Basu, B

Chatterjee, Tanuka

Paul, S N

Tah, Sabitri

IACS Institutional Repository

Indian J. Phys. 73B (4), 579-592 (1999)U P  B—  an in ternationa l jo u rn a lO n  th e  n o n - lin e a r  a n a ly s is  o f  th e  p r o p a g a tio n  o f  d e n s ity  w a v e s  th r o u g h  th e  d isk  o f  th e  g a la x y  : d e n s ity  a n d  v e lo c ity  p e r tu r b a t io nSabitri Tah\ S N Paul2, Tanuka Chatterjee' and B Basu41 D epartm ent o f  M athem atics, V idyasagai C ollege, C alcutta-700 006, India2 Seram porc G irls ’ College, P O  Seram pore, Dist. Hooghly, W est B engal-712 201, India 1 D epartm ent o f  M athem atics, Shibpur D inabandhu C ollege, H ow rah-711 102, India4 B irla  Institu te  o f  A stronom y and Planetarium  Sciences. C alcu tta-700  071, IndiaR e c e iv e d  17 F eb ru a ry  1999, a c c e p te d  3 0  M arch  199 9A h s tr a c t  Density waves m the galactic disk have been considered by many authors asthe real cause o f  generation o f  large scale spiral arms in the disk galaxies. Propagation o f  the density  w aves through galactic disk has been analysed by various authors over the last few decades, but so far, only linear perturbations have been considered In the present paper, we hav e  an a ly sed  the  p rob lem  co n sid erin g  the n o n lin ear e ffec ts It is found that n o n lin ea r  effec ts are im portant in analysing  the density  w ave phenom ena m the ou ter region o f  the galactic  disk.K e y w o rd s  * D isk galaxies, density w aves, non-linear perturbation P A C S  N o s. • 9 8 .5 2 .Nr, 98 35 H j, 98 .65 .C w1. IntroductionThe spectacular appearance of the spiral arms in the disk galaxies is now believed to be the results of propagation of density waves along the galactic disks. Density waves may be generated in the disk of a galaxy by gravitational disturbances in a number of ways under the influence of the differential rotation in the disk [1,2]. The generation of density waves in such a disk has been demonstrated by numerical simulation by Lindblad [3] and Hockney [4,5]. Lin and Shu (6, 7] and Lin [8,9] have established1 the original density wave theory of Lindblad [1] on a strong theoretical foundation by invoking rigorous mathematical treatment to explain various observed phenomena in our Galaxy. These authors adopted a wave solution of the linearized gas dynamical equations and could show among other things that (a) all components of the galaxy, including the gas and the stars, should form similar spiral patterns on the scale of the radius of the disk, and that (b) the galaxies that do not show prominent spiral patterns are mostly devoid of gas and the velocity dispersions of different stellar components in them are© 1999 LACS580 Sabitri Tah, S N Paul, Tanuka Chatterjee and B Basularge enough to suppress the instability completely. Basu [ 10] used the density wave model established by Lin and Shu [6, 7] to explain some observed phenomena in the solar neighbourhood and drew some plausible inferences of a general nature. Basu and Roy [11] extended the model to the inner region of the Galaxy and tried to interpret some of the observed dynamical behaviour of the gas in the central region. Basu eta l \  12] have derived the general dispersion relation from the wave solution of the linearized three-dimensional pressure free gas dynamical equations and deduced some useful conclusions by analyzing the density wave propagation in the outer and inner regions of the galactic disk. Similar analysis was made by Paul and Khan 113] including the pressure term in the gas dynamical equations. These latter authors found that the pressure has but minor role to influence the effects of the density wave propagation. The effects of density waves in the formation and maintenance of global spiral arms in disk galaxies have also been discussed in some details by Bcrtin [14] and by Toomre[15]. The fact that the density waves trigger star formation in spiral arms, has been confirmed by several authors [16-18]. Galactic magnetic field is also influenced by the propagation of density waves [19-21]. These latter authors have found among other things that the strength of the magnetic field is correlated with the strength of the density wave. This correlation again bears relation with the rate of star formation.Thus, density waves appear to have significant influence on various aspects of the manifestation and evolution of spiral galaxies. Different authors have investigated different aspects of the effects produced by the density waves. But for such investigations, linear theories have mostly been used. The higher order effects have rarely been considered. We have therefore, considered it worthwhile to examine the higher order effects on various physical parameters which are influenced by the density wave propagation. These effects have so far remained mostly unexplored. The higher order contribution might significantly change the simple behaviour of the field variables. New Physical interpretations might emerge for the observed phenomena in the galactic disk. In order to verify these probable changes, we have undertaken the project to study and analyze the non-linear behaviour of the field variables under the density wave model of the galactic disk. In particular, we shall first consider the effect of the contribution of the second order perturbations in the field variables on various observed phenomena in the galactic disk. However, in the present paper we have discussed such effects only on the density and velocity components using two-dimensional analysis. In subsequent works, we plan to discuss other aspects of the problem including the three-dimensional analysis and additional parameters such as gas pressure and magnetic field.2. The basic assumptions and equationsThe following assumptions have been made :(a) The effect of the galactic magnetic field is not significant and so can be ignored.(b) The dynamics of the galactic disk can be studied by using the pressure-free gas dynamical equations.(c) The density of gas in the disk is non-uniform and is a function of (r, 0).(d) The rotational motion of the galactic disk is perfectly circular and the velocity is a function of the radial distance r only.On the non-linear analysis of the propagation of density waves etc 581With these assumptions, the basic equations to be considered are :dp 1 d 1 d+  +  -  — ( p V) = 0 ,at  r d r  r dO 0 )du du v— +  M — +  -dt  d r  rdu, d 0  ,<90d rdv  dv  v dv  uv 1 d&-----  ^  U    h   — H = ----------- ,d t  d r  r dd  r r ddd 20 1 d(j> 1 d 20d r 2 r d r  r2 dO1= 4 /rGp,(2)(3)(4) .where p is the density of the gas, u and v are the radial and cross-radial components of the velocity, and 0 is the gravitational potential satisfying Poisson’s equation (4). Let the field variables p, u, v and 0 are purturbed asp = p0 + £ p |+ F 2p 2 + .....U =  0  + £W, + F 2M,+.. . ,v=r£2(r)  + tv, + t 2v'2+..., (5)* = * „ + £ * ,  + £ 2 <t>2 + . . . ,where Q  ( r ) is the variable angular velocity of the disk.3. The linear perturbationSubstituting the relations (5) in cq. (1) -  (4), and equating the coefficients of € on both sides, one ge ts the linear equations o f  perturbation  in field variables, asd P\ dll\ , PQ d v \d t  Pl) d r  r dQ+ C2 d p  id e=  o . (6)d u .  d u .  d< p .l- + Q  — ' - -2Q v ,  = ----1d t dd d r(7)B v > * a S vd tK~ _  ^ d $ t ~de+ 2 Q U' ~ ~  r do(8)d 2 <j>. ] dtp. 1d r■ + -  ra 2 0 , _d r d e -= AjtGpx (9)582 Sabitri Tah, S N Paul Tanuka Chatterfee and B BasuEq. (6) -  (9) have been solved by Lin and Shu [6,7], Lin [8,9], Basu [10], Basu et al [ 12], Paul and Khan [13], and by others, by assuming the wave solution of the field variables in the form (assuming two-dimensional case):w, s  expi (Q)t -  n6 + K xr),v, s c x p i ( ( O t - n d + K 2r),p, s  exp/ ( Q ) t - n 0  + K^r),  (10)s  exp/( ( Q t -n 6 + K ^ r ) ,where mis the wave frequency and w, Kt are respectively the wave number in Sand r-directions. The above authors have investigated the various properties of the density wave propagation through the galactic disk, using these solutions of the equations in linear perturbation. Using(10) in (6) -  (9), the first order perturbed field variables are obtained asrDx K , + 2 inQU. = -- \--:--------<1 rD2ni2D] - i r K yK 2 2rQD r(ID( 12)(K2 - i'/C, / r + n2 / r2)--------------0' '03)whereD = K 2 - ( (O -nQ )2 *0 ,Dx -CD -  nQ * 0. (14)Here, K is the epicyclic frequency defined byBasu etal[ 12] and Paul and Khan.[ 13] have derived the dispersion relation in the three- dimensional case and analyzed the density wave propagation in the galactic disk, in particular, in regions close to the galactic centre and far away from the centre. They have also calculated the magnitudes of the perturbation in velocity components in terms of perturbation in the density. (The latter authors [13] included the pressure term in their equations). In calculating the numerical values the authors have used the derived values given in Table 1 of Basu et al [ 12].4. The nonlinear perturbation (second order)Our principal aim in this paper is to make an assessment of the non-linear perturbation effects on the density wave propagation and on the resulting field variables. For the purpose we haveconsidered here the equations with second order perturbation of the field variables. These equations areOn the non-linear analysis o f the propagation o f density waves etc 583f o + n & l + a  d u 2 , P o ^ z  _  1 d  ,  . 1 B  . .a  d o  PoT T +T 1 7 - '  06)d t r d, r ,dS2 9 6- 2 . j O v ,  =  5  *!'d r  r d r  r dOdt d e (17)d v , <9v, dtp. dv. v. e)v. u.v.r ^ + f 2 - -  + - u , + - - f 2 = u, - 1 + 7 - - ! +  U- <9/ d0 2 2^ * r dQ 1 d r  r d6 (18)d 2 tp 1 d t p ,  1 d 2 t p ,7 T  + - ^ -  + — ^ 3 7 T  =  4 * G p 2 . d r  r d r  r 36 (19)These equations are obtained by using the cq. (1) to (5) and retaining the perturbation quantities upto the second order. The second order field variables in perturbation will then be obtained by using eqs. (11)-(13) in eq. (16) -  (19). We obtain the following relations :2i (co -  n Q)u2 -  212 v2 -  P<p^  = - 2 i K ,( K1 / 2Q)u2 + 2i(co -  n Q ) v 2 -Qty]  = (2w* / r) <p2 , { 2 K 2 - i K ,  / r  +  2 n 2 / r 2 )P 2 = - 2nGPn 2i(co — nC2)2/^1 p 0«2 - 2 / n —  V2 - / t y -  = ---- — ----(20)(21)(22)i'K, „ ! VT + 7 r (23)whereP  =InQD^-irK^K2 \ (rD^ + l i n Q )2rCiD rDiKJrPtK3+2inQ) d_ rP drrPx K3 + 2inQ \  rP J+ 2nnD, ‘rKi f L x ( -i„) (rP tK3 +2in£2) = P, +/P2 (say>> 212 r*D2(24)584 Sabitri Tah, S N Paul, Tanuka Chatterjee and B Basu'(rD, , + 2in£2) [iKy(2nQDx - irK}K2) d 2nf2D, -irKyK2 ^rD [ 2 rQD dr 2rQD n'2n£2D] -irKyK2 ' 2 1 [rD{Ky + 2in£2)(2nODl -irK^K2)2rf2D j ' r + 2£2riD2= Q] + iQ2 (say), (25)R =—  rr4nG* J . « -+ ~rD\Ky + 2inQTd  ,Xy +d_dr' rDtK?+2wQ < rDrD, Ky + 2inQ \75 JK 2 - ^  +  ~4 tcGiK y r' i  iK y  n 2 ^a 3 -------- +  “y4ttG Dr( v  2 \" 3  +_____ r r~4ttGiK, n2 ^- -  i  + -__r__r* f 2n£2D) -irKyK2 'AnGX2rQD j(-2 in) = R, + </?2 ( s y ) .(26)Solving for the system of eq. (20) -  (23), we get the second order perturbation quantitiesasU2 _ 2^ r GP2l - r r- JZ ’ (27)andJ_2_ _  __Pi2nG2 KI -  —  +IK 2 n2V z ’r r(28)On the non-linear analysis o f the propagation o f density waves etc 585where X  =  p \ - ^ p - - M c o - n i 2 ) 2r r~+ Q \ - 4 K-ni-■ + 2 i(Q) -  n£2) 2 K l J K ,  +  2 r r  m ]  r r~(u> — n £ 2 )  —4 in £2}•Y -  P \ . ^ l £ s L - H(0- n a ) _K£2_______r r2 nG(29)+ Q \ - 4 K i K ]p 0 - 4 ( ( O - n £ 2 ) 2 2 K j  - iKy | 2n~ > r  r 2, | 4 n+  /?•{—  ( 0 )  — n £ 2 ) ------ -^------r £2 (30)andZ = P { 2 y S r " ' 4^|Po(ft,"'jr2)}+ Q |  (Q) -  n£2) + 4/K, p0t f 1 + R {4( co -  n£2 )2 -  K 2}. (31)5. Numerical resultsFor computation of numerical results in the perturbation ol velocity components, both linear and second order cases, wc have adopted the values of the various galactic parameters from Table 1 given in Basu?/£i/[12].5. / Linear perturbation in velocity components :Using the relations (11) -  (13) we can writew, / p , = ax + /£, and v1 / p { = a2 + ib2, where4;rGrDrDx Ky ^K; + j ~{2n£2 K}) /2 AKl  + ^2r586 Sab itri Tah, S  N  Paul, Tanuka C hatterjee and  B  Basuandb , = -4 n O  2 » 0 ^ > y r 2 J + D ,K 3rD „2 V Kl* 3 + " t ] +  ~ 2 '  r ) ra*, =  +4 nG  2QrD(2 n£2 0 ,)  ^ ?  + nY r2 ^ j + K32KK j  +  -K tb j  — +4 k G 2Q rD(- rKi K 2) ( K j + n~ /2 )+- (2nQD,Ky)Kjwhere is the radial wave number in density perturbation for which the adopted value by Basuetal [ 12] was nKpc~x. The same value has been used here. Using other values from Basu etal  [12], the following Table 1 has been computed.Table 1. The first order velocity perturbation relative to the perturbation in density(K pc)u /p ,(K m M 0 ’. See '.P c ') v /p ,  (Km  M 0 ~‘.Sec-1. 1 4 7 .9 0 140 4 02 6 4 .4 0 1 10.653. 4 2 .7 7 1 3 054. 4 4 .7 3 16 355. 59 20 1 9 7 .5 96 8 3 .0 4 2 3 3 .4 07. 1 13 .68 2 7 0 .0 48. 148 .9 6 3 0 9 .7 19. 176 .12 4 6 7 .9 110. 2 0 7 .2 4 3 9 1 .4 211. 2 2 4 .0 8 4 3 8 .4 512. 2 5 7 .8 6 4 8 7 .2 2Figures 1 (a, b) show the run of the values of M,/pj and v^p, as given in Table 1. For comparison, the plot of the same parameters as obtained by Basu et al [ 12] is also presented (Figure 2 in that paper). It is found that the run of the plots are essentially similar, but the numerical values are somewhat higher in the present computation. This may be at least partiallydue to the difference in the adopted computation technique. It is found that the relative perturbation increases with the increasing distance from the centre of the Galaxy. But Figure 3 of Basu e t a l  112] shows that the absolute values I «, I and Iv I of the first order perturbation velocity systematically fall off with the increasing radial distance. This implies that the absolute density perturbation falls off faster with increasing radial distances. We sec therefore, that the first order perturbation in physical variables generated by propagation of density waves 300 —On the non-linear analysis of the propagation of density waves etc 587Figure lb . The plot of I i , / p ( I ; (1) for the present work, (11) for the previous work of Basu e t a l  112].5 88 Sabitri Tah, S N Paul, Tanuka Chatterjee and B Basudisk.5.2 Second order perturbation in velocity components : Using the relations (27) -  (31) we can writep2 P 2 B\ + lB 2whereAx = ( -2 k G)4 n Q  n  4 n p 0 K 3— — G . +-4 ° ^ g , - 4 * 2  £>, ^ 2 K 2 + ^ - j  G 2 - 4 D , 2 ^ 2 + ^ P I +  ^ 1 P 2 j>A ,  =  ( ~ 2 k G )_  _  A n d  „  4 / k ?n p 0 ^- 4 k 3 d , p 2 — — P i -----------— &f , 2m2>I 4*2D. P, ^  4n2p0+4*2 D, ( 2K2 + — J G, + ---- ^ 2 2 —  P2+  £ £ l £ i  -  4 D 2 f 2 / f 2 +  — 3- | P 2 | *B , = [ 2 P 2 + ^ - ['*,(4 D,2 -  ^  0,2, - « , P P , e ,2np0 K2 „ • 4 P | P 0 ^ i P |  ”  '  r  * 2+  ^ [ K ( 4 D 2 - K 2 ) r 1-4tf*2p0Gi + ^ -  AG2 P| ~ 4 K iP oD i P2On the non-linear analysis o f the propagation o f density waves etc 589r 212B2 = [/?2(4£>,2 -  K2)+ 4 K ]p 0Q Q l D &  + ^!£o . A ! Pt+ 7 ^ ] - 7 - K (4D ' - K2)AGi - 4 KiPo£2Q2 - 4 K Jp 0DlPi - ^ £ o  . * l />  r r 2£2 *C .= ( - 2 n G )  R 4 nKiPo K3 K 2 D2r £2 2nG+ pS ^ - \ { 2 K 2+ —£2 r2^ Q i \ ^ K , p 0 - 4 D 2 \ 2 K l ^+ G,4D2K,  ^ 4nD. K..K2--------- *— M + ------- L /e. + - 2 — R , ^ r J r 1 £2 2C , = - 0 * C )2 n G QP  4”V o  n 1 „ r 2AT, K 2 4 K^D:— ----------- D.P,+— i-Lr 2 n G Q  1 2 r G, - 4 ^ A P 0G24/zA■ ftFor numerical computations of u2/p2 and Vj/pj* we have usec* the parameter values from Table 1 of Basu etal[  12]. Also the following numerical values have been used :n is the wave number in 0-direction and this is actually the number of spirals ; so n = 2 has been used ; for density wave propagation in the plane of the disk, Basu [10] obtained the radial wave numbers K to be equal with the numerical value K( = nKpc~1; the values used for the basic density pQat different rare given in Column 6 of Table 1 of Basu etal [12].With these parameter values, the numerical values of ujp2 and v^p2 have been computed for different radial distances from the centre, using computer. These values are given in Table 2.A careful inspection of the values of the second order perturbation, reveals several important features. First, the second order perturbation values of u2/p2 are significantly higher than the corresponding first order perturbation values of u{/p] . This is shown in Figure 2a.590 Sabitri Tab, S N Paul Tanuka Chatterjee and B BasuTable 2. Values of the perturbed velocity relative to the perturbed density in the galactic disk (second order perturbations)r u y p /K m .M ,’ 1. Sec '.Pc ') v ,/p 3 (K m  M ^ .S e c(K p c)1. 3 2 2 9 5 1 1 2 .8 52 3 9 9 8 6 1 1 .1 7 03. 6 5 8 7 0 8 6 .0 7 74. 2 1 8 9 7 1 .4 3 7 95. 5 7 1 8 9 1 1 .4 9 86 9 9 6 0 9 5 1 .5 6 97. 148831 1 5 4 .1 98. 2 0 1 0 1 6 3 4 8 .3 09. 184081 8 1 1 .2 310 2 8 2 7 2 2 8 4 8 .5 311. 2 9 9 8 4 5 9 7 5 .8 312. 3 3 7 9 9 5 1250  76Because of the incompatibility of the magnitudes of numerical values, we have shown a logarithmic plots of u jp 2 and u]lpl in Figure 2b. The plots, however, show essentially similar trend of variations in values along the galactic disk. Figure 3 shows plots of Vj/pj and v2/p2 . Here we see that V2/p2 values cover a great range. While the values of u{/p ] are greater than those of u7/p2 in the range 1-8 Kpc of the galactic disk, u2/p2 rapidly increases beyond 8 Kpc and are much higher than u]/p] . We can therefore conclude that the second order perturbationFigure 2a. The plot of I u2 / p , I against radial distance r.591is important in analyzing the density wave phenomena in the outer region of the galactic disk. This may have a bearing on the abrupt termination of the molecular ring beyond 8 Kpc and also on the warping of the galactic disk beyond 10 Kpc. It is instructive therefore, to explore the idea that the large scale dynamics of the galactic disk will be better understood if the study is made on the basis of nonlinear analysis of the galactic dynamics. Better insight is likely to emergeOn the non-linear analysis of the propagation of density waves etcF ig u re  2b. The plot o f I m, / p I I and 1 u2 / p 2 I against r  shown in logarithmic scale.592 Sabitri Tah, S N Paul, Tanuka Chatterjee and B Basuwhen the corresponding dispersion relation is solved and analyzed. Our aim remains to do the same in a subsequent work. It may be noted that the introduction o f the third dimension and the galactic magnetic field, will undoubtedly make the things a lot more complicated, but is likely to give a more complete picture o f the large-scale galactic structure and dynamics.References[1] B Lindblad Stockholm O bs. Ann. 21 8 esp. sec. 5 “Spiral Structure" (1961)[2] J Silk The Big Bang (New York : W H Freeman) p299 (1989)[3] P 0  Lindblad in Interstellar Matter in Galaxies ed. L W altjer (New York : W A Benjamin) p.222 (1962)[4] R W Hockney Ap. J. 150 797 (1967)[5] R W Hockney Pub, Astron. Soc Pacific 80 662 (1968)[6] C C Lin and F H Shu Ap. J. 140 646 (1964)[7] C C Lin and F  H Shu Proc. Nat. Acad. Sci. (U.S.A.) 55 229 (1966)[8] C C Lin Lecturers in Applied Mathematics 9 66 (1967)[9] C C Lin IAU Symposium No. 31 313 (1967)[10] B Basu Bull. Astron. Instt. Czech. 22 30 (1971)[11] B Basu and A K Roy Bull. Astron Inst. Czech. 23 76 (1972)[12] B Basu, R Bandyopadhyay, S N Paul, A K Roy and Gopa Saha Indian J. Phys. 47 614 (1973)[13] S N Paul and M R Khan Bull. Astron. Inst. Czech. 28 300 (1977)[14] G Bertm Phys. Rep. 61 I (1980)[15] A Toomre Ann. Rev. Astron. Astrophys. 15 437 (1977)[16] C C Lm, C Yuan and F H Shu Ap. J. 155 721 (1969)[17] A J Kalnajs Ap. J. 212 537 (1977)[18] H Petit, C T Hua, D Bersier and G Courtes Astron. Astrophys. 309 446 (1996)[19] H Hanasz, H Lesch and M Krause Astron Astrophys. 243 381 (1991)[20] M Chiba and M Tosa Mon. Not. Roy. Astron. Soc. 238 621 (1989)[21] M Chiba and M Tosa Mon. Not. Roy. Astron. Soc. 244 741 (1990)

On the Non-Linear Analysis of the Propagation of Density Waves Through the Disk of the Galaxy : Density and Velocity Perturbation

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On the Non-Linear Analysis of the Propagation of Density Waves Through the Disk of the Galaxy : Density and Velocity Perturbation

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