1930 the Rotation of the Galaxy Being the Halley Lecture (Arthur S. Eddington)

8/13/2019 1930 the Rotation of the Galaxy Being the Halley Lecture (Arthur S. Eddington)

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THE R O TAT I ON OFTHE GALAXY

beingTHE HALL EY LECTURE

delivered on so May 1930BY

A. S. ED D I N GT ONM.A. D.Sc. LL.D. F.R.S.Plumian Professor of Astronomyin the University of Cambridge

O X F O R DAT T H E CLAR ENDO N PRESS

1930


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OXFORD UN IVER SITY PRESSA M E N H O U S E E . C . 4

L O N D O N E D I N B U R G H G L S G O WL E I P Z I G N E W Y O R K T O R O N T OM E L B O U R N E C P E T O W N B O M B Y

C LC U T T M D R S S H N G H IHUMPHREY MILFOR D

P U BL I SH E R T O T H EU N I V E R S I T Y


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STAR CL OU D I N S Wt ITTARIUSThe centre of our galaxy hidden by obscuring matter) lies near the

middle of the right-hand edge of the photograph


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FrontispieceSTAR CLOUD IN SAGITTARIUS

Photograph E E Barnard

P RI NT ED I N G R E A T B R I T A I N


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6 T h e Rota/ion o fdeterminations of their velocities in the line of sight byuse of the spectroscope. W e have therefore a mine ofmaterial from which we are trying to learn what we canof the nature of the motions of the stars as a system andto reach some kind of dynamical theory of what is goingon. A caution must be given at the outset. According tomodern views the dimensions of our galaxy are immense ;and although our survey of stellar motions extends overa region containing perhaps i o to Too mil lion stars, thisis but a small part o f the whole. W e have to take a riskin inferring the nature o f the complete system from thesmall sample within reach.Throughout the nineteenth century astronomers work-ing on stellar motions concentrated their attention on onemain themethe solar motion, or velocity of our sun asan individual star with respect to the system as a whole.For our present discussion of the system of the stars thishas no particular interest, being merely a distorting factorin our outlook which is sometimes troublesome to eliminate.We are concerned with the stellar motions remaining afterour own translational velocity has been allowed for ; theyare by no means those o f an unorganized crowd. B ylater researches four leading peculiarities have been dis-covered. I give them in historical order :(I) Star-streaming, i. e. a tendency of the stars to moveto and fro along one particular axis in space rather thanin directions at right angles to it.(2) A strong correlation between the velocity and thephysical characteristics of the stars. F o r example, starsclassed as of late spectral type have a higher averagespeed than those of early type.(3) Stars of exceptionally high velocity (greater than8o km. per sec.) are found to be moving exclusivelytowards one hemisphere of the sky.

(zj) An effect rather complicated to describe which weinterpret as evidence o f rotation o f the whole system.This is the main theme of my lecture.


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the Galaxy 7In conjunction with these results we have to considera matter of common knowledge inferred from the apparent

distribution (not the motions) o f the stars. O u r stellarsystem has a very oblate form. I t is believed to be almosta diskresembling the spiral nebulae seen abundantly inthe vast universe beyond the confines of our galaxy.Nature o f the Rotation

The discovery of the fourth effect and the interpretationplaced on it are due to J. H. Oort of Leiden. Amongother investigators should be mentioned especially B. Lind-blad, who had been developing the hypothesis of galacticrotation for other reasons, and J. EL Plaskett, to whom weowe the most convincing observational evidence.It will help us to understand what kind of indication ofrotation we might look fo r in a system of stars, i f wetransfer our attention for a moment to a phenomenonnearer home, namely Saturn s rings. These have a roughresemblance to the disk-like form attributed to our galaxy.At one time there was a division of opinion as to whetherthe rings were solid structures, or whether they consistedof swarms of small particles. I n a famous mathematicalinvestigation, which is one o f the classics o f celestialmechanics, Clerk Maxwell showed that the solid type ofring was dynamically impossible ; i t would be unstable.The only permissible constitution was a swarm of separatebodies. M a n y years later Maxwell s theory of the ringwas strikingly confirmed by Keeler ; and it is his methodof confirmation which especially interests us. I f a solidring rotates, its outer edge must necessarily travel fasterthan the inner edge ; on the other hand, i f the ring isa swarm of meteoric particles, they will follow the samerule as the planets i n the solar system, viz, the innerparticles must travel faster in order to counterbalance thestronger gravitational pull of the planet. Keeler found byspectroscopic observation that the inner edge of Saturn s


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8 T h e Rotation o f the Galaxyring travels faster than the outer edge, indicating thereforethat it is a swarm of particles and not a solid arch.

In the galaxy we know that we are dealing with a swarmof particlesstarsand not with a solid ring. Consequentlywe may expect that it will rotate after the manner ofSaturn s ring, the inner stars travelling faster than theouter stars. T h i s is fortunate for our hopes o f detectingrotation. F o r investigating this problem we are dependentalmost entirely on observed radial velocities. R a d i a lvelocity means the approach or recession of other particlesfrom our own particle (the sun) ; clear ly radial velocitymeasurements would be unaffected b y and would notdetect a rotation like that of a solid body in which allparticles preserve the same distance apart. I t is importantto bear in mind that the effect manifested by the radialvelocities, and detected and measured by Oort, is not theabsolute rotation but the differential rotation or Saturn sring effectthe increase of angular velocity as we gotowards the centre of the system.Fig. i shows a portion o f the galaxy rotating abouta centre situated far outside the diagram, the rotationbeing faster as we go towards the centre. W e must ask,How will this appear to an observer in the midst of theregion ? H e wi ll appreciate only the relative motion o fthe different parts o f the system. I n Fig. 2 we havereduced him to rest by applying to all parts of the regiona velocity equal and opposite to his own.The observer is armed with a spectroscope and measuresvelocities (relative to himself) in the line of sight. W e seefrom Fig. 2 that there are four directions in which thisline of sight velocity wil l be zero, viz, to the right and left(approximately) because there there is no relative motion,and up and down the page because there the relativemotion is entirely transverse to the line of sight. B u t indiagonal directions an effect wil l be observed ; the starsseen in both directions along one diagonal are recedingand those seen along the other diagonal are approaching.


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> > 4 < 4

> > 4

>i > >

FIG. I. F I G . 2.

\

/

1\

Ae

/ 0

0

\

FIG. 3. D ist ri bu ti on of Radial Velocity


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lo T h e Rotation o fFig. 3 shows the resulting distribution o f radial motionignoring the transverse motion which is not detected bythe spectroscope). I t will be seen that the distribution ofmotion is of the kind which distorts a square into a diamond.This distortion comes from the shearing effect when theinner part of a ring travels faster than the outer part.Mathematically we can describe th is distribution b ysaying that, when the stars are arranged according t ogalactic longitude /, their observed radial velocities containa term C sin 2 / - /0), w h e r e4i st h el o n g it u d eofthec en tr e

of the system. Moreover, i t is clear that the effect isgreater for greater distances being approximately pro-portional to the distance of the stars considered from theobserver. W e therefore express the term asA r s in 2 / -

where r is the distance of the stars examined, and A isa constant.The stars have their own individual motions super-posed on the general rotation of the system, and we canonly expect to discover this effect if we average out theindividual motions by taking means for a considerablenumber of stars. O w i n g to the increase o f effect withdistance it is best to search for it in the more distantclasses of objects. I t may be said at once that the searchis successful. T h e expected distribution o f velocity i sfound in al l classes o f objects that could be expected toshow it, and they agree among themselves both as to themagnitude of the effect and as to the direction in whichthe centre of the galaxy is situated.

Observational EvidenceThrough the researches of Harlow Shapley the centre ofour galaxy had already been located in the direction of thegreat star-clouds o f Sagittariusthe richest part o f theMilky Way. H e deduced this from the distribution ofthe most distant galactic objects observable, particularly


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the Galaxythe globular clusters, which may be supposed to outlinethe shape o f the system. T h e exact centre cannot befound with any high accuracy, but the position generallyadopted is in 325 galactic longitude. O or t s method ofdeducing it from the rotation effect is entirely independent ;it generally gives a rather higher longitude 336)-335 , butthe difference is within the probable uncertainty of thedeterminations.

As already stated, the magnitude of the effect increaseswith th e distance. F o r stars distant L000 parsecs 1 i tamounts to 17 km. per sec., that is to say the stars seenat this distance i n one part o f the sky are in the meanmoving towards us at 17 km. per sec. whereas those goaway are moving from us at the same rate. F o r otherdistances the effect is in proportion 81 km. per sec. for500 parsecs distance, 34 km. per sec. for 210 0 0 p a r s e c s ,a n dso on. T h i s provides what may ultimately prove to bea valuable means of finding the mean distance of a classof objects when it is not determinable by older methods ;for i f we measure the magnitude o f the rotational effectwe can at once wr ite down the corresponding distance.To illustrate this I will refer to a remarkable investigationby Plaskett and Pearce.Their research dealt with the radial velocities of about250 stars o f the most distant type known. T h e y wishedto sort these into groups according to distance ; but sincethe stars were far beyond the range of ordinary methodsof distance determination this separation presented somedifficulty. I t is not much use to sort them according toapparent brightness, because brightness is a poor criterionof distance. T h e authors availed themselves of a methoddeveloped recently by Otto Struve. W e are looking atthese stars through a th in ve il o f cosmical cloud. T h ecloud leaves its mark on the light, producing certainnarrow absorption lines in the spectrum of the star. I fthe absorption i s intense i t is a sign that we are looking

1p a r s e c = 3.26 light-years.


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12 T h e Rotation o fat the star through a great thickness of cloudthat thestar is very remote. B y this criterion Plaskett and Pearcedivided the stars into three groups showing low, medium,and high absorption, respectively, which must correspondpresumably to small, medium, and great distance.In the following table the third column gives the magni-tude of the rotation term for each of the three groups andthe four th column gives the deduced distance (the pro-portion being 17 km. per sec. per ,coo parsecs as alreadystated). I t wi l l be seen that Struve s criterion has beensuccessful ; or at least that Oort s and Struve s methodsof estimating distance (both o f which must be regarded ason trial) confirm one another.

STARS. C L O U D .Rotation Rotation

Absorption. No. ofStars. Effect.km. persec.Distance.parsecs.

Effectkm. persec.

Distance.parsecs.

low 90 10-2600 5o 9

medium 79 14-5 85o 6-9405igh 43 27-5 1 , 620

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8o5

Turn now to the fifth and sixth columns, in which thesame analysis is applied not to the stars but to the motionsof the cosmical cloud. T h e velocity of the cloud can bemeasured in the same way as that o f a star f rom theDoppler shift of the spectral lines which it absorbs ; but,of course, our measurement refers not to the whole cloudbut to the particular part of the cloud responsible for theabsorption. I f the absorption occurs uni formly i n thecloud the mean distance o f the stretch traversed b ythe star s li gh t should correspond t o half-way. T h edistance o f the veil ing cloud should therefore alwayscome ou t to be hal f the distance o f the correspondingstars. A glance at the table wi l l show how closely this


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The Rota lion o fmake the conclusion more convincing he made a test ofthe validity and efficiency of the method by applying it tothe apparent motions o f the asteroids, using them asa model for illustrating differential rotation as I have usedSaturn's ring . F r o m th e motions o f the asteroids hededuced the direction of the centre of the system, viz, thesun ; his error was about 6'. T h a t would be one way offinding the sun again i f ever i t ceases to be visibleReturning to the stellar system Gylden remarked thata definitive determination of the centre was not at presentpracticable because the common rotational motion waspresumably in the plane of the Milky Way, and propermotions for the part o f the Mi lky Way in the southernhemisphere were lacking. H e had to content himself withsuch indications of the centre as could be found fromanalysis in the plane of the equator. I t is true that thedirection provisionally given by Gylden for the centre ofthe system is opposite to that now generally accepted ;but that is because the double-period term fixes only theline to and from the centre, and does not decide betweenthe two possible antipodal positions. H e concluded : ' A tall events there remains an indication that the motions ofthe stars have something in common, and that they arenot so at random as many astronomers have been inclinedto assume.'

By these researches we find the change o f velocity ingoing towards or away from the centre ; we do not learnthe actual velocity at any point. A possible way of dis-covering this is by observing the globular clusters whichcan be seen at very great distances up to and beyond thecentre o f the galaxy. B y thei r great spread they w il lhave a mean motion fair ly representative of the system asa whole, whereas our stellar observations are limited toa comparatively small region and give the local motion.The difference represents the mean speed at which thestars in our neighbourhood are t ravel ling through thesystem. T h e result of this determination cannot at present


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the Galaxy 1 5be regarded as very accurate, but it is sufficient to showthat our orbital speed is large, probably between 200 and300 km. per sec.

Consequences of the Rotation.We have thus to recognize that for a broad cosmicalsurvey the standard of rest to which we have been in thehabit of referring all our measured velocities is an inappro-priate one. W e realized long ago that i t was too crude totake the sun as standard, and we have referred velocitiesto the mean of the stars meaning the stars which comewithin range of ordinary measurements. N o w we have torecognize that this also is a very local standard affectedby large orbital velocity, and we must apply a furthercorrection o f two or three hundred kilometres per secondto reduce to the centre of our galaxy. I am afraid it is toomuch to hope that this will be our final resting-place ; wesee in outer space some hundreds of thousands of othergalaxies which will claim a share in defining a universalstandard. Meanwhile the shift of our view-point to thecentre of the galaxy has produced one great improvement ;it has brought better order into the motions of the spiralnebulae. I t is the general rule that spiral nebulae arereceding from us at ve ry high speed ; the greater thedistance, the higher the speed. B u t the rule was marredby two notable exceptions. A s these two are the largestand almost the nearest of the spiral nebulae we do notexpect any decided recession in their case ; but it wasdisconcerting to find that they were approaching us withhigh velocity. W e now learn that this apparent approachis merely the reflection of our own high orbital speed intheir direction, and when we refer thei r motion to thecentre of the galaxy nothing very serious remains.

At this point we can weave into the picture anotherfeature of stellar motions mentioned in the list on p. 6.High velocity stars, i . e. stars w ith speeds greater than


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the Galaxy 1 7is at present badly determined, I wi l l give the result forseveral different adopted values. T h e mass of the systemwhich controls the orbital motion can also be calculated.AssumedOrbital Speed.km. per sec.

Distanceof Centre.parsecs.Mass ofSystem.Sun s Mass ,

150 6,600 33,000,000,0002 0 0 8,800 78, 000, 000,000250 11,000 150,000,000,000300 13,200 260,000,000,000350 15,400 4 2 090 0 010 0 010 0 0

These results may be compared with estimates arrivedat in an entirely independent way. T h e distance fromthe centre seems to be of the right order of magnitude.Thus Shapley from his work on globular clusters locatedthe centre of the galaxy at 13,000 to 25,000 parsecs distance.The mass also, although higher than most current esti-mates, is not unreasonably large. B y extrapolating theresults of actual counts o f stars Seares and van Rhijnobtained a to tal o f 30,000,000,000 stars i n the galaxy.Since dark nebulae hide our view, more especially in thedirection of the centre, it is doubtful whether their surveycomprehended the whole system and the number maywell be greater. T h e average mass of a star is probablynot more than hal f the mass of the sun, but there is inaddition the mass of the cosmic cloud and of the brightand dark nebulae to be brought into account.

How long does the galaxy take to make one completerevolution ? T h e answer is about 250 million years. W ecan state the figure fair ly definitely because i t does notdepend on any of the more doubtful estimates ; the onlydatum needed to determine it is the magnitude of the Oorteffect. I t should, however, be added that since the innerparts of the galaxy rotate faster than the outer parts, thereThe calculation is on the assumption that the main part of the massof the system is concentrated near the centre. I f the mass is moregenerally diffused the distance and controlling mass are somewhatreduced, but the order of magnitude is not greatly altered.

76


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18 T h e Rotation o fis no one period of revolution for the whole ; the period250 million years refers to the zone in which the sun lies.It is important to notice that the galaxy has made five orsix rotations within geological times. T h e sun and earthwere away on the far side of the centre Too million yearsagoa time which geologically does not seem to be veryremote.

We may now sum up the evidence for the hypothesis ofa rotation of the galaxy. A n effect resembling differentialrotation is observed in all classes of distant stars and alsoin the cosmic cloud pervading the system. These giveconsistent indications of the direction of the centre andthey agree also as to the amount of differential rotation.The evidence from proper motions has small weight butfor what it is worth it supports that derived from spectro-scopic radial velocities. T h e dimensions and total massof the galactic system inferred fr om t hi s effect arereasonably consistent with current estimates based onother data. O u r large orbital velocity o f 200-300 km.per sec. is confirmed to some extent by observations ofglobular clusters and spiral nebulae which are too remoteto partake of it. Further since stars with a large individualvelocity additive t o the general orbi tal velocity wouldescape from the system we have a simple explanation o fa well-known phenomenon viz that high velocity starsfavour a direction now identified as that opposed to theorbital motion. F i n a l l y the ve ry oblate shape o f thestellar system is strongly suggestive o f rapid rotation ;and in the spiral nebulae which are believed to be patternsof our galaxy the rotation can be directly observed andmeasured.

The evidence seems convincing ; nevertheless a threadof insecurity runs through the whole fabric. I t is the oldstoryour conclusions rest mainly on observations of thenorthern celestial hemisphere and the southern observa-tions make a poor counterweight. T h i s is a commoncomplaint in all discussions of stellar statistics ; but I think


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20 T h e Rotation o fmanent cluster of this k ind i f the hypothesis o f galacticrotation is accepted. Ta k i n g the minimum estimate o f700 parsecs diameter the differential rotation is such thatthe inner edge of the cluster will make eight revolutionswhilst the outer edge makes seven. Obviously a compactcluster w i l l be qui ck ly sheared in to elongated form,ultimately to be drawn out into a complete ring. I t is notlegitimate to reply that their mutual gravitation wil l helpto keep the stars of the local cluster together and perhapsoverride the forces of dispersal ; for it is from these verystars that the observational evidence of a dispersing motionis derived. T h a t the distribution of stellar motions aroundus is such as would elongate and disperse a local clusteris an immediate observational conclusionindependentof our interpretation of it as evidence for rotation of thegalaxy. I t would be contrary to observation to deny theexistence of irregularities of distribution like star-clouds,but I think they must be regarded as transitory eddies ina whirlpool, which form and dissipate continually.The results now before us raise an interesting dynamicalproblem ; but before entering on it , it is necessary to beclear as to our guiding principles. O n e possible aimwould be to develop a theory showing how the presentcomplexities of motion and distribution of the stars mighthave arisen by natural evolution from some simpler andmore uniform initial state satisfactory to our sense of fit-ness; but that is probably too ambitious a programme atpresent. I n most investigations the guiding idea has beenthat, whatever initial formation the stellar system mayhave developed from, it has at any rate been a very longwhile about it. Consequently i f we trace back its historya few thousand million years we ought not to find muchchange. Accordingly, the mathematical conditions of theproblem are assumed to be that the stellar system i s(approximately) in a steady state.It may perhaps be thought that too much of a fetish hasbeen made o f the steady state in the various mathe-


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the Galaxy 2 1matical theories of the galaxy, but that is a misconceptionof their aim. A l though formally the mathematician mayseem to be designing a model stellar system which wi l llast for ever, that is only his way of tackling the design ofa model which will last long enough to fulfi l the obviousrequirements o f the problem. Geolog ical time swallowsup at least 1,500 million years ; we must allow a reasonablemargin beyond that for the evolution of the solar system,say 3,000 million years altogether. T he minimum require-ment of our model system is that i t wil l keep going forthat time without collapse. Actua l ly we are hard put toit to invent a galaxy with even this limited degree ofpermanence, which shall at the same time embody themain features of stellar motion and distribution enumeratedon p. 6. A s for those who dabble i n the l ong t ime-scale of bill ions of years now fashionable (and I have toconfess myself one of them) we must simply ignore them.Whatever the study of individual stars may bring forthin its favour, the evidence of galaxies and of systems o fgalaxies is dead against so leisurely a rate of progress.The problem of the galaxies is unapproachable exceptfrom the standpoint that the material universe is a muchmore evanescent affair.

The term steady state is used with two distinct mean-ings, and we must further define which meaning concernsus here. Starting with an entirely irregular distributionof stars and stellar velocities, there are two stages in theapproach towards ultimate equilibrium. T h e first stage isaccomplished when the orbits described under the generalattraction of the whole mass have become so distributedas t o preserve the shape and density unaltered ; thedensity of population at any point then remains steadyalthough the individuals are moving to and fro. T h i s iscalled dynamical equilibrium. B u t these orbits are fromtime to time perturbed by chance approaches of the starsto one another, and the distribution slowly changes unt ila special form of dynamical equilibrium is reached with


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22 T h e Rota/ion o fthe additional property that these haphazard perturbationsproduce no change on balance. T h i s is called statisticalequilibrium. W e shall see presently that the steady stateof the stellar system is that of dynamical equilibrium, andnot the ultimate statistical equilibrium which belongs toa far distant future.

The rate of approach to equilibrium, whether dynamicalor statistical, may be measured by a 'time of relaxation 'a time in which deviations from the equilibrium distributiondecay to about half their original magnitude. F o r statisticalequilibrium the time o f relaxation is estimated by Jeansand Charlier at 1o" to i o " years. T h i s is so great com-pared with even the most extreme time-scale that we mayput statistical equilibrium outside our thoughts. B u t thetime of relaxation towards dynamical equilibrium is of theorder I,000 million years. Remembering that in our partof the system individual stars go righ t round their orbitsin 250 million years, any irregularity wi ll in general bedissipated all over the system in something like thatperiod. W e therefore expect to find dynamical equilibriumfairly complete.In a series of three papers in 1913-15 I discussed theconditions for a steady state (dynamical equilibrium) o fa system of moving stars, including the case of a flattenedsystem like our galaxy, both with and without rotation.Re-reading these papers I do not find anything to modifyin the mathematical investigations, except that later writershave found short cuts to some of the results ; nor do theyseem to need much extension or adaptation to cope withthe problems that have cropped up since then. B u t whenI turn to the efforts I then made to fi t the theory to theobserved properties of the system, i t is like a glimpse ofthe middle ages. I s it possible that only fifteen years agowe thought the stellar universe was like that W i t h somemisgiving I found I must place the sun at least soo parsecsaway from the centre of the system ; bu t I did not expectto be believed. Nowadays 7,000 parsecs is the minimum


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the Galaxy 2 3estimate. I was perturbed to fi nd that on the rotationtheory I must ascribe to our neighbourhood an orbitalvelocity as great as 25 km. per sec. ; in excuse I offereda suggestion as to how the differential effects o f so rapida rotation might happen to be concealed. N o w we admitan orbital velocity ten times greater, and claim that itsdifferential effects are not concealed but strikingly mani-fested. I t almost looks as though wi th a lit t le morecouragea little less reluctance to rend the then ac-cepted fabricthe purely dynamical theory might haveforecast the changes that have since been made as theresult of observation ; I doubt, however, if that would havebeen justifiable without some additional facts to build on.But I take warning not to be in too great a hurry to stretchthe dynamical theory to fi t even our present enlightenedideas. I seem to hear the voice of the Hal ley Lecturerof 1945 repeating my remark, ' I s i t possible that onlyfifteen years ago we thought the stellar universe was likethat 'In the past fifteen years the accepted dimensions of thegalaxy have been enlarged tenfold, and we have to startthe comparison of theory with observation anew. T h e r eis, however, one result which seems to have been able tosurvive all vicissitudes, viz, the period of revolution of thestars in their orbits round the centre. T h e period adoptedin 1914 was 300 million years, which is close enough tothe 250 million years deduced from the Oort effect.

Star Streaming and PermanenceIn 1915 the main stimulus to dynamical investigation camefrom the phenomenon of star-streamingthe tendency ofthe stars in our neighbourhood to move to and fro alongone particular line rather than at right angles to it. E v e rsince this was pointed out by Kapteyn in 1905, it has beenrecognized as the most conspicuous peculiarity o f theobserved proper motions. I t might be a merely local


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Me Galaxy 25regarded as an explanation of star-streaming in the sensethat Professor Turner s suggestion was. However, thefirst question i s no t wh ich scheme affords th e betterexplanation but which gives a correct representation o fthe facts. N o w that there is general agreement as to thedirection of the centre, we can answer at once ; the directionof star-streaming is radial, o r nearly radial, as Turnersupposed. B u t the observational decision in favour o fradial star-streaming is recent, and theory had first inningsin the contest Radial versus Transverse star-streaming.

Adopting the specification of star-streaming, introducedby Schwarzschild, I was able to show rigorously that ina steady system with star-streaming the preferential motionmust necessarily be radial. Sho r t l y afterwards J . H .jeans showed rigorously that in a steady system wi thstar-streaming the preferential motion must necessarily betransverse.2 Id o ub tw he t he rmanys pe ct at or softhe

game understood how to interpret the score of one allwhich was the apparent result o f the contest. Possiblythey thought i t was just a case o f two theorists con-tradicting one another as usual. B u t actually the twoinvestigations were complementary ; the one excluded al lsave radial star-streaming, the other excluded a l l savetransverse star-streaming. Between them they establishedthat neither radial nor transverse nor any other directionof star-streaming is compatible with strict dynamicalequilibrium. I t seems t o b e clearly established b yobservation that radial star-streaming exists ; but jeans sinvestigation indicates the price that must be paidthegalaxy cannot be in a steady state. N o r would a trans-verse direction of star-streaming have saved it ; the deter-mination of the direction merely decides which horn of thedilemma our galaxy shall impale itself on.

1Th e most trustwor thy determinations give a difference of aboutIoc) between the line of star-streaming and the radius : i t is difficult todecide whether the discrepancy is large enough to be significant.2Unless the system is sphericalan exception irrelevant to thestudy of our own highly oblate galaxy.3760


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26 T h e Rotation o fHere lies the ch ie f difficulty i n pursuing dynamicaltheories of the galaxy. A perfectly steady model having

been proved to be impossible, we must seek some com-promise among the incompatible conditions for perma-nence, which will give a semi-permanent model satisfyingour minimum requirements of duration. I do not thinkthat much serious progress has been made towards suchan adaptation of the steady state theory. O o r t and Lind-blad have obtained certain theoretical relations betweenthe intensity o f star streaming the prolateness o f theSchwarzschild velocity-ellipsoid) and the magnitude of thedifferential rotation and found a rather impressive agree-ment with observation ; but it may be urged in criticismthat they have selected one out of a number of incompatibleconditions for a steady state, and it is not at all clear whythis rather than the rejected conditions) should be retainedin the appropriate compromise. M y own impression isthat i t is the least essential of the conditions for longevity.

Rolation of the Cosmic CloudMy survey of the dynamics of the galaxy must necessarilybe superficial, since i t would be id le to enter into detailswithout recourse to mathematical formulae. I should like,however, to refer to an aspect o f the problem which hasnot as yet received much attention. T h e recognition of acosmic cloud of rarified gas extended through interstellarspace and sharing in the rotation of the galaxy opens upa new field of theoretical investigation ; and the dynamicalequilibrium o f the cloud is a problem equally importantwith, and considerably easier than, the dynamical equili-brium of the stellar configuration.We find that in our own neighbourhood the motion o fthe mean of the stars agrees with the motion of the cosmiccloud ; the difference cannot be put higher than 2 or 3 km.per sec. Moreover it appears that this agreement is notconfined to our immediate neighbourhood, but extends at


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the Galaxy 2 7least L000 parsecs away from us, since Plaskett s investi-gation of the differential rotation of the cloud covers thisrange. T h e coincidence was not unexpected so long asboth were regarded as a t rest ; bu t i t takes on a newsignificance when it is understood to mean that both starsand cloud are moving about the centre of the galaxy withthe same orbital velocity (of order 250 km. per sec.). T hatthey should agree to within one per cent. is indeed sur-prising; and I think that the explanation of this coincidenceshould take precedence over many of the other aims ofdynamical theory.It seems impossible to admit any kind of interactionwhich would tend to drag along the stars with the cloudor the cloud with the stars. W e have to regard them astwo intermingled systems independent in all respects ex-cept that the controlling gravitational field is the same forboth. I f there is any community o f motion i t must bebecause the same causes have operated in both and notbecause one has constrained the other. A t fi rst sight asimilarity seems plausible. W e often t reat the stellarsystem as a glorified gas wi th stars for molecules. I s i tnot then a case of two gases finding their own conditionsfor equilibrium independently ? W h y should we be sur-prised that they both h it on the same solution ? Ne ver-theless I admit that I am surprised ; the reason is that thetwo gases differ enormously in viscosity.

An atom in the cosmic cloud may in the course o f itswanderings expect a collision wi th another atom aboutonce a year ; in that time i t traverses a path about equalto the distance of the earth from the sun. T h i s is a longfree path according to ordinary standards, but it is in-significant in the scale of the stellar universe. O n theother hand the free path of a star is practically infinite ; i tcan go hundreds of times round its orbit from one side o fthe galaxy to the other without appreciable risk o f de-flection. T h e length o f the free path determines theviscosity of a gas. T h e viscosity o f the cosmic cloud is


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28 T h e Rotation o fnegligible for astronomical purposes ; the viscosity of thestar-gas is enormous. I n fact the stellar universe, regardedas a gas, is the stickiest thing you could possibly imagine.The theory of a rotating non-viscous gas is familiar,and it can be applied directly to the cosmic cloud. W ecan at once derive an important result ; the motion of thecloud must correspond almost exactly to that o f a particlerevolving in a circular orbit about the centre. A slightdeviationif only a few km. per sec.would set up anenormous density gradient in the cloud, so that in one orother direction the stars would be embedded in a solidjam o f cloucleatomsel T h e r e is perhaps a difficulty i nexplaining how this special distribution of motion couldhave been set up originally, but at any rate it is the onlydistribution which could survive fo r any length o f time.Since we find that the stars in the mean have the samemotion as the cloud i t follows that they also have thevelocity corresponding to a circular orbit. T h i s had beenassumed by investigators of galactic rotation without muchattempt at justification. B y reference to the cloud we arenow able to establish it firmly.I think i t i s not going too far t o say that the mereexistence of the cosmic cloud is in itself a proof of galacticrotation. W h a t has kept this gas distended through thegalaxy instead of collapsing long ago into a dense nebulaat the centre ? Some of the ways o f evading collapsepossible for particles with long free paths (the stars) arenot open to a regular gas ; so that the possible answers are

I ought to mention that the time of relaxation towards dynamicalequilibrium is longer for the cloud than fo r the stars owing to thelower velocity of the particles. W e ma y say that the velocity ofsound in the cloud (about 3 km. per sec.) is less than the veloci ty ofsound in th e star-gas ; and according ly the pressure-waves whichlevel out the distribution of cloud matter take a longer time to travelthrough th e galaxy than those which level o ut the distribution o fstars. I t is therefore possible that i f we adopt a short time-scale thecosmic cloud ma y not ye t have reached the dynamical equil ibriumassumed in my discussion.


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the Galaxy 2 9very limited. T h e distension could be maintained b ytemperature ; but the temperature of the cosmic gas (about15,000 ) is far too low. T h e on ly alternative i s rapidrotation sufficient to counteract the pull o f gravity. T h every existence of the cosmic cloud, therefore, depend s onrotation ; and although there is not the same theoreticalnecessity for rotation o f the galaxy of stars, the observedagreement of the motions shows that the latter also mustbe rotating.

The conception of the stellar system as a gas with starsfor molecules, and its comparison with the genuine cos-mical gas fill ing the same region and rotating in the sameway, helps us to see more clearly the difficulty in th e wayof constructing a strictly permanent stellar system. W i t hthe cosmic cloud w e have no diffi culty in fulfi lling thestrict conditions of equilibrium, but that is because weneglect viscosity. Viscositythe rubbing of one zone ofgas on another zone rotating at a sl ight ly different speed

must slowly change the distribution of rotation. Visco-sity objects to the existing differential rotation, and triesto set up a condition of its own. B u t it can never succeed ;it can only act as spoil-sport. A s soon as i t effects anyserious change the density gradients already mentionedmust be set upwhich means that the cloud falls to thecentre o f the system o r departs in to outer space. Idaresay that in the end viscosity triumphantly establishesthe law of motion it is striving foronly there is nothingleft to obey it.Similarly in the system o f the stars we have a tug-of-war between the viscosity conditions and the simple pres-sure conditions which must inevitably end in the collapseor disruption of the system. T h e only quest ion is how toarrange some ki nd o f balance which w i l l stave o ff thisfate fo r a reasonable time. I have already referred tocertain current solutions which seem to me to insist toostrongly on the fulfilment of what we here identify asviscosity conditions, perhaps not sufficiently recognizing


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30 T h e Rotation o f the Galaxythat by the time these conditions prevail there wil l be nogalaxy left.

Let us leave these deep waters o f theoretical investi-gation and for our last look at the galaxy use no othercriticism than o ur commonsense. W e have not muchdifficulty in imagining ourselves looking at it from outsidefor the telescope reveals multitudes of spiral nebulae seenfrom every aspect any one of which we believe may betaken as a pattern o f our own galaxy. Rotat ing ? O b -viously it is. I t is just like a Catherine wheel. Perma-nent? I t does not look ve ry permanent. E v e r y en-gineering instinct we possess protests that such disk-likearrangements of matter are precarious and their stabilityis not to be trusted. T o emphasize our sense of the transi-toriness of things the other galaxies are rushing away athigh speed as though our poor system were the plague-spot of the universe. I n a few thousand mil lion yearsour neighbourhood w i l l be nearly evacuated and o u rskies wil l have lost one of their chief telescopic glories.This is a rough-and-ready way of treating serious problemsbut it is not out of harmony with the results thrust uponus by stricter methods. Perhaps the lesson of the galaxiesis to wake us from our dream of leisured evolution throughbillions of years. I t is hard to credit our stellar systemwith so much age and endurance. I t is more like a youngman in a hurry.


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P R I NT E D I N G R E A T B R I T A I N A T T H E U NI V E R S I T Y P RE S S O X F O R DB Y J O H N J O H N SO N P R I N T E R T O T H E UN I V E R S I T Y

1930 the Rotation of the Galaxy Being the Halley Lecture (Arthur S. Eddington)

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