Angular Momentum - FAS

Angular Momentum

Angular Momentumthe cosmic pollutant

by Stirling A. Colgate and Albert G. Petschek

There seems to be too much angular momentum in the universe to allow theformation of stars or the accretion of matter onto variable x-ray sources. This

fundamental problem begs for solution.

w hen we have too much ofsomething and cannot finda way of getting rid of it, weoften think of it as a pollu-

tant. In the game of concentration andcollapse of matter in the universe fromclouds of gas to clusters of galaxies, galax-ies, stars, planets, and black holes, angularmomentum is the “pollutant” that pre-vents the game from being played to theabsolute limit, namely, collapse into oneawesome black hole for each cluster-sizedcloud condensed from the early universe.Only at the scale of the whole universedoes the energy in the Hubble expansionof the universe prevent collapse, inde-pendent of angular momentum con-straints. On smaller scales there seems tobe too much angular momentum to allowthe collapse of clouds into dense objects.

objects whose disk-like shapes indicate theyhave a net angular momentum. The centralfigure is Stephan’s Quintet, a group of fiveinteracting galaxies. Along the top from leftto right are an edge-on view of the spiralgalaxy NGC 4594, the star Beta Pictorissurrounded by a disk of dust, and thebarred spiral galaxy NGC 1300. Along thebottom from left to right are the galaxyM81, an artist’s conception of an x-raybinary, and the rings of Saturn. (Photocredits are given at the end of the article.)

LOS ALAMOS SCIENCE Spring 1986

Yet our universe is populated by planets,stars, and black holes. How does nature getrid of the cosmic pollutant?

Models proposed for variable x-raysources give a strong clue to this long-standing puzzle (see “X-Ray Variability inAstrophysics”). But before we restrict our-selves to x-ray variables, let’s look at theproblem more generally.

Angular Momentum, Weights,and Noncosmological Strings

The angular momentum of a weight ofmass m whirled at velocity vat the end of astring of length R is mvR. Under idealcircumstances, that is, a rigid support forthe string, no air friction, and no otherexternal torques, the weight will continueto circle at the same velocity forever. Inother words, its angular momentum J =mvR is conserved. If the string is short-ened, by pulling it through the support,then since the angular momentum mustremain constant, the weight will speed upto a higher angular frequency w = v/R.

The inward force necessary to keep theweight moving in a circular path, FR =mv2/R = mw2R, is supplied by the tensionin the string. Since FR = J2/mR3 in termsof the angular momentum, we see that thetension in the string increases very rapidly,as R –3, as the string is shortened. In thecosmic game of collapse, the analogue of

the tension in the string is the attractivegravitational force, which is proportionalto R–2. Since the required inward forcegoes as R -3, while the available gravita-tional force goes as R-2, there is bound tobe a point beyond which gravity is unableto cause further collapse. This is the basisof a stable Keplerian orbit, like that of theearth around the sun or that of accretingmatter around a compact star in an x-raybinary. Once in a stable orbit, the only wayfor matter to move farther inward is tolose angular momentum, but the puzzle ishow? We also would like to estimate howmuch angular momentum has to be lost bywhatever mechanism we devise.

Angular Momentumand the Universe

The universe as a whole does not seemto be rotating, as evidenced by the fact thatthe blackbody radiation believed to be arelic of the early universe is isotropic tobetter than one part in 104. Moreover,Tyson has found the orientations of a verylarge number of galaxies to be random.Since the net angular momentum appearsto be zero over very large scales, the pollu-tion is not as bad as it could be. Ourproblem is restricted to local patches of theuniverse where matter collapses to formrelatively dense rotating objects such asthose shown in the opening figure.

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Angular Momentum

Galaxies Are Not a Problem. Let usconsider the specific angular momentumJ s = vR (angular momentum per unitmass) of a typical, modestly sized, spiralgalaxy. The rotational velocity of matter atits outer edge, determined from the Dopp-ler shifts of spectroscopic lines, is about150 kilometers per second, and its radiusis about 10 kiloparsecs (-3 X 1022 cen-timeters),* so Js = 5 X 1029 centimeterssquared per second. Suppose that, beforecondensing, the galactic matter occupied aspace with a radius equal to one-half theaverage distance between galaxies, roughly3 megaparsecs, or 300 times the galacticradius. If angular momentum was con-served in the collapse to the 10-kiloparsecgalactic radius, the initial velocity of thematter must have been less by a factor of300, or about 5 X 104 centimeters persecond. This velocity, which is roughly thespeed of sound in hydrogen at 150 kelvins,seems to be a reasonable value for thevelocity of the turbulent eddies that musthave existed when galaxies began to form.In fact, theoretical calculations suggestthat density fluctuations in the early uni-verse may produce velocities of this order.Theoreticians thus regard angular mo-mentum in spiral galaxies not as a pollu-tant but as a much sought-after relic of anearlier history. This reasonable state ofaffairs is in sharp contrast to the problemangular momentum poses in the makingof a star. Angular momentum may also bea problem in the formation of nearlyspherical “elliptical” galaxies, which seemto have very little total angular momen-tum.

Collapse to Stars. The density of matterin our own galaxy before any of the mattercollapsed into stars was roughly 0.1 to 1hydrogen atom per cubic centimeter, or

“The unit of distance called a parsec is equal toabout 3 X 1018 centimeters; its name is derivedfrom parallax-second. A parsec is the distance atwhich the direction to an object, viewed from theearth at opposite phases of the earth’s orbitaround the sun, changes by 1 second of arc. Thepointing accuracy of a typical old-fashionedtelescope is about 1 second of arc.

about 10–24 gram per cubic centimeter. Toform a star, this dilute matter must havecollapsed to a density of about 1 gram percubic centimeter, an increase by a factor of1024. The radius would have decreased bya factor of 108, the cube root of the densityratio. The Keplerian velocity at the stellarradius is about 106 to 107 centimeters persecond, so the initial velocity needed toconserve angular momentum must havebeen 108 times smaller, or 10 -2 to 10-1

centimeter per second. This velocity isunreasonably small for the gas in aturbulent rotating galaxy. A more reason-able velocity for gas clouds, or even forgalactic rotation, over the radius of thespace from which the stellar matter oughtto have been drawn, would be 105 to 106

centimeters per second. With this valuefor the velocity, coordinated motion ofeven a small fraction of the matter willintroduce too much angular momentum,by a factor of 1O6 to 108, to allow collapse.This immense amount of excess angularmomentum must somehow have beendumped before collapse.

For collapse to a neutron star, 1014

times more dense than a normal star, theproblem would be worse by a factor equalto the sixth root of 1014. (Since theKeplerian velocity is proportional toR -1/2, Js = vR is proportional to R 1/2, o r

p -1/6.) Thus we have another factor ofabout 100, or a total of 1010 times toomuch angular momentum.

We have discussed only the initial andfinal states involved in star formation,both of which are spherical. It must not beimagined, however, that the collapse isspherical throughout. Angular momentumconservation prevents collapse only in thedirections perpendicular to the rotationaxis; collapse parallel to the rotation axis isnot inhibited and thus occurs first. Thisleads to formation of a disk whose radiusis almost as large as the initial radius of thecloud. But the disk still has the large initialangular momentum that seems to preventfurther collapse. Where and what are thegalactic dumping grounds for this angularmomentum?

Magnetic Fields

Only external torques can alter theangular momentum of a system. An ob-vious way to apply a global external torqueto dilute ionized matter is through mag-netic fields. Indeed this is an often-in-voked panacea for the problem. The dif-ficulty is that magnetic fields with thenecessary strength and dimension are notobserved in the universe. Furthermore,even if our estimates of magnetic fieldstrengths (obtained from observations ofthe Faraday rotation of the polarizationangle of radio waves caused by their pass-age through a magnetic field) are er-roneous, we are faced with the followingdilemma. If matter is to be strongly af-fected by magnetic fields, as is reasonablefor partially or fully ionized matter, it isalso reasonable that the matter is stronglytied to the field lines. Hence, as a not soextraneous conclusion, magnetic confine-ment fusion should be simple. The factthat it is not means that ionized matterescapes magnetic fields deceptively easily.Suppose, to the contrary, that matter andfield are strongly coupled. Then purelytwo-dimensional radial collapse by ourfactor of 108 would mean that a region ofuniform galactic field of 3 X 10–6 gausswould be compressed by a factor of 1016 inthe newly formed star, and the field wouldincrease to 3 X 10]0 gauss. This is toomuch field by many orders of magnitude.

1019 dynes per square centimeter, is largerthan the pressure inside the newly formedstar by a factor of 104 to 1O6. Hence, mag-netic field must escape easily from thecollapsing matter even though it cannotescape too easily if it is to remove the extraangular momentum. Such a balance be-tween field escape and field trappingseems most unlikely—although possible.

Thin Keplerian Accretion Disks

The hydrodynamics of thin accretiondisks provides a more plausible mecha-nism for getting rid of angular momen-

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Angular Momentum

AZIMUTHAL VELOCITIES OF ROTATING WHEEL AND KEPLERIAN DISK

turn, or at least for allowing its transport momentum (Fig. 1). The friction does justoutward as matter accretes toward a cen-tral point.

In variable x-ray sources and cataclys-mic variables, accretion disks formaround small, dense stars as matter from acompanion star is pulled toward the com-pact object and trapped into orbit by thestrong gravitational field. Such accretiondisks resemble the rings around Saturn,but, whereas the rings around Saturn areevidently composed of solid chunks ofmatter that occasionally bump into oneanother, an accretion disk is composed ofgaseous matter. The gaseous disks thateventually collapse to form isolated starsare thought to be quite similar.

Let us look at a likely state for matterthat has partially collapsed and run intothe angular momentum barrier. As ex-plained earlier, it will go into a stableKeplerian orbit. Now matter of slightlydifferent angular momenta will go intoorbit at slightly larger or smaller radii andhave slightly different velocities. If thismatter is in gaseous form, it will “rub”with this differential velocity, and the fric-tion will lead to a torque and hence achange in angular momentum. The direc-tion of the rub tends to make the gaseousdisk rotate more like a solid body or awheel. In other words, matter at the pe-riphery tends to speed up, increasing itsangular momentum, while matter near thecenter slows down, decreasing its angular

what we want—it transports angularmomentum from the inside to the outsideof the disk, allowing the inner matter tocollapse and the outer matter—a smallfraction—to be spun up and flung off,carrying with it all the excess angularmomentum. So what’s the rub?

The interaction between adjacent masselements moving at different velocitiescan be characterized by the kinematic vis-cosity D (the ratio of dynamic viscosity todensity). We use the unlikely symbol D forkinematic viscosity because this is really adiffusion coefficient. It describes how fasta viscous wave (velocity shear) relaxes dueto molecular motion. To carry out an or-der-of-magnitude calculation, we use thekinematic viscosity of hydrogen at roomtemperature, which (in centimeterssquared per second) happens to be almostequal to 10–3/p when the density p isexpressed in grams per cubic centimeter.As in any diffusion phenomenon, the re-

diffusion distance divided by the diffusioncoefficient, or R2/D. Combining theseequations with an expression for the den-

angular momentum to “diffuse” out of thedisk, a value of 3 X 1017 seconds timesthe mass of the central object (in solarmasses) and divided by the thickness ofthe disk (in parsecs). The thickness of thedisk is much less than its radius, which

Fig. 1. Distribution of azimuthal velocities

centrated at the periphery. In a Keplerian

shear in the disk tends to equalize thevelocities and therefore transport angularmomentum toward the periphery, that is,make the disk more like a wheel. In agaseous disk ordinary molecular diffusionis too slow to explain the transport ofangular momentum required for star for-

must become much smaller than 1 parsecin the course of star formation. Hence thetime required to form a star in this waywould exceed the age of the universe, atmost 6 X 1017 seconds, by several factorsoften. Yet stars abound. Clearly we need abetter rub or more viscosity.

The Rub, or a

A model that assumes a large viscositywas invented by Shakura and Sunyaev toexplain the apparently rapid accretion ofmatter from a disk onto a compact star inx-ray and cataclysmic variables. Thismodel invokes turbulence as the source ofthe viscosity but does not describe how theturbulence is driven. The strength of theturbulence is parametrized by a coeffi-cient a, which can be varied between O and1. Calculations based on this hypo-thetical turbulent viscosity have been verysuccessful in duplicating the apparent ac-cretion rates in x-ray variables. The valueof cc turns out to be quite large, implyingthat the accretion disk is highly turbulent.Such calculations, and even more detailedcalculations of accretion in cataclysmicvariables, strongly suggest the validity ofthe model. Thus the elusive friction inKeplerian disks may have been identified.If so, we know how nature gets rid of theexcess angular momentum that wouldotherwise prevent the formation of starsand hence us.

LOS ALAMOS SCIENCE Spring 1986 63

Angular Momentum

A Physical Interpretation of a. Tur-bulence is the enhanced transport of mat-ter due to relatively large-scale, randommotions of a fluid. If there are velocitygradients in the matter, then the effect ofturbulence is to transport momentumacross a mean velocity shear; it acts likeviscosity or friction. The maximum rate oftransport by turbulence is determined bythe maximum size of the eddies; that is,the diffusion caused by turbulence can beapproximated by a random walk with astep size equal to the diameter of the larg-est eddy. Since the largest eddy that can“fit” in the disk and transport matter inthe radial direction is a round eddy whosediameter is h, the half-thickness of the disk(Fig. 2), and since the maximum velocityof such an eddy is the local sound speed cs,the maximum possible random-walk dif-fusion coefficient, or turbulent kinematicviscosity D, is h cs. Thus Shakura andSunyaev parameterized the turbulent

between 0.03 and 1.

What is the Originof the Turbulence?

We are accustomed to the ubiquity of

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turbulence in fluids with velocity shearsand large Reynolds numbers. Since theseconditions are met in most accretiondisks, it seems reasonable to expect turbu-lence to supply the necessary fluid friction.But, as Lord Rayleigh pointed out morethan a century ago, the constraint ofangular momentum conservation is strongenough to stabilize the shear flow of aKeplerian disk against shear-produced, orHelmholtz, instabilities. Hence these in-stabilities alone cannot drive theturbulence. Another possibility is that theturbulence is driven by heat convection.Turbulence always produces friction, butnow we must ask, conversely, whether theheat produced by the friction from veloc-ity shear is enough to drive the turbulence.In the next section we will explore thispossibility as an example of how difficultit is to produce the large values of a re-quired to transport momentum outwardin Keplerian accretion disks.

Convection-Driven Turbulence. In anaccretion disk, friction from velocityshears should give rise to inhomogeneousheating concentrated near the midplane ofthe disk. This differential heating cancreate instabilities that lead to turbulentmotion. To transport angular momentum

A

Fig. 2. Cross section of a thin Keplerianaccretion disk around a compact object.Large eddies with radii equal to the half-thickness of the disk transport angularmomentum in the radial direction R. Thediffusion caused by these eddies can beapproximated by a random walk with stepsize h and velocity of the order of the soundspeed cs. Shakura and Sunyaev modeledthis transport by a turbulent kinematic vis-cosity hcs.

outward, the turbulent motion must beisotropic (or nearly so) rather than just inthe “easy” azimuthal direction. It is easyto create eddies whose axes are in theradial direction and whose velocities areazimuthal, but we need an instabilitystrong enough to overcome the stabilizingeffect of angular momentum and driveradial motions. If these instabilities exist,the turbulent motion provides an effectiveviscosity (an “eddy” viscosity) far largerthan the molecular viscosity and cantransport angular momentum at the raterequired for the evolution of the disk.

To see whether differential heating candrive the required instability, we mustlook at the structure of the accretion diskimplied by the Shakura and Sunyaevmodel (see Fig. 2). As indicated above, a

Spring 1986 LOS ALAMOS SCIENCE

Angular Momentum

value of a near unity implies that thediameter of the eddies that interchangematter in the radial direction must beclose to the half-thickness h of the disk,and their velocity must equal the soundspeed cs. Thus the interchange scale in theradial direction, if the eddies are round,will equal h. The internal energy in theeddy, which is determined by the soundspeed cs, will determine how much energyis available to drive the interchange. AsPringle has pointed out, the structure ofKeplerian disks is such that cs/v = h/R,where v is the azimuthal velocity and R isthe radial distance.

The disk is densest in the midplanebecause the pressure is greatest there. Thepressure is required to hold the materialnear the surface out against the compo-nent of gravity perpendicular to the disk(Fig. 3). Consequently frictional heatingfrom velocity shears and therefore thetemperature will be greatest at the mid-plane. The surface of the disk will becooled by radiation. This configuration isRayleigh-Taylor unstable in the directionperpendicular to the plane of the disk.Motions perpendicular to the disk and atthe same radius do not transport angularmomentum. On the other hand, motions

in the radial direction do transport angularmomentum and produce enhanced fri-ction and a. The force from differentialbuoyancy must be large enough to createan eddy that interchanges matter over a

Now let’s consider a single eddy andcalculate how much work is needed tointerchange two mass elements a distanceAR while conserving angular momentum.We will assume an initial laminar stablestate so that in the interchange of twoadjacent elements of equal mass, theangular momentum of each is conservedseparately (vR = constant). Then for eachmass element the change in azimuthal ve-

The change in specific kinetic energy of the

tum constraint, together with a little al-gebra, shows that the net change is equal to

energy will be zero because we have as-sumed the interchange of two equalmasses. Hence we need an energy equal to

must be provided by differential buoy-ancy.

The energy available from raising hot

Accretion Disk

Object

DIFFERENTIAL BUOYANCY IN A THIN ACCRETION DISK


fluid and lowering cold fluid is the internal

Since the work that must be done in theinterchange is the same as the energyavailable in buoyancy, there is barelyenough buoyancy to force an overturn or acircular eddy in the radial direction,especially at an eddy velocity near cs. Forthe eddy to develop we need a nearlyperfect heat engine that converts the heatof friction to potential and kinetic energy.

The Ideal Heat Engine. We can imaginean ideal heat engine driving the eddies.The heat is produced in our mass element

by turbulent friction due to the shear of theorbital velocity. The hot material expandsadiabatically as it rises to the surface of thedisk. There the remaining internal energymust be lost by radiation during the resi-dence time of the mass element at the

the diffusion coefficient for radiation,

tion energy density to total energy density

must be the same as the coefficient forturbulent mass transport. This implies

these restrictions our mass element wouldcool, and i t could then descendadiabatically with much smaller internalpressure, so less work must be done on it.When it reaches the midplane of the disk,

midplane, where the gravitational potentialis lower, than at points above and below it.The dense material will be heated by veloc-ity shears and rise to the surface of the diskwhere it will cool by emitting radiation. Thequestion to ask is whether the differentialbuoyancy is large enough to drive an eddyin the radial direction.

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can start the cycle over again. This cycle aswell as the alternating regions of hotterand cooler material that would result atthe surface are illustrated in Fig. 4.

The required condition on the opticaldepth, together with the opacity of thematerial, determines the mass per unitarea of the disk. Then the mass flow rate

I can be calculated from a and the soundspeed. It is not known whether this mecha-

nism leads to a self-adjusting disk in thesense that if the mass-injection ratechanges, then a and the other parametersvary to maintain a consistent disk struc-ture. It is also not known whether all astro-physical disks can be explained in thepa rame te r space j u s t ou t l i ned .Furthermore, nature does not like to makeideal heat engines, especially not in aturbulent environment because heat en-

gines must be so perfect and turbulence isso random. Thus, the above ideal cycle,although conceptually feasible, seems dif-ficult to justify as an explanation for theorigin of a. Convection-driven turbulencedoes not seem strong enough to overcomethe angular momentum barrier.

Thick Disks Beg the QuestionThe importance of angular momentum

Fig. 4. Convection-driven turbulence in a require a nearly perfect engine, that is, onethin Keplerian accretion disk creates large in which nearly all the heat was convertededdies that break the angular momentum to work as matter flows around the eddy.constraint by enhancing radial transfer of The hexagonal pattern of Benard-like cellsangular momentum. The energy available shown might reproduced by heating at thefrom differential buoyancy is barely enough midplane. The heat cycle that drives theto drive the eddies. Their formation would overturn of the eddies produces alternating

hotter (red) and cooler (blue) regions. The

figure suggests the required breakup of theeddies into smaller scale turbulence afterabout half a cycle. (A persistent eddy wouldnot produce any net transport.) Thisbreakup is a difference between the eddiesin the disk and those in a standard Benardcell, which are very slow (low Reynoldsnumber) and do not lead to smaller scaleturbulence.


constraints is illustrated in studies of thickaccretion disks by Wojciech Zurek andWiny Benz of Los Alamos. They haveperformed numerical simulations of theevolution of thick disks with specificangular momentum independent ofradius. Because the specific angularmomentum is constant, the interchange oftwo equal mass elements requires noenergy. The disks exhibit violentHelmholtz instabilities. As one would ex-pect, less constraint leads to greaterturbulence. The instabilities cause angularmomentum to redistribute itself very

(see “Redistribution of Angular Momen-tum in Thick Disks”). Thus the disk be-comes more Keplerian, but since theseinstabilities are damped the disk neverbecomes truly Keplerian (q = 0.5).

Such models invite the question of howa disk can be formed with small and nearlyuniform angular momentum. In the caseof quasars and active galactic nuclei pow-ered by the accretion of matter ontomassive black holes, these disks might beformed by the gravitational breakup ofstars scattered by interactions with otherstars in the strong gravitational field closeto the massive black hole. More specifi-cally, stars in a dense galactic nucleus scat-ter at random. Occasionally one of thesescattering events causes a star to approachthe black hole with an impact parameterso small (several Schwartzschild radii)that the star deforms tidally and a fractionof the star is captured. Other stars of thecluster then have a slightly greater angularmomentum because of its conservation.Jack Hills of Los Alamos calculated that athick disk of low angular momentum is areasonable outcome of such accretion.

Thus there may not be an angularmomentum problem in feeding a blackhole, but the original problem of making astar from tenuous gas remains. Either wemust posit an initial gaseous state of tinyand thus statistically unlikely angularmomentum, or we are left with, the im-perative to find a transport mechanism forthe cosmic pollutant. ■


Angular Momentum

AUTHORS

Stirling A. Colgate received his B.S. and Ph.D.degrees in physics from Cornell University in1948 and 1952, respectively. He was a staffphysicist at Lawrence Livermore Laboratoryfor twelve years and then president of NewMexico Institute of Mining and Technology forten years. He remains an Adjunct Professor atthat institution. In 1976 he joined the Theoreti-cal Division at Los Alamos and in 1980 becameleader of the Theoretical Astrophysics Group.He is also a Senior Fellow at the Laboratory, amember of the National Academy of Sciences,and Technical Director of the Gifthorse Pro-gram. His research interests include nuclearphysics, astrophysics, atmospheric physics, andgeotectonic engineering.

Albert G. Petschek is currently a Fellow at LosAlamos as well as being Professor of Physics atNew Mexico Institute of Mining and Tech-nology. He received his B.S. from MIT in 1947,his M.S. from the University of Michigan in1948, and his Ph.D. from the University ofRochester in 1952. His research interests are inastrophysics, cloud physics, and fracture me-chanics. The spring of 1978 found him in Israelas a Visiting Professor at Tel Aviv Univerity.He is a member of the American PhysicalSociety, the American Association for the Ad-vancement of Science, and the AmericanAstronomical Society.

Photo Credits

Stephan’s Quintet. A combination of best avail-able photographs in different colors taken withthe 200-inch reflector at Palomar Ob-servatory.© Halton C. Arp; reproduced withpermission.NGC 4594. Palomar Observatory photograph;reproduced with permission.Beta Pictoris. A color-coded map created fromoptical images of Beta Pictoris and the similarstar Alpha Pictoris, both taken with a corona-graph and a charge-coupled device at the LasCampanas Observatory in Chile. The disk sur-rounding Beta Pictoris was revealed by plottingthe ratio of the intensity of scattered lightaround Beta Pictoris to that around Alpha Pic-toris. Reproduced with permission of BradfordA. Smith, University of Arizona, and Richard J.Terrile, Jet Propulsion Laboratory.NGC 1300. Palomar Observatory photograph;reproduced with permission.M81. A color-enhanced image in which radia-

tion from gas in the spiral arms of the galaxyappears blue and radiation from older stars inthe disk appears orange. The image was com-posed from single-color photographs taken withPalomar Observatory’s 48-inch Schmidttelescope.© Halton C. Arp; reproduced withpermission.Rings of Saturn. An optical image in real colorcreated from data collected in October 1980 byNASA’s Voyager I satellite. The rings have beenenhanced with additional color. Reproducedwith permission of the NASA Jet PropulsionLaboratory.

Further Reading

Pringle, J. E. 1981. Accretion discs in astro-physics. Annual Review of Astronomy andAstrophysics 19:137-162.

Shakura, N. I. and Sunyaev, R. A. 1973. Blackholes in binary systems. Observational appear-ance. Astronomy and Astrophysics 24:337-355.

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Angular Momentum in Thick Disks

Redistribution of AngularMomentum in ThickAccretion Disks by Wojciech H. Zurek and Winy Benz

(a) (b)

CROSS SECTIONS OF THICK ACCRETION DISKS

Ases of the stability of the disks by Fig. 2. Isodensity contours of two tori

1 of angular momentum is il-lustrated in our recent numerical

simulations of thick accretion disks.These torus-like disks, in contrast to theflat, pancake-like Keplerian disks, havea height above the equatorial planecomparable to their extent in the radialdirection. Until recently the constantspecific angular momentum (Js = con-

Papaloizou and Pringle. Our numericalsimulations confirm those first suspi-cions. More important, we were able todemonstrate that growing instabilitiesin a constant-J, torus rapidly re-distribute angular momentum, causingthe torus to become thinner and moreKeplerian. Hence thick accretion diskswith constant Js cannot be regarded as

stant) variety of such non-Keplerian ac- models for astrophysical objects.cretion disks around massive black The equilibrium configuration ofholes was considered the best model for thick accretion disks, for constant Js andthe central “powerhouse” in quasars large pressure forces (sound speeds com-and active galactic nuclei. However, parable to rotational velocities), looksdoubts about the validity of that like a fat torus, or doughnut (Fig. 1a).hypothesis were raised in 1984 by analy- The sides of the torus form a funnel,

with approximately the same inner andouter radii, but with different distribu-tions of specific angular momentum: (a)a torus with Js = constant and (b) a toruswith Js - r0.27 Note the change in thefunnel opening angle from -10° in (a) to-30° in (b). The density decreases bytwelve orders of magnitude from the in-nermost to the outermost contours.

with a small opening angle 6, about therotation axis. As noticed by Lynden-Bell, this is a perfect shape for a deLavalnozzle,* that is, for accelerating theenormous supersonic jets observed tobe emanating from so many active ga-



lactic nuclei. Moreover, the steep wallsof the funnel allow the luminosity of thetorus to exceed the Eddington limit, aproperty that would be indeed useful inmodeling variable quasars with hugepower outputs.

The first question to ask is how suchnon-Keplerian accretion disks can exist,since matter at small radial distances rhas “too much” angular momentum(more than the Keplerian value) andmatter at large r has “too little” angularmomentum. The pressure in the disk isresponsible for maintaining this dis-tribution of angular momentum: thematter at large r is being prevented frommoving inward by the pressure, and thematter at small r is being kept frommoving outward by the pressure-mediated weight of the outer parts of thedisk.

Proponents of Js = constant accre-tion disks assume that the effectiveturbulent viscosity of such a disk is verysmall, so that it is all but impossible totransport angular momentum outwardas matter accretes toward the massivebody at the center of the disk. However,if the central massive body is a blackhole (as it is almost certain to be forquasars and active galactic nuclei,including the nucleus at the center ofour own Milky Way), then it is possibleto get rid of the angular momentum ofaccreting matter by pushing it into theblack hole. The necessary push can beprovided by applying pressure from faraway and “force-feeding” the black holewith gas.

Using an idealized model, Papaloizouand Pringle challenged the foundationsof the thick accretion disk theory byshowing that constant- J s tori areviolently unstable against nonaxisym-metric shear-driven perturbations. Theinstabilities revealed by their linearanalysis are Helmholtz instabilities andin some ways are analogous to “fire-

*The action of a deLaval nozzle is described by M.L. Norman and K.-H. A. Winkler in "SupersonicJets, ’’Los Alamos Science Number 12, 1985.


REDISTRIBUTION OF ANGULAR MOMENTUM

o 1 2 3 4 5

hose” instabilities. However, the gasdoes not stream out in random direc-tions, as would water from a hose leftunattended. Instead the gas deflectedfrom its equilibrium orbit by the in-stability is bound by the gravitationalpotential and so produces density in-homogeneities, pressure gradients, andsound waves, which, in turn, producemore deflections, which lead to moresound waves.

In a second paper Papaloizou andPringle extended the stability analysis toinclude very thin tori (like slender bicy-cle tires) with Js varying as rq. For q < 2

be unstable. For greater values of q alarge class of unstable modes isstabilized. Their linear analysis did not,however, reveal the ultimate fate of theoriginal configuration.

Fig. 2, The exponent q as a function oftime, where q is calculated from a power-

rq) to the specific angularmomentum distribution obtained fromthe numerical simulation. The results areshown for three simulations. Note that,for the two disks with initially constantspecific angular momentum, q increasesrapidly from O to about 0.27 within abouttwo rotation periods. After the critical q= qC -0.27 is reached, the redistributionof angular momentum slows down to therate observed in a disk with initial Js -r0.3, It is not yet known whether this slowrate of angular momentum transport iscaused in part by nonaxisymmetric in-stabilities or is totally explained by anumerical viscosity that is an un-avoidable artifact of such calculations.We are planning to study this problem

further.

69


We have extended such stability anal- (see Fig. 3, t = 0). These numerical ex- the angular momentum very quicklyyses to the nonlinear regime by adding a periments not only confirm that such (on the time scale of about a rotationsmall random density perturbation (of disks are unstable but also show that a period Js) from Js = constant to Js - r

qc,the order of 1 percent) to an initial equi- fat accretion torus is forced to undergo a where qc turns out invariably to be 0.27librium configuration with constant Js “crash diet”: instabilities redistribute (Fig. 2). Note that this value for the

Fig. 3. A computer-generated time se-quence showing the three-dimensionalevolution of the central region of a thickaccretion disk with initially constantspecific angular momentum. The upperpanels show isodensity contours in theequatorial plane of the central region; thelower panels show isodensity contours ina plane parallel to the rotation axis. Thedensity decreases by one to two orders of

TIME EVOLUTION OF A THICK DISK

t = o

magnitude over the region shown. Time isexpressed in rotation periods of the den-sity maximum. The velocity field is in-dicated by means of arrows whose lengthsare normalized to the maximum value ofthe velocity in each frame. Following theintroduction of a small nonaxisymmetricperturbation, the growth of instabilitiescauses a rapid redistribution of angularmomentum that, in turn, flattens the disk

t= 1

and fills in the central “hole.” This simu-lation was made with a three-dimen-sional hydrodynamics code that uses theso-called smoothed particle hydro-dynamics method (Lucy 1977). This freeLagrangian approach to solving the usualequations of hydrodynamics replaces thecontinuum by a finite set of spatiallyextended particles. Thus no mesh is re-quired, and the usual problems as-

t = 2



exponent is about the same as that ob-tained by Papaloizou and Pringle for thestabilization of “bicycle tire” tori. Fig-ure 3 shows the time evolution of thedisk.

sociated with its rezoning are bypassed.The simulation is made by computing thetrajectories of 1000 extended particlesthat interact through pressure forces in acentral gravitational potential. Since theparticles are allowed to move without anyconstraints in all three spatial directionsand since a mesh is not needed, thismethod is particularly suited for the sim-ulation of highly distorted flows.

t = 3


As the angular momentum is re-distributed, the fat torus becomes muchthinner and much more Keplerian inappearance. Moreover, the narrow fun-nel invoked to explain the formationand collimation of relativistic jets be-comes much wider (Fig. 1 b) and there-fore less effective in producing colli-mated jets and super-Eddingtonluminosities.

Regarding the question of angularmomentum transport discussed in themain text, our calculations show that, atleast for q < qc, shear-driven instabilitiesprovide a powerful source of the “rub,”that is, of a, the turbulent viscosity. Thenext obvious question—not addressed

Further Reading

Abramowicz, Marek A., Calvani, Massimo, andNobili, Luciano. 1980. Thick accretion diskswith super-Eddington luminosities. The Astro-physical Journal 242:772-788.

Fishbone, Leslie G. and Moncrief, Vincent.1976. Relativistic fluid disks in orbit aroundKerr black holes, The Astrophysical Journal207:962-976.

Lucy, L. B. 1977. A numerical approach to thetesting of the fission hypothesis, T h eAstronomical Journal 82:10131024.

Lynden-Bell, D. 1977. Gravity power. PhysicaScripta 17:185-191.

Papaloizou, J. C. B. and Pringle, J. E, 1984. Thedynamical stability of differentially rotatingdiscs with constant specific angular momen-tum. Monthly Not ices o f the RoyalAstronomical Society 208:721-750.

Papaloizou, J. C. B. and Pringle, J. E. 1985, Thedynamical stability of deferentially rotatingdiscs—IL Monthly Notices of the RoyalAstronomical Society 213:799-820.

Zurek, W. H. and Benz, W. Redistribution ofangular momentum by nonaxisymmetric in-stabilities in a thick accretion disk, Accepted forpublication in the 1 September 1986 issue ofThe Astrophysical Journal.

properly by the calculations performedto date—is whether the shear-driven in-stabilities will provide a mechanism fora when q > qc. Can these instabilitiesgenerate wave-like excitations and “in-teresting” a values in disks that are“barely” stable ( Js = r qc) or almostKeplerian ( Js = ro5)? We are now ex-ploring this question with one of thenew three-dimensional hydrodynamicscodes developed at Los Alamos. ■

AUTHORS

Wojciech Zurek earned a Ph.D. in theoreticalphysics from the University of Texas, Austin, in1979 and is now a J. Robert OppenheimerFellow in the Laboratory’s Theoretical Astro-physics Group. Winy Benz is a postdoctoralfellow in the same group, He earned his Ph.D.in astronomy and astrophysics from the Uni-versity of Geneva in Switzerland in 1984.

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