- Email: [email protected]
Magnetohydrodynamic density waves in a galactic disk system of stars and gas
May
1997
Communications
in Nonlinear
Science
& Numerical
Magnetohydrodynamic Density Waves Disk System of Stars and Gas ’ Yuqing LOU & Zuhui FAN (Department of Astronomy and Astrophysics, 60637, USA) Email: [email protected]
The University
Simulation ~--_---___
Vo1.2, No.2 -__
in a Galactic
of Chicago, Chicago, Illinois
Abstract: We study galactic magnetohydrodynamic (MHD) density waves in a composite system consisting of a stellar disk and a magnetized thermal gaseous disk. Perturbations in the two disks are coupled through gravitational interaction. In the tight-winding regime, dispersion relations for MHD density waves are derived under two different approximations for the stellar disk. This investigation clarifies the interrelation between spiral structures in the stellar disk and spiral synchrotron radio structures in the magnetized thermal gaseous disk. Key Words: galactic dynamics, density waves, magnetohydrodynamics, radio synchrotron emission
Introduction Prompted by total and polarized radio observations of normal spiral galaxies[‘~2~31(for example, NGC6946 and M51) which display coherent large-scale spiral magnetic field structures, a theory of magnetohydrodynamic (MHD) spiral density waves[435] was proposed very recently for disk galaxies containing magnetized thermal gas. In this earlier theoretical development[4T51, the emphasis was on the possible presence of fast and slow MHD density waves in the thin rotating gaseous disk component and the local dispersion relations in the tight winding approximation[6v7] were derived for both slow and fast MHD density waves. Since the stellar disk component is more massive than the gaseous disk component in a typical spiral galaxy and MHD density waves necessarily involve gravitational phenomena, it is of utmost importance to study MHD density waves in a composite disk system of stars and magnetized gas. Here we derive the local dispersion relations in such a composite disk system in the tight winding approximation and discuss their implications. The prospects of investigating global MHD density wave modes analogous to their hydrodynamic counterparts181 and of studying nonlinear MHD density shock wavesLgl are also outlined.
1. A brief review
for the basic scenario of galactic
dynamics
A luminous yet relatively thin rotating disk galaxy is the most prominent part of a typical galactic system consisting of a disk, a relatively old stellar halo, and the dark matter halo, the total mass of all of which determines the galactic disk rotation curveI1O>‘ll. Within a rotating disk galaxy, there are stars, gas and dust, magnetic field, and cosmic rays[‘2>13]. The stellar disk component is more massive than the gaseous disk component. The energy densities of thermal motions in the gas, gas turbulence, magnetic field, and cosmic rays are roughly comparable, all of which are much less than the kinetic energy density of the galactic rotation. The brilliant spiral arms in blue light are created by young 0, B stars and giant H II cloud complexes associated with high-density spiral gaseous arms and the ‘The
paper
was received
on Apr.
7, 1997
60
Communications
in Nonlinear
Science & Numerical
Simulation
Vo1.2, No.2 (May 1997)
large-scale spiral structures seen in nonthermal radio emissions are produced by synchrotron interactions between cosmic-ray electrons and galactic magnetic fields. Given the overall physical scenario sketched above, one needs to investigate various large-scale properties of spiral MHD density waves in a composite system of a stellar disk; a magnetized gaseous disk, and relativistic cosmic rays in order to understand the basic interrelation between optical and nonthermal radio observations. Complicated though, our formulation and approach to this problem are aimed at this eventual goal. Strictly speaking, a fully consistent treatment of the problem involves a set of nonlinear integro-differential equations. Before a brute-force onslaught, it is necessary and worthwhile to thoroughly understand the important linear aspect, a problem already of considerable complexity. Our recent analyses14t”l showed the possible presence of slow MHD density waves in an almost rigidly rotating disk portion and of fast MHD density waves in either almost rigidly rotating or differentially rotating disk portions using a thin magnetized gaseous disk model. The perturbation enhancements of surface mass density and magnetic field are roughly in phase for fast MHD density waves, whereas the perturbation enhancements of surface mass density and parallel magnetic field are significantly phase shifted (that is, a phase difference 2 r/2) for slow MHD density waves. We have further investigated 1141the self-excitation mechanism (that is, in the absence of tidal forces resulting from a central bulge or a satellite galaxy) of both fast and slow MHD density waves based on the same thin magnetized gaseous disk model from the perspective of a local swing amplification [15~161.The basic results are that slow MHD density waves are preferentially excited in the nearly rigidly rotating disk portion, whereas fast MHD density waves tend to be excited in the differentially rotating disk portion. Since the ‘grand-design’ spirals of the Whirlpool galaxy M51 (NGC5194/5195) extend far into its differentially rotating disk portion and the large-scale optical and magnetic spiral structures roughly coincide with each other[2t171, a case of fast MHD density wavesl435l can be made for the two-arm spiral structure of M51, although nonlinear MHD processes (such as fast MHD shockslgl and turbulence) are almost certainly involved in reality. In contrast, the multi-arm (e.g., m = 4) spiral structure of NGC6946 occupies the nearly rigidly rotating disk portionll*ls~lsl. Fu r th ermore, the magnetic spiral arms inferred from polarized radio emissionsl’l lie between optical spiral arms. We therefore suggested14151 that the nearby galaxy NGC6946 may carry features of slow MHD density waves. As a result of nonlinear MHD processes, these slow MHD density waves can also evolve into slow MHD density shock waves.
2. Dispersion
relation
for MHD
density
waves in a composite
disk
In the present paper, we study basic properties of MHD density waves in a composite rotating system of a stellar disk and a magnetized gaseous disk in the cylindrical coordinate system (T, 8, z); the gas turbulence, the finite disk thickness, and the influence of cosmic rays are ignored for the sake of simplicity. The coupling between the stellar disk and the magnetized gaseous disk is assumed to be purely gravitationa117~201 to a very good approximation. Using the distribution function formalism 17*201for the stellar disk and the MHD approach14>51 for the magnetized gaseous disk, it is straightforward to derive the following local dispersion relation
{[K” + k+C; - (w +k;[(w - d-I)2 x[n2 - (w - mR)2 2 k$[(w - VLR)~ -
- mfl)“](u - r&2)2 m2C;/r2][C; - 2nGp,/JkTI]} - 2rGp:lk(F&)] m2C~/r2]4n2G2~,~~1kl~“(~)/lk~l
(1)
for MHD density waves in the tight-winding or WKB approximation1213221, where exp[iwt im8 + i J’ k(s)&] dependence is assumed for all relevant perturbation variables, kg E k2 + m2/r2 defines the total wave number jkT[ in the composite disk plane, n(r) is the
No.2
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et. al:Magnetohydrodynamic
Density
Waves
in . . .
61
disk angular rotation speed taken to be the same for both stellar and gaseous disks, n2 E (2Q/r)[d(r2Q)/dr] d efi nes the epicyclic frequency K, CA is the Alfven wave speed associated with the background ring magnetic fieldl4’5l Be a r-l embedded in the gaseous disk, C’s is the sound speed in the gaseous disk, G is the gravitational constant, p0 is the background gas surface mass density, and ~5 is the background stellar surface mass density. The reduction factor7120 .F, (x) in equation (1) is given by
where G”(Z) = & LX e-++coss) r
cos(vs)ds
(3)
with Y E (w - m0)/n, 2 E ]lc]2Dg/~2, and Ds being the radial stellar velocity dispersion. In the limit of small Ds with z -+ 0, ,T,(x) + 1. By our sign convention, k > 0 and k < 0 correspond to leading and trailing spirals, respectively. In deriving dispersion relation (l), we used a higher order expansion of Poisson’s equation (see Ref. [22]), dropped out-ofphase terms121f231, and ignored the Tl E -[2mfl/(Kr)12dln R/dlnr (or J2 E [2rGp,/n212T~) term121v241. While the accuracy of some physical effects (such as growth rates of instabilities etc.) may be compromised by these omissions, our main theme of demonstrating possible types of MHD density wave modes in a composite disk system of stars and magnetized gas is prominently clear. It is worth noting that the Tl term can indeed be ignored in a nearly rigidly rotating disk portion. The Schwarzschild distribution function was also assumed for the background stellar disk as before1 r~201.In principle, perturbation analysis in the stellar disk should be carried out to higher orders than (Icr)-’ ; we have only kept leading order terms in the WKB approximation for the sake of simplicity. Information of both fast and slow MHD density waves is thus retained without too much mathematical complexity. The right-hand side of equation (1) represents the effect of gravitational coupling between perturbations in stellar and magnetized gaseous disks. In the absence of this coupling, the second factor on the left-hand side of equation (1) would correspond to the local dispersion relation for density waves in the stellar disk alone173201;and the first factor on the left-hand side of equation (1) would give rise to two local dispersion relations for fast and slow MHD density waves, respectively, in the magnetized gaseous disk alone14T51.In general, dispersion relation (1) contains three branches of MHD density waves in terms of (w - VZR)~. Based on equation (l), one can also study the local stability conditions[2”-2gl for MHD density waves in the composite disk system. In particular, local instabilities can appear as a result of gravitational coupling130-321, even if both stellar125*26] and magnetized gaseous14*5~281disks are locally stable separately. Given a specific branch of MHD density waves, it is fairly straightforward to derive magnitude and phase relationships among various perturbation variables in the stellar disk17~201and in the magnetized gaseous disk14t51. To the leading order of large tr, suffice it to briefly mention that for the first branch (fast MHD density waves), density perturbations in stellar and gaseous disks are in phase, and magnetic field and density perturbations in the gaseous disk are in phase; for the second branch (slow MHD density waves), density perturbations in stellar and gaseous disks are in phase, and magnetic field and density perturbations in the gaseous disk are significantly phase shifted (that is, a phase difference 2 n/2); for the third branch (MHD density waves outside the outer Lindblad resonance), density perturbations in stellar and gaseous disks are out of phase[331, and magnetic field and density perturbations in the gaseous disk are in phase.
3. Two-fluid
formalism
for a composite
disk
It is well known that the descriptions of spiral density waves in a thin stellar disk alone by using a distribution function formalism and by adopting a fluid approach are qualitatively
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in Nonlinear
Science & Numerical Simulation
Vo1.2, No.2 (May 1997)
similar to each other[“l, except near resonances. Since the fluid approach is much simpler, it is instructive to derive the analogous local dispersion relation for MHD density waves in the above composite disk system by also treating the stellar disk component as a fluid[30v321 (instead of using a distribution function). This kind of two-fluid formalism[30~321 has been utilized earlier; the main difference here is the inclusion of magnetic field in the thermal gaseous disk so that fast and slow MHD density waves in the magnetized gaseous disk are coupled to density waves in the stellar disk via gravitational interaction. The local dispersion relation in the tight-winding approximation is simply {[tc2 + k+C; - (w +k$[(w - w&)2 x[/c2 - (w - mR)2 2 k$[(w - mR)2 -
- mn)“](w - mR)2 m2C;/r2](C; - 2rGpU,/(kTI)} + k$(D; - 2nGp~/jkTI)] m2Ci/r2]4r2G2p:p,
(4)
The resemblance between the dispersion relations (1) and (4) is apparent. Again, the right-hand side of equation (4) represents the gravitational coupling effect; the second factor on the left-hand side is for density waves in the stellar disk alone using a fluid approach; and the first factor on the left-hand side is for fast and slow MHD density waves in the gaseous diskL4y51alone. In the absence of magnetic field (that is, CA = 0), dispersion relation (4) generalizes that derived earlier [30,32y33]by including several higher order terms. It follows from equation (4) that the composite system of stellar and magnetized gaseous disks becomes locally unstable when 47r2G2pop: > [K” + k$(D;
- ZrG,u:/lkzj)](C;
- 2nGpo/IkT()
(5)
even if both stellar and magnetized gaseous disks are locally stable separately. Furthermore, there always exists a wave number range around IkTJ 5 27rGp,/Cg such that inequality (5) is satisfied; this type of Jeans’ instability is mainly associated with slow MHD density waves[4*51 and its onset (but not its growth rate) is independent of magnetic field and disk rotation127-2gl.
4. A general perspective The results are derived under the tight-winding or WKB approximation[6~7~‘11. They contain important information regarding local stability and propagation of MHD density waves in a composite disk system. It is well known, however, that a local stability is no guarantee for a global stability for the present problem [34-3sl, because the long-range selfgravity effect is truncated in the WKB approximation for Poisson’s equation. In order to properly handle the problem on global scales, one must take into account the exact integral solution[3g1401 to Poisson’s equation relating perturbations of surface mass density and gravitational potential. This procedure of solving a set of linear integro-differential equations has been successfully carried out for density waves in a single-fluid stellar disk by implementing appropriate boundary conditions [8124*411.Besides a deeper understanding of global instabilitiesL34-381, it is now possible to numerically investigate open or bar modes in thin disk galaxies without the restriction of the WKB approximation. Given the MHD twocited, one can generalize fluid formalism presented here and the relevant references [30*32y331411 the global Auid approach to include a magnetized gaseous disk component. One expects which can be compared with generalized MHD counterparts of open or bar modes [*~24~40~411 optical and radio observations. Since bright young 0, B stars and H II cloud complexes are born continuously out of gas and nonthermal radio emissions are largely correlated with the magnetic field entrained in the gas, a global analysis of the set of integro-differential equations for MHD density waves[4151in a composite system of stellar and magnetized gaseous disks will lead to a much better understanding for the interrelation of density-wave patterns
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LOU, et. al:Magnetohydrodynamic
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in the old stellar disk (red light and infrared bands), in the gaseous disk (blue light), in dust (far-infrared), and in nonthermal radio emissions resulting from large-scale galactic magnetic fields and cosmic-ray electrons. Except for the very three-dimensional nature of gravitational potential, we still ignored the influence of cosmic rays, the finite disk thickness, the vertical stratification, and the vertical velocity and magnetic field perturbations in our present treatment for MHD density waves in rotating disk galaxies. An important physical effect, namely the Parker instability113~42-441 (or magnetic Rayleigh-Taylor instability) would arise when these neglected aspects are taken into account. This instability will lead to undulate magnetic field lines perpendicular to the galactic plane. The typical horizontal scale of magnetic field undulation is expected to be roughly comparable to the stellar disk thickness. While galactic magnetic field lines are anchored to the disk plane by conglomerations of dense clouds, the buoyant upper parts of magnetic field lines are inflated and swelled by relativistic cosmic rays. On the whole; a significant amount of magnetic flux remains retained to the disk, as evidenced by synchrotron radio emissions. On large-scale average, magnetic fields appear to lie within the disk. These relatively small-scale undulation phenomena are complicated in general and need to be considered for quantitative calculations of radio synchrotron intensities.
5. Concluding
remarks
The nonlinear aspects of galactic MHD density waves are multitudinous and challenging. Spatially extended fast and slow shocks associated with MHD density waves are expected to occur in magnetized gas-rich disk galaxiesl 2t91 Turbulence is yet another research area important for dynamo processes 1131 . In general, ‘the strength of magnetic field parallel to the shock front is enhanced across a fast MHD shock and is decreased across a slow MHD shock. These large-scale MHD shocks may trigger active star formation on smaller scales in high-density gaseous arms and newborn stars therein collectively lead to sharp brilliant spiral arms in blue light on large scales. One expects the overall environment of star forming regions across a fast shock is different from that across a slow shock, that is, the average magnetic field strength is stronger across a fast shock than that across a slow shock. This difference may lead to a systematic increase of more massive stars along a fast shock front12gl. Nonlinear effects may also produce the appearance of shredded multi-arm spiral structures in the gaseous disk, while the underlying ‘grand-design’ spiral structure in the stellar disk retains two broad arms. Perhaps, the most sensible first step is to numerically simulate the behavior of slow MHD density shock waves in an almost rigidly rotating gaseous disk alone to complement the earlier nonlinear study 191of fast MHD density shock waves. Acknowledgements University of Chicago.
This work was supported by grants from US NSF and NASA to the
References [l] R. Beck and P. Hoernes, Nature 379, 47 (1996). [2] D. S. Mathewson, P. C. van der Kruit and W. N. Brouw, Astron. Astrophys. 17,468 (1972). [3] N. Neininger and C. Horellou, in Polarimetry of the Interstellar Medium (eds Roberge, W. G. & Whittet, D. C. B.) 592 (Astron. Sot. Pacif. Conf. Ser. 97, Bookcrafters, San Francisco, 1996). [4] 2. H. Fan and Y.-Q. Lou, Nature 383, 800 (1996). [5] Y.-Q. Lou and Z. H. Fan, Mon. Not. R. Astron. Sot., submitted (1996). [6] C. C. Lin and F. H. Shu, Astrophys. 5.140, 646 (1964). [7] C. C. A. Rev. Astron. Astrophys. 5, 453-464 (1967).
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& Numerical
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(May
1997)
[B] C. C. Lin and F. H. Shu, Proc. Natl. Acad. Sci. USA 55, 229 (1966). [9] G. Bertin and C. C. Lin, Spiral Stmcture in Galaxies: A Density Wave Theory (MIT Press, 1996). [lo] W. W. Roberts, Jr. and C. Yuan, Astrophys. J. 161, 887 (1970). [ll] D. Mihaias and J. J. Binney, Galactic Astronomy 2nd ed. (Freeman, San Francisco, 1981). [12] J. Binney and S. Tremaine, Galactic Dynamics (Princeton Univ. Press, 1987). [13] L. Woltjer, in Galactic Structzlre (eds. Blaauw, A. & Schmidt, M.) 531 (The University of Chicago Press, Chicago, 1965). [14] E. N. Parker, Cosmical Magnetic Fields (Clarendon Press, Oxford, 1979). [15] Z. H. Fan and Y.-Q. Lou, Mon. Not. R. Astron. Sot., submitted (1996). [16] P. Goldreich and D. Lynden-Bell, Mon. Not. R. Astron. Sot. 130, 125 (1965). [17] A. Toomre, in The Structure and Evolution of Normal Galaxies (eds. Fall, F. M. & Lynden-Bell, D.) lll(Cambridge Univ. Press, Cambridge, Engl., 1981). [18] R. B. Tully, Astrophys. J. Supp1.27, 437 (1974). [19] L. J. Tacconi and J. S. Young, Astrophys. 5.308, 600 (1986). [20] C. Carignan, P. Charbonneau, F. Boulanger and F. Viallefond, Astron. Astrophys. 234, 43 (1990). [21] C. C. Lin and F. H. Shu, in Astrophsyics and General Relativity vol. 2 (Brandeis Univ. Summer Inst. in Theoretical Physics, 1968) 239 (Gordon & Breach, 1971). [22] Y. Y. Lau and G. Bertin, Astrophys. J. 226, 508 (1978). [23] G. Bertin and J. W.-K. Mark, SIAM J. Appl. Math. 36, 407 (1979). [24] C. B. Li, N. G. Han and C. C. Lin, Sci. Sinica 19, 665 (1976). [25] G. Bertin, C. C. Lin, S. A. Lowe and R. P. Thurstans, Astrophys. J. 338, 104 (1989). [26] V. S. Safronov, Ann. d’Astrophysique 23, 979 (1960). [27] A. Toomre, Astrophys. J. 139, 1217 (1964). [28] S. Chandrasekhar and E. Fermi, Astrophys. J. 118, 116 (1953). [29] D. Lynden-Bell, Observatory 86, 57 (1966). [30] Y.-Q. Lou, Mon. Not. R. Astron. Sot. 279, L67 (1996). [31] C. J. Jog and P. M. Solomon, Astrophys. J. 276, 114 (1984). [32] P. 0. Vandervoort, Astrophys. J. 383, 498 (1991). [33] B. G. Elmegreen, Mon. Not. R. Astron. Sot. 275, 944 (1995). [34] G. Bertin and A. B. Romeo, Astron. Astrophys. 195, 105 (1988). [35] R. H. Miller, K. H. Prendergast and W. J. Quirk, Astrophys. J. 161, 903 (1970). [36] F. Hohl, Astrophys. J. 168, 343 (1971). [37] J. P. Ostriker and P. J. E. Peebles, Astrophys. J. 186, 467 (1973). [38] J. M. Bardeen, in Dynamics of Stellar Systems (ed Hayli, A.) 297 (Int. Astron. Union Symp. No. 69, Reidel, Dordrecht, 1975). [39] S. Aoki, M. Noguchi and M. Iye, Publ. Astron. Sot. Japan 31, 737 (1979). [40] F. H. Shu, Astrophys. J. 160, 89 (1970). [41] C. C. Lin and Y. Y. Lau, Studies Appl. Math. 60, 97 (1979). [42] R. F. Pannatoni, Geophys. Astrophys. Fluid Dyn. 24, 165 (1983). [43] E. N. Parker, Astrophys. J. 145, 811 (1966). [44] T. Ch. Mouschovias, in Solar and Astrophysical Magnetohydrodynamic Flows (ed. Tsinganos, K. C.) 475 (NATO AS1 series, Kluwer Academic, Dordrecht, 1996). [45] B. G. Elmegreen, Astrophys. J. 378, 139 (1991).
1997
Communications
in Nonlinear
Science
& Numerical
Magnetohydrodynamic Density Waves Disk System of Stars and Gas ’ Yuqing LOU & Zuhui FAN (Department of Astronomy and Astrophysics, 60637, USA) Email: [email protected]
The University
Simulation ~--_---___
Vo1.2, No.2 -__
in a Galactic
of Chicago, Chicago, Illinois
Abstract: We study galactic magnetohydrodynamic (MHD) density waves in a composite system consisting of a stellar disk and a magnetized thermal gaseous disk. Perturbations in the two disks are coupled through gravitational interaction. In the tight-winding regime, dispersion relations for MHD density waves are derived under two different approximations for the stellar disk. This investigation clarifies the interrelation between spiral structures in the stellar disk and spiral synchrotron radio structures in the magnetized thermal gaseous disk. Key Words: galactic dynamics, density waves, magnetohydrodynamics, radio synchrotron emission
Introduction Prompted by total and polarized radio observations of normal spiral galaxies[‘~2~31(for example, NGC6946 and M51) which display coherent large-scale spiral magnetic field structures, a theory of magnetohydrodynamic (MHD) spiral density waves[435] was proposed very recently for disk galaxies containing magnetized thermal gas. In this earlier theoretical development[4T51, the emphasis was on the possible presence of fast and slow MHD density waves in the thin rotating gaseous disk component and the local dispersion relations in the tight winding approximation[6v7] were derived for both slow and fast MHD density waves. Since the stellar disk component is more massive than the gaseous disk component in a typical spiral galaxy and MHD density waves necessarily involve gravitational phenomena, it is of utmost importance to study MHD density waves in a composite disk system of stars and magnetized gas. Here we derive the local dispersion relations in such a composite disk system in the tight winding approximation and discuss their implications. The prospects of investigating global MHD density wave modes analogous to their hydrodynamic counterparts181 and of studying nonlinear MHD density shock wavesLgl are also outlined.
1. A brief review
for the basic scenario of galactic
dynamics
A luminous yet relatively thin rotating disk galaxy is the most prominent part of a typical galactic system consisting of a disk, a relatively old stellar halo, and the dark matter halo, the total mass of all of which determines the galactic disk rotation curveI1O>‘ll. Within a rotating disk galaxy, there are stars, gas and dust, magnetic field, and cosmic rays[‘2>13]. The stellar disk component is more massive than the gaseous disk component. The energy densities of thermal motions in the gas, gas turbulence, magnetic field, and cosmic rays are roughly comparable, all of which are much less than the kinetic energy density of the galactic rotation. The brilliant spiral arms in blue light are created by young 0, B stars and giant H II cloud complexes associated with high-density spiral gaseous arms and the ‘The
paper
was received
on Apr.
7, 1997
60
Communications
in Nonlinear
Science & Numerical
Simulation
Vo1.2, No.2 (May 1997)
large-scale spiral structures seen in nonthermal radio emissions are produced by synchrotron interactions between cosmic-ray electrons and galactic magnetic fields. Given the overall physical scenario sketched above, one needs to investigate various large-scale properties of spiral MHD density waves in a composite system of a stellar disk; a magnetized gaseous disk, and relativistic cosmic rays in order to understand the basic interrelation between optical and nonthermal radio observations. Complicated though, our formulation and approach to this problem are aimed at this eventual goal. Strictly speaking, a fully consistent treatment of the problem involves a set of nonlinear integro-differential equations. Before a brute-force onslaught, it is necessary and worthwhile to thoroughly understand the important linear aspect, a problem already of considerable complexity. Our recent analyses14t”l showed the possible presence of slow MHD density waves in an almost rigidly rotating disk portion and of fast MHD density waves in either almost rigidly rotating or differentially rotating disk portions using a thin magnetized gaseous disk model. The perturbation enhancements of surface mass density and magnetic field are roughly in phase for fast MHD density waves, whereas the perturbation enhancements of surface mass density and parallel magnetic field are significantly phase shifted (that is, a phase difference 2 r/2) for slow MHD density waves. We have further investigated 1141the self-excitation mechanism (that is, in the absence of tidal forces resulting from a central bulge or a satellite galaxy) of both fast and slow MHD density waves based on the same thin magnetized gaseous disk model from the perspective of a local swing amplification [15~161.The basic results are that slow MHD density waves are preferentially excited in the nearly rigidly rotating disk portion, whereas fast MHD density waves tend to be excited in the differentially rotating disk portion. Since the ‘grand-design’ spirals of the Whirlpool galaxy M51 (NGC5194/5195) extend far into its differentially rotating disk portion and the large-scale optical and magnetic spiral structures roughly coincide with each other[2t171, a case of fast MHD density wavesl435l can be made for the two-arm spiral structure of M51, although nonlinear MHD processes (such as fast MHD shockslgl and turbulence) are almost certainly involved in reality. In contrast, the multi-arm (e.g., m = 4) spiral structure of NGC6946 occupies the nearly rigidly rotating disk portionll*ls~lsl. Fu r th ermore, the magnetic spiral arms inferred from polarized radio emissionsl’l lie between optical spiral arms. We therefore suggested14151 that the nearby galaxy NGC6946 may carry features of slow MHD density waves. As a result of nonlinear MHD processes, these slow MHD density waves can also evolve into slow MHD density shock waves.
2. Dispersion
relation
for MHD
density
waves in a composite
disk
In the present paper, we study basic properties of MHD density waves in a composite rotating system of a stellar disk and a magnetized gaseous disk in the cylindrical coordinate system (T, 8, z); the gas turbulence, the finite disk thickness, and the influence of cosmic rays are ignored for the sake of simplicity. The coupling between the stellar disk and the magnetized gaseous disk is assumed to be purely gravitationa117~201 to a very good approximation. Using the distribution function formalism 17*201for the stellar disk and the MHD approach14>51 for the magnetized gaseous disk, it is straightforward to derive the following local dispersion relation
{[K” + k+C; - (w +k;[(w - d-I)2 x[n2 - (w - mR)2 2 k$[(w - VLR)~ -
- mfl)“](u - r&2)2 m2C;/r2][C; - 2nGp,/JkTI]} - 2rGp:lk(F&)] m2C~/r2]4n2G2~,~~1kl~“(~)/lk~l
(1)
for MHD density waves in the tight-winding or WKB approximation1213221, where exp[iwt im8 + i J’ k(s)&] dependence is assumed for all relevant perturbation variables, kg E k2 + m2/r2 defines the total wave number jkT[ in the composite disk plane, n(r) is the
No.2
LOU,
et. al:Magnetohydrodynamic
Density
Waves
in . . .
61
disk angular rotation speed taken to be the same for both stellar and gaseous disks, n2 E (2Q/r)[d(r2Q)/dr] d efi nes the epicyclic frequency K, CA is the Alfven wave speed associated with the background ring magnetic fieldl4’5l Be a r-l embedded in the gaseous disk, C’s is the sound speed in the gaseous disk, G is the gravitational constant, p0 is the background gas surface mass density, and ~5 is the background stellar surface mass density. The reduction factor7120 .F, (x) in equation (1) is given by
where G”(Z) = & LX e-++coss) r
cos(vs)ds
(3)
with Y E (w - m0)/n, 2 E ]lc]2Dg/~2, and Ds being the radial stellar velocity dispersion. In the limit of small Ds with z -+ 0, ,T,(x) + 1. By our sign convention, k > 0 and k < 0 correspond to leading and trailing spirals, respectively. In deriving dispersion relation (l), we used a higher order expansion of Poisson’s equation (see Ref. [22]), dropped out-ofphase terms121f231, and ignored the Tl E -[2mfl/(Kr)12dln R/dlnr (or J2 E [2rGp,/n212T~) term121v241. While the accuracy of some physical effects (such as growth rates of instabilities etc.) may be compromised by these omissions, our main theme of demonstrating possible types of MHD density wave modes in a composite disk system of stars and magnetized gas is prominently clear. It is worth noting that the Tl term can indeed be ignored in a nearly rigidly rotating disk portion. The Schwarzschild distribution function was also assumed for the background stellar disk as before1 r~201.In principle, perturbation analysis in the stellar disk should be carried out to higher orders than (Icr)-’ ; we have only kept leading order terms in the WKB approximation for the sake of simplicity. Information of both fast and slow MHD density waves is thus retained without too much mathematical complexity. The right-hand side of equation (1) represents the effect of gravitational coupling between perturbations in stellar and magnetized gaseous disks. In the absence of this coupling, the second factor on the left-hand side of equation (1) would correspond to the local dispersion relation for density waves in the stellar disk alone173201;and the first factor on the left-hand side of equation (1) would give rise to two local dispersion relations for fast and slow MHD density waves, respectively, in the magnetized gaseous disk alone14T51.In general, dispersion relation (1) contains three branches of MHD density waves in terms of (w - VZR)~. Based on equation (l), one can also study the local stability conditions[2”-2gl for MHD density waves in the composite disk system. In particular, local instabilities can appear as a result of gravitational coupling130-321, even if both stellar125*26] and magnetized gaseous14*5~281disks are locally stable separately. Given a specific branch of MHD density waves, it is fairly straightforward to derive magnitude and phase relationships among various perturbation variables in the stellar disk17~201and in the magnetized gaseous disk14t51. To the leading order of large tr, suffice it to briefly mention that for the first branch (fast MHD density waves), density perturbations in stellar and gaseous disks are in phase, and magnetic field and density perturbations in the gaseous disk are in phase; for the second branch (slow MHD density waves), density perturbations in stellar and gaseous disks are in phase, and magnetic field and density perturbations in the gaseous disk are significantly phase shifted (that is, a phase difference 2 n/2); for the third branch (MHD density waves outside the outer Lindblad resonance), density perturbations in stellar and gaseous disks are out of phase[331, and magnetic field and density perturbations in the gaseous disk are in phase.
3. Two-fluid
formalism
for a composite
disk
It is well known that the descriptions of spiral density waves in a thin stellar disk alone by using a distribution function formalism and by adopting a fluid approach are qualitatively
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similar to each other[“l, except near resonances. Since the fluid approach is much simpler, it is instructive to derive the analogous local dispersion relation for MHD density waves in the above composite disk system by also treating the stellar disk component as a fluid[30v321 (instead of using a distribution function). This kind of two-fluid formalism[30~321 has been utilized earlier; the main difference here is the inclusion of magnetic field in the thermal gaseous disk so that fast and slow MHD density waves in the magnetized gaseous disk are coupled to density waves in the stellar disk via gravitational interaction. The local dispersion relation in the tight-winding approximation is simply {[tc2 + k+C; - (w +k$[(w - w&)2 x[/c2 - (w - mR)2 2 k$[(w - mR)2 -
- mn)“](w - mR)2 m2C;/r2](C; - 2rGpU,/(kTI)} + k$(D; - 2nGp~/jkTI)] m2Ci/r2]4r2G2p:p,
(4)
The resemblance between the dispersion relations (1) and (4) is apparent. Again, the right-hand side of equation (4) represents the gravitational coupling effect; the second factor on the left-hand side is for density waves in the stellar disk alone using a fluid approach; and the first factor on the left-hand side is for fast and slow MHD density waves in the gaseous diskL4y51alone. In the absence of magnetic field (that is, CA = 0), dispersion relation (4) generalizes that derived earlier [30,32y33]by including several higher order terms. It follows from equation (4) that the composite system of stellar and magnetized gaseous disks becomes locally unstable when 47r2G2pop: > [K” + k$(D;
- ZrG,u:/lkzj)](C;
- 2nGpo/IkT()
(5)
even if both stellar and magnetized gaseous disks are locally stable separately. Furthermore, there always exists a wave number range around IkTJ 5 27rGp,/Cg such that inequality (5) is satisfied; this type of Jeans’ instability is mainly associated with slow MHD density waves[4*51 and its onset (but not its growth rate) is independent of magnetic field and disk rotation127-2gl.
4. A general perspective The results are derived under the tight-winding or WKB approximation[6~7~‘11. They contain important information regarding local stability and propagation of MHD density waves in a composite disk system. It is well known, however, that a local stability is no guarantee for a global stability for the present problem [34-3sl, because the long-range selfgravity effect is truncated in the WKB approximation for Poisson’s equation. In order to properly handle the problem on global scales, one must take into account the exact integral solution[3g1401 to Poisson’s equation relating perturbations of surface mass density and gravitational potential. This procedure of solving a set of linear integro-differential equations has been successfully carried out for density waves in a single-fluid stellar disk by implementing appropriate boundary conditions [8124*411.Besides a deeper understanding of global instabilitiesL34-381, it is now possible to numerically investigate open or bar modes in thin disk galaxies without the restriction of the WKB approximation. Given the MHD twocited, one can generalize fluid formalism presented here and the relevant references [30*32y331411 the global Auid approach to include a magnetized gaseous disk component. One expects which can be compared with generalized MHD counterparts of open or bar modes [*~24~40~411 optical and radio observations. Since bright young 0, B stars and H II cloud complexes are born continuously out of gas and nonthermal radio emissions are largely correlated with the magnetic field entrained in the gas, a global analysis of the set of integro-differential equations for MHD density waves[4151in a composite system of stellar and magnetized gaseous disks will lead to a much better understanding for the interrelation of density-wave patterns
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in the old stellar disk (red light and infrared bands), in the gaseous disk (blue light), in dust (far-infrared), and in nonthermal radio emissions resulting from large-scale galactic magnetic fields and cosmic-ray electrons. Except for the very three-dimensional nature of gravitational potential, we still ignored the influence of cosmic rays, the finite disk thickness, the vertical stratification, and the vertical velocity and magnetic field perturbations in our present treatment for MHD density waves in rotating disk galaxies. An important physical effect, namely the Parker instability113~42-441 (or magnetic Rayleigh-Taylor instability) would arise when these neglected aspects are taken into account. This instability will lead to undulate magnetic field lines perpendicular to the galactic plane. The typical horizontal scale of magnetic field undulation is expected to be roughly comparable to the stellar disk thickness. While galactic magnetic field lines are anchored to the disk plane by conglomerations of dense clouds, the buoyant upper parts of magnetic field lines are inflated and swelled by relativistic cosmic rays. On the whole; a significant amount of magnetic flux remains retained to the disk, as evidenced by synchrotron radio emissions. On large-scale average, magnetic fields appear to lie within the disk. These relatively small-scale undulation phenomena are complicated in general and need to be considered for quantitative calculations of radio synchrotron intensities.
5. Concluding
remarks
The nonlinear aspects of galactic MHD density waves are multitudinous and challenging. Spatially extended fast and slow shocks associated with MHD density waves are expected to occur in magnetized gas-rich disk galaxiesl 2t91 Turbulence is yet another research area important for dynamo processes 1131 . In general, ‘the strength of magnetic field parallel to the shock front is enhanced across a fast MHD shock and is decreased across a slow MHD shock. These large-scale MHD shocks may trigger active star formation on smaller scales in high-density gaseous arms and newborn stars therein collectively lead to sharp brilliant spiral arms in blue light on large scales. One expects the overall environment of star forming regions across a fast shock is different from that across a slow shock, that is, the average magnetic field strength is stronger across a fast shock than that across a slow shock. This difference may lead to a systematic increase of more massive stars along a fast shock front12gl. Nonlinear effects may also produce the appearance of shredded multi-arm spiral structures in the gaseous disk, while the underlying ‘grand-design’ spiral structure in the stellar disk retains two broad arms. Perhaps, the most sensible first step is to numerically simulate the behavior of slow MHD density shock waves in an almost rigidly rotating gaseous disk alone to complement the earlier nonlinear study 191of fast MHD density shock waves. Acknowledgements University of Chicago.
This work was supported by grants from US NSF and NASA to the
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