.400S .143. 66ApJ. 19 ON THE EVOLUTION OF GALACTIC NUCLEI Lyman Spitzer, Jr., and William C. Saslaw Princeton University Observatory Received July 2, 1965 ABSTRACT Successive evolutionary phases in the life of a compact, nearly spherical galactic nucleus or cluster are considered in this preliminary analysis. The first phase is the familiar contraction produced by evapora- tion, the more energetic stars escaping into elongated orbits in which they are essentially lost to the system. As the system contracts, collisions among stars become relatively more frequent, leading to liberation of substantial amounts of gas. The mass loss per collision, f, in solar units, is evaluated on a very simple model considering only the energy and momentum of colliding mass elements, and neglecting fluid interactions such as shocks which transmit energy and momentum transverse to the relative velocity, V; mean values of £ range from about 0.04 for a root mean square stellar velocity, vSy of 1300 km/sec up to about 0.10 for vs equal to 4400 km/sec. The gas given off in collisions cools rapidly by radiation and falls toward the center of the system. If any angular momentum is present, the gas will form a flat disk, whose density will increase until gravitational instability sets in. For typical parameters, the density in the disk will exceed about 109 atoms/cm3, and the thickness will be a few hundred astronomical units. Gravitational instability will lead to the formation of condensations in the disk, and we assume that normal stars are produced. These new stars will absorb kinetic energy from the stars in the nucleus, which will contract at an accelerated rate until the time of relaxation becomes substantially longer than the time required for a star to lose the bulk of its mass by collisions. The energy lost by the liberated gas yields a peak luminosity, which is reached at this phase, about equal to 1045 ergs/sec for a system with 108 stars of solar mass; the system radius is about 0.05 pc. The luminosity peak for the assumed system has a duration of about 1,000,000 years. For later times the new stars forming in the disk will tend to remain there, with relatively low random velocities, and the spherical system will be transformed into a flatter one, with a much reduced luminosity. The energies radiated at the luminosity peak are similar to those observed from quasi-stellar radio sources, but it is uncertain whether existing nuclei have evolved as yet to the degree of compactness required to account for these highly luminous objects. The evolution of an isolated, nearly spherical aggregation or cluster of stars poses many fascinating physical problems. It was pointed out several decades ago (Ambart- sumian 1938; Spitzer 1940) that in an isolated cluster some stars will occasionally gain through random collisions a velocity exceeding the escape velocity from the system. These stars will leave the system, and this process of “evaporation” leads to a contrac- tion of the cluster and to an acceleration of the rate of evaporation. Unless the stellar system has enough angular momentum to inhibit its contraction at a relatively low stellar density, it would seem that this process of accelerating contraction must lead inevitably to an increasing number of collisions between the stars in the cluster. Since apparently all stellar aggregations with sufficiently low angular momentum must in time reach a stage in which direct stellar collisions play a dominant role, an investigation of the effects which these collisions produce and their effects on the continuing evolution of the system is therefore of substantial interest. Additional interest in this investigation arises from the suspicion that stellar collisions in contracting galactic nuclei may, perhaps, account for the energy flux radiated by the radio galaxies in general and by the quasi-stellar radio sources in particular. It is reason- ably well established that these quasi-stellar sources radiate energy at a rate between 1044 and 1046 ergs per sec, and perhaps maintain this rate for some 106 years (Burbidge 1965; Schmidt 1965). The possibility that colliding stars in dense galactic nuclei might account for the vast release of energy in these objects has been discussed by Woltjer (1964), who suggests that the shocks produced by the collisions produce a large flux of relativistic particles, as in supernovae. Stellar collisions as an energy source in galactic nuclei have also been discussed recently by Gold, Axford, and Ray (1965), Spitzer (1965) 400 © American Astronomical Society • Provided by the NASA Astrophysics Data System .400S .143. EVOLUTION OF GALACTIC NUCLEI 401 . and van den Bergh (1965). If dense galactic nuclei are assumed to exist at the present time, there is no question but that large amounts of energy would be released in stellar 66ApJ. collisions. 19 The observational evidence for the existence of such dense nuclei, however, is still rather scant. At the center of M31 is a nucleus whose luminosity corresponds to 108 Lo, and whose diameter does not exceed about 5 pc. In M32 there is an even smaller nucleus, with a diameter of at most 2 pc, and with a mass estimated at 107 Mo. In NGC 4151, a prominent Seyfert galaxy, the light from the apparently stellar nucleus corresponds to a mass of 1010 Mo, while the upper limit on the diameter is 100 pc, The corresponding star densities in these various compact systems are in the range of at least 105 to 106 stars pc-3. To produce by stellar collisions the large radiative fluxes observed from quasi- stellar radio sources, the maximum star density must be assumed several orders of magnitude greater than these values. Whether the density in any existing galactic nucle- us has yet reached such high values is conjectural. Our approach here is not to propose that colliding stars in galactic nuclei are an ex- planation of any observed phenomenon. Instead we shall attempt to analyze what ap- pears to be a natural and reasonably unavoidable stage in the evolution of massive, nearly spherical systems in general and of galactic nuclei in particular. This stage may not in fact be reached in most galactic nuclei until long after most stars have exhausted their nuclear fuel and grown dim. On the other hand, it is possible that at least some of the very luminous radio sources may be galactic nuclei which are going through this stage at the present time. The following sections discuss various phases in the evolution of a nearly spherical stellar system, with particular emphasis on the more massive aggregations at the centers of galaxies. The first section reviews the rate of evaporation, the resulting evolution of the cluster, and the effects produced by an initial angular momentum. Section II treats the rate of stellar collisions in a dense cluster and analyzes, on a very simple model, the effects to be expected when two stars collide. The gas ejected from the two stars during a collision is treated in the third section, which treats the rate of cooling, of collapse toward the galactic center, and of star condensation from this gas. The final section dis- cusses the late phases of evolution of the galactic nucleus and the total luminosity of the hot gas liberated from the colliding stars. It should be emphasized that much of the analysis presented here is tentative and exploratory. Much additional study is required to delineate the complex physical processes occurring in compact stellar systems. I. STAR EVAPORATION AND EARLY EVOLUTION The rate of evaporation from a spherical stellar aggregation has been studied by many workers—see the recent survey by Michie (1964). The real phenomenon is remarkably complex. One simple approximation is that of Spitzer and Härm (1958), who consider the exact equation for diffusion in velocity space but who approximate the actual gravita- tional potential by a square well. On this basis the rate of escape from a cluster with stars all of the same mass is given by 1 dN 1 (i) N dt 887V where N is the total number of stars in the cluster, and TVs a reference time, which we shall call the “relaxation time” within the cluster, and which is given by g rDR= mi/2 2 2 h 7 . (2) 3TrG m n In ( A / 2 ) ’ n is the number of stars per unit volume, tn the mean stellar mass, and v8 the root-mean- square velocity in three dimensions; G is the gravitational constant. © American Astronomical Society • Provided by the NASA Astrophysics Data System .400S 402 LYMAN SPITZER, JR., AND WILLIAM C. SASLAW Vol. 143 .143. To relate this model to actual stellar systems, we define R as the root-mean-square radius. We then take 66ApJ. 19 3N n = ——. O) 47^R3, and from the virial theorem GNm fl*2 = — h<l> (4) 2R 1 where 0 is the mean potential energy per unit mass; the constant J in the right-hand expression of equation (4) gives a reasonably close approximation for polytropes with n in the range from 0 to 4 (Spitzer 1958). If equations (3) and (4) are substituted into equation (2), we obtain 7\7-l/2 P'3/2 = 8.3 x io5 7— r^j 7TT7 years, (5) (w/mo)7 1/2 (log Y — 0.3) where R' is the value of R expressed in parsecs. Since gravitational encounters among the stars change the stellar velocities by small increments, an escaping star will have nearly zero energy. Hence the total energy of the remaining system will remain constant, as will the total gravitational and potential energies. The system radius R will accordingly vary as N2, and the relaxation time, Tr, will vary about as N7/2, decreasing rapidly as the population of the system diminishes.