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Angular

Angular Momentum the cosmic pollutant by Stirling A. Colgate and Albert G. Petschek

There seems to be too much in the to allow the formation of or the of onto variable x-ray sources. This fundamental problem begs for solution.

hen we have too much of Yet our universe is populated by , the tension in the string is the attractive something and cannot find stars, and black holes. How does get gravitational , which is proportional a way of getting rid of it, we rid of the cosmic pollutant? to R–2. Since the required inward force often think of it as a pollu- Models proposed for variable x-ray goes as R-3, while the available gravita- tant.w In the game of concentration and sources give a strong clue to this long- tional force goes as R-2, there is bound to collapse of matter in the universe from standing puzzle (see “X-Ray Variability in be a beyond which is unable of gas to clusters of , galax- Astrophysics”). But before we restrict our- to cause further collapse. This is the ies, stars, planets, and black holes, angular selves to x-ray variables, let’s look at the of a stable Keplerian , like that of the momentum is the “pollutant” that pre- problem more generally. around the or that of accreting vents the game from being played to the matter around a compact in an x-ray absolute limit, namely, collapse into one Angular Momentum, , binary. Once in a stable orbit, the only way awesome for each cluster-sized and Noncosmological Strings for matter to move farther inward is to condensed from the early universe. lose angular momentum, but the puzzle is Only at the scale of the whole universe The angular momentum of a of how? We also would like to estimate how does the in the Hubble expansion m whirled at vat the end of a much angular momentum has to be lost by of the universe prevent collapse, inde- string of R is mvR. Under ideal whatever mechanism we devise. pendent of angular momentum con- circumstances, that is, a rigid support for straints. On smaller scales there seems to the string, no air , and no other Angular Momentum be too much angular momentum to allow external , the weight will continue and the Universe the collapse of clouds into dense objects. to at the same velocity forever. In other words, its angular momentum J = The universe as a whole does not seem mvR is conserved. If the string is short- to be rotating, as evidenced by the fact that objects whose -like indicate they ened, by pulling it through the support, the blackbody believed to be a have a net angular momentum. The central then since the angular momentum must relic of the early universe is isotropic to figure is Stephan’s Quintet, a of five remain constant, the weight will up better than one part in 104. Moreover, interacting galaxies. Along the from left to a higher angular w = v/R. Tyson has found the orientations of a very to right are an edge-on view of the spiral The inward force necessary to keep the large number of galaxies to be random. NGC 4594, the star weight moving in a circular path, FR = Since the net angular momentum appears surrounded by a disk of dust, and the mv2/R = mw2R, is supplied by the tension to be zero over very large scales, the pollu- 2 3 barred NGC 1300. Along the in the string. Since FR = J /mR in terms tion is not as bad as it could be. Our bottom from left to right are the galaxy of the angular momentum, we see that the problem is restricted to local patches of the M81, an artist’s conception of an x-ray tension in the string increases very rapidly, universe where matter collapses to form binary, and the rings of Saturn. (Photo as R –3, as the string is shortened. In the relatively dense rotating objects such as credits are given at the end of the article.) cosmic game of collapse, the analogue of those shown in the opening figure.

LOS ALAMOS 1986 61 Angular Momentum

–24 Galaxies Are Not a Problem. Let us about 10 gram per cubic centimeter. To Magnetic Fields consider the specific angular momentum form a star, this dilute matter must have

J s = vR (angular momentum per unit collapsed to a of about 1 gram per Only external torques can alter the mass) of a typical, modestly sized, spiral cubic centimeter, an increase by a factor of angular momentum of a . An ob- galaxy. The rotational velocity of matter at 1024. The would have decreased by vious way to apply a global external its outer edge, determined from the Dopp- a factor of 108, the root of the density to dilute ionized matter is through mag- ler shifts of spectroscopic lines, is about ratio. The Keplerian velocity at the stellar netic fields. Indeed this is an often-in- 150 kilometers per , and its radius radius is about 106 to 107 centimeters per voked panacea for the problem. The dif- is about 10 kiloparsecs (-3 X 1022 cen- second, so the initial velocity needed to ficulty is that magnetic fields with the 29 timeters),* so Js = 5 X 10 centimeters conserve angular momentum must have necessary strength and are not squared per second. Suppose that, before been 108 smaller, or 10-2 to 10-1 observed in the universe. Furthermore, condensing, the galactic matter occupied a centimeter per second. This velocity is even if our estimates of magnetic with a radius equal to one-half the unreasonably small for the gas in a strengths (obtained from of average between galaxies, roughly turbulent rotating galaxy. A more - the Faraday of the 3 megaparsecs, or 300 times the galactic able velocity for gas clouds, or even for of radio caused by their pass- radius. If angular momentum was con- galactic rotation, over the radius of the age through a ) are er- served in the collapse to the 10-kiloparsec space from which the stellar matter ought roneous, we are faced with the following galactic radius, the initial velocity of the to have been drawn, would be 105 to 106 dilemma. If matter is to be strongly af- matter must have been less by a factor of centimeters per second. With this value fected by magnetic fields, as is reasonable 300, or about 5 X 104 centimeters per for the velocity, coordinated of for partially or fully ionized matter, it is second. This velocity, which is roughly the even a small fraction of the matter will also reasonable that the matter is strongly in at 150 , introduce too much angular momentum, tied to the field lines. Hence, as a not so seems to be a reasonable value for the by a factor of 1O6 to 108, to allow collapse. extraneous conclusion, magnetic confine- velocity of the turbulent eddies that must This immense amount of excess angular ment fusion should be simple. The fact have existed when galaxies began to form. momentum must somehow have been that it is not means that ionized matter In fact, theoretical calculations suggest dumped before collapse. escapes magnetic fields deceptively easily. that density fluctuations in the early uni- For collapse to a , 1014 Suppose, to the contrary, that matter and verse may produce of this order. times more dense than a normal star, the field are strongly coupled. Then purely Theoreticians thus regard angular mo- problem would be worse by a factor equal two-dimensional radial collapse by our mentum in spiral galaxies not as a pollu- to the sixth root of 1014. (Since the factor of 108 would mean that a region of tant but as a much sought-after relic of an Keplerian velocity is proportional to uniform galactic field of 3 X 10–6 -1/2 1/2 16 earlier . This reasonable state of R , Js = vR is proportional to R , or would be compressed by a factor of 10 in affairs is in sharp contrast to the problem p -1/6.) Thus we have another factor of the newly formed star, and the field would angular momentum poses in the making about 100, or a total of 1010 times too increase to 3 X 10]0 gauss. This is too of a star. Angular momentum may also be much angular momentum. much field by many orders of . a problem in the formation of nearly We have discussed only the initial and spherical “elliptical” galaxies, which seem final states involved in , 1019 dynes per centimeter, is larger to have very little total angular momen- both of which are spherical. It must not be than the inside the newly formed tum. imagined, however, that the collapse is star by a factor of 104 to 1O6. Hence, mag- spherical throughout. Angular momentum netic field must escape easily from the Collapse to Stars. The density of matter conservation prevents collapse only in the collapsing matter even though it cannot in our own galaxy before any of the matter directions to the rotation escape too easily if it is to remove the extra collapsed into stars was roughly 0.1 to 1 axis; collapse to the rotation axis is angular momentum. Such a balance be- hydrogen per cubic centimeter, or not inhibited and thus occurs first. This tween field escape and field trapping leads to formation of a disk whose radius seems most unlikely—although possible. “The unit of distance called a is equal to about 3 X 1018 centimeters; its name is derived is almost as large as the initial radius of the from -second. A parsec is the distance at cloud. But the disk still has the large initial Thin Keplerian Accretion Disks which the direction to an object, viewed from the angular momentum that seems to prevent earth at opposite phases of the earth’s orbit further collapse. Where and what are the The hydrodynamics of thin accretion around the sun, changes by 1 second of arc. The pointing accuracy of a typical old-fashioned galactic dumping grounds for this angular disks provides a more plausible mecha- telescope is about 1 second of arc. momentum? nism for getting rid of angular momen-

62 Spring 1986 LOS ALAMOS SCIENCE Angular Momentum

AZIMUTHAL VELOCITIES OF ROTATING WHEEL AND KEPLERIAN DISK Fig. 1. Distribution of azimuthal velocities

centrated at the periphery. In a Keplerian

shear in the disk tends to equalize the velocities and therefore transport angular momentum toward the periphery, that is, make the disk more like a wheel. In a gaseous disk ordinary molecular is too slow to explain the transport of angular momentum required for star for-

, or at least for allowing its transport momentum (Fig. 1). The friction does just must become much smaller than 1 parsec outward as matter accretes toward a cen- what we want—it transports angular in the course of star formation. Hence the tral point. momentum from the inside to the outside required to form a star in this way In variable x-ray sources and cataclys- of the disk, allowing the inner matter to would exceed the age of the universe, at mic variables, accretion disks form collapse and the outer matter—a small most 6 X 1017 , by several factors around small, dense stars as matter from a fraction—to be spun up and flung off, often. Yet stars abound. Clearly we need a companion star is pulled toward the com- carrying with it all the excess angular better rub or more . pact object and trapped into orbit by the momentum. So what’s the rub? strong . Such accretion The interaction between adjacent mass The Rub, or a disks resemble the rings around Saturn, elements moving at different velocities but, whereas the rings around Saturn are can be characterized by the kinematic vis- A model that assumes a large viscosity evidently composed of solid chunks of cosity D (the ratio of dynamic viscosity to was invented by Shakura and Sunyaev to matter that occasionally bump into one density). We use the unlikely symbol D for explain the apparently rapid accretion of another, an is composed of kinematic viscosity because this is really a matter from a disk onto a compact star in gaseous matter. The gaseous disks that diffusion coefficient. It describes how fast x-ray and cataclysmic variables. This eventually collapse to form isolated stars a viscous (velocity shear) relaxes due model invokes turbulence as the source of are thought to be quite similar. to molecular motion. To carry out an or- the viscosity but does not describe how the Let us look at a likely state for matter der-of-magnitude calculation, we use the turbulence is driven. The strength of the that has partially collapsed and run into kinematic viscosity of hydrogen at room turbulence is parametrized by a coeffi- the angular momentum barrier. As ex- , which (in centimeters cient a, which can be varied between O and plained earlier, it will go into a stable squared per second) happens to be almost 1. Calculations based on this hypo- Keplerian orbit. Now matter of slightly equal to 10–3/p when the density p is thetical turbulent viscosity have been very different angular momenta will go into expressed in grams per cubic centimeter. successful in duplicating the apparent ac- orbit at slightly larger or smaller radii and As in any diffusion phenomenon, the re- cretion rates in x-ray variables. The value have slightly different velocities. If this of cc turns out to be quite large, implying matter is in gaseous form, it will “rub” diffusion distance divided by the diffusion that the accretion disk is highly turbulent. with this differential velocity, and the fric- coefficient, or R2/D. Combining these Such calculations, and even more detailed tion will lead to a torque and hence a equations with an expression for the den- calculations of accretion in cataclysmic change in angular momentum. The direc- variables, strongly suggest the validity of tion of the rub tends to make the gaseous angular momentum to “diffuse” out of the the model. Thus the elusive friction in disk rotate more like a solid body or a disk, a value of 3 X 1017 seconds times Keplerian disks may have been identified. wheel. In other words, matter at the pe- the mass of the central object (in solar If so, we know how nature gets rid of the riphery tends to speed up, increasing its ) and divided by the thickness of excess angular momentum that would angular momentum, while matter near the the disk (in ). The thickness of the otherwise prevent the formation of stars center slows down, decreasing its angular disk is much less than its radius, which and hence us.

LOS ALAMOS SCIENCE Spring 1986 63 Angular Momentum

A A Physical Interpretation of a. Tur- turbulence in with velocity shears Fig. 2. Cross section of a thin Keplerian bulence is the enhanced transport of mat- and large Reynolds numbers. Since these accretion disk around a compact object. ter due to relatively large-scale, random conditions are met in most accretion Large eddies with radii equal to the half- of a . If there are velocity disks, it seems reasonable to expect turbu- thickness of the disk transport angular in the matter, then the effect of lence to supply the necessary fluid friction. momentum in the radial direction R. The turbulence is to transport momentum But, as Lord Rayleigh pointed out more diffusion caused by these eddies can be across a mean velocity shear; it acts like than a century ago, the constraint of approximated by a random walk with step viscosity or friction. The maximum of angular momentum conservation is strong h and velocity of the order of the sound transport by turbulence is determined by enough to stabilize the shear flow of a speed cs. Shakura and Sunyaev modeled the maximum size of the eddies; that is, Keplerian disk against shear-produced, or this transport by a turbulent kinematic vis- the diffusion caused by turbulence can be Helmholtz, instabilities. Hence these in- cosity hcs. approximated by a random walk with a stabilities alone cannot drive the step size equal to the of the larg- turbulence. Another possibility is that the outward, the turbulent motion must be est eddy. Since the largest eddy that can turbulence is driven by convection. isotropic (or nearly so) rather than just in “fit” in the disk and transport matter in Turbulence always produces friction, but the “easy” azimuthal direction. It is easy the radial direction is a round eddy whose now we must ask, conversely, whether the to create eddies whose axes are in the diameter is h, the half-thickness of the disk heat produced by the friction from veloc- radial direction and whose velocities are (Fig. 2), and since the maximum velocity ity shear is enough to drive the turbulence. azimuthal, but we need an instability of such an eddy is the local sound speed cs, In the next section we will explore this strong enough to overcome the stabilizing the maximum possible random-walk dif- possibility as an example of how difficult effect of angular momentum and drive fusion coefficient, or turbulent kinematic it is to produce the large values of a re- radial motions. If these instabilities exist, viscosity D, is hcs. Thus Shakura and quired to transport momentum outward the turbulent motion provides an effective Sunyaev parameterized the turbulent in Keplerian accretion disks. viscosity (an “eddy” viscosity) far larger than the molecular viscosity and can Convection-Driven Turbulence. In an transport angular momentum at the rate between 0.03 and 1. accretion disk, friction from velocity required for the evolution of the disk. shears should give rise to inhomogeneous To see whether differential heating can What is the Origin heating concentrated near the midplane of drive the required instability, we must of the Turbulence? the disk. This differential heating can look at the structure of the accretion disk create instabilities that lead to turbulent implied by the Shakura and Sunyaev We are accustomed to the ubiquity of motion. To transport angular momentum model (see Fig. 2). As indicated above, a

64 Spring 1986 LOS ALAMOS SCIENCE Angular Momentum

value of a near unity implies that the in the radial direction do transport angular fluid and lowering cold fluid is the internal diameter of the eddies that interchange momentum and produce enhanced fri- matter in the radial direction must be ction and a. The force from differential close to the half-thickness h of the disk, buoyancy must be large enough to create and their velocity must equal the sound an eddy that interchanges matter over a

speed cs. Thus the interchange scale in the Since the that must be done in the radial direction, if the eddies are round, Now let’s consider a single eddy and interchange is the same as the energy will equal h. The in the calculate how much work is needed to available in buoyancy, there is barely eddy, which is determined by the sound interchange two mass elements a distance enough buoyancy to force an overturn or a

speed cs, will determine how much energy AR while conserving angular momentum. circular eddy in the radial direction,

is available to drive the interchange. As We will assume an initial laminar stable especially at an eddy velocity near cs. For Pringle has pointed out, the structure of state so that in the interchange of two the eddy to develop we need a nearly

Keplerian disks is such that cs/v = h/R, adjacent elements of equal mass, the perfect heat that converts the heat where v is the azimuthal velocity and R is angular momentum of each is conserved of friction to potential and . the radial distance. separately (vR = constant). Then for each The disk is densest in the midplane mass element the change in azimuthal ve- The Ideal . We can imagine because the pressure is greatest there. The an ideal heat engine driving the eddies. pressure is required to hold the material The change in specific kinetic energy of the The heat is produced in our mass element near the out against the compo- nent of gravity perpendicular to the disk by turbulent friction due to the shear of the (Fig. 3). Consequently frictional heating tum constraint, together with a little al- orbital velocity. The hot material expands from velocity shears and therefore the gebra, shows that the net change is equal to adiabatically as it rises to the surface of the temperature will be greatest at the mid- disk. There the remaining internal energy . The surface of the disk will be energy will be zero because we have as- must be lost by radiation during the resi- cooled by radiation. This configuration is sumed the interchange of two equal dence time of the mass element at the Rayleigh-Taylor unstable in the direction masses. Hence we need an energy equal to perpendicular to the plane of the disk. the diffusion coefficient for radiation, Motions perpendicular to the disk and at must be provided by differential buoy- the same radius do not transport angular ancy. tion energy density to total energy density momentum. On the other hand, motions The energy available from raising hot

must be the same as the coefficient for turbulent mass transport. This implies

Accretion Disk these restrictions our mass element would cool, and it could then descend adiabatically with much smaller internal Object pressure, so less work must be done on it. When it reaches the midplane of the disk,

midplane, where the gravitational potential is 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 disk where it will cool by emitting radiation. The question to ask is whether the differential DIFFERENTIAL BUOYANCY IN A THIN ACCRETION DISK buoyancy is large enough to drive an eddy in the radial direction.

LOS ALAMOS SCIENCE Spring 1986 65 can start the cycle over again. This cycle as nism leads to a self-adjusting disk in the gines must be so perfect and turbulence is well as the alternating regions of hotter sense that if the mass-injection rate so random. Thus, the above ideal cycle, and cooler material that would result at changes, then a and the other parameters although conceptually feasible, seems dif- the surface are illustrated in Fig. 4. vary to maintain a consistent disk struc- ficult to justify as an explanation for the The required condition on the optical ture. It is also not known whether all astro- origin of a. Convection-driven turbulence depth, together with the opacity of the physical disks can be explained in the does not seem strong enough to overcome material, determines the mass per unit parameter space just outlined. the angular momentum barrier. of the disk. Then the Furthermore, nature does not like to make I can be calculated from a and the sound ideal heat , especially not in a Thick Disks Beg the Question speed. It is not known whether this mecha- turbulent because heat en- The importance of angular momentum

Fig. 4. Convection-driven turbulence in a require a nearly perfect engine, that is, one figure suggests the required breakup of the thin Keplerian accretion disk creates large in which nearly all the heat was converted eddies into smaller scale turbulence after eddies that break the angular momentum to work as matter flows around the eddy. about half a cycle. (A persistent eddy would constraint by enhancing radial transfer of The hexagonal pattern of Benard-like cells not produce any net transport.) This angular momentum. The energy available shown might reproduced by heating at the breakup is a difference between the eddies from differential buoyancy is barely enough midplane. The heat cycle that drives the in the disk and those in a standard Benard to drive the eddies. Their formation would overturn of the eddies produces alternating cell, which are very slow (low Reynolds hotter (red) and cooler (blue) regions. The number) and do not lead to smaller scale turbulence.

66 Spring 1986 LOS ALAMOS SCIENCE Angular Momentum

constraints is illustrated in studies of thick AUTHORS accretion disks by Wojciech Zurek and Winy Benz of Los Alamos. They have Stirling A. Colgate received his B.S. and Ph.D. performed numerical simulations of the degrees in physics from Cornell University in evolution of thick disks with specific 1948 and 1952, respectively. He was a staff physicist at Lawrence Livermore Laboratory angular momentum independent of for twelve and then president of New radius. Because the specific angular Mexico Institute of Mining and Technology for momentum is constant, the interchange of ten years. He remains an Adjunct Professor at two equal mass elements requires no that institution. In 1976 he joined the Theoreti- cal Division at Los Alamos and in 1980 became energy. The disks exhibit violent leader of the Theoretical Astrophysics Group. Helmholtz instabilities. As one would ex- He is also a Senior Fellow at the Laboratory, a pect, less constraint leads to greater member of the National Academy of , turbulence. The instabilities cause angular and Technical Director of the Gifthorse Pro- momentum to redistribute itself very gram. His research interests include nuclear physics, astrophysics, atmospheric physics, and geotectonic . (see “Redistribution of Angular Momen- tum in Thick Disks”). Thus the disk be- comes more Keplerian, but since these instabilities are damped the disk never becomes truly Keplerian (q = 0.5). Albert G. Petschek is currently a Fellow at Los Such models invite the question of how Alamos as well as being Professor of Physics at a disk can be formed with small and nearly New Mexico Institute of Mining and Tech- uniform angular momentum. In the case nology. He received his B.S. from MIT in 1947, his M.S. from the University of Michigan in of and active galactic nuclei pow- 1948, and his Ph.D. from the University of ered by the accretion of matter onto Rochester in 1952. His research interests are in massive black holes, these disks might be astrophysics, cloud physics, and fracture me- formed by the gravitational breakup of chanics. The spring of 1978 found him in Israel as a Visiting Professor at Tel Aviv Univerity. stars scattered by interactions with other He is a member of the American Physical stars in the strong gravitational field close Society, the American Association for the Ad- to the massive black hole. More specifi- vancement of Science, and the American cally, stars in a dense galactic nucleus scat- Astronomical Society. ter at random. Occasionally one of these scattering events causes a star to approach the black hole with an impact parameter Photo Credits tion from gas in the spiral arms of the galaxy appears blue and radiation from older stars in so small (several Schwartzschild radii) Stephan’s Quintet. A combination of best avail- the disk appears orange. The image was com- that the star deforms tidally and a fraction able photographs in different colors taken with posed from single-color photographs taken with of the star is captured. Other stars of the the 200- reflector at Palomar Ob- Palomar Observatory’s 48-inch Schmidt servatory.© Halton C. Arp; reproduced with telescope.© Halton C. Arp; reproduced with cluster then have a slightly greater angular permission. permission. momentum because of its conservation. NGC 4594. Palomar Observatory photograph; Rings of Saturn. An optical image in real color Jack Hills of Los Alamos calculated that a reproduced with permission. created from data collected in October 1980 by thick disk of low angular momentum is a Beta Pictoris. A color-coded created from NASA’s Voyager I . The rings have been optical images of Beta Pictoris and the similar enhanced with additional color. Reproduced reasonable outcome of such accretion. star Alpha Pictoris, both taken with a corona- with permission of the NASA Thus there may not be an angular graph and a charge-coupled device at the Las Laboratory. momentum problem in feeding a black Campanas Observatory in Chile. The disk sur- rounding Beta Pictoris was revealed by plotting Further Reading hole, but the original problem of making a the ratio of the of scattered star from tenuous gas remains. Either we around Beta Pictoris to that around Alpha Pic- Pringle, J. E. 1981. Accretion discs in astro- must posit an initial gaseous state of tiny toris. Reproduced with permission of Bradford physics. Annual Review of Astronomy and and thus statistically unlikely angular A. Smith, University of Arizona, and Richard J. Astrophysics 19:137-162. Terrile, Jet Propulsion Laboratory. momentum, or we are left with, the im- NGC 1300. Palomar Observatory photograph; Shakura, N. I. and Sunyaev, R. A. 1973. Black perative to find a transport mechanism for reproduced with permission. holes in binary . Observational appear- the cosmic pollutant. ■ M81. A color-enhanced image in which radia- ance. Astronomy and Astrophysics 24:337-355.

LOS ALAMOS SCIENCE Spring 1986 67 Angular Momentum in Thick Disks

Redistribution of Angular Momentum in Thick

Accretion Disks by Wojciech H. Zurek and Winy Benz

(a) (b)

CROSS SECTIONS OF THICK ACCRETION DISKS

A ses of the stability of the disks by Fig. 2. Isodensity contours of two tori of angular momentum is il- Papaloizou and Pringle. Our numerical with approximately the same inner and 1 lustrated in our recent numerical simulations confirm those first suspi- outer radii, but with different distribu- simulations of thick accretion disks. cions. More important, we were able to tions of specific angular momentum: (a)

These torus-like disks, in contrast to the demonstrate that growing instabilities a torus with Js = constant and (b) a torus with J - r0.27 Note the change in the flat, pancake-like Keplerian disks, have in a constant-J, torus rapidly re- s a height above the equatorial plane distribute angular momentum, causing funnel opening angle from -10° in (a) to comparable to their extent in the radial the torus to become thinner and more -30° in (b). The density decreases by direction. Until recently the constant Keplerian. Hence thick accretion disks twelve orders of magnitude from the in- specific angular momentum (Js = con- with constant Js cannot be regarded as nermost to the outermost contours. stant) variety of such non-Keplerian ac- models for astrophysical objects. cretion disks around massive black The equilibrium configuration of with a small opening angle 6, about the holes was considered the best model for thick accretion disks, for constant Js and rotation axis. As noticed by Lynden- the central “powerhouse” in quasars large pressure (sound com- Bell, this is a perfect for a deLaval and active galactic nuclei. However, parable to rotational velocities), looks nozzle,* that is, for accelerating the doubts about the validity of that like a fat torus, or doughnut (Fig. 1a). enormous supersonic jets observed to hypothesis were raised in 1984 by analy- The sides of the torus form a funnel, be emanating from so many active ga-

68 Spring 1986 LOS ALAMOS SCIENCE Angular Momentum in Thick Disks

lactic nuclei. Moreover, the steep walls of the funnel allow the of the torus to exceed the Eddington limit, a that would be indeed useful in modeling variable quasars with huge outputs. The first question to ask is how such non-Keplerian accretion disks can exist, since matter at small radial r has “too much” angular momentum (more than the Keplerian value) and matter at large r has “too little” angular momentum. The pressure in the disk is responsible for maintaining this dis- tribution of angular momentum: the matter at large r is being prevented from moving inward by the pressure, and the matter at small r is being kept from moving outward by the pressure- mediated weight of the outer parts of the disk.

Proponents of Js = constant accre- tion disks assume that the effective turbulent viscosity of such a disk is very small, so that it is all but impossible to transport angular momentum outward REDISTRIBUTION OF ANGULAR MOMENTUM as matter accretes toward the massive body at the center of the disk. However, if the central massive body is a black o 1 2 3 4 5 hole (as it is almost certain to be for quasars and active galactic nuclei, hose” instabilities. However, the gas Fig. 2, The exponent q as a of including the nucleus at the center of does not stream out in random direc- time, where q is calculated from a power- our own ), then it is possible tions, as would water from a hose left rq) to the specific angular to get rid of the angular momentum of unattended. Instead the gas deflected momentum distribution obtained from accreting matter by pushing it into the from its equilibrium orbit by the in- the numerical simulation. The results are black hole. The necessary push can be stability is bound by the gravitational shown for three simulations. Note that, provided by applying pressure from far potential and so produces density in- for the two disks with initially constant away and “force-feeding” the black hole homogeneities, pressure gradients, and specific angular momentum, q increases with gas. sound waves, which, in turn, produce rapidly from O to about 0.27 within about Using an idealized model, Papaloizou more deflections, which lead to more two rotation periods. After the critical q and Pringle challenged the foundations sound waves. = qC -0.27 is reached, the redistribution of the thick accretion disk theory by In a second paper Papaloizou and of angular momentum slows down to the showing that constant- J s tori are Pringle extended the stability to rate observed in a disk with initial J - 0.3 s violently unstable against nonaxisym- include very thin tori (like slender bicy- r , It is not yet known whether this slow q shear-driven perturbations. The cle tires) with Js varying as r . For q < 2 rate of angular momentum transport is instabilities revealed by their linear caused in part by nonaxisymmetric in- analysis are Helmholtz instabilities and be unstable. For greater values of q a stabilities or is totally explained by a in some ways are analogous to “fire- large class of unstable modes is numerical viscosity that is an un- stabilized. Their linear analysis did not, avoidable artifact of such calculations. *The of a deLaval nozzle is described by M. L. Norman and K.-H. A. Winkler in "Supersonic however, reveal the ultimate fate of the We are planning to study this problem Jets, ’’Los Alamos Science Number 12, 1985. original configuration. further.

LOS ALAMOS SCIENCE Spring 1986 69 Angular Momentum in Thick Disks

We have extended such stability anal- (see Fig. 3, t = 0). These numerical ex- the angular momentum very quickly yses to the nonlinear regime by adding a periments not only confirm that such (on the time scale of about a rotation qc small random density perturbation (of disks are unstable but also show that a period Js) from Js = constant to Js - r , the order of 1 percent) to an initial equi- fat accretion torus is forced to undergo a where qc turns out invariably to be 0.27 librium configuration with constant Js “crash diet”: instabilities redistribute (Fig. 2). Note that this value for the

Fig. 3. A computer-generated time se- magnitude over the region shown. Time is and fills in the central “hole.” This simu- quence showing the three-dimensional expressed in rotation periods of the den- lation was made with a three-dimen- evolution of the central region of a thick sity maximum. The velocity field is in- sional hydrodynamics code that uses the accretion disk with initially constant dicated by means of arrows whose so-called smoothed hydro- specific angular momentum. The upper are normalized to the maximum value of method (Lucy 1977). This free panels show isodensity contours in the the velocity in each frame. Following the Lagrangian approach to solving the usual equatorial plane of the central region; the introduction of a small nonaxisymmetric equations of hydrodynamics replaces the lower panels show isodensity contours in perturbation, the growth of instabilities continuum by a finite of spatially a plane parallel to the rotation axis. The causes a rapid redistribution of angular extended . Thus no mesh is re- density decreases by one to two orders of momentum that, in turn, flattens the disk quired, and the usual problems as-

TIME EVOLUTION OF A THICK DISK

t=o t= 1 t=2

70 Spring 1986 LOS ALAMOS SCIENCE Angular Momentum in Thick Disks

exponent is about the same as that ob- As the angular momentum is re- properly by the calculations performed tained by Papaloizou and Pringle for the distributed, the fat torus becomes much to date—is whether the shear-driven in- stabilization of “bicycle tire” tori. Fig- thinner and much more Keplerian in stabilities will provide a mechanism for

ure 3 shows the of the appearance. Moreover, the narrow fun- a when q > qc. Can these instabilities disk. nel invoked to explain the formation generate wave-like excitations and “in- and collimation of relativistic jets be- teresting” a values in disks that are qc comes much wider (Fig. 1 b) and there- “barely” stable (Js = r ) or almost o5 sociated with its rezoning are bypassed. fore less effective in producing colli- Keplerian (Js = r )? We are now ex- The simulation is made by computing the mated jets and super-Eddington ploring this question with one of the of 1000 extended particles . new three-dimensional hydrodynamics that interact through pressure forces in a Regarding the question of angular codes developed at Los Alamos. ■ central gravitational potential. Since the momentum transport discussed in the particles are allowed to move without any main text, our calculations show that, at constraints in all three spatial directions least for q < qc, shear-driven instabilities and since a mesh is not needed, this provide a powerful source of the “rub,” method is particularly suited for the sim- that is, of a, the turbulent viscosity. The ulation of highly distorted flows. next obvious question—not addressed

Further Reading AUTHORS Abramowicz, Marek A., Calvani, Massimo, and Wojciech Zurek earned a Ph.D. in theoretical Nobili, Luciano. 1980. Thick accretion disks physics from the University of Texas, Austin, in with super-Eddington luminosities. The Astro- 1979 and is now a J. Robert Oppenheimer physical Journal 242:772-788. Fellow in the Laboratory’s Theoretical Astro- physics Group. Winy Benz is a postdoctoral Fishbone, Leslie G. and Moncrief, Vincent. fellow in the same group, He earned his Ph.D. 1976. Relativistic fluid disks in orbit around in astronomy and astrophysics from the Uni- Kerr black holes, The Astrophysical Journal versity of Geneva in Switzerland in 1984. 207:962-976.

Lucy, L. B. 1977. A numerical approach to the testing of the fission hypothesis, The Astronomical Journal 82:10131024.

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

Papaloizou, J. C. B. and Pringle, J. E, 1984. The dynamical stability of differentially rotating discs with constant specific angular momen- tum. Monthly Notices of the Royal Astronomical Society 208:721-750.

Papaloizou, J. C. B. and Pringle, J. E. 1985, The dynamical stability of deferentially rotating discs—IL Monthly Notices of the Royal Astronomical Society 213:799-820.

Zurek, W. H. and Benz, W. Redistribution of angular momentum by nonaxisymmetric in- stabilities in a thick accretion disk, Accepted for publication in the 1 September 1986 issue of The Astrophysical Journal.

t=3

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