ANALYSIS OF LARGE SCALE DISTURBANCES IN SPIRAL GALAXIES

by ROBERT E. DICKINSON (*)

Summary - - A scale analysis is made of large scale eddying motions superimposed upon the state of mean rotation of spiral galaxies. By this means a balance relation- ship is derived between the fields of potential on the one hand and the horizontal eddy ve- locity on the other. This should provide a useful kinematical relationship for observa- tional astronomy. It is obvious to the meteorological reader that the large scale irreg- ular motions of galaxies are analogous to th~se which occur in our own atmosphere.

1. Introduction - - I t is always encouraging to one who believes in the basic un i ty of scientific methodology to find physical phenomena which are seemingly unrelated to previously acquired knowledge, bu t which in reali ty can be elucidated by familiar techniques. Furthermore, it is especially interesting to find tha t phe- nomena whose unders tanding would add materially to our knowledge of the uni- verse, are amenable to certain familiar research methods which have as yet been bu t little employed for such purposes. This seems to be the si tuat ion for the me- teorologist confronted by the large scale motions which occur in our own galaxy and in the so-called extragalactic nebulae.

I t has been evident for many years tha t large scale asymmetries are present in the luminosi ty distr ibution of the majori ty of galaxies as ascertained by photo- metric measurements, or more simply by visual inspection (see VAUCOULEURS 1959 for a general discussion and an extensive bibliography on measurements). This certainly seems suggestive of irregularities in the mass distr ibution but there are some difficulties in such an interpretat ion. One can optimistically hope tha t observational astronomy will learn to make use of the many complex electromag- netic forms of observation now becoming available, so as to make more direct measurements of the mass distributions of galaxies. At present there have been only indirect measurements of the symmetric mass distributions implied by meas- urements of rotational velocities. Thus unt i l rather detailed new relationships are worked out, one can estimate bu t roughly the asymmetric mass dis tr ibut ion from the asymmetric luminosi ty distribution. The aesthetic reader at this point might tu rn to SAND&GE'S Hubble Atlas of Galaxies or some other photographic collection of galaxies for a momentary f irsthand perusal of the objects of this paper.

(*) Massachusetts Institute of Technology, Cambridge 39, Mass. (U.S.A.).

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Very recently, spectrographic measurements of Doppler shifts have been used to discover large scale i rregular motions superimposed upon the general :rotation in several galaxies (see for instance BURBIDGE, BURBIDGE & PI~EIXDER- aAST 1962). These studies have been of a survey nature , and a more sys temat ic s tudy concentrat ing on one single ga laxy has not as ye t been made. I t is never- theless evident from the da ta tha t is a l ready available t ha t the basic ingredients are i rregular eddy velocities and eddy potent ia l fields superimposed upon a s ta te of mean ro ta t ion - - a s ta te of affairs a l together familiar to the meteorologist . The analogy between galactic and meteorological motions was perhaps first poin ted out b y STARtl & PEIXITO (1962).

Stars and other gravi ta t ing ma t t e r can be assumed to be smeared out into a fluid when considering the mot ion of regions much larger than the distance be- tween stars. Thus the equations of fluid dynamics can be derived for a s tar gas in a fashion similar to the der ivat ion of fluid equations for the molecules of our atmo- sphere. The replacement of discrete part icles by a cont inuum would be expected to lead eventua l ly to a devia t ion of the fluid solution from the exact solution of part icle dynamics. Needless to say, such an error has not received much a t ten t ion from meteorologists developing the science of numerical weather predic t ion be- cause of the manifold difficulties which occur much earlier. Yet i t is f requent ly tbund in theoret ical as t ronomy tha t the models inves t iga ted are so simple t ha t the only behavior t hey exhibi t is this deviat ion of the motions of individual stars fom those predicted b y fluid mechanics (see for instance KURTIt 1957).

I t thus happens t ha t if a meteorologist 's in teres t is sufficiently s t imula ted by the analogy of galactic to a tmospheric motions to make him seek out the current and classical l i te ra ture on galactic dynamics, he would be l ikely to come to the conclusion t ha t the subject has not as ye t been developed sufficiently for this purpose. In view of this circumstance i t is encouraging to note the work of P~E~C- DE~GAST (1960, i962) who made an appl icat ion of the concept of the gradient wind to the flow in the gravi ta t ional field of b ina ry stars and bar red spirals.

The absence of invest igat ions of galactic dynamics similar to the studies of our a tmosphere ' s general circulation may be a t t r ibu ted to the following difficulties, I t is not known how to relate the pressure tensor due to random mot ion of s tars to the parameters of mean flow, since there is no known equat ion of state. Thus the fluid equations are not mathemat ica l ly closed. Fur the rmore the as t ronomer is ve ry re luc tant to even consider the re levant equat ions governing such large scale motions as are observed, because great difficulties are inherent in analysing irregular, nons teady fluid motions. Moreover, observat ional studies of these mo- tions are still in their infancy and bu t l i t t le more is known about them than the i r existence.

Many research workers will p robab ly ned in assent, agree t ha t the problem is wholly impossible and re turn to other ones which, while lacking much relevance to observat ional as t ronomy, nevertheless possess nea t answers, are probably l inear and most cer ta inly are well posed in a mathemat ica l sense. On the other hand the meteorological invest igator will undoubted ly remember, i f he is seriously inter- ested, t ha t for the last two decades the major effort in meteorology has been devoted to the large scale dynamics of the ear th ' s a tmosphere which had been neglected earlier for exac t ly the same reasons which now seem to impede progress in the field of galactic dynamics. This is to emphasize the poin t t ha t the difficul- ties inherent in realistic galactic dynamics are comparable and similar to, indeed

176

almost identical with, the difficulties of a realist ic theory of the large a tmospheric motions. There are of course a wide var ie ty of phenomena which should be incor- pora ted into the ultimate theory, bu t unless a fluid dynamic foundat ion is created within which other physical effects, magnet ic fields, exploding supernovae, random motions of stars, and other per t inen t factors may be included, there can be but l i t t le hope tha t an adequate unders tanding of the large scale i rregular motions in galaxies will be obtained and tha t a realistic discussion of galactic evolut ion will be possible.

I t is well known to the meteorologist t ha t his dynamic equations are not mathemat ica l ly closed in pract ice because mot ion on a scale too small to be incor- pora ted into init ial conditions and too small to be computed in a numerical integ- ra t ion of the governing equations is unavoidable. I f one considers only the large scale motions and incorporates turbulence by a crude parameter iza t ion , i t is still possible to unders tand the large scale motions, and remarkab ly useful results have thus been obtained. In considering the large scale motions of a galaxy, one will have present momentum transfers due bo th to the motions of individual stars and to eddy motions on a scale too small to be considered. The components in the galactic plane of these la t te r forces can in general be expected to be smaller than gravi ta t ional forces so tha t a rough parameter iza t ion of thei r effect, or in some cases even thei r complete neglect, will not obl i terate the major features of the galactic dynamics. I t is ve ry impor t an t t ha t nonlinear models of the free motions of a self g rav i ta t ing fluid be formulated, and t ha t galactic models be inves t iga ted b y numerical integrat ion. Much effort has been devoted in recent years to the analogous meteorological problem, so t ha t a ve ry extensive amount of pract ical knowledge of the in tegrat ion of nonlinear fluid motions has been gained which has ye t to be shared with other sciences. For example, ve ry recent ly a funda- menta l obstacle to long t ime integrat ions, namely the presence of nonlinear com- pu ta t iona l instabil i t ies, has been overcome.

In this paper the wri ter presents a pre l iminary formulat ion of the equations governing the free eddying motions of a gravi ta t ing fluid whose basic s ta te of mot ion is differential rota t ion. Scale analysis is used to isolate the most impor t an t effects. Thus, to use a t e rm coined by STARR, i t is formally shown tha t a s ta te of (( galactostrophic balance ~ exists between the field of eddy mot ion and the poten- t ia l field (and consequently the mass field). As discussed by STARX~ (1963), one m a y through this means qual i ta t ive ly conclude t ha t when galaxies ro ta te with thei r spiral arms trai l ing, there wil~l be an inward t r anspor t of zonal angular momen tum within them. In general there exists away from the nucleus of spiral galaxies a dif- ferential angular ro ta t ion which varies roughly as the inverse of the radia l distance from the center. Thus the ro ta t iona l l inear veloci ty is roughly a constant , usual ly of the order of 200 km/sec. This introduces a fundamenta l difference between galactic and atmospheric flows on a large scale, inasmuch as the galactic eddy flow. is then divergent to a zero order of approximat ion and lines of equipotent ia l are no longer streamlines. When the mean s tate of motion is near solid-body rota t ion, as perhaps occurs in the flow field nearer to the nucleus of galaxies, one may choose a ro ta t ing frame of reference appropr ia te to the given ro ta t ion and leave zero order nondivergent flow.

I t is to be expected t ha t many of the observat ional mysteries which confront astronomers in their s tudy of galactic s t ructure may be be t te r unders tood wi thin the f ramework of a sys temat ic s tudy of the eddy dynamics of a grav i ta t ing fluid.

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The fundamental importance of the irregular nonsteady motions was first empha- sized by HrlSE~BZX~G & WEIZSACKER (1949) and WEIZSACKER (1948), bu t little progress has been made toward their understanding. The most extensive investi- gations of galactic dynamics up to present have been made by LII~DBLAD. One m a y refer to B. LINDBLAD (1959) for a summary and listing of references. He has obtained m a n y interesting results by particle dynamics considerations, but it is often unclear how well these results correlate with the actual field behavior of gravi ta t ing galactic fluids. Recently numerical models using a number of gravi- ta t ing mass points have been integrated by P. O. LI?CDBL_~D (1961). At first these particles undergo motions which make them appear similar to spiral arms. Later they break up into random motions, reminiscent to the meteorologist of the non- linear computat ional instabilities first encountered by PnlLLirs (1956) in his pioneering numerical s tudy of the long te rm integration of the quasigeostrophic equations of the atmospheric general circulation.

In ordinary spirals the eddy potential field created by the mass in the spiral arms has the same appearance as tha t of atmospheric long waves. I f these la t ter baroclinic waves were assumed to be advected with the shearing flow in which they form, they would be rapidly torn apart . However it is fairly well known in the theory of geophysical fluid dynamics how these waves form and persist, the fluid moving through them. On the other hand in as t ronomy it has been some- thing of a mys te ry why spiral arms are not stretched out and torn apar t by the differential rotat ion of the mean galactic flow, and m a n y complicated phenomeno- logical explanations have been offered, whereas what is needed more is probably a systematic fluid dynamics approach.

2. E q u a t i o n s - - The Eulerian equations of motion, the continuity equation, and Poisson's equation may be wri t ten in linear cylindrical coordinates (x', y ' , z') in the form

8u' 8u" v' 8 8u ' 8q~ ' (1) a t ' + u ' - - + - - - - ( , ' u ' ) + w ' = Fx"

- - 8x" r' 8y' ~ + 8x'

8v' 8v" 8v" 8v" u '~ 8 9" (2) ~t" + u ' - - + v " w ' - - + - - + - - = F v '

- - ax' ~ + az' ~' ay '

8w' 8w" 8w' 8w" 8 9" (3) 8t ~ + u ' - - + v ' - - + w' - - + Fz '

8x" 8y ' 8z' 8z'

8p' 8 1 8 8 , t

(4 ) at ~ + 8x ' ( p ' u ' ) + r ' 8y ' ( r 'p 'v ' )+ ~ ( p w ) = O

8zq~ ' 1 8 ( 89' / 82qJ (5) 8x ' ~ + r' 82" r' - - -- 4~Gp'

- - - - 8 y ' ] + 8z'

where (u', v', w') denote velocities in the zonal (0), inward radial ( - - r'), and axial (z/) directions so tha t

8x' dy" dz" V p W t

St' d~" dr'

- - . 178

The q u a n t i t y r ' is the d i s tance f rom the cy l indr ica l axis , t ' is the t ime , 9 ' is the g r a v i t a t i o n a l po ten t i a l , p' is the dens i ty , G = 6.670 �9 10 - s cgs un i t s is the un ive r sa l g r a v i t a t i o n a l cons tan t . Also (Fx ' , Fy ' , F / ) are n o n g r a v i t a t i o n a l forces inc lud ing magne t i c forces, pressures due to r a n d o m mot ions of s ta rs or o ther smal l scale microscopic p h e n o m e n a (e.g., supe rnovae explosions) , gas pressure inc lud ing cosmic r a y pressure , l ight pressure , or any of the o the r possibi l i t ies t h a t have been or m i g h t be discussed in the a s t ronomica l l i t e ra tu re . We define

27r 1 J '

[( )1 ~ 2r~ ( )dO" 0

Le t

r%o' (r') ~ [u' (r ' , O, z = O, t ' = to')]

be the zona l ly ave raged di f ferent ia l r o t a t i o n of the g a l a x y in the ga lac t ic p l ane a t some i n s t a n t in t ime. Also l e t

~y' ~ [Fy' (r', O, z" = O, t' = to')]

where to', a cons tanG is some in i t ia l t ime. W e t h e n define

U H ~ U / - - r / o ) t

F j , " ~ F S - - ,.r .

The equa t ions (1-4) now become

~u" ~u" ~u" v' ~ ~u" (6) - - + ,o'r ' - - + u " - - + - - - - (~'u") + w ' - -

~ ' ~x' ~x' r ' ~y' ~z'

q- - - _ _ v' a (co,r,2) § __~9' = Fx' r ' ~y' ~x'

Ov' r cqv' cqv' cqv' (7) Ot" + o ~ ' r ' - - + u " - - + v' w ' - - +

- - Ox' Ox' ~ § Oz'

(o,'r ' + u") ~ ~9' + + = %' + Fy"

/-" Gqy j

3w' Ow' 3w' 3w' Ow" (8) 3t~ V- + o Y r ' - - § u " - - + v ' - - + w' - -

0x' ~x' By' ~z'

r q- = F / cqz ~

~p' 0p' ~ 1 a (9) ~t~ G- q- r ~x' + ~ (p 'u" ) § --r, --~y' (r' p'v') q-

+ ~z' (p'w') = 0

which cons t i t u t e a new sys t em when cons idered t o g e t h e r w i th (5) which r ema ins unchanged .

- - 1 7 9 - -

3. Scale analysis of galactic motions - - We choose scales appropriate to the large irregular motions in a giant spiral such as our own galaxy so tha t the nondi- mensional variables to be defined are of order un i ty in the system considered. We now define the following quanti t ies:

H ~ a characteristic radius of system, ~ 10 kpc, L = a characteristic horizontal scale of eddying motions, ~ 1 kpc, H --~ a characteristic vertical scale, ~ 1 kpc, f20 = a characteristic mean rotation, ~ 20 km/sec/kpc, c = a characteristic velocity of large scale horizontal eddy motions,

5 km/sec.

We define unprimed, nondimensional variables by the following relationships :

p~l X U t U

We let

where

Defining

cH r" ~ Lr ; o~' ~ f2oO) ; w' ==- w .

L

1 dy') + f2ocLq~ + ' ~ ~II ' ( fo)2rdr + - - f "~y'

~o c

4~G

/

p - r ( 9 2 r . r 6qr

f~0II ~

- - - ~ 6 ) / � 8 4 ~ t ' cqO - - S t '

the local rate of change for a point moving with the mean rotation, and assuming tha t t ime changes moving with the mean rotat ion take place on an advective scale characteristic of the eddies, we may let

L 8 c 8 t - - ~ t ' ;

c St" L 8t We define z to be the small nondimensional parameter

c 1

~0L 4

We let cr = L/zII ,~ 1. The resulting nondimensional equations are:

( ~ ~U~x ~u __UVr ~ ) ~ ) a + u + v ~ - - z z + w v 2 ~ o + r J

(lO) ~9 F~'

~x f20c

- - 1 8 0

(11)

Sv 8v 8v av u 2 ) ~ + ~ + v ~ - y +u a - V + ~ T +2~.

aq Fy" + - - - - - - F y

8y ~o c

( ~w aw aw w aw I 8r Fz" (12) ~\ at 4- u ~ 4 - V ~ y y 4- -~-z / 4- az -- ~o c

8,o 8 8 8 a~ (13) a%- + ~ ( ~ ) + ~ (p~) + ~-~ ( ~ ) - - ~v a~- +

T y - - __ - - ~ + 4- 8z r

(14) ax 2 @~ az 2 r ay P "

We fo rmal ly e x p a n d u, v, w , lax, F~, Fz , ~, 9 in an a sympto t i c power series in r and equa te coefficients:

U = U 0

V ~ V 0

W ~ W 0

Fx = ~Fxl

Fy = ~ 1

Fz = Fzo

= ~o

-~- Z U l 4 - . ~

4-~ v14- . .

4 - z w l + . .

4- ~2Fx2 § . .

+ &Fy2 + . .

4 - ~ F z l 4- . .

+ z qz4- . .

P = Po + z Pz 4- . .

W e e x p a n d (o and r (~6a/ar) in a Tay lo r series abou t some rad ius r o ~ 1 :

( a w ) ( r - - to) 4- ... ~ = ~ o + - ~ - r o

We define

ar = r T F o + T 7 ~ ~ F l f o ( ~ - to) + ...

( 8 w ) 2 L .

~ ~T-~ o zl~

so t h a t

(~ = 6)o + -~- (Y - - Yo)

L

eH

r ar = r o + zv ( y - y ~

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I t is to be noted t ha t the balance of averaged forces in the radial direct ion has a l ready been el iminated b y our definition of ~. This includes pressure due to random star motions and magnet ic fields. I f a t some la ter t ime a mean zonal flow develops with respect to the ro ta t ion t h a t was present a t the ini t ia l t ime t o , our results will still be valid provided this flow remains an order of magni tude smaller t han ~or. Our procedure is analogous to subt rac t ing off the earths mot ion from the to ta l so t ha t the dominant effect is no longer the mean rotat ion. Because the ga laxy is in a s ta te of differential ro ta t ion one cannot obta in a useful single ro ta t ing coordinate system but our development yields most of the benefits of such a system.

We shall assume tha t the asymmetr ic nongravi ta t iona l forces in the horizontal are an order of magni tude smaller t han the asymmetr ic gravi ta t ional forces. This is no t a res t r ic t ion on the method as one can easily formulate a quasi-galactostro- phic balance between gravi ta t ional forces plus other forces and the Coriolis-like forces due to eddying motions. I t seems, however, a reasonable assumption with regard to known da ta t ha t gravi ta t ional forces dominate at least the flows of non- ionized galactic gas and stars.

Our zero order solutions for (10), (11), (12), (13), and (14) are

(15) v 0 -- l 890

~x

(16) u 0 I 890

2o)0 8y

ago (17) F~0 - ~z

(18) ~u 0 8% Dw 0 + = 0

(19) ~29~ § ~29~ + ~29~ ~x ~ ~y~ ~z 2 - P0.

The galactostrophic relat ionship, equations (15) and (16) are analogous to the geostrophic relat ionships of meteorology. There is an approximate s ta te of balance between the eddy veloci ty and the eddy galactic potent ia l field. I t is impor t an t to note, however, t ha t differential ro ta t ion produces an asymmetr ica l relat ionship between the two flow components so tha t lines of constant po ten t ia l are not approximate streamlines. For cer tain radi i near the nucleus of galaxies, the mean ro ta t ion is approx imate ly independent of radius so t ha t the eddy mo- t ions will be hor izontal ly nondivergent to zero order and the zero order ver t ical velocities will vanish. The zero order cont inui ty equation, (18), is t ha t of incom- pressible flow. This m a y be useful for k inemat ical considerations and short t e rm dynamical considerations, bu t in considering the evolut ionary aspects of the ga laxy compressibi l i ty effects will be very impor tan t , as t hey are necessary for the con- version between potent ia l and kinetic energy. I n par t icular , bar red spirals appear to be in a s ta te of lower potent ia l energy than ord inary spirals as discussed b y

.-- 182

I{EISE31BEnG & WEIZSACKER (1949) and may be in a later evolutionary state than ordinary spirals (see discussion by STAR~ & PEIXOTO i962).

The hydrostatic equilibrium in the vertical given by equation (17) is an as- sumption well known in the literature. I t is usually considered to represent a balance between the potential field and random star motion pressure. One may here refer to the discussions of CHAI'qDRASEKttAR (1960), KVRTK (1957) and LI:ND- nLAD (1959). The consequence of a vertical magnetic pressure gradient has been discussed by WOLTJER (1962). I n our derivation, hydrostatic balance follows as a necessary consequence of the state of quasi-galactostrophic balance.

I n our present analysis, the equations which may be used to first order in investigating the dynamics of galaxies are the following:

{~u ~u ~u ~u } - - - - - - v (~ -4- , ) (y Yo) (20) z ~ .§ U-~x d- v ~Y § w ~z

- - v , 2 ( o 0 d- \ ~r ]0J d- ~ - = Fz

(21) { S v ~v ~v ~v }

r ~ t § +v~--y § ,§ ,§

~9 . § ~y

(22) (~w u ~ w ~w ~w ) ~9 ~ t d + ~x + V ~-y + w ~ ? +~=r~

(23) 3p ~ ~ ~ ~v

P-/ + ~ u + + = 0

(24) ~2~ + ~ -4- ~29 x ~ ~y~ ~z ~ = p �9

These are similar to the so-called p-plane equations of meteorology. I t must be cautioned that if one wishes to consider the flows in the nucleus of a galaxy or eddying motions of the scale of a galactic radius, then the curvature terms that have been eliminated by scale analysis will be the same magnitude as the terms obtained. Nevertheless, as the possibility of treating the Cartesian equations of motion often leads to simplifications because of the resultant symmetry, it may be permissible to consider the curvature terms as introducing only quantitative differences so that for many theoretical purposes the Cartesian equations may be used.

I t is a mathematical simplification to consider two-dimensional motions rather than three-dimensional. Furthermore, we lack an equation such as the thermodynamic equation which relates the vertical flow field to the field of eddy potential. I t therefore becomes necessary to make certain assumptions as to the fashion in which the fields relevant to the dynamics vary in the vertical before

- - 1 8 3

one can obta in a suitable equation. Thus a two dimensional approach, seemingly ,consistent wi th the observat ional l imita t ions of as t ronomy, will be the subject of immedia te fur ther work b y the writer. Perhaps one will never see more than a two dimensional cross section of external galaxies.

4. C o n c l u s i o n s - - I t is believed tha t the equations derived are a ten ta t ive foundat ion for fluid dynamic invest igat ions of galactic motions. I t seems highly desirable t ha t as t ronomy have a be t t e r knowledge of the behavior of a self gravi ta t - ing fluid masses where eddying motions are superimposed upon a s ta te of ro ta t ion , before more complicated explanat ions are sought for the observed asymmetr ies of the mass and mot ion fields observed in our own and in o ther galaxies. No ade- quate Euler ian fluid models of such motions have ye t been ut i l ized by astrono- mers, a l though the Euler ian equations are much simpler to handle in realistic problems. I t is of course necessary to parameter ize the effects of small scale mo- t ions in order to solve ini t ial value problems numerically.

The technique of scale analysis which we have employed is f requent ly used in meteorology to isolate the most impor t an t dynamica l terms. I n our a tmosphere and likewise in galaxies, one will always be l imited to a finite observat ional accur- acy. I t is not always necessary to employ equations which are more accurate than observations. I t is a paradoxia l t r u t h t ha t just if ied simplifications of the govern- ing equations m a y allow considerat ion of far more realist ic s i tuat ions than other- wise possible, and thus lead to a far greater unders tanding of observed behavior than could be accomplished with more complicated equations.

Most extragalact ic as t ronomical observat ions relate p r imar i ly to the gas and dus t present or young giant blue stars which form within the gas so t ha t observed motions m a y be a typical of the flow of the ma t t e r as a whole. However, if gravi- t a t iona l forces are dominant , all the mass may be considered to move as a single fluid, the observed motions being a suitable t racer of this flow.

A c k n o w l e d g e m e n t s - - The wri ter is great ly indebted to Dr. V . P . STARR for his great enthusiasm for the subject of this paper and for many p leasant conver- sations exchanged with him. He is indebted to Dr. L. WOLTJEn for a formal course on the dynamics of galaxies which allowed him to reach an apprecia t ion of the present s ta te of knowledge concerning the subject. Dr. R. NEW~LL of M.I.T. has also given encouragement and assistance. This research has been sponsored by the Air Force Cambridge Research Center under Contract No. A F 19 (628) - 2408.

REFERENCES

BURBIDGE E . M . , BURBIDGE G . R . & K . H . PRENDERGAST 1962: The rotation and velocity field of N.G.C. 253 Ap. J . 136, 339. - - CHA~DI~ASEK~AR S., 1960: The Principles of Stellar Dynamics, D o v e r 313 pp . - - HEISENBEI~G W. & WEIZSACKER C. F. y o n , 1949: Die Gestalt der Spiralnebel, Z. P h y s i k 125, 290. - - KUI~TH R. , 1957: Intro- duction to the Mechanics of Stellar Systems. P e r g a m o n P re s s 174 pp . - - LII~DBLAD B., 1959: Galactic dynamics, in H a n d b u c h der P h y s i k L I I I , p. 21-99. - - L~NDBLAD P. O., 1961: Gravitational resonance effects and the formation of spiral structure. Sov ie t A s t r o - n o m y , A J , 5; 3, 376. - - PHILLIPS N. A. , 1956: The general circulation of the atmosphere: a n u m e r i c a l experiment. Q.J.R.M.S., 82, 123. - - PRrNDE~CAST K. H., 1960: T h e m o t i o n

184 --~

of gas streams in close binary systems. Ap. J. , 132, 162. - - 1962: The motion of gas in barred spiral galaxies. In c~ The Distr ibution and Motion of Inters te l lar Matter in Ga- laxies~. L. WOLTJEI~ ed., W. A. Benjamin, Inc. New York 217-221. - - SANDAGE A., 1961: The Hubble Atlas of Galaxies, Carnegie Ins t i tu t ion of Washington, Washing ton D.C., 50 pp. - - STARi~ u P. & PEIXOTO J. P., 1962: Certain basic atmospheric processes and their counterparts in celestial mechanics, Geofisica pura e applicata, 51, 171. - - STAR~ V. P., 1963: Galactic Dynamics in the light of meteorological theory. Geofisica pura e ap- plicata, 56, 163 - - VAUCOULEURS G. DE,, 1959: General physical properties of external galaxies, Handbuch der Physik LI I I , pp. 311-373. - - WEIZSACKER C. F. yon, !948: Die Rotation Kosmischer Gasmassen, Z. ~aturs 3a, 524, - - WOLTJEX~ L., 1962: Magnetic fields and the dynamics of the galactic gas. In ~ The Distr ibut ion and Motion of Inters te l lar Matter in Galaxies ~. L. Wolt jer ed., W. A. Benjamin, Inc. New York, p. 247-255.

(Received 10 August 1963)