In this paper the general gas dynamical equations have been solved in the wave form and the general dispersion relation has been deduced. This dispersion relation has been used with simplifying assumptions plausible for special regions of the Galaxy, and results obtained have been shown to be able to interpret some observed dynamical behaviours as well as the distributional property of the gas in those special regions. For example, the analysis has yielded the interpretation of (a) the absence of any wave-pattern in the central region of the Galaxy, (b) the largescale deviation of the gas from the galactic plane in the outer regions of the Galaxy and (c) probably, the large-scale outflow of gas in the central region, as well as the large outward motion of the 3kpc arm. The analysis further indicates that in the solar neighbourhood the rotation curve of the Galaxy may possess a local maximum

Bandyopadhaya, R

Basu, B

Paul, S N

Roy, A K

Saha, Gopa

English

IACS Institutional Repository

Indian J. Phu9. 47, G14-626 (1973)B. B a s u , R. B a n d y o p a d h a y a , S. N. P attl, A. K. R oy AND G o pa  Sa h aDepartment of Mathematics, Jadavpur University, Calcutta-32 {Received 29 June 1972, revised 13 September and 11 December 1972)In this paper the general gas dynamical equations have been solved in the wave form and the general disxJersion relation has been deduced. This dispersion relation has been used with simplifying assumptions plau­sible lor special regions of the Galaxy, and results obtained have been shown to be able to interpret some observed dynamical behaviours as well as the distributional property of the gas in those special regions. For example, the analysis has yielded the interpretation of (a) the absence of any wave-pattern in the central region of the Galaxy, (b) the large- scale deviation of the gas from the galactic plane in the outer regions of the Galaxy and (c) probably, the large-scale outflow of gas in the central region, as well as the large outward motion of the 3kpc arm. The analysis further indicates that in the solar neighbourhood the rotation curve of the Galaxy may possess a local maximum.1. I nthodtjotionThe formation of spiral arms in disk galaxies and their maintenance over a long period of time, inspite of the presence of differential rotation in these galaxies, have been intriguing questions to the astronomers for a long time Lindblad suggested many years ago that the phenomena might bo explained .in terms of the density waves Lindbald’s original suggestion has recently been developed and put on a firm mathematical basis by Lin & Shu (1964, 1966) and by Lin (1967a, b). Basu (1971) has used the model to explain some observefl phenomena in the solar neighbourhood and also to draw some plausible inferences of a general nature. Basu & Roy (1972) have extended the model to the inner region of the Galaxy and tried to interpret, on the basis of their calculations, some of the observed dynamical behaviours of the gas in the central region. All these works suggest that the density wave model of the spiral structure of the Galaxy helps to explain many peculiar observed phenomena regarding the motion and distribution of the gas in it. Tn the present paper we have derived the general dispersion relation from the wave solution of the linnearizied three-dimensional gas dynamical equations and deduced some conclusions, on the basis of the ana­lysis of the dispersion relation under particular conditions.2. Linearized equations derivedThe three-dimensional gas dynamical equations in cylindrical coordinates (r, 0,z), for a gas rotating about the »-axis, areA  n ote on  the den sity  w a v e  m od el o f the g alaxy(1)614A  note on the density wave model etc. 6 1 5_  50dt dr r 60 6z r dr • (2)d t^ ^ d r ^ r  d0^'^dz^~r ~  60 • (3)+ S .3Wdt dr r 60 6z 6z • (4)together with the Poisson’s equation5^ 2 ' r d r^  dO^   ^ 47r^ r/3.5^ 2 (5)Avhoro 0 is the gravitational potential and G the eoiiRtant of gravitation. Before purturbation, let the density, potential and velocity ooinponents be given by,P =  Po 0 =  00 u — 0 V =  rf2(/‘) w; =  0( 6)wJicre Q(r) is the angular velocity at a distance r from the cenU^ r. After pertur­bation these becomeP =  P^ -VP 0 — 00 +  0' u =  u'V — rCl{r)-\-v' w =  w'(7)vvln^ re the dashed quantities are those due to perturbation. Using (7) in (l)-(5) and linearizing in porturbtion quantities, we get the following sot of linearized equations :^P' I c\^P' ^  _L^ o \n — nm +  “  dr- +  r- ae +^» ^dt dO dr+  n  ^  +  -  ^dt ^  60 ^  2Q, ^  _ 2r 60( 8 )(9)(10)6 1 6 B a su , B an d yo p a d h ay a , P au l, R o y  and Sahadw' , Q dw' _' W ^  dd ~  ~dz (11)5r» ^  r dr dz^where k is the epicyclio frequency defined byf - ]The third term in equation (8) is actually\[rp,u') =  +u'^^{rp,)]( 12)(13)N(jw, u' is a small quantity and rp^  changes slowly with r compared to the rapid change of u' with r. Thus the second term appearing in the third brackets above is small compared to the first term. We therefore haveI d , 5m''^ ° -d V3. Solution ojt’ equations and dispbbsion relationsLot us assume the following wave solutions for the equations (l)-(5) . (Lin & Shu 1964, Basu 1971, for two-dimensional case) 'm' =  Gxpi\wt—nd-i~k^ r-\-ljZ\ v' expw* Gxpi\^ u)t—nd-\-h^ r-\-l^ z\  ^ ... (14)p' -rr Gxpi\wt—nO-\-h r^- l^ z^\4>' =  oxpiYuit—nO-\-kgr-\-l^ z\whore w is the wave frequeoy, fct, n and li are the wave numbers in the directions of the coordinates (?, 0,z), respectively n essentially represents the number ol spirals and we shall use n — 2 Using (14) and (8)-(5), the solutions are given byrD_  _  2nQ,D^—irkJc  ^  ^2r£lD ^(15)(16)A(17)A  note on the density wave model etc. 6l1p = 47tG -5i' (18)(19)whereD =  k^-{w ~nQ .)^^0'] and Di =  co—nQ, ^  0Using (]6)-(18) in equation (8) we obtain the dispersion relation asV  _ / f c i  _L  » _  U  _ 2 M i ,  I _  M  (20^Da \d  i)^r 2ClDl\r I * Dl\r ' i » j i V  ~ D V ' '  'Avhore ilg — AnGp  ^ The dispersion relation (20) characterizes the propagation of density waves through the medium of int-crstellar gas. We note that the dis- ])orsion relation (20) involves the wave numbers in radial, tangential and axial directions implying that the density waves propagating in radial, tangential and the axial directions are coupled. The dispersion relation (20) also indicates that the waves do not propagate in the region of the Galaxy whore D and T>i as given by (19) vanish. The same conclusion was arrived at by Lin (1967a) from a quite different approach He called the region where 1) 0 as the principal part ofGalaxy.4. D ispersion  relation in  spboial oasesWe shall now study the dispersion relation (20) under special conditions and analyse the results in order to draw plausible inferences.Case 1. We consider the dispersion relation at large distances from the centre of the Galaxy, that is, as r—> oo. The relation (20) becomesD® D I>i"(21)If wo now assume that h-^  == ■= 1^ , then the relation (21) simplifies to... (22a)A + _ L =  1i),* Dwritten explicitly this becomen ny/%(22b)which shows that the relation between the angular velocity of the pattern oyjn and that of the material £2 is essentially dependent on the opicyclic frequency618 B asu , B an d yo p a d h ay a , P aul, B o y  and Sahaand tho basic density of matter If  we furtlior assume tliat the density waves propagate along the galactic plane, then ~  0. With k^  =  wo nowgeto) =  47r6rpo)i. (23)This shows that there are two modes of propagation of density waves along the galactic plane both of which will be stable in time ifk^  >  4:7tGpqis satisfied. The modes will both be unstable, however ifk^  <  i^rGpQ ... (24)Thus at largo distances from the galactic centre, tho stability of density waves essentially depends on tho mutual relation between the epi cyclic frequency and the basic density of matter.Wo can now study the behaviour of the density waves at large distanco.s from tho galactic centre, using tho numerical values of the parameters involved, as have boon given by Basu (1971, 1972) and by Basu & Roy (1972) Tho last column of table 1 shows that 4^ nGpy^  ^  k'^ , at all distances from tho galactic centre We are concerned with tho outer region of the Galaxy for wliich the inequality (24) has boon derived Since this holds hi the outer regions, tho density waves propagating there along tho galactic disk will be unstable. Tho relation' (23) reduces to the form(j) = (25)where both a and are positive. Relation (25) shows that under the. conditions we have specified, there will be two modes of propagation of the density waves, one of which will decay while the other will be amp.ified. This amplifioztion of density waves in the outer parts of the Galaxy may be physically associated with tho largo distortion of tho galactic disk observed by many authors (Kerr 1957, Burke 1957), in those regions. Tho important observed facts of the dis­torted region arc that it contains a number of spiral features as if the spiral-arm condensations are superposed and that the gas layer greatly deviates from tho mean plane of tho Galaxy (Kerr & Westorhout 1905) Although a number ol explanations have boon bUggosted for those phenomena by various authors, wo like to empliasize that both of the above features find fairly satisfactory ex­planation as being the manifestation of tho amplification of density waves.Case 2. We consider the dispersion relation very close to the centre, that is, for very small values of r. The relation turns out to bo o> =  nO.±{k^— T^TQpJ which is the same as (23). But hero no restrictions have been made regarding thewavo numbers in the tangential and axial directions. The last column of 'table 1 sIjows that the relation ^nGp  ^ >  jg satisfied for small values ofj also. Thus in the central region of the Galaxy all density waves will be unstable loading to the diffusion of any spiral pattern i.hat might be genoratod here. This fact is relevant to observation in the central region, whore no spiral pattern has so far been detected.Cdsc 3. Let us now discuss the nature of the density wave propagating along perpendicular to the galactic disk under the assumption that the radial propagation is not important, that is, k =0 , r ^  1, 2, 3, 4. The dispersion rola- lion (20) then takes the formA  note on the density wave model etc. 61912 — -^) (26)Tlu! relation (26) shows that such density waves do not propagate at large distances iioin the galactic centre, as well as in the region wherek^—{ai—nQ,)^  — 4:7TOp„. ... (27)liolds. But columns 5 and 8 of table 1 shows that the relation (27) holds nowhere in the main part of the Galaxy (1-12 kpc). Thus in the main part of tluj Galaxy Ihe propagation of density waves perpendicular to the galactic plane is important. The velocity of propagation is given byReal LI rw{k^ - { w-nQ.)^]i\ n\{ni'l—(x})\ (28)Tlu^  values of | | for different values of r are plotted in figure 1. We see that thevalues are very largo in 2 5-4 kpc region with a sharp maximum at 3 kpc. The kpc region of the Galaxy should have, therefore, a large-scale disturbance in the axial direction. The validity of this conclusion may be a reality in view of tlio observed facts of large-scale outflow of gas in the central region and the largo motion of the 3 kpc arm. The large-scale outflow of matter over the entire region of the Galaxy may bo dynamically related with the large-scale disturbances in the axial direction in the 2.5-4 kpc region, as has boon envisaged hero.Case 4. Wo now consider a more general case of the density waves for which all the wave numbers in the radial and axial directions are equal, that is,1 “ ~  2^ 3^ =  3^ — 4^ 4^ •'I’he dispersion relation (20) then takes the form.. (29)^2 _  '■^ 21 + 1 .)- vj l  1 , 2n a  Izjj 2 a  DDi1 1-  r^ -\ D i>j620 B asu , B an d yop ad b aya , P au l, R o y  and SahaFigixro 1* The phase velocity m the z -dirootion shown against the distance from the galactic centre. The sharp peak in the :j-kpo region indicates an unusual disturbance in t ho axiul direction in this regionTable 1 shows that throughout the region 2-12 kpc of the Galaxj’^ the inequalitieMD « D ^i .e ., «  (to — nD.)^-\-47TOp^^and «  D,i.e., 2(to--nQ)® < <  k^ .are satisfied. Using these inequalities, the dispersion relation (29) becomes ^ '* 2Qri)l)V "" r^DThis yields5 i !  r ■2r I * 2ClI)D,|2)7i r 4n2 _(fcQ2_P)2naUl^ 1 DDi® J JRelation (31) shows that the density waves oeaso to propagate whore(30)(31)4?t3 ^  (4Q2_J5j2)2^ 2M l®  4Q*£)2i)i2i.e. where (4Q2-P)2 >  16Q2J9 holds.A  note on the density wave model etc. 621On the otherhand, in the region where <  leQ^D holds, there must he two unstable modes of propagation of density waves, one of which will bo damped while the other will be amplified. If, on the otherhand, the wavelengths are such thatV > >  ^2£lrDD^that is, < < ~ 2nrl>is satisfied, then the dispersion relation (30) simplifios to4^ — di(32)(33)Tliis dispersion relation (33) clearly shows that stable modes of density waves wliose wavelengths do not exceed certain critical value characterized by the rela­tion (32), can propagate in the region of the Galaxy where the inequalityP - (c o -w Q )2 > 0  ... (34)holds. Table 1 shows that this inequality holds throughout the region l-12kpc of the Galaxy. Tn Schmidt’s (1965) model, the inequality holds in the region 2-12 Iqjc. The region of validity of the inequality (34) may bo called the principal part of the Galaxy.Tho values of the right hand side of (32) are 8.8, 21.4 and 15 kpc respectively, at r — 2, 6 and 10 kpc. Thus tho modes of density waves in the solar neighbour- hf)od with wavelengths far less than 15 kpc will be stable. The mcjdes of longer wavelength may be stable at the middle region of tho Galaxy. Toomro (1964) arrived at similar conclusions from his analysis of the gravitational instability of tho galactic disk. Ho, however, has limited his analysis to axisymmetric distur­bances.5. P e b t u r b a t io n  op v e l o c it y  co m po n en t  Using equations (15)-(18) we getu' _  rh {^o}—nQ)-\-2inQ 47tGr D^  2 n a [a i-n a )~irh^hp *  2 Q r D  ■ “47tG^  _  ?4 _________p '  u , - n O  ■6 2 2  B a s u , B a n d y o p a d h a y a , P a u l, R o y  a n d  S a h ac^ 'i*H^ 5To SQ.O;*I « 3 SBMOCM§  :? l-H .-( us§  ^ in oCO o oCO oo ot-- ooO i-i O -Hi£>CO IMCO ooo OSCO OSCO ooI11ooIRiaeo o3 >> 0 ■a ©g'S5si ri JII ^  I!S 3 GA  note on the density wave model etc. 623To simplily the diBOussioii wc restrict ourselves in the region whore <i <^t' holds. Basu (1971) obtained the radial wave number to be w-kpo"  ^ from his discussion of the two-dimensional density waves. So the relation <  <.r may bo considered to represent the regions at some fair distances from the galactic centre. In those regions we haveI = 47rQfc^|(nn-gj)| r ^  y... (35)I _  iinnG\{<ii—‘n£i) |... (36)4:7rGli... (37)Since all the parameters involved on the right hand sides of the expressions (35)-(37) arc functions of the central distance r, the ratios of the velocity to density pertur­bations are different at different regions of the Galaxy This dependence is shown in figures 2 and 3. To draw those figures, wo have neglected the term r/,®/r2from the denommators of the relations (35)-(37), since this term is relatively small, Tho figures show the ratios u' and v' increase steadily with r and that the latter is always greater than the former except at r — 2 kpo. This result is consistent with that obtained by Basil & Roy (1972) with their two-dimensional analysis of the density waves. The ratio w'jf)' , on tho otherhand, shows curious behaviour. It has a very sharp peak at 3 kpo, then falls rapidly to a minimum at 7 kpo and again steadily rises to high values till tho boundary of tho region considered. This large perturbation velocity in the axial direction near tho boundary may be associated with tho observed largo deviation of tho gas Irom the galactic plane in this region. Wo can compare the curves in figure 2, with those in figure 4, where wo have plotted \u' \, | v' 1 against r with the assumption that p7p — 0.6 for gas everywhere in tho Galaxy (Basu & Roy 1972) and the values of have been taken from Schmidt’s (1966) model, Figure 4 also compares the curves for | w'fp' \ and \ w'\. Similar curves will be obtained from tho perturbed stellar velocity components on the assumption that p' — O.l for stars. Only tlie magnitudes of the velocities will be each depressed by a factor of 6. We SCO that the curve for ] | steadily falls all the way down from 2 to 12 kpo, while that for v' has a second peak at 10 kpc. This may be regarded os merely a local phenomenon and may be interpreted as tho local region possessing a relatively higher angular velocity. Tho curve for | w' | again shows a very sharp peak at 3 hpc and then falls all tho way down to the boundary. The sharp peaks foi624 Baeu, B an d yopadh aya, P aul, R o y  and SahaIw'Ip ’ I and \w'\ in figure 4 are consistent with our prediction in the previous section in the region between 2.4-4 kpc from the centre, the Galaxy may have an unusual degree of disturbance in the axial direction.i^ ’ uf^ Lirn 2, Tlio i-uiio« of perturbod velocity to perturbed doiisity in the radiul and tangential diroctioijH shown against tho diatancoa from the gulactico centre.Figuro 3. Tho radial and tangential velocity perturbation shown agamst the distance from tho galactic centre. The peak of v' in 10 kpc region probably signifies a highei rotation of material m the solar neighbourhood.A. note on the density wctve inodet etCt mFigaro 'I Tlio uppor curve in the ratio oJ' the porturbod velocity in the axial direction to the perturbed enaity Tho lowei* curve in the perturbed velocity m the axial direction. Both ourvea show peaks at tho .1 kpc region.6. SummaryW g have coiiBidered here tho donaity wavea of a very gonral patt-Grn and how they propagate in different regiona of tho Galaxy. Uang the values of tho various parameters aoross the entire galactic plane, as have been used earlier by Basu (1971) and by Basu & Roy (1972) we have been able to interpret (a) tho ab- aemse of any density wave pattern in the central region of the Galaxy, (b) the large-scale deviation of the gas from the galactic plane in its outer regions. Wo have also tried to interpret the large-scale outflow of gas from tho inner regions, and tho outward motion of the 3 kpc arm, as dynamically related to the very large-scale disturbances in tho axial direction in the 2.5-4 kpc region, as has been suggested from our present analysis. It has also been demonstrated that within tho principal part of the Galaxy given by P —(oi—wQ)® >  0, that is within 2-12 kpc region, stable modes of density waves can propagate provided their wavelengths are limited to certain values. The perturbed axial velocity to density ratio curve as woU as the perturbed axial vel< city curve in figure 4, both confirm tho very high degree of axial disturbances in the 2.5-4 kpc region. Tho perturbed tangential velocity curve suggests that in tho solar neighbourhood tho rotation curve may have a local maximum.626 Basu, B an d yop ad h aya , P au l, K o y  and SahaWe express our thanks to the referee for his valuable oommonts m improvinjr this paper.ItEFEaSNOESBasu B . 1071 Bull. Aalron. Inal Cztxh. 22, no 1, 30Basu B . & R oy, A . K . 1972 Bull, Aalron. Inal Czech. 23, I^o. 2rBurko B . F. 1057 A.J. 62, 90.Korr F . .1 1967 A J. 62, 93.Korr F . J. & Wosterhout G. 1965, Stars arid Stellar Syatc?na, 5, Ch. 9, (Tho Univ. of ChicagoProBs, 111,).Lin C. 0 . 1967a Lecture in Applied Mathemahca 9, 66.Lin C. C. 1967b I.A.U. Sympoaium No. 31, p. 313.Lin C C & Shu F. II. 196-1 Ap. J. 140, 646.Lm  C C. <fc Shu F. H  1966 Proc Nat. Acad Sci, U.S., 55,Schmidt M, 1965 Stars and Stellar Systems 5, Cli. 22 (The Univ. of Chicago Proas, 111.) Toomro A . 1964 Ap. J. 140, 1217.

A Note on the Density Wave Model of the Galaxy

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A Note on the Density Wave Model of the Galaxy

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