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THEORIES OF SPIRAL
STRUCTURE
z2119
Alar Toomre
Department
of Mathematics,
Massachusetts
Institute of Technology,
Cambridge,
Massachusetts
02139
Much
as the discoveryof these strangeformsmaybe
calculatedto exciteour curiosity,andto awaken
an
intensedesireto learnsomething
of the lawswhichgive
orderto thesewonderful
systems,as yet, 1 think,we
havenofair groundevenfor plausibleconjecture.
LordRosse(1850)
A beginninghas beenmadeby Jeans andother mathematicians
on the dynamical
problems
involvedin the structureof
thespirals.
Curtis(1919)
Incidentally,if youare lookingfor a goodproblem...
Feynman
(1963)
1
INTRODUCTION
Theold puzzleof the spiral armsof galaxiescontinuesto taunt theorists. Themore
they manageto unravelit, the moreobstinate seemsthe remainingdynamics.Right
now,this sense of frustration seemsgreattst in just that part of the subject which
advancedmostimpressivelyduring the past decade--the idea of Lindbladand Lin
that the grandbisymmetricspiral patterns, as in M51and MS1,are basically compressionwavesfelt mostintenselyby the gas in the disks of those galaxies. Recent
observationsleave little doubtthat suchspiral "densitywaves"exist andindeedare
fairly common,
but no one still seemsto knowwhy.
To confoundmatters, not eventhe N-bodyexperimentsconductedon several large
computerssince the late 1960shaveyet yielded any decently long-lived regular
spirals. Bycontrast, quite a fewsuch modeldisks havedevelopedbar-like or oval
structures that haveenduredalmostindefinitely in their interiors. This findingis
undoubtedly a major step toward understanding various bar phenomenathat
complicatethe observedspirals, but it also remindsus rudely just howlittle we
really comprehend:
Toavoid the rapid nonaxisymmetric
instabilities associatedwith
this bar-making,the experimental"stars" require (or otherwisethey acquire) random
velocities proportionately muchlarger than those observedamongstars near the
437
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438 TOOMRE
Sun, or than those suggestedby edge-onviewsof other galaxies. It is this dilemma,
of course, that has fueled muchof the recent speculationabout massivebut largely
unseenhalos--thoughthe alternative that somevery hot inner disks or spheroids
mightalready cure those instabilities seemsnot yet to havebeenexcludedfirmly.
Whateverthat answermaybe, it is clear that the spiral dynamicsgets no easier
when even the most rapid but least visible membersof a galaxy need to be
suspectedof helpingdistort the axial symmetry.
Still, one mustbegin somewhere.
Happily,this remainsa subject whereit makes
sense to start almostat the beginning.
2
SOME OF LINDBLAD’S
IDEAS
AsOort (1966) remarkedin a graceful obitnary, Bertil Lindblad’s"long series
papers, beginningalready in Uppsalain 1927, and continueduntil his death, bears
witnessto the struggle withthe spiral problem.It is natural that in this field, on
whichat that time nothing was ripe for harvesting, he did not immediatelyfind
the right path." Oort wasreferring mostly,of course, to Lindblad’slong obsession
with leadingspiral arms(cf his 1948Darwinlecture). Lindbladbelievedsuch arms
to consist of material shed recently fromunstable circular orbits just outside a
spinningcentral nebula that was shrinkingand flattening with time in the manner
envisagedby Jeans (1929)figr the entire Hubblesequence.In that setting, unlike
Chamberlin
(1901) or Jeans himself with tidal forces fromoutside, Lindbladsought
an intrinsic reason for the fi’equent two-armed
symmetry
of spirals. Alreadyin his
earliest papers he thoughthe had foundone in the well.known
dynamicalinstability
of the Maclaurinspheroids, whichgenerates a growingtriaxial shapethat rotates
in space,in a wavelike
fashion,onlyhalf as fast as the fluid itself. Lindbladexpected
the spillage to occurmostlyfromthe twoends of the longer equatorial axis, which
he felt wouldin reality rotate evenmoreslowly.
Suchwasthe origin of Lindblad’slifelong interest in wavesin galaxies. During
the manyyears that he continued to believe in the transient leading arms, this
interest developedmostlyinto a fascinationwith bars. Heoften wroteof the latter
in terms like "vibrations" or "modes"or "density waves"--although
the last phrase
for himusually meantonly motionslike the rotation of a lumpyrigid bodyor of a
Jacobiellipsoid, hardlytrue wavesat all. In retrospect,noneof Lindblad’sextensive
labors to calculate such bar modesseemsworthrecountinghere. Still quite remarkable, however,are his repeatedguesses such as "the mostpromisingapproachto a
solutionof the problemof s, piral structure appearsto be by wayof the barredspiral
nebulae," or that "in the ordinaryspirals ... a plainly developedbarred structure
maynot appear,but there is likely to be a characteristic densitywaves =- 2, which
causes a departure from rotational symmetryand whichincites the formation of
spiral structure by its disturbing action on the internal motionsin the system"
(Lindblad1951).
Muchof this early work obviously dates from an era before 21-cmastronomy
andevenBaade’sexplicit recognitionof the twostellar populations.Yet that alone
does not explain why,as Oort put it, the attempts at spiral-making via gravity
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SPIRAL STRUCTURE439
Figure1 Lindblad’s(1964) sketch of his %irculationtheory?’ (Reprinted with permission
fromAs~rophys.Norv.)
"remained for 35 years practically a monopolyof the StockholmObservatory." A
subtler reason was that by the 1950s it had becomea little too obvious, to Baade
(1963, p. 67) and manyothers, not only "that the primary phenomenonin the
spiral structure is the dust and gas," but also "that we could forget about the vain
attempts at explaining spiral structure by particle dynamics.It must be understood
in terms of gas dynamics and magnetic fields." Amidst such a tide of facts and
opinion, Lindblad stood almost alone in not renouncing his hopes for basically
gravitational explanations of spiral structure--though he did finally concede, in a
review article (Lindblad 1962) still worth reading, that his old leading-arm models
are "not reconcilable with modernevidence." Rather more reconcilable, it seems
now, are two further notions that occurred to Lindblad during his last decade.
Oneof these, stemmingfrom the mid-1950s, was the idea of "dispersion orbits,"
which we return to in the next section. The other notion is pictured in Figure 1,
taken from the second of two papers in which Lindblad (1963, 1964) speculated
explicitly "on the possibility of a quasi-stationary spiral structure."
To give "the typical spiral structure ... a certain steadiness in time" despite the
differential rotation, Lindblad proposed that the two gaseous arms in Figure 1 are
simply a pattern that
rotates rigidly in space with a clockwise angular speed ~o
r.
Outwardof the two (corotation) points F, where the orbital speed equals F, gas
elements movein the retrograde direction, as viewed in the rotating frame; inside
F they tend to overtake the pattern. Nevertheless, Lindblad argued, the pattern
persists. Theself-gravity of the upperarmlI, for instance, ensuresthat anyinterstellar
matter that joins it at, say, the 12 o’clock position will drift slowlyoutwardin radius
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z~lO
TOOMRE
alon9the armas its differential rotation carries it a quarter-turnto the leftward
point G, whereat last it will part company
with armII/III. Then,"thanks to the
attraction from[the other arm]II, [such material] will return towardsthat arc in
a showerof orbits, of whichone is indicated by the dotted curves." Inside the
corotation radius, Lindbladenvisagedanalogouslythat the gas woulddrift inward
whilesliding clockwisealongeither of the spiral arcs I. Subsequently
it wouldtend
"to be assimilated in an outwardmotion" towardthe next arm, to completethe
cycle. Alsodrawnin Figure 1 are someinclined interarm"branches,"whichLindblad
suspectedto be characteristicof the materialin transit.
With this simple sketch Lindblad displayed an intuitive grasp of the gas
trajectories involvedin spiral shock wavesthat wasnot to be surpassedmarkedly
until the workof Fujimoto(1968) and Roberts(1969). Norwas his reasoning
entirely qualitative: If only in a roughand laborious manner,Lindbladreckoned
that enoughextra force to n:taintain armsof 18° pitch angle could be providedby
an arm masstotalling one-tenth of the entire massof a galaxy. In short, this
"circulationtheory" wasa big step forwardfromanythingthat precededit, including
evenOort’s (1956) suggestionthat "perhapsone side of an arm collects material
whilethe other side evaporatesit, so that the armholds a constantposition."
Asit happened,apart fromthose gas orbits, this hypothesisof quasi-stationary
spiral structure was in turn outdistanced soon by the QSSShypothesis of Lin &
Shu (1964, 1966) deliberately bearing the samename.Alreadythese two papers
furnished muchmoredetail and clarity in support of their ownconjecture that
somehow
"the matter in the galaxy can maintain a [spiral] density wavethrough
gravitationalinteractionin the presenceof the differential rotation," andthat "this
density waveprovides a spiral gravitational field whichunderlies the observable
concentrationof youngstars and the gas." Lin and Shu took the big extra step of
including--and from their 1966 paper onwards, of also calculating far more
plausibly--the gravitational forces to be expectedfromtightly wrappedcollective
wavesamongordinarydisk stars. Contraryto whatis often supposed,however,these
majorimprovements
madeor inspired by Lin had remarkablylittle to do with any
"density waves"ever consideredby Lindblad.Instead, they relate muchmoreto that
other idea whichLindbladtermed"dispersionorbits." It is just those orbits, as we
will nowsee followingmostlyKalnajs(1973),that canbe turned(literally) into a quick
introductionto the basic mechanics
still envisagedby Lin(e.g. 1971,1975)and Shu
(1973)for the bisymmetric
spiral waves.
3
NEARLY KINEMATIC
WAVES
Theclassic paper in whichMaxwell(1859)exploredthe stability of Saturn’s rings
focusedespecially on the topic of infinitesimal disturbancesto a configurationof
N equal small massesrotating in centrifugal equilibrium at the vertices of an
N-sided regular polygon, around a heavy mass imagined fixed at the center.
Maxwellseparated this problem into N/2 simpler problems, each involving a
sinusoidal [sin (toO) or cos (m0)]dependence
of the radial and tangential displacementsof the various ring particles upontheir longitudes 0. For each integer wave-
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SPIRAL STRUCTURE441
number0 < m < N/2, he demonstrated that such a polygon can exhibit four distinct
modesof distortion confined to its ownplane--and that all four are pure waves
travelling aroundthe ring, providedthose satellite massesare sufficiently small.
Lindblad (1958, 196l) extended this analysis, with some minor extra approximations, to the m= 1 and m = 2 perturbations of an essentially continuous ring
of particles rotating with angular speed f2 in the fixed, non-Keplerianforce field of
an idealized galaxy. H~likewise obtained four basic modesfor each m, all of which
are indeed wavelike whenthe ring mass is almost nil. Twoof these modes, the ones
he again called "density waves," then reduce to nearly frozen sinusoidal disturbances
to the ring density, rotating in space with almost exactly the material speed f~. As
such they need not concern us further here (though the reader may wish to emulate
Maxwellby resolving the little paradox of whythe obvious tangential attractions
from such orbiting mass concentrations are not immediately destabilizing). Much
more relevant is the slower of the two remaining modes, or "wavesof deformation":
As Figure 2 illustrates for wavenumber
m--- 2, the limiting zero-massversion of that
moderotates in space with angular speed f~- ~/m, where~ is the so-called epicyclic
frequency, or the angular rate of radial oscillation of each individual particle in the
given central force field.
It is essentially this slowly advancingkinematic wave(rather than its fast partner
with speed f~+ h-/rn), composedof manyseparate but judiciously-phased orbiting
test particles, that Lindblad meantby his term "dispersion orbit." That nameitself
now seems only a quaint relic of his hope from the mid-1950s that the gas layer
~t--O
Figure 2 Slowm= 2 kinematicwaveon a ring of test particles, all revolving clockwise
(like the 12shown)with meanangularspeedf~ in strictly similar andnearly circular orbits.
Thesmall elliptical "epieycles," traversed counterclockwise
in the abovesequenceof snapshots separatedin time by exactly one-quarter of the period 27z/~of radial travel along
each orbit, depict the apparentmotionsof these particles relative to their meanorbital
positions or "guiding centers." Drawnfor the case x = x/~D--or one wherethe rotation
speed V(r)= r~(r)= const at neighboring radii--the diagramemphasizesthat the oval
locus of such independentorbiters advancesin longitude considerablymoreslowly than
the particles themselves.That precessionrate equals f~- ~/2, as onecan verify at onceby
comparing
the last framewith the first.
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~2
TOOMRE
a=5
a= 10
a=16.7
Fi.qure 3 Threekinematicdensity wavesof spiral form(Kalnajs 1973), obtained by merely
superposingsomekinematic ring wavessuch as shownin Figure 2.
in a galaxy might spontaneously tend to aggregate or "disperse" into just a few such
distorted rings (an idea that might still apply somewhatto the shearing awayof star
associations born into eccentric orbits). But not at all quaint is the mainreason for
Lindblad’s interest in this dispersal: He had just noticed that, whereas the circular
speed t~(r) varies markedly with radius r in a typical galaxy, the wave speed
~(r)-rc(r)/m, in the bisymmetric case m = 2 only, remains fairly constant over a
considerable range of intermediate radii. This fact greatly intrigued Lindblad--who
did not need to be told that strict constancy would banish wrapping-up worries
or that the nicest spirals tend to have two arms. Yet astonishingly, that is about
as far as he ever got. Per Olof Lindblad agrees that somehowit never occured very
explicitly to his father--despite writing in 1958that "it is likely that the spiral arms
are largely osculating to dispersion orbits"--to combinealready those "orbits" into
any long-lived spiral patterns such as he himself later proposedin Figure 1 !
This logical gap bet~veen Lindblad’s orbits and the density wavesof Lin and Shu
was closed emphatically otlly by Kalnajs (1973), who presented the three striking
geometric examplesreproduced in Figure 3. Each of these pictures consists of just
11 ellipses shifted systematically in longitude according to the rule 20 = ~ In (major
axis). Noneof them pretends to contain any actual dynamics, nor any trace of the
helpful extra densities expected near all the major axes already from Figure 2.
Kalnajs devised these three diagramssimply to dramatize howeasy it is, in retrospect,
to concoct trailing or leading spiral loci of modest or even very pronounced
crowding from nothing more than ad hoc superpositions of individual finiteamplitude ring waves resembling Lindblad’s m = 2 dispersion orbits. Such spiral
patterns are inherently wavelike and thus, quite apart from any deeper questions of
origin or the preference for one shape over another, it is clear that these beautiful
patterns would last forever ~f all the separate ovals could somehowbe persuaded
to precess not just with the small and nearly equal speeds ~q- x/2 but (a) at exactly
the same rate and (b) without change of amplitude. The big task is to do all this
persuading.
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~IRALSTRUCTURE
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4
LIN-SHU
4~3
DENSITY WAVES
WhenLin and Shu almost succeededin providing such assurances with their WKBJ
estimates a decade ago, they concentrated largely--and understandably--upon
condition (a), that of avoiding all waveshear. Evenmoreunderstandably, they
focusedupontightly wrappedwaves,as a majoranalytical shortcut. UnlikeLindblad
with his fewor single rings, Lin and Shu reasonedthat the disturbance gravity
forces should be predominantlyradial in any real two-armed
spiral that is wound
aboutas tightly as the one shownin the last frameof Figure3. Andthese forces,
they reckoned,oughtto affect the collective oscillations of the disk stars and gas
in any given orbiting neighborhood
very nearly as muchas they wouldthere have
affected a strictly axisymmetricdensity waveof the same(large) radial wavenumber.
4.1 Axisymmetric Vibrations
Eventoday, this reviewer knowsof no analytical theory for interactive radial
vibrations with amplitudesapproachingthose in Figure3. However,
the theory for
short axisymmetric
wavesof infinitesimal amplitudegoes backat least to Safronov
(1960). Among
other things, Safronovrecognizedalmost explicitly that the selfgravity of any thin, cold disk of surface massdensity # tends to reducethe local
frequency~o of short axisymmetricwavesof wavenumber
k fromthe epicyclic value
~ to that impliedby
o~2 =~2(r)-2~6~(r){/~l.
(1)
In providingthis formula,Toomre
(1964)in essenceonlyrepaireda coefficient error
madeby Safronov.Thereupon,he very nearly contributed one of his own:Toavoid
the severe Jeans instabilities impliedby Equation(1) for all wavelengths2 2~/k
shorter than
2~,~t(r)4n2G~t )/K2(r),
(r
(2)
the root-mean-square
radial velocity~ru of anythin disk consistingonlyof stars with
a Schwarzschild
distribution of randomvelocities mustexceed
Ou,mi.(r)3.358G#(r)/~(r),
(3)
as Kalnajskindly correctedthis writer (by about30~)before publication. It then
emerged, ironically, that Kalnajs in 1961 had omitted a factor 2~ from an
analogousestimate--and this had dampened
his owninterest in such collective
effects!
Kalnajs(1965, p. 50) redeemedhimself with a flawless first derivation of the
(implicit) mathematicalrelationship betweenthe local wavcnumber
k and frequency
o for an axisymmetric
waveinvolvingstars with an arbitrary rmsspeed~ru =
rather than just ~r, = 0 as in the simpleEquation(1). Lin&Shu(1966)soonrederived
the sameas the crucial m= 0 subcase of their own--far morewidely known-dispersionrelation. In its original integral form,their versionlooksrather different
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444
TOOMRE
~~-~
__ 0.8
Q=2
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o.o
L \
0
/
\o.,/
0.4
08
1.2
1.6
Wavelength,
Figure 4 The Lin-Shu-Kalnajs dispersion relation for axisymmetric density waves of local
frequency ~o ~ v~ and modest radial wavelength 2 in a thin, rotating disk of stars endowed
with Qtimes the minimum
randommotionsrequired by Equation(3).
from the series expressionby Kalnajs [cf Toomre1969, Equation(tl)], but both
implythe identical curvessummarized
in Figure4 for several values of the stability
parameterQ >- 1 at whichthe Jeans instabilities are no longer possible.
The key dynamicalfact in Figure 4 is that the local self-attraction can again
only reduce the frequencyto of the small-amplitudecollective vibrations to some
fraction I v[ of the epicyclicfrequencyx, whichitself continuesto characterizethe
finite-amplituderadial and tangential excursionsof the individual stars whosenet
density changescomprisethat wave.This lowering of the wavefrequency, as all
three authors recognized,is greatest at intermediatewavelengthsand tends to zero
at bothextremes.Theimportantreturn of ~o to ~cas 2 --* 0 occursbecause,although
all stars obviouslysense the sinusoidal force field, only those whosemaximum
radial speeds U are less than about x/k manageto contribute to the wavedensity
a net amountgreater than 60~ of what they wouldhave contributed if they had
merelyorbited in concentric circles before the wavecamealong. That ratio of
cooperativenessindeed reaches zero already at U = 1.841 x/k for Iv I= 1, and at
2.405 ~c/k for v = 0; moreover,from stars with U = ~/k--or ones whoseapocenter
andpericenter distances differ by exactlyone wavelength--one
evenexpectsa bit of
cussedness amountingroughly to a neyative one-tenth contribution to the wave
density (Kalnajs1964, private communication).
4.2 Matching of Precession Rares
Togetherwith various kin involving gas and/or thickness corrections, the axisymmetricLSKdispersion relation from Figure 4 has played a fundamentalrole
in nearly all of the "asymptotic"explorationsof tightly wrapped(i.e. almostaxisymmetric) m = 2 spiral wavesthat Lin, Shu and several collaborators have
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SPIRAL
STRUCTURE
445
conductedduring the past decade. That role has been remarkablydirect: As Lin
&Shu(1964)first illustrated using the muchless plausible Equation(1), the above
reductionof that essential frequencyfromKto I v I K impliesan analogouschange,
fromt2 +__~c/2to t2 +__I v I ~/2,in thespeedsof precession
of the netdensitycontributions
expectedfrom the manyseparate stars belongingto any given meanorbital radius
but gyrating aroundvarious epicycles. Andit wasprecisely in the seemingnuisance
that v dependsuponthe local radial wavelength2 (or uponthe local pitch angle
of the trailing or leadingspiral) that LinandShu,alreadyin 1964,spottedthe grand
opportunityof simplyadjusting those morerealistic precessionrates to the strict
constancywith radius that had eluded Lindblad.
Among
muchelse, it neededto be demonstrated, of course, that this desired
"tuning" of f~+ Ivlh/2 could actually be accomplishedsimultaneously, for any
single patternspeedt2p, overlongenough
stretches of radii andwithplausiblespiral
armspacings. That this is indeed possible without runningbadly afoul of observationswasfirst shownfor the Galaxyby Lin &Shu(1967;see also Lin et al. 1969)
and for M33,M51,and MS!by Shu, Stachnik&Yost(1971 ; see also Tully, 1974,
Robertset al. 1975,Rots1975,Rogstadet al. 1976).
Theessential points of those feasibility studies maybe gleanedfromFigure5b.
Thefive curves in that diagramdescribe, for N = 2, 3, 4, 5, 6, the usual signed
fractions
v(r ; ~p) ~- 2[~p--
t2(r)]/x(r)=vu(r)
(4)
of the local epicyclic frequencywherebyan intendedrn = 2 waveof hypothetical
pattern speedf~e = f~(Nro)wouldappear"Dopplershifted" to an observerorbiting
with speedf~(r) in one rather typical modelgalaxy. That modelis a thin disk
surfacemassdensity/~(r) =/~oexp(- r/ro) analogousto the simplelaw that Freeman
(1970a)amongothers has stressed to be characteristic of manyobservedligtht
distributions. Its dimensionless
rotation curve~’(r) in Figure5a, basedon a formula
given by Freeman,mainlyjust corroborates that the Sun belongssomewhere
near
r = 3to in this crudefacsimileof our Galaxy.
Giventhat Iv I< 1, Figure 5b implies first that only pattern speeds from an
intermediaterange, roughlybetweenf~(3ro) and f~(5ro), makeit evenplausible
contemplateany single m= 2 spiral structure extendingfromthe deepinterior well
beyondour make-believe
Sun.Thehigher speedlimit is here set by the unwillingness
of the self-gravitating disk to carry wavesthat precess evenmorerapidly than the
fast kinematicwavescorrespondingto v = + 1, whereasthe lowerlimit arises from
the similar expectationthat the collective effects near each radius can onlyraise
the slowwavespeedsaboveLindblad’svalue t2- t¢/2. As usual, the radii at whicha
prescribed~p exactly equals f~+__tc/2are here referred to as the outer and inner
Lindbladresonances.
A secondtype of basic restriction is representedby the shadedI v l < 0.44 strip
in Figure5b. It warnsthat already whenthe stability index Qexceedsunity by as
little as 20~, there exists a sizable forbiddenannulusaroundeach proposedcorotation radius. Lin-Shuwavesof obviously low apparent frequencymight still
"tunnel"across any such region (cfToomre1969,Figure 3), but they will there not
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446 TOOMRE
find any local hospitality in the WKBJ
sense. Clearly that gap only widensfor
Q= 1.5 and2.0, since Figure4 thenforbids I v l < 0.61and0.74, respectively.
This secondrestriction has been muchunderplayedin all of the w~vefittings
cited above, whichhavein essencejust postulatedthat Q= 1, uponincludingthose
gas corrections,thicknesseffects, etc. Observations
of nearbydisk stars suggestvery
roughly that 1.2 ~< Q ~< 2 (of Toomre1974a); elsewhere Q remains totally unmeasuredand well-nigh unmeasurable.Onthe theoretical side, quite apart from
anyglobal instabilities, one interesting difficulty withpresuming
that Q ~ 1 in the
gas-rich parts of a galactic disk is that the effective massesof any large chunksof
interstellar matter--or,looselyspeaking,the variousbits and piecesof armspresent
in even the grandest spirals--should then be strongly enhancedby cooperative
density"wakes"frompassingstars (cf Figure13). Theresult, as Julian (1967)perhaps
overcautioned,can be a growthof the stellar randommotions(or Qitself) at a rate
well in excessof any estimatedby Spitzer &Schwarzschild
(1953;see also Woolley
&Candy1968)fromencounterswith "cloudcomplexes."If only for the last reason,
it seemsprudent to concentrate mostlyon those features of Figure 5b that would
surviveevenif Q= 1.5, or at the veryleast if Q= 1.2.
Contraryto whatLindblad(1964) surmisedin Figure 1, such features evidently
include next to nothing frombeyondany of the corotation radii r = Nro, since the
upper strip 0.44 < v < 1 in Figure 5b indicates for each N that one can hopeto
tamperwith the fast kinematicwavesonly over very limited stretches of radii, of
order ro/2, whenQ > 1.2. This nominalextent seemsabsurdly small, comparedeven
with the "short" wavesof lengths 2 of order 2to impliedthere by Figures4 and 5a.
Whatis more, even whenQ = 1, the ranges of radii correspondingto 0 < v < 1
remainmodestalso in modelssuch as shownin Figures 2 and 4 of Shu, Stachnik
& Yost (1971)--whothemsdvesproceededto ignore those modified fast waves
the moreenigmaticgroundsthat "if Lin (1970) is correct, only the region where
~q- ~:/2 < f~p< l) showsan organizedspiral pattern." In a sequel, Roberts,Roberts
&Shu (1975) likewise took no active interest in that short outer segment,now
apparentlybecause"the theoretical behaviorof spiral density wavesnear corotation
is morecomplicated,evenin the linear regime, than has beenpreviouslyassumed."
Onewayor another, this revieweragrees that the v > 0 wavesdo not seemdestined
for muchglory.
This leaves only the bottomhalf of Figure 5b--or moreexactly, only its lowest
quarter or so--if we are to continue this exercise in pessimismwith Q = 1.2 and
1.5. Downthere at last wemeet one very reassuring fact in the sizable range of
radii over whichespecially the v5 curve still managesto remainwithin eventhe
narrow Q = 1.5 corridor - 1 < v < -0.61. Of course, this welcomenews leans
heavily uponthe familiar near-constancyof f~-~/2. After all, not muchhelp is
neededanywayfrom the self-gravity when(a) those slow kinematic speeds are
nearly uniform as Lindbladnoted them to be near the "turnover" radii of most
rotation curves, and providedalso that (b)one contemplatespattern speeds like
f2(5r0) that exceedthose elementaryspeedsonly slightly. In such circumstances,the
comparatively, modesthelp available from the LSKdispersion relation when
Q= 1.2 or 1.5 canstill be wortha great deal.
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SPIRAL STRUCTURE 447
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Rot
n.
Speed,
~ ~"~--~~
a~’"
"~crit/ro"
or/
I
I
I
II
I
I
0
1
2
3
4
5
6
Io
Radius,
r/r
o
I
b
Frequency,
0.44
-0.44
-0.74
-1
-1
ILR
1.2
2
1.5
Drift
5
1
()LR
Time, t
E
Q=
o
I
1
0
Figure5 Theoreticaldata for a thin exponentialdisk of scale length r0 : (a) Rotationspeed
~" andstability length 2,~,, (b) dimensionless
local .frequenciesv2,..., v6, (c) groupvelocity
drift timests(r ; Q), all plottedas functionsof r. Theshadedfrequenciesare locally unattainable whenQ = 1.2. CRmeanscorotation ; ILRand OLRare someinner and outer Lindblad
resonances. Thepoints markedA through E appear also in Figure 4.
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448
TOOMRE
In short, at least whenQ =>1.2, the onlyreasonableoverall wavespeedstip for
two-armedspiral structures as extensive as those seen in MS1and MSl--andless
obviously, perhaps also in our Galaxy-seemto be those that are slow enoughto
placecorotationon, but not beyond,the outskirts of thosesystems.[This conclusion,
by the way,appearsto be completelyindependentof any "conceptof the initiation
of density wavesby the Jeans instability in the outer regions of normalspiral
galaxies (Lin 1970)" that Shu, Stachnik &Yost (1971) felt they were testing.]
Almostas a corollary, however,the exact radius of inner Lindbladresonance(ILR),
if indeedany, seemsmuchmorejittery. It mustalso be quite sensitive to modifications
of the rotationcurveitself in the interior.
As for other angular wavenumbers,
notice that neither such ILRsensitivity nor
any extensivenear-resonancewith ~- ~/m couldhavearisen for m> 2, since a mere
rescaling of the v axis in Figure 5b reveals that the ILRand OLRradii wouldlie
muchcloser to each other alreadywhenm--- 3 or 4. Thisis the simplebut compelling
reason whyLin (1967)urgedthat no one should expectmulti-armedstructures to be
capable of rotating rigidly as very extensive wavepatterns--though he obviously
slippedin phrasingit that "onlytwo-armed
spirals can be expected."In fact, just a
halving of all labels on the v axis in Figure 5b explains whyLin, Yuan&~hu
(1969)wrotelater that "the possibility that one-armed
galaxiesexist cannotbe ruled
out by the present discussion"for Q -- 1. Curiously,however,at least the Q >~1.3
exclusionsoughtindeedto rule out all local prospectsfor slowm= 1 waves.
Finally, as regardswavelengths,
onlyone thing seemsfairly certain. It is that the
large but typical valuesof the referencelength.~cr~,(r) shownin Figure5a implythat
the observedinterarm distances can be accommodated
in disks of dominantmass
only with wavesbelongingto the short side of Figure 4--or to the side dominated
by the randomepicyclic motions,rather like the I v [ = 0.6 choice C instead of A
in that diagram.This importantinsight stemsuniquelyfromLin &Shu(1966,1967).
Indeed the analogousaxisymmetricwaveswere knownalready to Kalnajs (1965),
but it was Lin and Shu whofirst appreciated and pointed out vividly that just
those short collective wavescan be remarkablyuseful.
A contrary view was voiced by Marochnik,Mishurov& Suchkov(1972). These
authors stressed, in effect, that the original long-wavehopesof Lin &Shu(1964)
mayyet deserve resurrecting. Toillustrate that evensuch wavescan be madetight
enough,Marochniket al. immobilizedeverything in the Galaxyexcept a uniform
-~. That use of the old trick of reducing
"population I" disk with /~ = 40 M~pc
).¢~, seemsa bit extreme(and indeed it wastechnically weakin admittingneither
any shock dissipation nor the refraction of someyet shorter wavesback from
corotation), but it does provide one good closing reminder: Given our poor
knowledgeof galactic halos and/or hot disks, no one should be hugelysurprised
if the effective length 2~,i~ fromexamplessuch as in Figure 5 shouldneed to be
reducedby factors of order 2, at least for tightly wrappedwavesand especially
toward the interior. This wouldstill not upset the impression that the most
prominentreal wavesare apt to be "short" ones. However,it does makeit almost
meaninglessto debate nowwhetherone modelor function Q(r) or pattern speed
f~ results in a photogenicdiagramthat fits observationsbetter than another.
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SPIRAL
STRUCTURE
449
4.3 GroupVelocity
After this longdiscussionof whatmightbe feasible, it needsto be said plainly that
none of these WKBJ
studies of what are sometimesstill called "self-sustained"
density waveshas yet comeclose to demonstratingthe long-term survival of any
such spiral patterns. Eventoday,it is not just the origin but eventhe persistence
(cf Oort 1962)of these wavesthat remainsdistressingly unexplained;all weknow
for sure nowis that the twotopics are nearly inseparable!
In an early volumeof this series, Lin (1967) wroteinstead that his workwith
Shu"essentially amountsto establishing the possible existenceof certain density
wavesof the spiral form,sustainedby self gravitation. Theseare collective modes
in the combined
gaseousand stellar system." Evenat that time, it wasclear that
the WKBJ
approximationsunderlyingthe basic Lin-Shutheory becomeinvalid near
(and beyond)any radius where Iv[= 1--and that there, as Prendergast (1967)
remarkedin an excellentshort review,"the solution gets verydifficult, and onehas
not yet succeeded in guiding it properly through the Lindblad resonance."
Nevertheless,it seemedto Lin that his success with Shuin adjusting the speeds
f2+lvl~c/2 over most of the radii betweenthose resonanceshad in large measure
already established a "theory free fromthe kinematicaldifficulty of differential
rotation."
Thefallacy of such reasoningbecameevident in 1969, whenit was noticed that
all this tuning of precessionspeedshad concernedonly the phasevelocity of what
mightlogically be just a traveling wave(in the usual technical sense) rather than
a genuineglobal mode.Lin and Shuhad completelyoverlookedthat in repairing one
serious defect they had actually created another: Aninevitable price for altering
those speedsof precessionin a wavelength-dependent
mannervia the (very sensible)
radial forcesis a ~Troup
velocity,likewisedirectedradially. Andit is this secondkind
of wavetransport, familiar enoughfromcountlessother situations, that sooner or
later even here plays havocwith the amplitudes--whichthe kinematic ovals in
Figures2 and 3 at least had the decencyto keepinvariant. Lin (1970) agreedthat
this "consideration... brings the problemeveninto sharper focus," and Shu(1970)
concededthat the "amplification of trailing waves"announcedby Lin &Shu(1966;
see also Lin 1967)had resulted froma similar misunderstanding.
The demonstration that the waveamplitudes--or more exactly, the action
densities--areindeedconveyed
radially is fairly complicated,andseemsbest left to
the papersof Toomre
(1969), Shu(1970), Dewar(1972), and Mark(1974a). To
any needlessmystery,however,let us quicklyrederive the parallel effect of the
groupvelocity uponthe slowprecessionrates
f~(r, t) =- f~(r)- I v l~’(r)/2
(5)
for a trailing, two-armed
spiral wavepattern in whichf~p varies mildlyfromradius
to radius. In that case, the slight waveshear ~f~p/Ornear someradius r will slowly
loosen or tighten that part of the spiral pattern, at a rate Ok/Or= -2 Of~p/Or,
whereagain k(r, t) is the radial wavenumber.
Moreto the point, as k changeswith
time, so mustf~ itself at that radius, accordingto Equation(5), at the rate
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450 TOOMRE
where
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cg(r,t)-~(r)[01vl/~kJ,,Q
......,.
(7)
Thelatter is just the groupvelocity, herereckonedpositive wheninwards.It deserves
that namebecauseEquation(6) implies that a traveler movingwith the (gradually
varying) speedcg radially towardthe center wouldfind f~p changingnot at all in
his vicinity. Conversely,
the f..~,’s expectedin the not-too-distantfutureat anyfixed
radius are just the current values at nearbyexterior radii, all as if prerecordedon
a magnetictape that is aboutto be played.
Theseverity of this radial drifting of all local propertiesof a trailing m= 2 wave
pattern of approximatespeedfl~ ~ ~(5ro) within the exponentialdisk is indicated
by the four pairs of"characteristic"curvesin Figure5c. For eachchoiceof Q, these
curvesreport the radii whichthe imaginedtraveler wouldhavegotten to at various
times ta reckonedin multiples of the full rotation period 2n/~(Sro). Giventhat
tip is conservedalong each curve, the dimensionlessfrequencyv and wavelength
)./2,~t at successiveradii are impliedby Figures5b and4 exactlyas before.
Sincethese curvesare so similar, let us concentratenowon just the inner member
of the Q = 1.2 pair. Starting off, for instance, at the point markedC (with
v = -0.6, 2 = 0.42 2~rit, and in termsof Figure3, a = 11) at r = 3.8 to, notice that
just one rotation period seemsto be required for all this informationto drift to
point E (with v = -0.9, 2 = 0.21 2erit and e = 16) at r = 2.5 ro. Ahalvingof 2~,it
everywherewouldobviouslydoublesuch drift times, but otherwisethis relatively
tight spiral actually errs on the tranquil side: Hadwe kept J.¢rit(r) unchanged
but
selected instead the generally moreopenwavewhose~qp--- fl(4ro), similar WKBJ
estimateswouldhaveyieldeda total elapsedtimeof 1.9 in the present units for the
entire drift fromthe equivalent of point B to the very center. Exceptversus the
old straw manof material arms, such a "persistence" seemsnot very impressive.
Eventhe kinematicspirals fromFigure3 wouldlast aboutthat long with the speeds
~- x/2 suggested by Lindblad.
4.4 Forcino and/or Feedback
Seriousthoughthey were,these groupvelocity criticisms weredirected only at the
faulty demonstration
of longevity, not at the basic idea of nearly permanentspiral
waves. Asthis reviewerhas remarkedbefore, no such radial propagationactually
excludesreally long-livedspiral wavepatterns in a galaxy. If those patterns are to
persist, the abovesimply meansthat fresh wavesmust somehow
be created to take
the place of older wavesthat drift awayand presumablydisappear.
That these short wavesmust indeed decay eventually--by Landaudampingin
an ILRregion, at least in linear theory, and providedsuch a resonanceis both (a)
accessible and (b) unsaturated--wasestablished firmly by Mark(1971, 1974a)
Lynden-Bell&Kalnajs (1972), whoalso corrected Toomre’s(1969) inept remark
that there should be similar damping"already in transit." At larger amplitudes,
however,as Kalnajs (1972a; see also Simonson
1970)seemsto have beenthe first
to notice, most of the short-wavedampingmaywell occur en route, via the gasdynamicalshocksyet to be discussedin Section5.
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SPIRAL
STRUCTURE
451
Incidentally, either form of dampingargues that these density wavesmustbe
trailing. Theshort leading wavesadmittedequallywell by the original analysesof
Lin and Shu wouldhave the opposite (or outward) group velocity, and we would
then be stuck with a task analogousto trying to create waterwavesthat travel away
froma beach!
It is hard enoughto tell whatcauses the incomingwavesof a trailing sense.
Wherecould such fresh and relatively openspiral wavesconceivablyoriginate in
an isolated galaxy?Broadlyspeaking, the safest answertoday remainsas obvious
(andas painfullyvague)as it did eight years ago: Theystill seemmostlikely to
a "by-productof sometruly large-scale distortion or instability involvingan entire
galaxy," or the "consequencesof someyet morebasic density asymmetries--e.g.
disturbancessuch as mildly bar-like wavesor oval distortions whichmaybe hard
to detect but which cannot even remotely be approximatedas tightly wrapped
waves"(Toomre1969; see also Lin 1970, especially p. 389, and Feldman&Lin
1973for somerelated opinions).
Theseunconsciousechoes of Lindblad’s remarkableinsight from 1951stemmed
largely froma single hint conveyedby diagramssuch as Figure 5c: At least in a
formal sense, the shorter and shorter trailing wavesC, D, and E seemto be the
descendantsof other, muchlooser and superposedtrailing wavessuch as A from
the long-waveside of Figure 4. The latter have an outwardgroup velocity, and
they seemto refract into the short wavesonly near the "caustic" or turn-around
radius B, wherethe groupvelocity changessign and the amplitudeno doubttends
to be larger than usual. That hint remainsvery murky,however,owingto the large
wavelengths,~. = 0.69 and 1.28 2~r~t, expectedalreadyfor B and A. Usingthe full
density of the exponentialdisk, these lengths correspondto would-bec~ -- 7 and
3.5 logarithmic spirals from Figure 3. For waveB, such WKBJ
estimates might
still be crudely trustworthy(after the usual extra care to be taken near such
caustic), but no a = 3.5 spiral can be deemedtightly wrapped.Indeed, it is not
even clear that the latter can be modeledadequately with any modified local
dispersionrelations (cf Lynden-Bell
&Kalnajs1972).It is no disaster, of course,
that these antecedentwavesseemto pass so soon beyondthe assured competence
of WKBJ
analyses.Butit certainly is a pity.
It seemsonly fair to add that such skepticism has not been universal. For
instance, Shu, Stachnik &Yost (1971) claimed that even the long wavesseem
self-evident in M51.Lately, the samekinds of waveshave figured anewin the
three-waveamplificationprocesses(cf the solid Q= 1 curvebranchesin Figure5c)
near corotation that have beenstudied extensively by Mark(1974b,1976a,b), and
in the unstable spiral modesannouncedby Lau, Lin &Mark(1976), using just
such"amplifiers."
Mark(1976c) and Lau, Lin & Mark(1976) madeone very interesting point
noting that /f the stability index Q(r) can be arrangedto rise steeply (and yet
gently!) enoughtowardthe center of a disk, then their asymptotictheory predicts
a secondrefraction just outside the inhospitableinterior, withthe incomingshort
trailing wavesreverting there into the outgoinglongones. In effect, those authors
required the "forbidden" strip in Figure 5b to widenrapidly enoughtowardthe
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452 TOOMRE
left to intercept again a given vs(r) curve, and yet mildly enoughnot to violate
the assumptionsof WKBJ
theory. This maynot be easy, but if it can reasonablybe
arranged,sucha secondrefraction at last promisesa simplefeedbackcycle for which
Lin (1970; see also someapt criticisms by Marochnik&Suchkov1974 on pp.
94-95 of their review) has long been searching. The dampingfrom the gas would
continue, but hardly any from the ILR. Unfortunately, the big question is
not whether the WKBJ
idea gets stretched out of all proportion by the long
wavesinvolved in that scheme--anissue that Lau, Lin, and Markdid not resolve
by offering an examplein whichthe active density is everywhereless than one-half
of the full surface density implicitly required by their adoptedrotation curve. The
real dilemmais that one just cannot tell, without embarkingupongenuinelyfullscale analysessuchas wehaveyet to discuss, whetherthe special conditions(including
Q~- 1 near the CR,and gooddissipation near the OLR)that seemto permit these
weakly unstable WKBJ
modesmight not also admit other, more powerful nonaxisymmetrieinstabilities that wouldcompletelyswamp
the former.
Speaking of waveamplifiers, it should not be forgotten that Goldreich &
Lynden-Bell(1965;see also Julian &Toomre
1966)showedlong ago, via a different
sort of local analysis in whichazimuthalforces were retained and played a vital
role, that the rapid swingingof open leading wavesinto trailing ones by the
differential rotation has one remarkableconsequence:Largelyfor kinematicreasons
arising from temporarynear-resonancewith ~, this swingingmanagesto tap the
abundantenergy of differential rotation--and it amplifies those waveseasily a
hundredtimes moreper full cycle than the typically two-foldgrowthsobtained by
Mark(essentially from a ’,;low and almost axisymmetricshrinkage of already
tightly wrappedwavesthroughtheir most nearly Jeans-unstablewavelengthswhile
driRing withthe groupvelocity.) Andthat is just one reasonwhyother conceivable
but not-entirely-WKBJ
feedback cycles--including, for instance, even a slight
reflection of ingoingshort trailing wavesinto short leading ones that propagate
awayfromour internal "beach"--cannotbe dismissedlightly.
Lau &Mark(1976)recentlyannounced"a newphysically significant enhancement
of the growthrate," itself indebted"to the combined
effects of differential rotation
and of azimuthalforces." It is easily shown,however,that the key amplifyingterm
labeled T~ in Equations(41 and (5) of that paper exactly matchesthe only sheardependentdestabilizing term evident in the importantEquation(76) of Goldreich
and Lynden-Bell.
All things considered, only cumbersome
"global" modeanalyses and/or numerical
experimentsseemto offer any real hopeof completingthe task of providingthe wave
idea of Lindbladand Lin with the kind of firm deductive basis that one likes to
associate with problemsof dynamics,Beforeproceedingto such studies and their
ownjoys and frustrations in Section7, however,it seemswell to reflect that another
famousstructural problemwouldnot haveprogressednearly as far as it has during
the past decadeor twoif geologists had insisted that one should first establish
theoretically from basic laws that the Earth’s mantle must convect in such and
such patterns. The sameis true of our subject: As reviewedbriefly in the next
section, the galactic analoguesof magneticstripes in mid-ocean,chains of fresh
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SPIRAL STRUCTURE
453
volcanoes,and zonesof intense crushingof relatively superficial material have
lately donejust about all the persuadingone couldpossiblydesire that somegrand
drivingforces indeedare at work.
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5
SHOCKS AND OTHER CONSEQUENCES
Thefirst inkling that the interstellar materialin certain galaxiesmaybe repeatedly
experiencingshockwaveson a very large scale predates eventhe sketch by Lindblad
(1964)reproducedin Figure 1. In the early 1960s,Prendergastoften expressedthe
viewthat the intense, slightly curvingdust lanes seen within the barsof such SB
galaxies as NGC
1300and 5383are probablythe result of shocksin their contained
gas, whichhe believed to be circulating in very elongatedorbits. In such "geostrophic" flows of presumedinterstellar clouds with randommotions,Prendergast
(1962)wrotethat "it is not clear whatis to be taken for an equationof state," but
he knewthat "weshould expect a shock waveto intervene before the solution
becomesmultivalued."
5.1 Spiral Shock Waves
Asregards the normalspirals, Lin &Shu(1964)stressed fromthe start that since
the gas has relatively little pressure,its densitycontrast "maythereforebe expected
to be far larger than that in the stellar components"
whenexposedto a spiral force
field such as they had just postulated. That hint remainedlargely dormant,however,
until Fujimoto(1968, printed tardily and in Russian; see also Prendergast1967)
combinedthese last two lines of thought. He showedthat, whena supposedly
isothermal gas layer slides past whatPrendergasttermeda "gravitational washboard," its steady response includes shockwavesalready at a modest(although
finite) amplitudeof that periodic forcing. To be sure, Fujimotomadeone fairly
serious error of analysis (spelled out by Roberts1969,p. 140)and perhapsanother
of judgment--thelatter in placin~corotation in our Galaxyat a mere5 kpc, rather
as Lindblad had imagined. But Fujimoto’s goal was commendable:he suspected
that not just "the high-densityhydrogengas in the spiral arms"but also "the dark
lanes observedon the concavesides of the bright armsof external galaxies ma.ybe
due to the present shockwave."
This inspired guess was followedup by Roberts (1969). Togetherwith Lin,
improvedthe analysis and added the further suggestion that the shock wave
somehow
also "triggers" the formation of the numerousyoungstar associations
that accentuate optical photographsof spiral galaxies. Both these themesare
summarized
neatly in the twooften-copied diagramsreproducedanewin Figure 6.
Robertshimself (see also Lin, Yuan&Shu 1969, pp. 734, 737) was very vague
to howthe suddenincrease of the meandensity and pressure of the interstellar
material could actually provokethe required gravitational collapse of densepreexisting gas clouds. But this hardly matters. Thecrucial point is that before the
shockidea there hadbeenno defensibleexplanationat all for the striking geometrical
fact, first noticed by Baadein the late 1940s, that the mainHII regions in large
spirals tend to define considerablycrisper and narrowerarmsthan the rest of the
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454
Too~
Figure6Large-scalespiral shockwavesaddedby Roberts(1969)to a density-wave
pattern
muchlike that favoredby LinandShufor our Galaxy.(a) Typicalkinked-oval
streamlines,
viewedfroma framerotating withthe presumed
pattern speedf~p. (b) Impliedrelative
locationsof the shockfronts, the brightestnewstars, andthe maingaseousarms.
visible material, Oftenthese emissionnebulosities"are strung out like pearls along
the arms" (Baade1963, p. 63 ; see also Carranza, Crillon &Monnet1969and Arp
1976for two fine modernexamples). Andlike the dust, these highly luminous
chains seembiased towardthe inner edges of arms, thoughperhaps not quite as
much.
Asboth Lin (1970) and Roberts(1970) rightly emphasized
at the Basel symposium,
the occurrence of dust lanes and O and B stars typically (though by no means
always)on the inner or concavesides of (presumably)trailin9 spiral armsoffers
strong corroborationnot just for the shockwavesbut indeedalso for the theoretical
expectation that the basic wavepatterns should rotate moreslowly than most of
the disk material.
So muchfor the "volcanoes"and folded sedimentsof our subject. Its magnetic
stripes are the bright spiraMikeridges in the radio-continuum
mapof M51obtained
by Mathewson,van tier Kruit & Brouw(1972) using the Westerborktelescope.
Theseridges coincide muchbetter with the major dust lanes of that galaxy than
with its optical arms. Hereindeed was "first-class observational evidence" for
widespreadshock-wavecompressionof a gaseous mediumcontaining both cosmic
rays and a weakmagneticfield--a circumstancethat should lead naturally (el
Pooley1969)toa spiral pattern of enhancedsynchrotronemission.Somedifficulties
withthis picture, resulting mainlyfroman oversimplifieddescription of the compression by Mathewson
et al., have been reviewedalready by Kaplan&Pikel’ner
(1974) and by van der Kruit &Allen (1976). Hardlymoreneeds to be said about
this wavetest. It speakseloquently
for itself.
Thelast of the three principal confirmationsto date of the spiral-waveidea came,
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SPIRAL STRUCTURE
455
in this reviewer’sopinion,also fromWesterbork,
this time via 21-cmline studies of
M81.Asreported by Rots &Shane(1975), the two mainoptical armsof that galaxy
indeed agree well with unmistakablemajorarmsof neutral hydrogen.In the words
of perhaps the most vehementcritic of the density-wavehypothesis, whohimself
favoreda magneticexplanation,"if so, then it is generallyagreedthat gravitational
forces must be invoked and the density-wave or somesimilar theory appears
inevitable" (Piddington1973).
Thesemajoritems of evidencein supportof spiral shockshavebeenaccompanied
by a host of other observational tests and "applications" of the density-wave
hypothesis. Their conclusions have ranged from very encouraging to highly
optimistic; none has provedseverely contradictory. Muchof this (and the above)
workhas been described at length in several other reviews, including those by
Burton(1973), P. O. Lindblad(1974), and Widen(1974). Rather than repeat
let us simplyclose with the reminderthat evenin such readingone mustconstantly
bewareof the old trap of treating often-repeatedpresumptions
as establishedfacts.
Perhaps the best warningcomesfrom the confusingsituation in our ownGalaxy,
wherethe viewis far fromperfect: it is not at all clear yet whetherthe famous
3-kpcarm is to be regarded as an ILRterminusof an inward-travelingwave, let
aloneas the result of violentactivityin the nucleus.At least this writerstill wonders
seriously whetherde Vaucouleurs(1964, 1970;see also Lindblad1951, Kerr 1967)
did not comecloser to the truth in suggestingthat it mayinstead be "gas streaming
along a bar tilted by about 30 to 45° to the sun-centerline" in the inner regions
of our possibly SAB-typeGalaxy.
5.2 Pendulum Analogy
In rushing throughthese importantobservations,wedid not dwell in simple terms
uponwhyone should expect the gas shocks to develop. Roberts (1970, Figure 3)
offered an analogyin whicha particle gets caught and is draggedsomedistance
radially by the potential well of a given spiral arm. Lin and Shu have often
remarkedthat the density contrast in eachsubsystemshouldbe roughlyproportional
to the inverse of its mean-squarerandomvelocity. AndKalnajs (1973), somewhat
like Prendergastoriginally, has stressed that the answerlies insteadin the attempted
crossings of elementaryrings or orbits such as shownin the e = 16.7 frame of
Figure3. Which,if any, of these helpful hints shouldonetrust?
The first explanation turns out to be essentially the sameas what Lindblad
wroteaboutFigure 1--but it doesnot correspondclosely to the sinusoidal gravity
forces that Robertsusedin his calculations. Thesecond,thoughfine at large speeds,
wouldclearly be wrongif carried to the zero-pressureextreme.Andeventhe third
reason, thoughmostnearly right, does not distinguish plainly enoughbetweenthe
free andforcedtrajectories of variousparticles.
A closer analogy to the tightly wrappedwavesadopted by Roberts (1969) and
Shu, Milione&Roberts (1973) is the (supposedlyendless and continuous)row
identical pendulain Figure7 devised largely by Kalnajs. If weassumethat these
pendulaare long enoughto act like harmonicoscillators and that they do not yet
bumpinto eachother, then of coursetheir displacements
~(r, t) fromthe equilibrium
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456
TOOMRE
positions r obey the equation
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~2 ¢/0t2
-F K2¢ = C
sin [k(r+ ¢)- cot],
(8)
whenthat systemis exposedto the traveling sinusoidal force field given on the right.
Equation (8) is exactly equivalent to the zero-pressure limit of the locallyapproximated Equations (ll) of Shu et al. Obviously, it also mimics the radial
dynamicsof stars and other collisionless objects in a galaxy subjected to strictly
axisymmetric forcing of a wavelengthshort enough that variations of the epicyclic
frequency n can be ignored.
Requiring~(r + ()/Sr > 0 everywhere,the linearized forced solutions
~(r,
t)
~---
C(/( 2 - (/)2)-
si n (k r- ~ot),
(9)
obtained after omitting ¢ from the right-hand side of Equation (8), imply that one
should not expect collisions as long as the forcing amplitude C < (K2 -~2)/k. This
inference is corroborated by the accurately calculated Figure 7a, where C was set
to one-half that nominal critical value, and where I co/~l = 3/4, about as inferred
by Lin and Shu for their presumedwavein the solar vicinity.
By definition, at larger amplitudes whereconflicts or "shocks" should first arise,
Equation (9) is no longer accurate, since k¢- O(1) or the response is already
nonlinear. [The same criticism, by the way, applies also to any use of the LSK
b
Figure 7 Forced traveling wavesin a "curtain" of pendula subject to Equation (8) and
inelastic collisions. Thesinusoids depict, with 5-fold exaggeration,the respective forcing
potentials amounting
to (a) 0.5 and(b) 1.5 times the nominalcritical value from the text.
Thefrequencyis ~ = 3K/4.
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SPIRAL
STRUCTURE
457
dispersion relation wheneverstron9 gas shocksare expected. The fact that the
perturbation to the overall star density maythen still be only of order 5~ is
misleading. The slow-moving
stars expectedto contribute the bulk of that extra
density wouldalready oscillate in a distinctly nonlinearmanner,as illustrated
beautifully by Widen(1975) in his discussionof spiral-forced periodic orbits near
the Sun.] Nevertheless,for the abovefrequencyratio--which lies about as far as
possible from either the Lindbladresonanceco = + ~c or the so-called "ultraharmonic"resonances ~ = +~/2, _+~/3 .... discussed by Shu et al. and also
Woodward
(1975)--the accurate critical amplitudeCcrit = 0.430 ~c2/k happensto
agreefortuitously wellwiththe linear estimate0.438.
Beyond
such values, of course, there is no substitute for numericalintegration.
Figure 7b showsone such result, obtained fromforcing steadily with an amplitude
1.5 times the latter estimate. It wassupposedin this calculation that the pendula
(almostlike lead balls, or exactlylike gas particles in an isothermalshockof infinite
Machnumber)undergoperfectly inelastic collisons. Contraryto a first impression,
however,these individualoscillators donot stick to eachother in the intensepile-ups
seen in Figure7b. It is simplythe rampressureof the newarrivals fromthe right
that persuadesthe early comersto linger for a while(19.6~of the full cycle, to be
exact) in these peculiar traffic jamsnear the minima
of the imposedpotential, until
at last theycanslip awayquietlyto the left.
Compared
with this naive rederivation, the inclusion of a uniform"effective
speedof sound,"a = typically 8 kmsec- l, in the analysesof Fujimoto,Roberts,and
especially Shu, Milione&Roberts(1973) seemsto havehad twomaineffects. One
is that, understandably,sucha finite intrinsic speedsoftens the density contrast
across a shockand also widensthe zoneof crowdingbehind it. In particular, if
o~/ka < l--or wheneverthe forcing appears subsonicin a meansense--shocksarise
onlywith difficulty (cf Figure6 of Shuet al.) andtend to yield "broadregions
relatively low gas compression."As Roberts, Roberts&Shu (1975) stressed, this
maywell account for the thick and "massive"arms seen in smaller spirals like
M33.
The other and perhapsmoresurprising effect arises whenthe speedc = co/k of
the imposedforce sinusoid appearsmoderatelysupersonic: Thefinite gas pressure
then makesit possible to obtain shocksfrom considerablyweakerforcing than in
our a--, 0 hypersoniclimit. It also causes the shocksfroma given forcing to be
markedlystronger in termsof impactvelocity. For example,as Shuet al. reported
for r = 10 kpc, v = -0.72, and Mach~aumberc/a = 1.63, the first cusp or zerostrength shockoccurs in their isothermalgas whenthe forcing amplitudeC is just
undera fraction F = 1.0~of the central force f~Zr. Bycontrast, the pendulain the
samesetting wouldnot have bumpeduntil F = 4.0~. Moreover,using F = 5~ as
again recommended
by those authors for the nearby Lin-Shuw~ve,their a = 8 km
sec- ~ gas enters eachshockwitha relative speedu~ --- 26kmsec- ~ andexits with
speed u: = aZ/ul, whereasour a -- 0 pendula wouldhave managedto crash only
with speedux = 12. Withan intermediatesoundspeeda = 4, by the way,one would
already find that u~ = 22--whichunderscoresthat the extra shock-making
declines
only slowly with increasing Machnumber.
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458 ~’OOr~RE
Technicallyspeaking, there is no real mysteryabout this perverse supersonic
behavior. Thosestronger shocks seemto occur essentially because the pressure
term in Equation(11) of Shuet al. raises the natural frequenciesof our pendula,
so to speak, markedlycloser Co the first ultraharmonicresonance--andno doubtit
also contributessomethingakin to the classical Riemann
steepeningof free acoustic
waves(whichis a process discussed briefly by Woodward
1975. amidst his nice
time-dependent
shockcalculations). Nevertheless,this reviewercannot help wondering whetherthe shock-makingmight not have been seriously exaggeratedby the
isothermal modelitself. Can we be sure that this "bouncy" inviscid gas of
Fujimoto and the later workers mimics well enoughthe variously dampedand
heated multi-phasestew that is the real interstellar medium?
Thequestionmatters
becausethe increased impactspeedul seemsa very mixedblessing in the context
weare aboutto describe.
5.3 Damping1
If the forcing of the very in.elastic pendulain Figure7b wereto cease abruptly,
we wouldknowinstinctively that their present strong collisions (as opposedto
manylesser "aftershocks") wouldendureonly perhaps another half-cycle. And,if
instead of stopping suddenly,this forcing stemmed
fromthe collisionless but not
entirely randomswinging to-and-fro of other, muchmoremassive pendula (not
shownhere) that interact gravitationally with ours, then it wouldbe just as clear
that the steady drain of mechanicalenergyvia the visible crashes wouldin the long
run causethose unseencollective oscillations to decayas well.
Thefirst of these thoughl: experimentsobviouslymimicsthe demiseof free axisymmetricshock wavesin a purely gaseous disk. Their rapid dampingmayseem
irrelevant, since it is the secondcase that at first sight imitates muchbetter any
tightly wrappeddensity/shockwavesin a real galaxy, wherethe gas is well known
to compriseonly a small fraction of the total mass. Kalnajs (1972a)pointedout,
however,that a serious fallacy maylurk evenhere.
Thecrucial point, in the contextof our Galaxy,is that relatively fewdisk stars
manageto take part significantly in wavesof such short, 2 ~ 4 kpc length as Lin,
Shuand Robertsadoptedfor the solar vicinity. Usingthe conventionalvalue of x,
one calculates easily for 2 = 4 kpc that only about 12~(or 40~)of the stars from
a Schwarzschilddistribution with a~ = 40 kmsec- 1 wouldhave randommotions
small enoughto ensure morethan the 60~ (or 0K) participation discussed at the
endof Section4.1. Thicknesscorrectionswouldseverelyreduceeventhat. Thishelps
explain whyLin (1970) reported that the nearby waveamplitude~ = 11 M~pc-z
correspondingto a disturbance gravity F = 5~ should consist almost equally of
excessgas and star density. Conversely,however,this argumentcautions--ignoring
the group transport of fresh wavesfor the moment--thatthe Roberts shocks
mightstill be dampingthe local Lin-Shuwavealmost as fiercely as one wouldhave
guessedfor the gas byitself!
Muchparaphrased, that is what worried Kalnajs--whohimself, like Roberts &
Shu(1972)in a not veryconvincingrebuttal, couchedit all in the moreexpert but
also moreopaqueterms of net torques and negative densities of waveangular
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SPIRAL STRUCTURE
459
momentum.
To reassess this vital matter, it is simpler to pretend again that we
are dealingwithexactlyaxisymmetric
densitywaves/~’(r,t) =/~wcos k(r- ct). Let us
also suppose,althoughperhapsmoredubiously,that their amplitudes/~w
are small
enoughfor infinitesimal estimatesof the density Ewof waveenergyper unit area to
remainreasonablyaccurate.
For such self-consistent wavesin an extremelythin isothermal gas layer of
2" (t~w/t~)
2,
unperturbed
surface density#, this energydensitywouldequal Ew=½1~c
quite independentof the (horizontal) soundspeed a. The analogousformulafor
the LSKwavesin a razor-thin stellar sheet is moreponderous[cf Toomre1969,
Equation(33)], but it can be recast as just a multiple H(~,/~.etit; Q)>1 of that
elegantgas result. Sometypical valuesof/-/are 1.17, 1.55, 2.66, 4.74, 16.86for the
. Q = 1.2 points A, B, C, D, E in Figure4; the steep rise of that correction factor
towardthe smallestwavelengths
is helpful, for it meansthat suchshort stellar waves
tend to contain considerably morekinetic energy (and hidden commgtion)than
one mighthavesupposedfromtheir phasespeedc and net density/~walone.
To be comparedwith the available Ewis the loss AEof mechanicalenergyper
unit area and per time interval T~ = n/(f~-~t,) betweensuccessive shocks, here
imaginedto be steady. Asfollowsstrictly fromtime-averagingover one suchcycle,
the variable v(q) in Equation(11) of Shu,Milione&Roberts(1973),this loss amounts
z z -2aZ ln(u~/u2)] = _~#gue,
to AE= 171~[u~_u2
~ where/~gis the meandensity of the
supposedlyisothermalgas. At that rate of depletion, the waveenergywouldlast a
time
Te=(E,,/AE)T~=(l~/kta#)(c/ue)2H
(10)
Asimpliedalready, c = co/k = 13 kmsec- ~ and/~w= 11 Mepc- 2 for the specific
2 = 3.7 kpc, v = - 0.72, F = 5~owaveat r = 10 kpc definedin Table1 of Roberts
and Shu. AssumingQ= 1 and ignoring thickness and gas corrections for brevity,
-2, and T~= 2.7 × 10s yr. Locally,the
those data also implyH = 5.30,/~ = 89 M~,pc
-2
choice /zo = 5 Mope seemsconservative; together with the above items, and
ua = 26, a = 8 kmsec- ~ discussedearlier, it yields a remarkably
short TE= 2.0 × 108
yr. This roughappraisal seems, if anything, a little harsher tha~a TE= 0,8 to
1.5 × 10~ yr claimed by Kalnajs--whoused a perhaps slightly too pessimistic
~ta = 10, and whoselow estimate of 0.8 also suffered almost twofold from an
erroneousneglect of the In (u~/u~) term in AEmentionedonly parenthetically in
his paper.
The counterproposal by Roberts and Shu that the relevant decay time works
out morelike 6 to 9 × 10s yr referred in fact to the entire interior annulusbetween
r = 5 and 10 kpc, whereat lea_st the unperturbedstars seemmoreabundant.More-z, a value
over, those authors adopted a meangasdensity ~to = only. 3 M®pc
strangely at oddswith the 6 M,pc-z waveamplitude(cf Yuan1969, p. 897) in the
gas alone imaginedto providehalf of the local F = 5~ forces in the first place. In
retrospect, that meandensity also seemslow in comparisonwith the dramatically
increasedinterior amountsof molecularhydrogeninferred somewhat
tentatively by
-~, as
Gordon& Burton (1976) from COdata. Using instead/~ = 8 and 12 M~pc
hinted by Gordonand Burtonfor r - 8 and 6 kpc without even including any He,
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460
TOOMRE
it also appears that 7~ = 1.8 and 1.9 x 10~ yr for the F = 5.1 and 5.7~ waves
considered by Roberts and Shu at those two interior radii.
Like the earlier criticism involving the group velocity, all this maynot be quite
as devastating as it sounds. For one thing, the increased stellar participation (and
faster cg) in moreopen spirals should help considerably. Second, as hinted already,
the isothermal model may exaggerate the shock losses. And third, one would
suppose in any case that those losses should be largely self-correcting upon some
reduction of amplitude--given that the shock-making must be a threshold
phenomenonthat sets in only beyond some minimumlevel of forcing. The pendula
make these last two points very nicely: If only we could be sure that the nearby
gas behaves in that naive fashion under the F = 5~ forces, we would reckon that
T~ = 5.1 × 10~ yr, whereas below F = 4~ there would be no damping at all. More
annoyingly, however, the idealiz~ a = 8 km sec- 1 isothermal gas under F = 4~
forcing wouldstill imply T~.. = 2.1, 2.2, and 3.6 × 10s yr at r = 10, 8, and 6 kpc.
Alas, it is simply not knownyet whichstory is moreto be believed.
Whatdoes seem broadly clear, nonetheless, from this insight by Kalnajs is that
at least the tighter Lin-Shu waves in any fine "normal" spiral should--during the
very process of creating their own luminous froth while getting shorter and thus
moreintense--be almost as capable of dissipating most of their arriving mechanical
energy as any foaming water waves incident upon a gently sloping beach. Better
still, just as with the ocean wavesthat start breaking well offshore, it looks as if
this almost insatiable dampingmaynot even "scour" very badly the vicinity of an
[LR itself. Ironically, though, Kalnajs was wrongin his closing remark that "it
takes about 10~ yr for help to arrive" in the solar region via the group velocity.
Another look at Figure 5c near its point D should persuade that his estimate is too
large, probably at least by a factor 2. In this respect, Roberts and Shu scored a
good point in replying tha~ it maybe "no accident that the ’dampingtime’ and the
’propagation time’ should be comparable." Indeed not. That is exactly what one
finds with those breakers off the beach.
Not to be confused with this remarkably potent wavedampingis the related fact
that any orbiting gas that persists in overtaking a pattern of trailing spiral shocks
must on the average drift inward as well. Such a net transfer of angular momentum
from the gas to the stars ~eemsto have b~n noticed first by Pikel’ner (1970; see
also p. 1"/2 of Simonson1970, Kalnajs 1972a, Shu et al. 1972, and Section 6 of the
review by Kaplan & Pikel’ner 19"/4). It meant not only that Roberts (1969)
slightly mistaken in claiming the gas orbits to be closed in the present Figure 6a.
Muchmoreimportant, this inexorable radial evolution implies that even the neatest
spiral structures can at best be only quasi-steady. As Pikel’ner noted also, such
inward drifting of the gas in very open spirals should not even require many
revolutions, given strong shocks.
5.4
Spiral Shocks without Spiral Forcing
It wouldbe regrettable if all this well-deserved emphasis upon, first, the tightly
wrapped waves of density and force alike, and then the very nonlinear gas
dynamicswhichthe latter can soon provoke, were to leave the impression that it is
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SPIRAL
STRUCTURE
461
the only conceivableroute to makingspiral shocks in a galaxy. Suchit definitely is
not, as Figure 8 nowdramatizes.
To produce this diagram, Sanders & Huntley (1976) began their computer
simulation with a uniform disk of almost cold "gas" without any self-gravity. It
merelyrevolveddifferentially in an inverse 5/4-powercentral force field. Then,rather
gradually, somefairly modest(up to _+ 13~ tangential, _+ 3.25~ radial) extra forces
were introduced, as if comingfrom an oval distortion to the density of the unseen
backgroundstars. These imposedrn --- 2 forces without any spirality of their own
were presumedto rotate at a steady rate, placing corotation and the two Lindblad
resonances at the radii indicated. The resulting spiral wavestirred up amidst the
shearing gas is pictured in Figure 8, approximatelyone revolution after the forcing
had becomesteady. That response itself, however, remains only quasi-steady-owingessentially to the radial drifting (and its analogue outside CR)knownalready
to Pikel’ner. The shocks as such are none too clear in this reproduction, but at
least they can be located by comparing the concave and convex sides of the arms
between the ILR and CR.
Artificial though it is, Figure 8 stresses beautifully that the mannerin which
Lindblad’s "characteristic density wave.., incites the formation of spiral structure
by its disturbing action on the internal motions" maynot even be so subtle as to
require, as an intermediary, the spiral gravity forces from any vaguely Lin-Shutype
density waves apt to be excited at the same time. Similar warnings have often been
uttered by Kalnajs (1970, 1971, 1973)--e.g. "the density wave[in an unstable mode
illustrated in his 1970paper] is essentially a bar-like distortion of the central region
--
OLR
CR
ILR
ILR
CR
OLR --
Figure 8 Quasi-steadydistribution of density in a shearing gas disk exposedto a mildly
oval gravitational potential (Sanders&Huntley1976). Boththat forcing andthe gas again
rotate clockwise.
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462 TOOMRE
of the galaxywhichdrives the gas." Suchalarmsignals werealso fairly explicit (cf
Figure 11) in the N-bodyexperimentsof Miller, Prendergast&Quirk(1970) to
discussedbelow.Lately, Lynden-Bell
(1974, 1975;see also 1972,with Kalnajs)has
stressed anewthat "it is not at all clear to methat the gravity of the ’spiral’ is
’important’for the formationof spiral shocks.Myadviceto observersis to believe
those parts of the theory basedaroundpropagatingshocksand shock-inducedstar
formation,and to be very waryof those parts that rely heavilyon the self-gravity
of anythingbut a slight oval distortion or a bar shape."
Facedwith such advice, this reviewer remainsambivalent.Onthe one hand, it
seemseminentlysensible to presumethat the gas forcing mightbe rather direct in
SB/SAB
spirals like NGC
!.097 (cf Arp 1976again, thoughwhereis corotation?)
and perhaps even in such milder SABtypes as NGC
4321 = M100.Yet the same
warningsseemoverdonefor the "best" SAspirals, essentially for tworeasons. One
is that, as Sandersand Huntleyexplainednicely, their kind of immediateforcing
can shift arm longitudes only 90° per resonanceradius--and the three familiar
resonancesare simply too few to meet such hopes in NGC
5364or M51or perhaps
evenMS1.The other reason is that Zwicky(1955; see also Sharpless&Franz1963)
really seems to have been onto somethingwith his "compositephotography"of
M51.As Schweizer(1976) reported from ~ detailed photometricreexamination
this and several other well-knownspirals, the thick "red arms" discovered by
Zwickyindeedrepresent broadspiral variations of surface brightness(thoughnot of
color!), as if from pronouncedwavesof similar spirality amongthe background
disk stars as well.
Classic exampleslike those continue to argue strongly for somethingmuchmore
like the Lin-Shu-Roberts
wavesthan the picture in Figure 8. Yet, as we nowknow,
such wavesthemselvesseemdesperately in needof help from larger scales. Hence
the mainquestionto be distilled fromthe last fewparagraphsmightjust be whether
the much-desired"crushing" of the gas by forces stemmingultimately from major
bars, ovals, long waves, or whateveris best regarded in any given instance as
direct, indirect, or someshadein-between.Differences of opinion here amongus
theorists maymerelybe akin to those expressed by the blind menwhoexamined
the elephant.
It seemsright to end this long discussion of galactic shocksalmost whereit
began. A fascinating computer study of (again admittedly impermanent)gas
motionsin somestrongly bar-like potentials--with the shocksthere falling indeed
mostlyat the observedlocations of the dust lanes that tantalized Prendergastlong
ago--has recently been publishedby Sq~rensen,Matsuda&Fujimoto(1976).
6
A QUICK SYNOPSIS
Lin (1967) has long stres,~;ed that he and Shu--withtheir QSSShypothesisthat
wavelike spiral force field somehowexists and keeps on rotating--jumped
deliberately right into the middle of a difficult problem,hopingthat fromthis
"focal point in the development
of a theory"it mightbe possibleto progressfaster
by working"in both directions." Bythis technique, muchmoredefensible here than
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SPIRAL
STRUCTURE
463
in mountaineering,
these and quite a fewsubsequentexplorers landedessentially at
point B of Figure 9. Asdiscussedin Section4, the route "downhill"fromB toward
first principles at A has since provedsurprisinglytreacherous;at least, no one has
yet managed
to descendsafely all the way.Onthe other hand, the going"uphill"
fromB towardthe gas armsand spiral shocksat C has, as seenin Section5, turned
out to be delightful despite somecrevasses. Of course, a nasty segmentfromC to
Dstill lies ahead--though
evenit has lost someof its formerterror, chiefly to the
helpful notion of a two- or multi-phaseinterstellar medium
favoredsince the late
1960s(cf Shuet al. 1972,Salpeter1976).Happily,that task belongsmostlyto other
specialists. Forwavetheorists, the big challengeremainsmuchmorethe logical gap
fromAto B. It had better get closed firmly lest those reconnaisanceparties far
ahead becomestranded.
This thumbnail sketch says nothing about the manyragged or patchy or
"multi-armed"spiral features that both outnumberand complicatethe generallyagreed two-armed"grand
designs" amongwhat Hubblecalled normalspirals. Worse
still, our little summary
seemsevento haveforgotten that "barred spirals are not
rare oddities of nature. Among
994 bright spirals, about 2/3 showbar structure:
31~ SA, 28~o SAB,37~ SB, 4~ S," to quote Freeman(1970b; see also Sandage
1975)and indirectly de Vaucouleurs.
Thefirst flawis a real one, to be scarcelyremedied
in Section8. Asfor the bars,
however,
it is increasinglyclear that they, too, belongin Figure9 as closely related
wavephenomena.Theyare basically to be thought of, it seemsnow,not as the
stiffly revolvingJacobi ellipsoids that one still tends to see quoted,nor as the
Dedekind
ellipsoids in whichthe fluid circulates along ellipses but the shapestays
put, but morelike the in-betweenRiemann
ellipsoids with both rotation and circulation that Chandrasekhar(1969, for a goodsummary)
did muchto resurrect and
for whichFreeman(1966) devised someelegant stellar-dynamical analogues.
course, the real bars are not isolated entities but tend to be immersed
deepwithin
reasonablynormaldisks, as Freeman(1970b,1975)stressed in his reviews. Aneven
morevital clue, as he wrotealreadyin 1970, is that unlike the air gap in Hubble’s
tuning fork, "the transition SA-SAB-SB
is so smooththat it seemsinconceivable
STAR FORMATION
STELLAR DYNAMICS,
GAS RESPONSE
spir~al (?) force
,- HI ----- light
Figure9 Statusof the wavetheories.Asyet, onlythe thickenedportionsof this schematic
route seemreasonably
secure.
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464 TOOMRE
that the structure-producingprocessesshouldbe essentially different for SAandSB
systems." Theseremarksmuchdeserve to be kept in mindas we nowexaminebriefly
in Section7 the difficult andas yet inconclusivestruggle upwardfromAvia various
large-scale computations.
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7
GLOBAL INSTABILITIES
With the sole exception of the completely flattened and uniformly rotating
Maclaurin-Freeman
disks--for which Bryan (1889), Hunter (1963), and Kalnajs
(1972b) together showedhowto obtain the entire spectrumof modes,even when
"hot," using only pencil, paper, and Legendrepolynomials--allexisting studies of
the verylarge-scaleor "global"behaviorof disk-likemodelgalaxieshavebeenheavily
numerical.Someof these studies, devotedagain to linear instabilities and/or modes,
haveat least madea mild pretense of analysis. Others have amountedfrankly to
brute-force time-integrations of the nonlinearequations of motionof up to about
l0s gravitationally interacting particles, with their only eleganceconsisting of
various "checkerboard"schemesfor rapidly solving the Poissonequation. As might
be expected, the modesearches have tended to be muchless expensiveand, within
their limited domain,also moreaccurate than these large N-bodyexperiments.Yet
it is indeed the latter that have taught us far moreof what we really neededto
know.
7.1
Tendency toward Bar-Making
Threeof those importantlessons are summarized
well in Figure 10, whichis due to
Hohl (1971). Onelesson was that disks of stars with randommotions barely
sufficient to avoid the Jeans instabilities still tend to be grossly unstablein an
m= 2 sense. Secondly,there is nothinglong-livedabout the spiral structures that
ensue amongsuch "stars." Anda third point was that the muchhotter stable
configurations into whichthese experimentaldisks rather quickly convert themselves tend often--though not always--to include somesort of oval shapes that
continueto revolveindefinitely in their interiors. Essentiallythe sameconclusions
werereachedalready in the numericallycoarser but physicallyyet moreinteresting
experimentsof Miller, Prendergast&Quirk(1970), whichtypically includedalso
subsystemof dissipative "gas" whoseownstrikingly different behavioris shown,as
yet a fourth topic, in Figure11. Let us considerthese four topics in 3-4-2-1order.
The tendencytowardhat-makingevident in both figures deserves exceptional
emphasisbecauseit is the one corner of our subject wherethe quest for durable
wavelikedistortions of theoretical galaxies can with somejustice be said to have
succeeded already. This does not meanthat such a finite-amplitude lock-in
phenomenon
is at all well understoodyet. However,it has nowarisen in enough
situations [including the much-softened1000-particle integrations by Dzyuba&
Yakubov(1970) and--in a milder sense--also the pioneering N-bodyexperiments
of P. O. Lindblad(1960)]that eventhis skeptic feels convincedthat it represents
the basic recipe for the survival of the bars and oval shapesseen in the interiors
of actual SBand SABgalaxies.
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SPIRAL
STRUCTURE
465
t = 1.0
t= 1.5
Fi#ure 10 Evolution of a uniformly rotating disk of 10~ naass points (Hohl 1971). Its
surface density began like that of a cold Maclaurin disk; time is reckoned in rotation
periods of the latter. Gaussian rar~dom velocities were preassigned from Equation (3);
hence that model started only roughly in equilibrium, but it was nevertheless essentially
stable in an axisymmetric sense, as Hohl also showed.
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466
TOOMRE
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SPIRALSTRUCTURE
467
Themainthing to notice is that the "bars" in Figures I0 and 11 revolvein the
samedirectionbut moreslowlythan the computerparticles that undoubtedly
circulate
(andgyrate) withinthem.It is, of course, hopelessto try andtrace anyof the latter
in these diagrams,but at least Figure 11 showsclearly enoughthat someof the
materiallumpsorbiting outside the bar actually manageto overtakeit. For Figure
10, Hohlreported and the stroboscopiceffect confirmsthat the rotation period of
the oval blob seenin the later framesequals 2.25 times the original spin period-whereasthe orbital period of any relevant interior particle probablyremainsless
than 1.5 even then. Whatwe havehere, in short, is again somethingvery akin to
the slowadvanceof Lindblad’s"dispersionorbit" in Figure2. Tobe sure, there are
nowmanyplayers, plus the hot and massive"star" disk that is not even shownin
Figure1 l. All these interact with eachother in an appallinglynonlinearand nonlocal manneramidst muchrandommotion.Yet, remarkably,the outcomefalls short
of completeaxial symmetry.Instead, it seemsthat the various precessingorbits
evenhere, helpedno doubtby the familiar near-constancyof ~- x/2, still manage
well enoughto "trap" each other--for reasons that Contopoulos(1970, 1975),
Vandervoort(1973), and especially Lynden-Bell& Kalnajs (1972) maderather
plausible--to yield the oval shapesthat persist near the center. Characteristically,
the angular speeds of eventhese special kinds of density waves,althoughslow,
againexceedslightly the typical valuesof f~-~c/2(cf Quirk1971,Figure6).
A second but already considerably vaguer sort of encouragementfor wave
theorists camefromthe spiral structure in the "gas" outsidethe obviousbar shape
in Figure11. A lot of this structure is chaotic, andeventhe nice "third" armseen
shearing rapidly past the 3 o’clock position at step 88 can in essence be only a
transient materialfeature. Besidesall that "noise," however,these pictures together
with the magnified frames 101-103 and the yet later steps 121-130shownby
Miller et al. conveyalso a fuzzybut persistent imageof a reversedS, withthe bar
serving only as its middlesegment.This must indeed be a wave.Similar remarks
apply to the barred and vaguelytwo-armedshapes obtained by Hohl(1971, Figure
17) upon"cooling" one previously featureless hot stellar disk in a gentle but
admittedlyartificial manner.
Miller, Prendergastand Quirkwrotethat "the spiral pattern [or did they chiefly
meanjust a strong impression of spirality?] lasted for about three complete
revolutions"--that is, approximatelyfrom step 60 till 160, by whichtime the
"pattern ... finally becametoo close to be seen in our pictures" and eventhe bar
had mostlydisappeared.It wouldbe delightful to infer fromthis half-successthat
the bar wassomehow
essential for preservingthat spiral. Regrettably,these authors
and especially Quirk(1971)concludedafter muchthoughtful discussionand Fourier
analysis that such an interpretation (amongothers} wouldbe muchtoo simplistic.
Theyagreedthat somesort of a stellar "asymmetry
seemsto ’drive’ the pattern."
Yet the mild, warped-ovalm~- 2 shapeanalyzedin detail in Quirk’sFigures3 and
4 plainly involvedcomparable
disturbancedensities fromthe gas and stars alike,
and it also extendedat least 3 times as far in radius as the obviousbar in our
Figure~ l. Thus,despitesomeresemblance
to the situation later idealizedin Figure8
by Sandersand Huntley,it washard to tell here just whathad causedwhat.
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4~8
TOOMRE
Thetotal lack of spiral shapesof respectabledurationnot only in Figure10 but
also in every other purely stellar-dynamical experimentconductedwith sizable
fractions of"mobile"mass(e.g. someby Hockney&Brownrigg1974, or others by
Hohlusing initially Gaussianand exponentialdisks) is one of those results that
almostspeaksfor itself. It is conceivable,of course, that somemilderinstabilities,
whichmight themselveshave led to moreenduringspirals, were thwartedin these
experimentsby a kind of overheatingfrom the fierce initial behavior. This seems
unlikely, however,becauseof Hohl’sextra tests with that artificial cooling:Whenever it was switched off again, the spiral features soon disappeared, no doubt
speededby the reasonthat follows.
7.2 Ostriker-Peebles Criterion
Asis well known
by now,whatremainedevenafter the efforts at cooling, by Miller
(1971, Figure 6) as well as by Hohl, weresomeastonishingly hot disks of makebelieve stars, with morerandommotionthan actual rotation. Speakingnot only
personally, the trickle of such reports from about 1968onwardscameindeed as a
rude surprise, especially whenaugmentedby the similar analytical findings of
Kalnajs (1972b) as to what it takes to stabilize fully those Maclaurin-Freeman
disks. It wasof little comfortto recall that "a questionwhichthe presentdiscussion
[of the short and mostlyaxisymmetricinstabilities] leaves completelyunanswered
is... whetheror not a givendisk mightprefer to developinto a bar-like structure"
(Toomre1964). Thetrouble withsuch face-savingwasthat this writer also believed
then (and still does now,thoughmuchmoretimidly!) that temptations like the
bar-makingouoht to belongmainlyto the inner and moreuniformlyrotating parts
of a disk, Out here in this Galaxy--ashinted by a probably exaggerated nearagreement(cfToomre1974a) betweenthe observed and required randommotions
of the nearby K and Mdwarfs, and also by the absence of true nonaxisymmetric
instabilities of a local sort (cf Julian &Toomre
1966)--it had seemeda safe bet that
a stability index like Q := 1.5 should curb all lesser misbehavior,and yet not
excludethe Lin-Shuwaves.Instead, the experimenterskept on reporting Q = 2, 3,
and upwards.
In retrospect, Miller (1971)wasclearly too modestin still entertainingthe idea
that this severe heatingarose mostlyfromproceduralflaws in the simulations.Since
then, as Miller (1975; see also Hohl1975)has rediscussed,fears of much-increased
relaxationeffects in these verythin disks--.thoughquite seriousin principle(Rybicki
1971)--were
in practice largely laid to rest not onlyby noting that the near-gravity
wasmuchsoftenedin the actual computercodes, but also by checkingwith further
experiments(Hohl 1973, Miller 1974) either knownor constrained to remainaxisymmetric,in addition, Hohl(1972) confirmedexperimentallyfor the uniformly
rotating exact modelswhatKalnajs already knewfrom his analyses: they must be
hot indeedto avoid the bar-making.
Mostconvincingof all wasthe discoveryby Ostriker &Peebles (1973; reviewed
thoroughlyby Bardeen1975)that the ultimate and/or stable formsof these disks of
collisionlessparticles sharethe empiricalrule or criterion
T/W=_(organizedK.E.)/(negativeP.E.) ~<
(11)
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SPIRAL STRUCTURE
469
that Ostrikerhadcometo expectfor the secular stability of variousrotating models
of stars. Withthat, it wassuddenlybeyondall reasonabledoubtthat the various
investigationsweretelling essentially the samestory--andthat it wasnot a story of
numericalerrors.
Eventoday, however,it remainsmuchless certain whatthis surprising unanimity
actually demands
of seeminglydisk-like galaxies like our own.A lot of "heat" is
clearly needed,but mustit be hiddenin faint, massivehalos, as Ostrikerand Peebles
suggested?Or mightsomevery hot inner disks or "spheroidal components"
suffice
already?Hereprobablythe mostimportantthing to remember
is that the OstrikerPeeblescriterion is still not a theorembut just a fine summary
of the existing N-body
experiments.Reasonablethoughthey seem,those experimentshavein fact not been
very diverse. Onestrong bias has beentowarddisks initially in fairly uniform
rotation. Anotherand possibly evenmoreserious limitation mayhavebeenthe fact
that nearly all constituentsof those experimentaldisks wereeither prearrangedor
"cooled"again to rotate in the samedirection. Thisconcernmayseembizarre until
werecall (a) that no onewouldset out to construct a stable spherical galaxyfrom
stars with only one sense of angular momentum,
and (b) that the average disk
galaxy indeed contains a spherical component.Almostnoneof the existing N-body
simulations,it seems,has either includedor managed
to developany sizable fraction
of stars orbiting backwards
near the center. As Kalnajs(1977)has just explained,
such mobileretrograde stars oughtto interfere considerablywith any bar-making
tendency.
In short, the needfor properhalos has still not beenestablished firmly in our
subject--though this is neither to imply that they wouldbe unwelcome,nor to
dispute the growingevidencefromflat rotation curves (cf Roberts1975, Krumm
Salpeter1977)infavorof muchextra massof lowluminosity.In fact, halos of interest
to disk theorists wouldnot evenhaveto be inordinately massive,Roughlyspeaking,
it seems(cf Kalnajs 1972b,Bardeen1975, Hohl1976)that already the adoption
a rigid halo--or any other frozen background
distribution--of total masscomparable to that of the mobiledisk within it cures at least the violent bar-making,Of
course, immobilizing
all but 10 or 20~oof the massdoesevenbetter. AsHohl(1970)
first illustrated and Hockney
&Brownrigg(1974) confirmed,experimentsstarting
withcold disks then yield not exactly"a spiral densitywave,unchanged
for at least
10 rotations" (as the latter authors claimedextravagantly) but somethingelse
almost as interesting: Whatone tends to find are multi-armedstructures that
surviveonly in a (graduallyevolving)statistical sense--thoughindeedfor quite a
fewrevolutions--while their detailed features windup and yet somehow
keepon
reappearing!
It is doubtful,however,whetherany postulatedhalos, especiallyif they rotate to
someextent, can be relegated safely to an altogether passive role. For one thing,
as several authors includingMark(1976c)havesuggestedin the logical footsteps
Sweet(1963), some"two-stream"instabilities mightwell arise betweensuchsystems
and the cooler disks embedded
within them. Andmoreimportant, even those very
hot collections of stars might, like the one shownin Figure 10 or hopedfor in
Figure8, still preferovaldistortionsof their own.
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470
TOOMRE
7.3
Unstable Spiral Modes
The mainalternative to these fascinating but expensiveN-bodyexperimentshas
been to pretend that a galaxy consists not of a finite numberbut a continuumof
stars and/or gas particles, andto try and determinethe fates and shapesof various
large-scale but small-amplitude
perturbationsto disk-like equilibriumconfigurations
of suchmaterial. Seriousnonaxisymmetric
efforts to explorethe overall stability and
assorted global modesof shearingdisk galaxies in this mannergo backat least to
Hunter (1965) and Kalnajs (1965). Since then, as summarizedin part by Hunter
(1972) and Bardeen(1975), a lot of additional cleverness and computingmoney
has been expended--indeed far more than is immediately evident from the
literature. Yetthe overall progressin this area has beenpainfullyslow.
That situation should improvewhenseveral majorongoingstudies, particularly
the ones by Kalnajs (1971, 1976) and Bardeen(1975), reach print in final form.
These promise manyinteresting comparisons with WKBJ
estimates, stability
criteria, resonantparticle e~cts, and ideas about angularmomentum
transfer. Here
it seemsfoolish to try and anticipate such conclusions. Instead, it maybe more
instructive to look backwardfor a moment
and to ask whyall this laudableglobalmodecalculating has taken so long.
The reasons involve both principles and complexity. For one thing, it has
gradually dawneduponmostworkersthat stable self-gravitating disks (especially
ones composedof collisionless stars) maynot evenpossess manydiscrete normal
modesof the sort that one associates with church bells and Cepheids. The
vibrational spectra of our systems seem, by and large, to be continuous. The
associated modesmust look disagreeable indeed near any resonanceradii--somewhat like the van Kampen(1955) modesfrom plasma physics--as Hunter (1969)
demonstratedfor certain m¢ 0 wavesnear their corotation radii, and as Erickson
(1974) showedeven for someseeminglyinnocuousm--- 0 vibrations. Hunteralso
cautionedof similar complicationsnear edgesof lowdensity.
A seconddifficulty has beenthat, evenif strictly oscillatory discrete m= 2 modes
do exist, it remainsunproventhat they include any of spiral form. Indeed, such
truly permanentspiral wavesof infinitesimal amplitudenowseemhardly possible.
This personal opinion stems muchmorefrom the formidableresonanceeffects and
gravitational torques discussed,for instance, by Lynden-Bell
&Kalnajs(1972)than
from the weak"anti-spiral theorem"of Lynden-Bell&Ostriker (1967)--which
essence states only that the existence of a steady trailing modeimplies a twin of
leadingshape,andvice versa, becauseof time-reversibility.
Theabovefrustrations have not referred especially to unstable spiral modes.
(Quite reasonably,it is the milderversions of just such modesfromnondissipative
linear theory that are widelythought nowto offer the best hopefor somesort of
conversioninto quasi-steadyspiral patterns of finite amplitude--andthis is where
thresholdeffects suchas the nonlinearshockdamping
stressed in Section5.3 not only
assumea special importancebut mayharbor a secondvital reason as to whygood
spirals seem to need gas.) The problemhere has been muchmorethat one has
tendedto find such unstablemodes,evendiscrete ones, far too fiercely and far too
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SPIRAL
STRUCTURE
471
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often! Just as in the N-bodystudies, muchof this nuisancehas doubtlesshad good
physical causes, but also as with those experiments,somemayeven havestemmed
fromingenioustricks that backfired.
Onesuch examplecomesfroma labor-savingidea that occurredto Miller (1971,
1974)among
others. Henoticedthat the use of the "softened"gravitational potential
z q-d2) - 1/2 rather than the precise q~ = - GM/dfor the interaction of
c~ = - GM(b
twoparticles separatedby a distanced wouldyield the dispersionrelation
~2 = tc2(r)_ 2nGl~(r)I k l e-lkl~
(12)
instead of Equation(1) for short axisymmetric
vibrations of a cold thin disk. The
exponentialmimicsthe "reductionfactor" of Lin &Shu (1966), and so this simple
relation indeed resemblesthe one in Figure 4. Towardthe short wavelengthsin
particular it seemsmuchmorerepresentativeof the ~ --, x behaviorof wavesin a
stellar disk than wouldbe implied by the analogousformula
~
~o2 =K2(r)-2nG#(r)
(13)
I k } +aZk
(cf Safronov1960, Hunter1972)for a thin gas layer with "honest" gravity but
imaginedsoundspeed a.
Theirony here is that Erickson(1974),using just this presumably
superiorgravity
shortcut (to avoidall the stellar-dynamicalbother with the Boltzmann
equationin
his global-modecalculations for someGaussiandisks) indeed obtained numerous
spiral instabilities resemblingthe m= 2 modein Figure12. Unfortunately,he also
foundhimself unable to turn off all such growingmodes,no matter howlarge he
chose that cutoff length b. By contrast, Bardeen(1975) adopted the seemingly
weakergas idealization and managed
to stabilize such disks altogether, at least for
energyratios T/W<~0.26 corresponding
to dynamical(rather than secular) stability
of Maclaurinspheroids. Withthese experiences in mind--andalso noting that
Kalnajs(1970onwards)has not yet succeededin obtainingfully stable stellar disks,
apart from his tour de force with the Maclaurin-Freeman
models--it is no wonder
that modecalculators remaina little unsureof just wherethey stand.
To concludethis litany of surprises, somesigns of a counterexampleto the
Ostriker-Peeblescriterion havesurfaced recently in the workof Zang(1976)
the unstable modesof a class of exact stellar disks with randommotions.These
disks themselves,whichwere noticed also by Bisnovatyi-Kogan
(1975), epitomize
"flat" rotation curvesin assuming
V(r) -~ const= V0at all radii; this meansthat the
surface density/~(r) Vd/2nGr,orthat the total massgrows
infinite linear ly with
the radius r. Thelatter property,and also the central singularityof/~ itself, need
not concern us too much,since both can doubtless be remediedwith plausible
cutoffs. Onebig advantageof this modelis that it is everywhereself-similar.
Anotheris that its Velocitydistributionfunction
f(u, v, r) = F(E, J) = const- Jq e-E/a~.,
(14)
z
2)
z
withE = ½(u + v + V0 In r, J = rv, andq + 1 = VoZ/az.,canbe written so compactly
for J > 0 (withf -~ 0 otherwise).
Asfar as axisymmetric(or m= 0) modesare concerned,Zangfoundthis class of
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472
TOOMRE
models to becomeglobally stable when cr~ ~ 0.378Vo, in good agreement with the
"local" estimate given by Equation(3). Alreadyfor that critical value of the velocity
dispersion, however, Zangwas unable to locate numerically, despite muchsystematic
searching, any two- or multi-armed(i.e. m> 2) instabilities. This important result
needs to be qualified in two ways. Oneis that Zangliterally examinednot the full
models given by Equation (14), but ones where he had immobilized the central
regions--as if the central "bulge" were particularly hot and therefore unresponsive-by multiplying that f by the factor [l÷(Jo/J)"]-~; he did this mostly to avoid
immenseangular speeds as r ~ 0, but also to provide a length scale Jo/Vo. The other
proviso is that Zang managed to interrogate his difficult integral equation,
patterned closely after the one by Kalnajs (1971), only for discrete unstable modes,
which he assumedto grow like est while rotating with an arbitrary pattern speed
~. An unstable continuum seems unlikely, but Zang could provide no proof.
Thus, to be precise, Zang’sinability to locate any m= 2 instabilities referred only
to discrete modes of modified disks with n = i or 2--or ones whose centers had
been "carved out" fairly gently. As that truncation was made more abrupt, some
unstable modes actually emerged; one such mode, for index n = 4, is pictured in
Figure 12. Modeslike that were definitely indebted, however, to the artificial
sharpness of the inner boundary. Althoughsuperficially similar to the modesinferred
by Lau, Lin &Mark (1976) from WKBJ
theory, the attractive but unrealistic m =
spiral wave in Figure 12 grew an order of magnitude faster than then expected by
those authors.
......
m=2
"...
m=l
Figure 12 Twomost unstable spiral modesobtained by Zang(1976) for centrally cut-out
versions of the V(r) = Vodisks. In both cases, only the positive parts of the disturbance
density are shownhere, using contour lines set at 80, 60, 40, 20, and 10~of peakvalues;
dots markthe nodes. The shadedareas have radius r = Jo/Vo. Them = 2 mode(referring
to the modelwith velocity ,dispersion tr, = 0.378Voand cut-out index n = 4) has pattern
speed flu = 0.439 and growthrate s = 0.127, both expressed as multiples of Vo~/Jo. The
analogousdata for the m= 1 mode(obtained for n = 2) are ~, = 0.141 and s = 0.066.
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SPIRAL STRUCTURE
473
Eventhis story, however,proved to contain a nasty surprise. Regardless of
whetherhe tried cut-out indices k = 1, 2 ..... Zangremainedplaguedwith numerous
one-armedunstable modes!Oneof themis also shownin Figure 12. Unlike the
fewtwo-armed
modes,this m= 1 instability could not be attributed veryplausibly
either to the frozen center or evento the fact that such a core--in a deliberate
rerun of a rare old error by Maxwell--was
here kept mostlyfixed. Zangconcluded
ruefully that with this curious swapof difficulties "wemaymerelyhave jumped
fromthe frying paninto the fire."
8
SHEARING BITS AND PIECES
As our last topic, it needsto be said briefly but aloud that there seemsnothing
fundamentallywrongwith the simple idea that muchof spiral structure--which,
after all, is often very raggedand confusing,as in most of M101--consistsmore
nearly of things rather than waves, to use Prendergast’s (1967) apt word. The
difficulties with this notion are mainlyjust quantitative, onceone excludesthe
"granddesigns."Obviously
all shearing"things" in a galaxywill briefly exhibit some
spirality, like creampouredinto a newly-stirredcupof coffee. Hencethe real task
remains instead to decide howand whetherthe material in question can keep on
makingsuitable fresh bits and pieces to replace those older fragmentsthat wrapup
and disappear,
Theonlypromisingsite for suchrecurrentinstabilities is, of course,the gas layer
of a galaxy. AlreadyyonWeizs/icker(1951) thought so, with his vaguetheory
"turbulence... producedby nonuniformrotation." Jeans (1929, p. 379) did almost
the samewhenhe wrote of the "chaos of movingclusters" in the outer parts of
spiral nebulae, andremarkedthat "the theory of gravitational instability makesit
easyto understandhowthese large clusters cometo exist."
In fact, the regenerationof large newfragmentsin a gas disk is not quite so easy.
VonWeizs/ickerin effect forgot that our kind of shear flowis supportedmuchmore
by gravity forces than by pressuregradients, and Jeans did not realize that rotating
disks also havea maximum
critical lengthfor gravitational instability, suchas 2¢rlt
from Equation(2), whichneed not be at all comparableto overall dimensions.
Indeedthat lengthcanbe distressinglyshort if the "active"surfacedensity/~oof the
subsystemof interest is onlya small fraction of the total, and if the stellar disk
-2 for the
cooperateshardly at all. For example,evenif we adopt l~o = 10 M®pc
nearbygas in this Galaxy,its 2trot worksout as only1.7 kpc--andthen an effective
soundspeeda _->4 kmsec- 1 shouldalreadycurball Jeans instabilities [cf Equation
(13)].
All this waswell appreciatedby Goldreich&LyndenoBell
(1965) in whatremains
the only extensivemodernexplorationof the themethat Sc galaxies consist largely
of "a swirling hotch-potch of pieces of spiral arms." To overcomethe above
difficulties, these authors suggested that "spiral arm formation should not be
regardedas an instability in [just] the gas but rather as an instability of the whole
star-gas mixturewhichis triggered by an increase in gas density" in a volumetric
sense--or, whatamountsto the samething, by any cooling or dampingthat lowers
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474
TOO~R~
the sound speed a in the gas. Even those plans, however,had two serious flaws. One
was the fact that the fiercely amplified shearing waves studied by Goldreich and
Lynden-Bellwere, strictly speaking, no true instabilities. The other flaw was the
presumption that strong help from the undampedstars would continue despite all
their repeated heating by the postulated clumpingsof the gas; this idea rested mostly
upon wishful thinking. Nevertheless, in that severe amplification, Goldreich and
Lynden-Bell offered one real nugget of a discovery that greatly softens the last
criticism. It is shownin a morerefined form in Figure 13.
This old diagramreemphasizesthat a shearing stellar (or gas) disk that is locally
quite stable can still respond with remarkable vigor to the nonaxisymmetric
gravity forces from any fortuitously bound, orbiting gas lumps. Notice that the
excess mass that pauses in the trailing "wakes"illustrated here exceeds the imposed
mass by about an order of magnitude, even though Q = 1.4 plus a modest softening
due to thickness were assumedin this example. Notice especially that this arm-like
shape does not undergo any shearing. It is itself a steady density wave, albeit a
forced one. In short, Figure 13 renews hope that here may be a reason why some
of the observedspiral irregularities extend, as in M101, over quite sizable fractions
of the radius despite the fact that the gas content tends to be hardly of order 10 per
cent. It also cautions that even the "hotch-potch" is apt to be a little more wavelike
in character than one might perhaps have supposed.
Any reader who(rightly) finds these remarks unpersuasive should look again
Figure 11 and especially at the experiments of Hohl (1970) and Hockney
Brownrigg (1974), all conducted with active masses totaling only 10 or 20~.
Figure13 Wavelikeridge of excess density in a Q= 1.4, V(r) = const disk of stars shearing
past a small orbiting masspoint (Julian & Toomre1966). By comparison,that massitself
wouldyield unit density if spreaduniformlyover the little square. This diagramstems from
small-amplitudetheory with several "local" approximations;all subsequentovertakingsof,
andby, stars havebeenneglectedhere. Theself-gravity of stars wasincluded,however,and
the assumptionthat 2erlt = R0, madeonly in plotting, possibly underestimatesthe true
extent of this quasi-steadyforced wave.
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SPIRAL STRUCTURE 475
should ask howhe wouldexplain those examplesof "material clumping[that] is
periodically destroyedby differential rotation and regeneratedby gravitational
instability" (Lin &Shu1964),if not by somethingakin to the amplificationprocess
of Goldreichand Lynden-Bell.Of course, the bigger question remains:Whyshould
enoughdisk material still be %oo1"enoughin its randommotionto act even as
cooperatively as shownin Figure 13? Are we, for instance, underestimatingthe
amounts
of gas present, or else the recent historyeither of infall of gas fromoutside
or its return fromdyingstars, includingthose in any halos?
All in all, this topic of various "secondary"spiral features seemsmuchoverdue
for renewedattention. NotevenvonWeizs/icker’sidea canyet be dismissedentirely:
someinstabilities mightindeedbe largely hydrodynamic.
It still needsto be explored,
for instance, whetherthe well-knownlumpinessof the major arms themselveson
about a 1 or 2 kpc scale mightnot stemmainlyfrom somesort of Kelvin-Helmholtz
instability (rather than Jeans instability, or eventhe Parker = magneticRayleighTaylor kind advocatedby Shu 1974)of the strong shear flows that are inevitable
whengas froma wholerange of radii crowdstogether behinda spiral shock.
Finally, what about spiral structures provokedfromoutside? Aquick answeris
that this old tidal idea indeed seemsto workfine--in a small handful of obscure
cases like Arp 295 (Stockton 1974)and NGC
2535/36or 3808. In those instances
just as in somesimple computations(cf Toomre1974b), the two"material arms"
are rather open and the one toward the companionoften looks grotesquely
straighter and longer than the other. It remainsfar less certain, however--despite
muchspeculationgoingback at least to Chamberlin
(1901) in the case of M51,and
again to Lindblad(1941) with MS1/82--thatthe familiar "granddesigns" of M51,
M81,and NGC
5364are indebted in any serious wayto the (admittedly worrisome)
proximity of major companions.
For reasons of tightness and extent of spiral structure, plus the Westerbork
detection of ampleH I also betweenthe mainarmsof M81,those famousexamples
cannotpossiblybe tidal in the naive kinematicsense impliedabove.Herethe tidal
process, if indeed any, musthavebeena lot subtler. Thefirst step wouldhavehad
to be somemajordistortion of the outer disk or halo. This in turn mayor maynot
haveinduceda relatively transient spiral wavein the properdisk, traveling inward
with the groupvelocity. If only somedecentlycool but stable modelsof galaxies
existed already, it wouldbe fairly easy to checkwhethersuch a schememakesany
sense dynamically.Until then, however,claims such as by Tully (1974) that °’a
density wavehas been generated [in MS1]by the innermost material clumping
arising from the encounter"with NGC
5195strike eventhis sympatheticreviewer
as very premature.In fact, judging from the broadouter contours of MS1toward
the south--especially evident in one very deepphotographand in various HI data
that Lyndsand Shanehave kindly shown--it seemsthat Toomre&Toomre(1972,
Figure20a) erred in reckoningthe present epochof that encounterto be as late as
"t = 2.4." Abetter guessappearsto be t = 1.6 or 1.8, in the sameunits. Giventhat
those outer distortions becomepronouncedonly from t = l onward,this shorter
elapsedtimegravelyaggravatesa difficulty notedalreadyby Lin(1975),namelythat
"the time requiredfor the propagationof suchan influenceinto the central regions
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476
TOOMRE
is tOO long." Hence it looks more and more as if even in M51 the main spiral
structure is in essence just a badly-dented version of what would have been there
anyway.
To end on a more positive note, it has emerged recently that density/shock waves
of roughly circular
shape--now indeed caused by a second galaxy--seem to be
present in the rare but very striking oddities known as ring galaxies. As Fosbury
& Hawarden (1977) have just illustrated
for the fine example of the "Cartwheel"
discovered by Zwicky (1941), it seems there again that simple mechanical crushing
has evoked intense star formation in the knots of H II regions which mark that
ring itself--and also that the residue of this indelicate processing includes the usual
"feathers"
or "interarm branches" or secondary spiral
arms!
ACKNOWLEDGMENTS
Among several kind friends,
I am most indebted here to Agris Kalnajs--and
to
editors Burbidge, Goldberg, and Layzer for much patient nudging. I am also very
grateful for all the support of the NSF.
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