Stationary perturbation configurations in a composite system of stellar and coplanarly magnetized gaseous singular isothermal discs

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1 Mon. Not. R. Atron. Soc. 35, 5 () doi:./j x Stationary perturbation confiuration in a compoite ytem of tellar and coplanarly manetized aeou inular iothermal dic Yu-Qin Lou,,3 and Yue Zou Phyic Department, The Tinhua Centre for Atrophyic, Tinhua Univerity, Beijin 8, China Department of Atronomy and Atrophyic, The Univerity of Chicao, 56 Elli Avenue, Chicao, IL 6637, USA 3 National Atronomical Obervatorie, Chinee Academy of Science, A, Datun Road, Beijin, China Accepted 3 December. Received 3 October 8 ABSTRACT We contruct alined and unalined tationary perturbation confiuration in a compoite ytem of tellar and coplanarly manetized aeou inular iothermal dic (SID) coupled by ravity. Thi tudy extend recent analye on (manetized) SID by Shu et al., Lou and Lou & Shen. By thi model, we intend to provide a conceptual framework to ain iniht for multiwavelenth lare-cale tructural obervation of dic alaxie. Both SID are approximated to be razor thin and are in a elf-conitent axiymmetric backround equilibrium with power-law urface ma denitie and flat rotation curve. The aeou SID i embedded with a coplanar azimuthal manetic field B θ (r) of a radial calin r / that i not force-free. In comparion with the SID problem tudied earlier, there are three poible clae of tationary olution allowed by more dynamic freedom. To identify phyical olution, we explore parameter pace involvin three dimenionle parameter: ratio λ of Alfvén peed to ound peed in the manetized aeou SID; ratio β of the quare of the tellar velocity diperion to the a ound peed; and ratio δ of the urface ma denitie of the two SID. For both alined and unalined piral cae with azimuthal periodicitie m, one of the three olution branche i alway phyical, while the other two branche miht become invalid when β exceed certain critical value. For the onet criteria from an axiymmetric equilibrium to alined ecular bar-like intabilitie, the correpondin T / W M ratio, which varie with λ, β and δ, may be coniderably lower than the oft-quoted value of T / W., where T i the total kinetic enery, W i the total ravitational potential enery and M i the total manetic enery. For unalined piral cae, we examine marinal intabilitie for axiymmetric ( m =) and non-axiymmetric ( m > ) diturbance. The reultin marinal tability curve differ from the previou one. The cae of a compoite partial manetized SID ytem i alo invetiated to include the ravitational effect of an axiymmetric dark matter halo on the SID equilibrium. We further examine the phae relationhip amon the ma denitie of the two SID and azimuthal manetic field perturbation. Our exact lobal perturbation olution and critical point are valuable for tetin numerical manetohydrodynamic code. For alactic application, our model analyi contain more realitic element and offer ueful iniht into the tructure and dynamic of dic alaxie conitin of tar and manetized a. Key word: MHD wave ISM: eneral alaxie: kinematic and dynamic alaxie: piral alaxie: tructure. INTRODUCTION In alactic context, we venture to formulate a theoretical manetohydrodynamic (MHD) dic problem to explore poible lare-cale tructure and dynamic of tationary MHD denity wave in a compoite ytem of tellar and manetized intertellar medium (ISM) a dic. The two ravitationally coupled dic are treated a fluid and manetofluid repectively and are both expediently approximated a razor-thin inular iothermal dic (SID), with the aeou SID bein embedded with a coplanar azimuthal manetic field. For the ravitational effect of a maive axiymmetric dark matter halo, we precribe a backround compoite ytem of two coupled partial SID (Syer & Tremaine louyq@mail.tinhua.edu.cn and lou@oddjob.uchicao.edu Downloaded from on September 8 C RAS

2 Confiuration of compoite MSID 996; Shu et al. ; Lou ; Lou & Shen 3; Shen & Lou 3). In our model analyi, we contruct tationary alined and unalined loarithmic piral MHD perturbation confiuration in a compoite ytem of two SID with flat rotation curve, and attempt to relate variou morpholoie of dic alaxie, includin barred and lopided, barred and normal piral tructure. For poible obervational dianotic, we derive phae relationhip amon perturbation pattern of the tellar urface ma denity, the a urface ma denity and the azimuthal manetic field (Lou & Fan a,b). Thi introduction erve two purpoe. The firt one i to provide eneral backround information relevant to the problem at hand and the econd one i to ive reaon for puruin thi MHD dic problem. In a pioneerin tudy of a compoite ytem of tellar and a dic coupled by ravity, Lin & Shu (966, 968) ued a tellar ditribution function and a a fluid dic decription to derive and analye the local diperion relation of alactic piral denity wave. Since then, there have been extenive theoretical tudie on the perturbation confiuration and tability propertie of a compoite dic ytem, mainly in alactic context. Kato (97) invetiated ocillation and overtabilitie of denity wave uin a formalim imilar to that of Lin & Shu (966), 968). In a two-fluid formalim, Jo & Solomon (98a,b) examined the rowth of local axiymmetric perturbation in a compoite dic ytem. Bertin & Romeo (988) tudied the influence of a a dic on piral mode in a two-fluid model framework. Vandervoort (99a,b) tudied the influence of intertellar a on ocillation and tabilitie of pheroidal alaxie. The two-fluid approach wa alo adopted in a tability tudy of a two-component dic ytem with finite dic thickne by Romeo (99). The analyi for morpholoie of dic alaxie wa performed by Lowe et al. (99). For the tability of a compoite dic ytem, different effective Q eff parameter (Safronov 96; Toomre 96) have been ueted uin a two-fluid formalim by Elmereen (995) and Jo (996). Recently, Lou & Fan (998b) ued the two-fluid formalim to tudy the propertie of open and tiht-windin piral denity-wave mode in a compoite dic ytem. Lou & Shen (3) tudied tationary lobal perturbation tructure in a two-fluid ytem of SID and, intead of a redefinition of a different Q eff parameter, Shen & Lou (3) offered a more practical D criterion for the axiymmetric intability in a compoite SID ytem. A rich cla of dic problem involve the tability propertie of SID. There have been numerou tudie on thi ubject ince the pioneerin work of Metel (963) (e.. Zan 976; Toomre 977; Lemo et al. 99; Lynden-Bell & Lemo 993; Goodman & Evan 999; Chakrabarti, Lauhlin & Shu 3). Specifically, Syer & Tremaine (996) made an important breakthrouh to derive emi-analytic olution for tationary perturbation confiuration in a cla of SID. Their work ha been eneralized in two important apect recently, that i, the incluion of a coplanar manetic field (Lou et al., in preparation) and the eneralization to a compoite ytem of two ravitationally coupled SID (Shen & Lou ). Shu et al. () obtained tationary olution for perturbation confiuration in an iopedically manetized SID (MSID) with a flat rotation curve. Throuh numerical exploration, they interpreted thee tationary alined and unalined loarithmic piral confiuration a onet of bar-type and barred piral intabilitie (ee alo Galli et al. ). Different from yet complementary to the analyi of Shu et al. (), Lou () performed a coplanar MHD perturbation analyi in a inle backround SID embedded with an azimuthal manetic field, from the perpective of tationary fat and low MHD denity wave (FMDW and SMDW; Fan & Lou 996; Lou & Fan 998a). Lou () alo derived a form of manetic virial theorem for an MSID and ueted the ratio of rotation enery to the um of ravitational and manetic enerie to be crucial for the onet of bar-like intability in an MSID ytem. In alactic context, it would be more realitic to conider lare-cale tructure and dynamic in a compoite ytem of tellar and manetized ISM dic. A a firt tep, Lou & Shen (3) made a foray into thi model problem, contructed tationary alined and unalined loarithmic piral confiuration in uch a compoite SID ytem, and further examined axiymmetric intability propertie (Shen & Lou 3). In dic alaxie, the ISM dic i manetized, with the manetic enery denity bein comparable to the enery denitie of thermal a and of relativitic comic ray a (e.. Lou & Fan 3). Information of alactic manetic field can be etimated by ynchrotron radio emiion from piral alaxie. For uch a manetized compoite ytem, MHD will play an indipenable role and reveal more realitic apect of dynamic and dianotic information. Thee important problem (Shu et al. ; Lou ; Lou & Shen 3) are not only interetin by themelve, but alo erve a neceary tep for etablihin an even more realitic model. Motivated by thi propect (Lou & Fan 998b; Lou ; Lou & Shen 3), we contruct here tationary perturbation confiuration for alined and unalined loarithmic piral cae in a compoite ytem of a tellar SID and a coplanarly manetized aeou MSID, and dicu their tability propertie. We adopt a relatively imple formalim for a compoite ytem involvin fluid and manetofluid dic coupled by ravity. We provide in Section an MHD decription for the coplanarly manetized aeou MSID, obtain condition for the backround axiymmetric equilibrium tate for both tellar SID and aeou MSID, and derive linearized equation for coplanar perturbation. There are alined and unalined clae of lobal MHD perturbation olution; they are analyed in Section 3 and Section, repectively. The exact olution of tationary perturbation, their tability propertie and their correpondin phae relationhip amon perturbation variable are examined and ummarized in Section 5. Detail are included in Appendice A E. FLUID MAGNETOFLUID FORMALISM It would be phyically more precie to adopt a ditribution function formalim in dealin with a tellar dic epecially in term of inularitie and reonance (e.. Lin & Shu 966, 968; Binney & Tremaine 987). For the preent purpoe of modellin lare-cale tationary perturbation tructure and for mathematical implicity (Lou & Shen 3; Shen & Lou 3), it uffice to tart with the fluid manetofluid formalim, includin an MHD treatment for the aeou MSID (Lou ; Lou & Fan 3). In thi ection, we preent the baic equation for the fluid manetofluid ytem conitin of a tellar SID and a aeou MSID. For flat rotation curve, condition on the backround rotational equilibrium with axiymmetry can be derived. We then obtain the linearized equation for coplanar MHD perturbation in the compoite MSID ytem. C RAS, MNRAS 35, 5 Downloaded from on September 8

3 Y.-Q. Lou and Y. Zou. Baic non-linear MHD equation The two SID, located at z =, are both approximated a infiniteimally thin. In our fluid manetofluid treatment, the two SID are coupled throuh mutual ravitational interaction. For lare-cale tationary perturbation, diffuive procee uch a vicoity, ambipolar diffuion, thermal diffuion, etc., are inored. For the phyical variable under conideration, we hall ue upercript or ubcript to indicate an aociation with the tellar SID and upercript or ubcript to indicate an aociation with the aeou MSID. In cylindrical coordinate (r, θ, z), the baic fluid manetofluid equation for a compoite MSID ytem can be readily written out. In the fluid approximation for a tellar SID, the ma conervation, the radial component of the momentum equation and the azimuthal component of the momentum equation are, repectively, t u t + r (r u ) r + ( j ) =, r θ + u u r + j u r θ j r = φ 3 r r, j + u j t r + j j r θ = θ φ θ, (3) where u i the radial component of the tellar bulk velocity, j rv i the tellar pecific anular momentum alon the ẑ direction, v i the azimuthal component of the tellar bulk velocity, φ i the total ravitational potential, i the vertically interated (effective) preure (ometime referred to a the two-dimenional preure), and i the vertically interated tellar ma denity (i.e. tellar urface ma denity). In the manetofluid approximation for the aeou MSID, the ma conervation, the radial component of the momentum equation and the azimuthal component of the momentum equation are, repectively + (r u ) + ( j ) =, () t r r r θ u + u u t r + j u r θ j = φ r 3 r r [ dzbθ (rbθ ) B ] r, (5) πr r θ j t + u j r + j j r θ = θ φ θ + [ dzbr (rbθ ) B ] r, (6) π r θ where u i the radial component of the a bulk velocity, j rv i the a pecific anular momentum alon the ẑ direction, v i the azimuthal component of the a bulk velocity, i the vertically interated a preure (ometime referred to a the two-dimenional a preure), i the vertically interated a ma denity (i.e. a urface ma denity), and B r and B θ are the radial and azimuthal component of manetic field B. The lat two term on the riht-hand ide of equation (5) and (6) are the radial and azimuthal component of the Lorentz force due to the coplanar manetic field. The couplin of the two et of fluid and manetofluid equation () (3) and () (6) i effected by the total ravitational potential φ throuh the Poion interal, namely G( + )ζ dζ Fφ(r,θ,t) = dψ, (7) [ζ + r ζr co(ψ θ)] / where F i a contant ratio with < F, Fφ i the ravitational potential from the tellar SID and aeou MSID toether, and the fraction ( F)φ i attributed to an axiymmetric dark matter halo that i unreponive to coplanar MHD perturbation in the compoite MSID ytem (e.. Shu et al. ). The diverence-free condition for the coplanar manetic field B = (B r, B θ,)i (rb r ) + B θ r θ =, (8) and the radial and azimuthal component of the manetic induction equation are B r t = r θ ( u B θ v B r ), () () (9) B θ = ( ) u B θ v B r. t r Equation () () form the bai of our theoretical analyi. (). Rotational MSID equilibrium For a flat rotation curve in a tellar SID, we write the correpondin anular rotation rate in the form of (r) = a D /r, Downloaded from on September 8 () C RAS, MNRAS 35, 5

4 Confiuration of compoite MSID 3 where a, mimickin an effective iothermal ound peed, repreent the velocity diperion of the tellar SID and D i a dimenionle parameter for tellar SID rotation. We invoke the expedient polytropic approximation = a () to relate the two-dimenional preure and the urface ma denity. The epicyclic frequency κ of the tellar dic i defined by κ d ( ) r = r dr. (3) In parallel, we have the anular rotation rate of the aeou MSID a (r) = a D /r, () where a i the iothermal ound peed of the aeou MSID and D i a dimenionle parameter for MSID rotation. The polytropic relation and the definition of the epicyclic frequency κ for the aeou MSID are imply = a, (5) κ =. To avoid the manetic field windin dilemma in a rotatin dic (e.. Lou & Fan 998a), the backround coplanar manetic field, which i not force-free, i taken to be purely azimuthal about the ymmetry ẑ-axi: B θ (r) = Fr /, where F i a contant (Lou ) proportional to the encircled manetic flux within r, and B r = B z =. In alactic model application, one need to invoke a central bule or other procee to avoid the diverence of B θ a r. From the radial momentum equation () and (5) in a rotational equilibrium and the fact that F φ/ r = π G( + ) by Poion interal (7), one derive the followin expreion for the backround urface ma denitie: ( ) = F a + D πgr ( + δ), (9) = F a ( + D ) C A / δ πgr ( + δ), () where δ / i the urface ma denity ratio of the two coupled backround SID and C A i the Alfvén wave peed in the MSID defined by C A / dzb θ (π ). () From equation (9) and (), it then follow that ( ) ( ) a + D = a + D C A/. Phyically, condition () reult from the baic fact that the ame total ravitational force φ/ r, includin the contribution from the dark matter halo, act on both the tellar SID and manetized aeou SID, and i very ueful in our analyi below. It hould be noted that the rotation rate of the two SID are different in eneral (Lou & Shen 3; Shen & Lou 3). In dimenionle form, condition () can be written in the form of either D = β( + D ) + λ / (3) or D = β ( + D λ / ), where parameter β a /a tand for the quare of the ratio of the tellar velocity diperion to the ound peed of the MSID, and parameter λ C A /a tand for the quare of the ratio of the Alfvén peed to the ound peed in the MSID. In dic alaxie, the tellar velocity diperion a i uually hiher than the ound peed a, o we naturally focu on the cae of β (e.. Jo & Solomon 98a,b; Bertin & Romeo 988; Elmereen 995; Jo 996; Lou & Fan 998b; Lou & Shen 3; Shen & Lou 3). In thi β reime, it follow from condition () that + D + D λ / < + D, (5) implyin that, for β, D < D. (6) Inequality (6) i very important to identify phyically valid mathematical olution of D for tationary MHD perturbation. For pecified parameter β and λ, D and D are related to each other linearly by condition (3). We emphaize that mathematical olution of D and D become unphyical for either D < ord < or both. The key here i that, by inequality (6), we only need to conider D > becaue D C RAS, MNRAS 35, 5 Downloaded from on September 8 (6) (7) (8) () ()

5 Y.-Q. Lou and Y. Zou mut alo be poitive. In our analyi, we mainly ue equation (3) to derive a cubic alebraic equation in term of D and examine olution propertie..3 Perturbation in a compoite MSID ytem For mall coplanar MHD perturbation in a compoite MSID ytem, the baic non-linear equation () () can be linearized in a traihtforward manner, namely: + ( ) r u + t r r θ + j r θ =, (7) u u + t θ j = ( ) a + φ r r, (8) ( ) j + rκ u t + j θ = a + φ θ, (9) for coplanar hydrodynamic perturbation in a tellar SID; + ( ) r u + t r r θ + j r θ =, (3) u u + t θ j = ( ) a r r + φ [ dzbθ (rbθ ) b ] r + C A πr r θ r dzbθ (rb θ ), (3) πr r ( ) j + rκ u t + j θ = a θ + φ + dzbr (rb θ ), (3) π r for coplanar MHD perturbation in a aeou MSID; ( Fφ = G dψ + ) ζ dζ (33) [ζ + r ζr co(ψ θ)] / for the linearized Poion interal; and (rb r ) + b θ r θ =, (3) b r t b θ t = r θ ( u B θ r b r ), ( u B θ r b r ), = (36) r for the linearized diverence-free condition and the linearized manetic induction equation. We do not conider vertical variation alon the z direction acro the compoite MSID ytem. With a harmonic exp[i(ωt mθ)] dependence for all perturbation variable, we introduce complex radial variation µ (r), µ (r), U (r), U (r), J (r), J (r), V (r), R(r) and Z(r) for,, u, u, j, j, φ, b r and b θ, repectively. Thu, hydrodynamic equation (7) (9) can be reduced to the form of i(ω m )µ + r r ( r U ) im J r =, (35) (37) i(ω m )U J r = r, (38) i(ω m )J + rκ U = im, for the tellar SID, where a µ / i(ω m )µ + r r ( r U ) im J r =, i(ω m )U J = + C A µ r r r i(ω m )J + rκ U = im + dzr π where a µ / + V, (rr) imz =, r Downloaded from on September 8 + V. Similarly, MHD equation (3) (36) can be cat into the form dzz (rb θ ) [ ] dzbθ (rz) πr r + imr, () πr r (rb θ ), () r (39) () (3) C RAS, MNRAS 35, 5

6 Confiuration of compoite MSID 5 i(ω m )R + imb θ U =, r () iωz = r (r R) r (B θu ), for coplanar MHD perturbation in the aeou MSID. By ettin anular frequency ω = in coplanar MHD perturbation equation (33) and (37) (5), we can contruct lobal tationary MHD perturbation confiuration in a compoite ytem of MSID without invokin the Wentzel Kramer Brillouin Jeffrey (WKBJ) or tiht-windin approximation and analye their propertie. 3 ALIGNED MHD CONFIGURATIONS Coplanar perturbation in a compoite MSID ytem can be claified a alined and unalined olution (e.. Kalnaj 973; Shu et al. ; Lou ; Lou & Shen 3). For alined confiuration, all treamline and manetic field line are alined in a compoite MSID ytem. For unalined piral confiuration, neihbourin treamline hift relative to each other in a ytematic manner (Kalnaj 973); the ame phyical cenario hold true for neihbourin manetic field line (Lou & Fan 998a). In thi ection, we obtain the tationary diperion relation for alined coplanar MHD perturbation (both full and partial MSID) by perturbation equation (33) and (37) (5) in the precedin ection. The olution behaviour and the correpondin phae relationhip are analyed, mainly in the context of a full SID ytem (i.e. F = ). At the end of thi ection, we derive the MHD virial theorem for a compoite MSID ytem and uet the onet criterion for ecular bar-like intabilitie in a compoite MSID ytem (Otriker & Peeble 973; Binney & Tremaine 987; Shu et al. ; Lou ). 3. Diperion relation for alined perturbation To contruct tationary perturbation confiuration of MSID that are alined, we et ω = in equation (37) (5). Let u firt et / t = or ω = in equation () (5) for coplanar MHD perturbation in the MSID to obtain ) m + m µ + r r m iu + J r ( r iu = r r J =, C A µ r + C A miu C A r r / ( ) iu C A r m r / r / (5) (6) [ r ( )] iu, (7) r r m r / m J + rκ iu = m + C A iu r, ir = B θiu r, Z = i (rr). m r For alined perturbation, we take the followin potential denity pair (Shu et al. ; Lou ; Lou & Shen 3) µ /r, µ /r, V = πgr (µ + µ ), m uch that a µ / + V = contant. For a contant iu, a will be hown preently, combination of equation (6) and (8) and expreion (5) (53) ive J = r µ, (8) (9) (5) (5) (5) (53) (5) (55) iu = m r µ / C A /( r) r. A i contant by equation (5), it i clear that iu i another contant. Conequently, equation (55) and (7) ive [ m r C ] A (m /) iu r µ m r + C A µ =. (57) Subtitution of expreion (53), (55) and (56) into equation (57) ive a relation between µ and µ, namely ( a r r πg m + K A ) µ = πg m µ, C RAS, MNRAS 35, 5 Downloaded from on September 8 (56) (58)

7 6 Y.-Q. Lou and Y. Zou where the two coefficient K and A are defined by K C A / r r and A m r C A (m /) C A / r, repectively. In parallel, we et ω = in equation (37) (39) for coplanar perturbation in the tellar dic and ue relation (5) (53) to obtain another relation between µ and µ, namely ( m a r r πg m + r ) r µ = πg m µ. (6) Combinin equation (6) and (58), we derive the tationary diperion relation r [ (m m m r ) CA (m /) r + πg m r ] m a [ ( ) C {m + A + a πg m C ] [( )( A m r m r r + m CA a r r πg m ) C ]} A m r r = π G (6) for alined coplanar MHD perturbation in a compoite MSID ytem. In the abence of mutual ravitational couplin between the MSID and the tellar SID, repreented by the term on the riht-hand ide of equation (6), the left-hand ide of equation (6) would ive rie to two eparate diperion relation, one for the tellar SID and one for the aeou MSID. The firt one would be (m ) r + πg m r m a =, which ive the tationary diperion relation for alined coplanar perturbation in a tellar SID alone, that i, a inle SID without manetic field. Subtitutin expreion () and (9) for and (with F = ) into equation (63), we obtain ( m ) [ D ( m +) m ] =, which i imply equation (6) of Shu et al. (). The econd factor in the curly brace on the left-hand ide of equation (6) i [ ( ) C m + A + a πg m C ] [( )( A m r m r r + m CA a r r πg m ) C ] A =, (65) m r r which i the tationary diperion relation for alined coplanar MHD perturbation in the aeou MSID alone, that i, a inle MSID with a coplanar manetic field. Equation (65) i imply equation (3..6) of Lou (). By the above reult, it i clear that equation (6) repreent the diperion relation for tationary MHD denity wave in a compoite MSID ytem. A both SID and MSID rotate, the tationarity of MHD denity wave pattern in an inertial frame of reference impoe condition on dimenionle rotation parameter D (or equivalently, D ). For the followin analyi, we ubtitute expreion (), (), (9) and () into equation (6) to yield another form of tationary diperion relation in a compoite full or partial MSID ytem, namely [ m m D (m )D λ (m /) + F m + ] D + δ m ( {m D [D + λ + F + D ] λ / δ )m λ m D m + δ [( + F + D )( λ / δ m ) } ]m λ λ F ( + D)( + D λ / ) δ ( ) m + δ =. (66) + δ Subtitution of expreion (3) for D into (66) ive a cubic alebraic equation in term of D. Equivalently, ubtitution of expreion () for D into (66) would yield a cubic alebraic equation in term of D. A noted earlier, we focu on the cubic equation of D to identify phyical olution of D, becaue D < D a a reult of a > a in typical dic alaxie. For a later examination of the patial phae relationhip between azimuthal manetic field perturbation b θ and the urface ma denity perturbation of the aeou MSID, we combine equation (9) and (5) to obtain (59) (6) (63) (6) Z = iu B θ m r, (67) In view of the exchane ymmetry between the tellar and a SID, relation (6) can alo be obtained by imply ettin C A = in equation (58) and witchin ubcript and. Downloaded from on September 8 C RAS, MNRAS 35, 5

8 Confiuration of compoite MSID 7 relatin the θ component of the manetic field perturbation Z and the radial a flow peed perturbation iu. Uin equation (57) and (67), one can eliminate iu to obtain ( Z = B θµ r CA [ /) m r CA (m /) ], (68) which can be further reduced to [ µ Z = m D λ (m /) ] B θ D. λ / For real µ /Z, the in of the riht-hand ide of equation (69) will determine the phae relationhip between the azimuthal manetic field perturbation b θ and the a urface ma denity perturbation in the aeou MSID. That i, b θ and are in and out of phae for a poitive and neative riht-hand ide of equation (69), repectively. (69) 3. Axiymmetric diturbance with m = For alined axiymmetric diturbance with m =, it would be inappropriate to ue relation (6) or (66) directly. One hould carefully examine equation (37) (5) with ω = m =. With U = U = R =, equation (37), (39), (), (), (3) and (5) can be atified, and equation () i identically zero. By chooin Z r /, µ r, µ r, J r, J r, V ln r, ln r + contant and ln r + contant, the two remainin equation (38) and () are conitent with a recalin of the axiymmetric backround. In the preent context, thi recalin i omewhat trivial. We turn to cae of m below. 3.3 Non-axiymmetric diturbance with m 3.3. Behaviour of D olution We firt come to the alined cae of m =. For a full compoite MSID ytem with F =, it i eay to verify that equation (66) can be atified for arbitrary D. Thi i quite imilar to cae of a inle SID tudied by Shu et al. (), of a inle MSID tudied by Lou (), and of a compoite SID ytem tudied by Lou & Shen (3). However, we note that for a inle partial MSID (Lou ) and for a compoite ytem of two coupled partial SID (Lou & Shen 3), uch tationary alined eccentric m =perturbation are not allowed for arbitrary D. Likewie, in our cae of a compoite partial MSID ytem with F <, it i eay to ee that equation (66) can no loner be atified for arbitrary D by imply ettin m =. In the followin, we analye cae of m and focu on the pecial cae of F = for a full compoite MSID ytem. A mentioned earlier, we work in the parameter reime of β with a typical dic alaxy in mind. By an extenive numerical exploration of alined cae from m =to 5, we note empirically that cae of m > are fairly imilar to the cae of m =. For thi reaon, we hall mainly conider the cae of m =. A noted earlier, we can ubtitute expreion (3) for D into equation (66) to derive a cubic alebraic equation of y D a A + By + Cy + Dy 3 =, where m = and F =, and the four coefficient A, B, C and D are explicitly defined by A 3β 3λ + λ δ + 8δ 8βδ 6βλ δ + 8β + 6λ + 3β δ 6λ δ, B λ 3βλ δ 8 + 3β + 8β δ βδ δ + 6λ 6λ δ + 3λ δ, C 3βλ δ + βδ + 6β β, D 6β δ 6β. Formally, three mathematical olution for y D can be written out analytically from cubic equation (7) or from the more eneral cubic diperion relation (66) with m > and F <, althouh the olution expreion are fairly involved (ee Appendix C for detail). Practically, we explore numerically variou parameter reime for the three D olution. For pecified value of parameter δ and λ,we how three mathematical olution y D veru β in Fi 3. For typical parameter, the three olution branche of D do not interect with each other. For the convenience of dicuion, we ue y, y and y 3 to denote the upper, middle and lower olution branche, repectively. Generally peakin, the upper two olution branche, y and y, are qualitatively imilar to thoe in a compoite unmanetized SID ytem (Lou & Shen 3) that i, the y branch alway remain poitive, while y decreae monotonically with increain β and become neative when β exceed a certain critical value β c. A expected for mall λ (i.e. weak manetic field; ee Fi. ), y and y repectively obey the limit of thoe in compoite SID without manetic field (ee equation 5 and 55 of Lou & Shen 3), howin that, a λ, y and y conitently approach the upper and lower branche repectively of Lou & Shen (3). One novel feature i the lowet olution branch y 3 due to the preence of manetic field. In mot cae, y 3 i neative when β varie from to +, but for ome pecial parameter y 3 may become poitive when β become maller than a critical value β c. Analytical expreion of β c and β c for the cae of m = can be derived from a quadratic equation and are iven by the followin pair: C RAS, MNRAS 35, 5 Downloaded from on September 8 (7)

9 8 Y.-Q. Lou and Y. Zou m =, δ=., λ =.9.5 y D.5 y βc β c.5 y β Fiure. Three olution curve of D veru β for the alined cae with m =, δ =. and λ =.9. m =, δ=., λ =.5 y D.5 y β c.5 β c y β Fiure. Three olution curve of D veru β for the alined cae with m =, δ =. and λ =. 3.5 m =, δ=., λ =3.6 D.5 y.5 y β c.5 y β Fiure 3. Three olution curve of D veru β for the alined cae with m =, δ =. and λ = 3.6. [ β c δ λ δ + (6δ + 8δ λ δ + 6 9λ δ + λ δ + λ + λ δ 8λ ) /], 8 + 3δ (7) [ β c δ λ δ (6δ + 8δ λ δ + 6 9λ δ + λ δ + λ + λ δ 8λ ) /], 8 + 3δ (7) where β c alway remain larer than β c. We note the followin. Firt, there i no eential mathematical difficulty of obtainin more eneral form of β c and β c for arbitrary m value with < F <. Secondly, by ettin λ = in expreion (7) for β c, we obtain ( ) β c = 3 + δ +, (73) Downloaded from on September 8 C RAS, MNRAS 35, 5

10 Confiuration of compoite MSID y 3 m =, δ=., λ =.9.5 µ /µ.5 y.5 y β Fiure. Curve of µ /µ veru β for the alined m = cae with δ =. and λ =.9..5 m =, δ=., λ =3.6.5 β c y µ /µ.5 y Fiure 5. Curve of µ /µ veru β for the alined m = cae with δ =. and λ = 3.6. The uppermot branch related to y 3 i only dicernible for β< and i not hown here. Specifically, thi i caued by a neative y 3 when β> (ee Fi. 3). conitent with the cae of m = in expreion (56) for β c by Lou & Shen (3). Finally, while β c i real in many cae [i.e. a poitive determinant in both expreion (7) and (7)], β c may be too mall to be dicernible in the parameter reime of β. For ome pecial parameter pecified, β c become noticeable. For example, in Fi. with δ =. and λ =, we have β c =.667 > accordin to expreion (7). In thi cae, for β maller than.667 (till quite retrictive), there are three poitive olution of D.Forβ c <β<β c, the upper two branche y and y are poitive, correpondin to two poible tationary perturbation mode. When β exceed β c, only y remain poitive, correpondin to one poible tationary perturbation mode. For the uual cae of β c <, there are at mot two poible tationary mode when β (ee Fi. 3) Phae relationhip amon perturbation variable We now examine phae relationhip amon the azimuthal manetic field and the urface ma denity perturbation, becaue they may provide clue for manetized piral alaxie throuh optical and ynchrotron radio obervation (e.. Mathewon, van der Kruit & Brouw 97; Beck & Hoerne 996; Fan & Lou 996; Lou & Fan 998a,, 3; Frick et al., ; Lou ; Lou et al. ). For the phae relationhip between the two urface ma denity perturbation µ and µ, a combination of expreion (), (9) and (6) ive µ = [y(m ) m ]( + δ), (7) µ m (y + ) where y D. Subtitution of expreion (7) into equation (7) lead to a cubic alebraic equation of µ /µ. We how different curve of µ /µ veru β in Fi and 5 by pecifyin different value of δ and λ. In parallel, we examine the phae relationhip between the azimuthal manetic field perturbation b θ and the urface ma denity perturbation in the aeou MSID. By mean of equation (69), we introduce a dimenionle q parameter q [m D λ (m /)] D, (75) λ / whoe in determine the phae relationhip between Z and µ. Uin equation (75), we may expre D a a function of q, namely D = λ (q + m ) q + m. C RAS, MNRAS 35, 5 Downloaded from on September 8 β (76)

11 3 Y.-Q. Lou and Y. Zou.5 y 3.5 m =, δ=., λ =.9 µ /Z.5 y y β Fiure 6. Curve of µ /Z veru β for the alined m = cae with δ =. and λ =.9..5 m =, δ=., λ =3.6 β c µ /Z.5 y y Fiure 7. Curve of µ /Z veru β for the alined m = cae with δ =. and λ = 3.6. The uppermot branch related to y 3 appear omewhat flat at about and i not hown here. A combination of equation (66) and expreion () ive a cubic alebraic equation of D ; and ubtitution of expreion (76) into the reultin cubic equation of D lead to a cubic equation in term of q (with m =). We preent different curve of q veru β for different value of δ and λ in Fi 6 and 7. From expreion (7) and (75), one can how that both larer µ /µ and µ /Z correpond to maller y. The lowet branche of µ /µ or µ /Z are then related to the uppermot y ; the middle branche of µ /µ or µ /Z are related to the middle y ; and the uppermot branche of µ /µ or µ /Z are related to the lowet y 3 a hown in Fi 7. Several interetin feature are noted here for thee curve. Firt, when λ i mall (ee Fi. ), the lowet and middle branche of µ /µ repectively follow thoe in a compoite SID ytem without manetic field (ee equation 58 and 59 of Lou & Shen 3), howin conitently that, a λ, the middle and lowet branche of µ /µ correpond to the lower and upper branche of Lou & Shen (3), repectively. Secondly, µ /µ and µ /Z of each olution branch of y bear definite in in mot cae. For the cae of δ =. and λ =.9 a hown in Fi, and 6, the tationary perturbation mode of y branch ha a phae relationhip between the urface ma denity of the two SID (i.e. µ /µ ) and a phae relationhip between the urface ma denity of the MSID and the azimuthal manetic field (i.e. µ /Z), both alway bein out of phae. In mot cae, the tationary perturbation mode of y branch ha µ /µ bein in phae (ee Fi. ) and µ /Z bein out of phae (ee Fi. 6). The tationary perturbation mode of y 3 branch ha both µ /µ and µ /Z alway bein in phae (ee Fi and 6). Thirdly, for the tationary perturbation mode of y branch, both µ /µ and µ /Z have a common zero-point at β c that increae with increain λ. For a ufficiently lare λ (e.. λ = 3.6), β c become reater than. Thu, µ /µ and µ /Z may become zero or even carry oppoite in (ee Fi 5 and 7). Our analyi how that the middle y branch for µ /µ and µ /Z ha a common root at β c = (6 + 3δ)λ + 8δ + 6. ( + δ) For a fixed δ, β c increae with increain λ. Specifically for δ =. and λ =.9 in expreion (77), we have β c =.637, maller than. Therefore, the middle y branch alway remain poitive (ee Fi. ). On the other hand, for λ = 3.6 in expreion (77), we have β c =.38, larer than (ee Fi. 5). β (77) Downloaded from on September 8 C RAS, MNRAS 35, 5

12 Confiuration of compoite MSID 3 It i not urpriin that the middle y branch of µ /µ and µ /Z ha a common root a a reult of µ =. The pecial µ = cae may be a poible ituation in our model. By equation (55), it follow that J vanihe. A µ and Z do not vanih, we ee that V, and conequently U do not vanih by expreion (53), (5) and (56). Finally, our analyi how that the uppermot branch of µ /µ and µ /Z correpondin to the y 3 branch ha a root at β =. It follow that y 3 alway remain poitive. 3. Secular bar-like intabilitie For the onet of alined ecular bar-like intabilitie in a compoite MSID ytem, we here derive the MHD virial theorem for a compoite MSID ytem (either full or partial) from the backround radial equilibrium condition () and (5): r = d ( ) a dr dφ (78) dr and r = d ( ) a dr dφ dr C A r. (79) Addin equation (78) and (79), we readily obtain ( + ) d ( r = a dr + ) ( a + + ) dφ dr + C A r. (8) Multiplyin equation (8) by πr dr and interatin from to a finite radiu R, that i allowed to approach infinity eventually, we have the followin MHD virial theorem: (T + U) + W M = πr [ a (R) + a (R)] (8) within radiu R, where T R ( r ) πr dr + R i the total rotational kinetic enery of the two SID, R W r ( + ) dφ πr dr dr i the ravitational enery in the compoite MSID ytem, R ( U a + a ) πr dr (r ) πr dr (8) i the um of tellar and a internal enerie, and R M CA r πr dr (85) i the manetic enery contained in the aeou MSID component. For a full compoite ytem with F =, we combine relation (), (), (9) and () with expreion (8) (85) to obtain R R T = (r ) πr dr + (r ) πr dr = a (D + ) D + [D + + λ /(β) /β]δ R, (86) G( + δ) W = M = R R r ( + ) dφ CA dr πr dr = a (D + ) G r πr dr = λ a ( + D )δ Gβ( + δ) R. R, (87) Baed on early numerical imulation (Miller, Prenderat & Quirk 97; Hohl 97), it wa known that a thin elf-ravitatin dic in rotation may rapidly evolve into bar confiuration (e.. Binney & Tremaine 987). Otriker & Peeble (973) ueted an approximate criterion T / W. ±., neceary but not ufficient, for tability aaint bar-type intabilitie, on the bai of their N-body numerical exploration involvin 3 particle. In the preence of a coplanar azimuthal manetic field with a radial calin of r /, Lou () propoed that the ratio T / W M may play the role of T / W in an unmanetized SID to determine the onet of intability in a inle MSID. By the above analoy and a a natural extenion, we ue T / W M intead of T / W to examine the onet criterion of intability in a full compoite MSID ytem in the preence of coplanar non-axiymmetric alined MHD perturbation. Uin expreion (86), (87) and (88), we arrive at T W M = δ + δβ + δλ + (δβ + β)d δλ + β + δβ + (β + δβ)d C RAS, MNRAS 35, 5 Downloaded from on September 8 = β + δ δλ + β( + δ) (, (89) D + ) (83) (8) (88)

13 3 Y.-Q. Lou and Y. Zou.6 β=.5. T/ W M...8 β=.6. β=.5 m =, λ = δ Fiure 8. Curve of ratio T / W M veru δ for different value of β =.5,,.5 for the alined m = cae with λ = alined cae δ=., β=.5, λ = D m Fiure 9. The D olution of the y branch veru m with parameter δ =., β =.5 and λ =.9. The mallet D occur at m =9. indicatin that the value of T / W M fall between and / a in the unmanetized cae (Lou & Shen 3) and increae with increain D. Therefore in a compoite MSID ytem, the three poible value of D correpond to three different value of T / W M ratio; and larer value of D correpond to hiher T / W M ratio. We here examine a few cae to illutrate the utility of our propoed onet criterion for intability. Firt, we et m =, δ =., β = and λ =. From equation (7), one obtain three D olution D =.86,.553 and.553. Inertion of thee three value of D into expreion (89) would ive T / W M =.38,.9 and.963, repectively. A another example, we et m =, δ =., β = and λ =. From equation (7), we have D =.958,. and.3; the third branch of the olution hould be inored a D <. The correpondin value of T / W M for the firt two D olution are.7 and.986, repectively. From thee numerical etimate, it i then clear that the ratio T / W M can be much maller than. (mainly throuh the perturbation mode of the y branch), analoou to an unmanetized compoite SID ytem (Lou & Shen 3). Shu et al. () ueted a correpondence between T / W ratio and the onet of ecular bar-like intabilitie in a inle fluid SID. It eem natural to extend thi uetion to a inle MSID (Lou ) and to an unmanetized compoite ytem of two SID (Lou & Shen 3). By thi analoy, we now propoe a further extenion of thi correpondence of ratio T / W M and the onet of ecular bar-like intabilitie in a compoite ytem compoed of a tellar SID and a aeou MSID. It then appear, a in the cae of Lou & Shen (3), that the threhold of T / W M ratio can be coniderably lowered in a compoite SID or MSID ytem. Qualitatively, thi illutrate that mutual ravitational couplin tend to make a dic ytem more untable (Jo & Solomon 98a,b; Bertin & Romeo 988; Romeo 99; Elmereen 995; Jo 996; Lou & Fan 998b; Lou & Shen 3; Shen & Lou 3, ). We now explore trend of variation for the ratio T / W M of the y branch. Settin m =, we preent curve of T / W M veru δ when parameter β and λ are pecified (ee Fi. 8). A there are three D olution branche involvin three parameter δ, β and λ, value of ratio T / W M can be diverified. Conequently, the intability propertie of a compoite (M)SID ytem are more complex than in a inle SID. Moreover, poible perturbation mode of the D olution of y branch can vary with different value of azimuthal wavenumber m a hown in Fi 9 and. For parameter δ =., β =.5 and λ =.9, D attain it minimum value around m =9, while for parameter δ =., β =. and λ = 3.6, the minimum value of D occur at m =. In aociation with the mallet value of D, the mot vulnerable tationary perturbation Downloaded from on September 8 C RAS, MNRAS 35, 5

14 Confiuration of compoite MSID D alined cae δ=., β=., λ = m Fiure. The D olution of the y branch veru m with parameter δ =., β =. and λ = 3.6. The mallet D occur at m = in thi cae. confiuration varie with choen parameter. In other word, a lihtly different choice of parameter may lead to entirely different MHD perturbation confiuration. Aain, thi example how the diverity and complexity in a compoite MSID ytem and indicate that bar-like intabilitie (i.e. m = ) may not necearily be the dominant intability in a compoite MSID ytem. UNALIGNED MHD PERTURBATIONS OF LOGARITHMIC SPIRALS In thi ection, we analye unalined perturbation confiuration of loarithmic piral tructure in a compoite MSID ytem. We firt derive the tationary diperion relation for both full and partial compoite MSID ytem. We then focu on the full cae (F = ) and addre the problem of axiymmetric marinal intabilitie with m =.. Stationary diperion relation for loarithmic piral To contruct tationary MHD perturbation confiuration of unalined loarithmic piral in a compoite MSID ytem, we pick the potential denity pair (Lynden-Bell & Lemo 993; Syer & Tremaine 996; Shu et al. ; Lou ; Lou & Fan ; Lou & Shen 3; Shen & Lou ), namely µ = σ r 3/+iα, (9) µ = σ r 3/+iα, (9) where σ and σ are two mall contant coefficient and α i the radial calin parameter related effectively to the radial wavenumber, toether with V = v r /+iα + v r /+iα, (9) where the two contant coefficient v and v are related to σ and σ repectively by v = πgn m (α)σ, (93) v = πgn m (α)σ, (9) with N m (α) K (α, m) bein the Kalnaj function (Kalnaj 97). Conitently, we may write U = u r /+iα (95) and U = u r /+iα, (96) where u and u are two contant coefficient. Uin expreion (9), (9) and (95), equation (6), (7), (9) and (5) lead to m σ + iu (iα + C A r 3 ) m ( ) a σ r r + v + v =, (97) { [ )]} ( C m r κ r C A m A r 3 )( C A r iu ( = C A iα + C )[ A m σ r m ( )] ( a σ r r + v + v + m r iα + 3 )( ) a σ r + v + v m CA σ, (98) ir = r / B θ r iu r +iα = B θiu r, (99) C RAS, MNRAS 35, 5 Downloaded from on September 8

15 3 Y.-Q. Lou and Y. Zou Z = iαr / B θ m r iu r +iα = iαb θ m r iu. A combination of equation (97) and (98) toether with relation (93) and (9) ive {( )[ ] ( Km a r A r πgn m (α) C + Km )} ( ) Km σ = r A πgn m (α)σ, () where the three coefficient K, A and C are defined by ( ) K m m r (α + m )CA, m r ) A ( 3 C A r 3 α, (3) C C ( A C A r r 3 ). () For tationary coplanar perturbation in the tellar dic in parallel, we et ω = in equation (37) (39) and ue equation (9) (9) and (96) to obtain {( m + α + )[ a πgn r m (α) ] () () ( (m ) r }σ r = m + α + ) πgn m (α)σ. (5) Equation (5) can alo be obtained by ettin C A = in equation () and exchanin ubcript and. By equation (5) and (), we derive { ( r (m ) m r r m + α + )[ ]} a r πgn m(α) { [ ( C m m + A + a πgn )( m(α) m + α + ) C ] A r r r [( + m CA a r r πgn )( m(α) m + α ) C ]} A r r ( = π G N m (α) m + α + )[( m + α + ) C ( A r m + α )]. (6) In the abence of ravitational couplin between the tellar SID and the aeou MSID repreented by the riht-hand ide term of equation (6), the left-hand ide of diperion relation (6) would be reduced to two eparate diperion relation factor. The firt diperion relation would be (m ) r r ( m + α + )[ a r πgn m(α) ] = (7) for tationary loarithmic piral in a inle tellar SID without manetic field. Subtitutin expreion () and (9) of and with F = into equation (7), we have ( m + α + ) [ ( + D )N m (α)] =, (8) D (m ) which i imply equation (37) of Shu et al. (). The econd diperion relation would be [ ( C m + A + a πgn )( m(α) m + α + ) C ] A m r r r [( + m CA a r r πgn )( m(α) m + α ) C ] A = (9) r r for tationary loarithmic piral in a inle aeou MSID with coplanar manetic field. Equation (9) i imply equation (3..5) of Lou (). Without manetic field with C A =, equation (6) i equivalent to the diperion relation (86) for tationary loarithmic piral confiuration in a compoite SID ytem of Lou & Shen (3), a it hould be. For numerical computation, there are two ueful formulae of N m (α), namely, the recurion relation N m+ (α)n m (α) = [(m + /) + α ] () and the aymptotic expanion N m (α) (m + α + /) / () when m + α (Kalnaj 97; Shu et al. ; Shen & Lou ). Downloaded from on September 8 C RAS, MNRAS 35, 5

16 Confiuration of compoite MSID 35 For the purpoe of examinin the phae relationhip between the azimuthal manetic field perturbation b θ and the urface ma denity perturbation of the aeou MSID in the full cae of F =, we derive the followin relation. From equation (97) and (5) with F =, we readily obtain [ ( ) πgmσ ( + δ) D iu = [iα + C A /( r ) 3/ ] ( r) δ ( + D λ / ) AN ] m(α), () (A B) where the two coefficient A and B are defined by A ( m + α + / ) (m )D (3) and B N m(α) ( + δ) (m + α + /) ( ) + D. () It follow from expreion () and () that [ ( Z λ α + iα 3 )][ ( + δ)( D ) µ D δ ( + D λ / ) AN ] m(α), (5) (A B) which will be dicued later in our analyi.. Marinal tability for axiymmetric coplanar diturbance While the m = cae i omewhat trivial in the alined cae a a recalin of the axiymmetric backround, it i of coniderable interet in term of tability with radial ocillation (i.e. α ). In thi ubection, we analye thi ituation for a full compoite MSID ytem with F =. However, it would be very mileadin imply to et m = in the zero-frequency diperion relation (6) derived for the non-axiymmetric cae to analye the axiymmetric cae. The proper way to derive the tationary diperion relation for axiymmetric perturbation would require retainin ω and ettin m = in expreion (37) (5) with potential denity pair (9) (9). A remarkable difference between the m = cae with radial ocillation and the piral m cae i that in order to keep equation (37) (5) elf-conitent with ω and m =, U and U carry the followin form: U iωr /+iα and U iωr /+iα, which are ditinctly different from expreion (95) and (96) for the non-axiymmetric m cae in term of radial calin (ee Appendix B). A ubtitution of expreion (6), (7) and (9) (9) into equation (37) (5) lead to ( [ α + )( a r πgn ( (α) )][ r α + )( a + CA πgn )] ( (α) = α + ) π G N r r r (α) (8) in the limit of ω with an exact cancellation of the imainary part. Equation (8) i the tationary diperion relation for axiymmetric diturbance, and differ from the equation obtained by imply ettin m = in equation (6) derived for non-axiymmetric perturbation. By expreion (), (), (9) and (), diperion relation (8) can be cat into the dimenionle form of ( [D + α + )( + λ N (α)( + D ( λ /)δ )][D + δ + α + )( N (α)( + D ) )] + δ ( = α + ) N (α) δ( + D ) ( ) + D ( + δ) λ. (9) By recurion formula () and aymptotic expreion () for the Kalnaj function, we have an approximate expreion for N (α), namely N (α) = (α + 9/)/(α + /)N (α) = (α + 9/)/[(α + /)(α + 7/) / ]. () A ubtitution of approximate expreion () for N (α) and expreion (3) for D into diperion relation (9) yield a quadratic alebraic equation in term of D ; and imilarly, a ubtitution of approximate expreion () for N (α) and expreion () for D yield another quadratic alebraic equation in term of D. A noted earlier, we conider the quadratic equation of D imply becaue of the important fact that D < D when β. For different value of parameter δ, β and λ, we how curve of D veru α in Fi 5. For the poitive portion of olution D with typical parameter, it turn out that the baic feature of a D veru α profile are qualitatively imilar to thoe of a inle SID [ee fi. of Shu et al. ()] and to thoe of a compoite SID ytem (ee fi 5 of Lou & Shen 3). In particular, we emphaize that the lower rin framentation curve dicued in a inle MSID (Lou ) doe not appear here. Thi can be readily een from Fi. 5 where the neative portion of the D curve i alo hown explictly. The appearance of a lower rin framentation curve in Lou () arie from an improper limitin procedure that imply et m = in the zero-frequency diperion relation, derived C RAS, MNRAS 35, 5 Downloaded from on September 8 (6) (7)

17 36 Y.-Q. Lou and Y. Zou m =, δ=., β=.5, λ = rin framentation 6 D 5 MHD denity wave 3 collape α Fiure. The marinal tability curve of D veru α with m =, δ =., β =.5 and λ =. 9 8 m =, δ=., β=.5, λ =3.6 rin framentation 7 6 D 5 MHD denity wave 3 collape α Fiure. The marinal tability curve of D veru α with m =, δ =., β =.5 and λ = 3.6. See Fi. 5 for complete curve m =, δ=., β=, λ = rin framentation D 5 3 MHD denity wave collape α Fiure 3. The marinal tability curve of D veru α with m =, δ =., β = and λ =. for m perturbation a dicued above. The corrected analyi for the tationary diperion relation of axiymmetric perturbation in a inle MSID i preented in Appendix B, where the lower rin framentation curve do not appear (ee Fi B B). A parameter δ, β and λ vary, we note everal trend of variation in the profile of D veru α. For example, when λ increae for fixed δ and β, the rin framentation curve of D appear to be raied, and the collape reime tend to be uppreed omewhat by comparin Fi and (thi trend i more apparent in the cae of a inle MSID dicued in Appendix B). In other word, the preence of coplanar manetic field tend to decreae the chance of collape and reduce the daner of rin framentation. A β increae for fixed δ and λ, the Downloaded from on September 8 C RAS, MNRAS 35, 5