Angular Momentum

Angular Momentum Angular Momentum the cosmic pollutant by Stirling A. Colgate and Albert G. Petschek There seems to be too much angular momentum in the universe to allow the formation of stars or the accretion of matter onto variable x-ray sources. This fundamental problem begs for solution. hen we have too much of Yet our universe is populated by planets, the tension in the string is the attractive something and cannot find stars, and black holes. How does nature get gravitational force, 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 point beyond which gravity is unable clouds of gas to clusters of galaxies, galax- Astrophysics”). But before we restrict our- to cause further collapse. This is the basis ies, stars, planets, and black holes, angular selves to x-ray variables, let’s look at the of a stable Keplerian orbit, like that of the momentum is the “pollutant” that pre- problem more generally. earth around the sun or that of accreting vents the game from being played to the matter around a compact star in an x-ray absolute limit, namely, collapse into one Angular Momentum, Weights, binary. Once in a stable orbit, the only way awesome black hole for each cluster-sized and Noncosmological Strings for matter to move farther inward is to cloud 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 weight of how? We also would like to estimate how does the energy in the Hubble expansion mass m whirled at velocity vat the end of a much angular momentum has to be lost by of the universe prevent collapse, inde- string of length 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 friction, and no other Angular Momentum be too much angular momentum to allow external torques, the weight will continue and the Universe the collapse of clouds into dense objects. to circle 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 disk-like shapes indicate they ened, by pulling it through the support, the blackbody radiation 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 group of five remain constant, the weight will speed up better than one part in 104. Moreover, interacting galaxies. Along the top from left to a higher angular frequency 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. galaxy NGC 4594, the star Beta Pictoris 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 spiral galaxy 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 SCIENCE Spring 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 density 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 system. An ob- galaxy. The rotational velocity of matter at 1024. The radius would have decreased by vious way to apply a global external torque its outer edge, determined from the Dopp- a factor of 108, the cube 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 second, 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 dimension are not squared per second. Suppose that, before been 108 times 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 field space with a radius equal to one-half the unreasonably small for the gas in a strengths (obtained from observations of average distance between galaxies, roughly turbulent rotating galaxy. A more reason- the Faraday rotation of the polarization 3 megaparsecs, or 300 times the galactic able velocity for gas clouds, or even for angle of radio waves caused by their pass- radius. If angular momentum was con- galactic rotation, over the radius of the age through a magnetic field) 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 motion 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 speed of sound in hydrogen at 150 kelvins, 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 neutron star, 1014 Suppose, to the contrary, that matter and verse may produce velocities 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 gauss -1/2 1/2 16 earlier history. 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 magnitude. a problem in the formation of nearly We have discussed only the initial and spherical “elliptical” galaxies, which seem final states involved in star formation, 1019 dynes per square centimeter, is larger to have very little total angular momen- both of which are spherical. It must not be than the pressure 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 perpendicular to the rotation escape too easily if it is to remove the extra collapsed into stars was roughly 0.1 to 1 axis; collapse parallel to the rotation axis is angular momentum.

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