B δp + δp CR + δa 8π B 4π t δ = v t δp = v P γp 2 2 δa + y2 δa 2 + δg gas, δ u2 1 + α + β, 22 y v y + v 23 y v y + v. 24 Thus we have obtained the linearied equations describing the evolution of infinitesimal perturbations. 3.2 Dispersion relation Now we take perturbations about the mid-plane: δ = δ exp cos ξ 2, 25 v y = iv y exp + cos ξ 2 v = v exp 26 + sin ξ ere = iω/u, ν = k ξ = k. ω is the angular frequency k is a wave vector of a Fourier mode. k is the component of the wave vector along the -axis. Note that the chosen density perturbation profile decays exponentially along the -axis, once we fix the form of the density perturbation, the form of the velocity perturbations follows directly from equation 23. Also it should be noted that exp /2 is a natural choice for the density perturbation because of the pillbox structure centred at the origin. When equations 25, are substituted into the pure Parker instability, δa becomes sinusoidal in : δa = δa exp sin ξ. 28 In this work, we have allowed the perturbations to have only y spatial dependence. One of the reasons behind this is of course analytical convenience. But we can also argue the following: since our primary aim is to underst the formation of large-scale structures, we can ignore the growth of perturbations along the x-axis. It is well known that the Parker instability has growing modes of short wavelength along the x-axis long wavelength along the y-axis, while the Jeans instability has unstable modes of long wavelength along both the x-y-axes. ence along the y-axis, the long wavelength growth is affected by both the Jeans the Parker instabilities. But along the x-axis, due to the Parkerinstability-induced short-wavelength mode, fragmentation occurs. Because our primary aim is not to study the fragmentation of clouds, we may safely ignore the growth along the x-axis. Also we believe this picture will not significantly change on inclusion of tides hence the particular choice of the form of the perturbation. It can be shown that for an infinitely thin layer having mass density 4δ,then δg gas,y y,,t = 8πiGδy,,tƔ y, k, k 29 δg gas, y,,t =±8πGδy,,tƔ, k, k. 3 ere Ɣ y 1 2 Ɣ 1 2 A ± [A + + A e 2k 1/2 cosk [A + A e 2k 1/2 cosk ] 2B k tank 31 ] 2B + k tank, 32 2k + 1 2k k ± 2k k k 2 1 B ± 2k k ± 1 2 2k k In δg gas, y,, t, + isfor< for>. Also, we obtain: B J δa + J 2 B δa = u 2 δggas, 2 + iνqδg gas,y, 35 J 1 2αγν α + γ, 36 J 2 rν 2 4 2α + γ ν αγν 4, 37 4 q 1 + α + β γ 38 r 1 + α + β q αγ MNRAS 45,

4 The intermediate steps leading to equation 35 are as follows. First, we substitute equations in equations Next we substitute δp δ into equation 21 find an expression for v y. Then we replace δp, δ v y in equation 22. We eliminate v δp CR using equation 18 equation 19. This results in a single equation with δa, δg their derivatives. We can simplify equation 35 after putting in the explicit forms of the perturbations to get ξ 2 J 1 + J 2 ae sin ξ = s δ 2 Ɣ νqɣ y e / E cos ξ, 4 a δa B, 41 s 8πG 2 42 u 2 E exp From equation 4 we get ξ 2 ξ J 1 + J 2 ae sin = s δ 2 Ɣ νqɣ y e / E cos ξ. 44 We write δ/ in terms of v y substitute it in equation 26 after eliminating P B. Then we obtain δ 2ξaν 2 q = γν 2 s νg y ere π/k1 Ɣ y e / E cosξ/d G y = π/k1. 46 E cosξ/d Similarly, π/k1 Ɣ e / E cosξ/d G = π/k1. 47 E cosξ/d Substituting equation 45 into equation 44 integrating it over the height of the perturbation, we obtain the main result of this paper: ξ 2 s ν 2 q J 1 + J 2 = 2 G νqg y, 48 D D 2 + γν 2 s νg y. 49 In Fig. 2 we show the variation of the imaginary part of ω/u with k for different values of. Smaller values of imply there is a compressive tidal force. So from Fig. 2 it is clear that a compressive tidal force helps in the growth of the Parker Jeans instability. We can clearly see that the rate of collapse increases on application of a compressive tidal field. As a check of the correctness of our result, we will now compare it with some of the results from the literature. First, one can confirm that dispersion relation 48 boils down to what has been obtained Effect of a tide on Parker Jeans instability 1877 Figure 2. Variation of iω/u versus k for different values of T. The solid line corresponds to T /g = 2 the dashed one is for T /g =. Curves acareforα = β =.5. Curves b d are for α = β = 1.. Note that positive T implies a compressive tidal force. by Jog 213 in the absence of a magnetic field. Secondly, from Fig. 2 it is clear that in the absence of any tidal force but for different α β, the results obtained in fig. 7 of Elmegreen 1982 are unambiguously reproduced. owever, it should be carefully noted that the abscissa is k not ν = k as used in Elmegreen The main result of this paper, however, is that the tidal force enhances the growth rate for a given wavenumber as shown by Fig. 2. Last but not least, our result should match with Chou et al. 2. That paper deals with an external pressure, which can, in principle, originate from an external tidal field. It established that the maximum growth rate of instability shown to occur at a finite k x infinitesimally small k y decreases when one incorporates external pressure. Before we comment on our results vis-à-vis Chou et al. 2, note that k x = in the present paper. Indeed, along the line k x = in fig. 2 of Chou et al. 2, one finds that for a given k y, the growth rate increases with the external pressure. Thus, our result is validated. In passing, it may be noted that although our calculations follow Elmegreen 1982 closely, one must keep in mind the difference that in this paper is different on the two sides of the x y-plane. Due to this, while on one side the tidal force assists gravity in compressing, on the other side it resists a similar compression. This essentially amounts to different growth rates for the perturbations on the two different sides of the mid-plane. 4 DISCUSSION It is highly unlikely that a molecular cloud or 1 cloud is completely isolated. There is always some form of external force acting on it. Jog 213 pointed out that this tidal mechanism can be an additional trigger for star formation can explain the formation of ultraluminous galaxies. Bekki & Couch 23 showed that starbursts can occur in giant molecular clouds due to the static ram pressure of the intracluster medium. Detailed knowledge of these scenarios can only be obtained from accurate numerical simulations. But some analytical understing goes a long way in formulating the numerical problem. Our work provides such a relevant analytical insight. Jog 213 obtained a dispersion relation in the absence of magnetic fields. But magnetic fields are omnipresent in astrophysical scenarios. So one must use the dispersion relation obtained herein, which also includes the effect of a magnetic field. Although our work is MNRAS 45,

5 1878 S. Mondal S. Chakraborty based on an analytically tractable simplistic model, we hope that an indication of the time-scale of the tide-modified Parker Jeans instability can be obtained from our work. Another relevance of the result obtained in this paper can be observed from the following argument: Mouschovias, Kun & Christie 29 proved that a pure Parker instability can form giant molecular clouds as it can enhance the density sufficiently for a phase transition to a cold denser cloud phase. Again Kim & Ryu 21 point out that a rom magnetic field component, as in Parker & Jokipii 2, if sufficiently strong can stabilie the undular mode. Our work indicates that since a compressive tidal force helps the Parker instability to grow, compressive tidal fields will counteract the effects of the rom magnetic field allowing the phase transition mechanism proposed by Mouschovias et al. 29 to form molecular clouds. In all our calculations we have used a continuum picture for gas we conducted a linear stability analysis. While this is customary, for completeness let us check if our results validate our assumptions. Then for >, = 1 + T 1, 5 g is the scale height in the absence of tides. Solving equation 5 plugging in values of = 16 pc T /g = 1, we find that 14 pc. The mean free path of our system is around 1 7 m, which is again much less than the scale height. This conclusion obviously holds for <as at >islessthan at <. In this work, we have taken into account only the tidal acceleration in the -direction T. This is possible only when the forces in the other directions are very negligible. As an example of such a situation, consider a galaxy collision scenario. Also we have not taken into account that tidal forces can strip a cloud of hydrogen. We did not take into account the effects of rotation as this will be negligible for moderate to high densities, as in a mildly compressed medium Elmegreen In a future paper, we hope to address these additional issues in detail. ACKNOWLEDGEMENTS The anonymous referee is thanked for raising pertinent issues that helped the authors to make the paper more transparent. SC gratefully acknowledges financial support received through the INSPIRE faculty fellowship awarded by the Department of Science Technology Government of India. REFERENCES Bekki K., 212, MNRAS, 422, 1957 Bekki K., Couch W. J., 23, ApJ, 596, L13 Bekki K., Forbes D. A., Beasley M. A., Couch W. J., 22, MNRAS, 335, 1176 Blit L., Shu F.., 198, ApJ, 238, 148 Chou W., Matsumoto R., Tajima T., Umekawa M., Shibata K., 2, ApJ, 538, 71 Dobbs C. L., Bonnel I. A., 26, A&AS, 27, Elmegreen B. G., 1982, ApJ, 253, 634 Falgarone E., Lequeux J., 1973, A&A, 25, 253 ennebelle P., Falgarone E., 212, A&ARv, 2, 55 Jog C. J., 213, MNRAS, 434, L56 Kim J., Ryu D., 21, ApJ, 561, L135 Kuwabara T., Ko C.-M., 26, ApJ, 636, 29 McKee C. F., Ostriker E. C., 27, ARA&A, 45, 565 Mouschovias T. Ch., 1974, ApJ, 192, 37 Mouschovias T. Ch., Shu F.., Woodward P. R., 1974, A&A, 33, 73 Mouschovias T. Ch., Kun M. W., Christie D. A., 29, MNRAS, 397, 14 Oort J., 1965, in Blaauw A., Schmidt M., eds, Stars Stellar Systems Vol. 5, Galactic Structure. University of Chicago Press, Chicago, p. 455 Parker E. N., 1966, ApJ, 145, 811 Parker E. N., Jokipii J. R., 2, ApJ, 536, 331 Pasetto S., Cropper M., Fujita Y., Chiosi C., Grebel E. K., 215, A&A, 573, in press Renaud F., Boily C. M., Naab T., Theis Ch., 29, ApJ, 76, 67 Renaud F., Bournaud F., Kraljic K., Duc A. P., 214, MNRAS, 442, L33 Sadavoy S. I., Francesco Di J., Johnstone D., 21, ApJ, 718, L32 Shu F.., 1974, A&A, 33, 55 Wada K., Koda J., 24, MNRAS, 349, 27 This paper has been typeset from a TEX/LATEX file prepared by the author. MNRAS 45,

Analysis of Jeans Instability of Partially-Ionized. Molecular Cloud under Influence of Radiative. Effect and Electron Inertia

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