Proceedings of the NATIONAL ACADEMY OF SCIENCES

Proceedings of the NATIONAL ACADEMY OF SCIENCES Volume 55 * Number 1 * January 15, 1966 DYNAMICS OF SPHERICAL GALAXIES, II* BY PHILIP M. CAMPBELL LAWRENCE RADIATION LABORATORY, LIVERMORE, CALIFORNIA Communicated by George Gamow, November 22, 1965 This report is concerned with some results of a study of galactic structure in which numerical calculations of dynamical evolution are used in conjunction with theories of star formation to obtain final equilibrium states consistent with observation. The calculations presented here are restricted to collisionless, selfgravitating systems with spherical symmetry. A preliminary investigation, based on a model with nearly radial orbits, has already appeared.l Principles of Galactic Structure and Evolution.This study is based on the postulate that spherical and elliptical galaxies were formed by the condensation and fragmentation of an initially gaseous state, the protogalaxy. The initial conditions for the dynamical problem are determined by the properties of the protogalaxy immediately prior to the formation of stars. Initially, the positions of the stars correspond to the mass distribution in the protogalaxy. If local density variations caused by turbulence in the gaseous medium initiate the fragmentation process, the initial velocities of the stars are strongly correlated with the turbulent velocity spectrum. The initial distribution of stellar masses is given by the theory of star formation. The structure of the individual stars depends on the initial composition but will be altered by subsequent development of nuclear burning. The present massluminosity relation for the stellar population must be obtained from the initial mass spectrum in conjunction with the theory of stellar evolution. The galaxy thus formed undergoes mechanical development as a selfgravitating system of mass points. The system may evolve to a state of dynamical equilibrium a state of motion whereby the stars maintain a steady macroscopic configuration while moving in their own collective potential field. Two principal mechanisms occur in the evolution to the steady state: (1) "mixing" or randomization of the relative phases of the orbits, and (2) net macroscopic mass flow resulting from incomplete cancellation of velocities. The steady state is achieved when the orbits are so distributed that the velocities of all stars in every small volume cancel. Necessary and sufficient conditions for the attainment of a steady dynamical state are not known. However, it is apparent from the nature of the steady state that only for certain special cases of symmetry is such a state possible, and that the required symmetries must be present initially. The final steady state obtained from the dynamical calculations must be com 1

2 ASTRONOMY: P. M. CAMPBELL PROC. N. A. S. 2 X +1 4.* YELLOW LIGHT +1 (RIGHT SCALE) l0 0 _ *4I(YELLOW) le I a BLUE LIGHT 0 (LEFT 9 SCALE)) Ie(BLUE). re* 70.6 2_N*. _3 0*.\ ~~~8 ~ 3 2 3 4.4 ( RADIUS. r) 1'4 FIG. 1.Photoelectric observations of the luminosity distribution in NGC3379, a normal elliptical galaxy.3 The empirical luminosity law (1) appears as a straight line. The distributions for both colors are the same, except for r < 0.2 r,. Radial distance is measured in units of 0.74 sec of arc. pared with the observed properties of galaxies in order to test the correctness of the assumed initial conditions. Observations.The most important observed property of spherical and elliptical galaxies is the distribution of luminosity in the image projected on a plane perpendicular to the line of sight. All ellipticals show the same relative distribution of luminosity, which is well represented by de Vaucouleurs' law:2 log I 3.33 where re is the radius within which half the total luminosity is emitted, and Ie is the intensity along the isophote defined by r,. For all objects so far measured, the general validity of de Vaucouleurs' law has been directly verified from r 0.03 r. out to r c 5 re; the photoelectric measurements of Miller and Prendergast3 are shown in Figure 1. In order to compare dynamical properties with observation, the space distribution of luminosity, J(r), must be obtained from the projected distribution, I(x), by means of the relation J(r) = dx(x2 r2)12dx Ir _ x (X I

VOL. 55, 1966 ASTRONOMY: P. M. CAMPBELL 3 where R is the radius of the system. The distribution of mass can be inferred from the space distribution of luminosity if the mass/light ratio is known throughout the system. The mass/light ratio is determined by the mass spectrum of the constituent stars, together with their massluminosity relation. If the mass spectrum is uniform throughout the system, the mass/light ratio is constant, and the distribution of luminosity is a direct measure of the mass distribution. The photoelectric measurements shown in Figure 1 reveal that the distribution of color in a normal elliptical is quite uniform, except for r < 0.2 re. The uniformity of color is a strong indication of the uniformity of the stellar population. In Figure 2 the cumulative mass distribution obtained from the empirical luminosity law (1) with a constant mass/light ratio is shown in comparison with the mass distributions of several gaseous systems. The distributions are scaled to unit mass and radius, the scale factor for the galaxy being 20 re (the extreme limit of detectability). For spherical systems the slope of the cumulative mass relation, Pr' M(r') = J p(r) dr, is just the spherical density, p(r) = 47rr2k(r), where +(r) is the space density in gm/cm3. It is clear from the figure that stellar systems are much more concentrated than gaseous systems. The mass distribution alone is not sufficient to determine the state of a stellar system uniquely; the velocity distribution remains unspecified. A system with the observed mass distribution could be constructed out of any combination of orbits of different form. The only observation directly pertaining to the velocity distribution is the broadening of spectral lines in the light from the nucleus. This observation has been used to estimate the dispersion in a Gaussian velocity distribution, assumed valid for the stars in the central regions.4 Results of the dynamical calculations (see below) indicate that spherical galaxies are composed of 1.0 GALAXY TROPE / O0.6 POL. 0 FE ~ //ISTHERMAL U) 4 0Q2 / / p=const. 00 0.2 0.4 0.6 0.8 1.0 RADIUS,r FIG. 2.Cumulative mass distribution in spherical galaxies (assuming a constant mass/light ratio) compared with mass distributions in polytropic and isothermal gas spheres. The distributions are scaled to unit mass and radius. The scale factor for the galaxy is 20 r., the extreme limit of detectability.

4 ASTRONOMY: P. M. CAMPBELL PROC. N. A. S. highly elongated orbits, and that the velocity distribution in the nucleus is far from Gaussian.1 Although the broadening of spectral lines is an important observational result, the significance of a related velocity dispersion is obscure and cannot be considered as a firmly established dynamical property. The most reliable estimates of the masses of galaxies are those obtained by Page,5 who finds Ml 4.1011 solar masses (8.1044 gm) for the mean of 45 pairs of elliptical galaxies. The average effective radius of 20 large ellipticals studied by Fish6 is re 3.2 kiloparsecs (1.1022 cm). These values are based on a distance, redshift relation of H = 100 km/sec/mpc. Numerical Computation of Stellar Dynamics. The stellar system is represented by several hundred distinct groups of stars, each group composed of stars with common orbital elements. If the initial state of the system is spherically symmetric, the stars of each group are distributed uniformly about the center in thin spherical shells. The initial velocity of a star determines the orientation of its orbital plane, and each group can maintain its symmetry only if the initial velocities are oriented uniformly in all directions. In a spherical system, the transverse component of force vanishes everywhere, and the angular momentum, L, of each orbit is constant. The evolution of the system is computed by following the motion of each shell by numerical integration as it moves in the timedependent potential field of the whole configuration. Individual orbits are computed by Hamming's predictorcorrector method,7 with starting values determined by a RungeKutta procedure.8 The use of a predictorcorrector method is preferred, since it provides a reliable estimate of the error at each step of the calculation. Each orbit is computed over a major cycle, At, on its own time interval, bt(bt << At), which is determined at each step according to the error. At the end of each major cycle, the cumulative mass distribution in the system is redetermined. Since only a few hundred orbital groups can be used, the acceleration function, d2r L2 M(r) dt2 r3 G 2 contains small random fluctuations that tend to reduce bt unnecessarily. The effect of these fluctuations is eliminated by smoothing the 111(r) distribution with a thirdorder, 9point leastsquares interpolation formula.9 Since the acceleration experienced by a star is independent of its mass in a collisionless system, the dynamical evolution cannot effect a segregation of different masses into separate regions, a fact verified by the calculations.' If the mass spectrum is initially uniform, it is therefore not necessary to consider the subdivision of each group into its various mass components. If the star formation process results in a nonuniform distribution of masses, the mass spectrum must be included in the calculation before a comparison with the observed luminosity distribution can be made. Results of the Numerical Calculations.Several different initial mass and velocity distributions have been investigated in an effort to obtain final states which agree with the mass distribution obtained from de Vaucouleurs' law (1) under the assumption of a constant mass/light ratio. The cumulative mass distributions of these states are shown in Figure 2 on the same scale as the observed final state

VOL. 55, 1966 ASTRONOMY: P. M. CAMPBELL 5 (ey = ca/c,, the ratio of specific heats).10 It is assumed that the turbulent energy density in the gas is everywhere proportional to the pressure necessary to support the protogalaxy in hydrostatic equilibrium. The proportion of thermal and kinetic energy in each state is varied to achieve final states which match the empirical distribution as closely as possible. (1) Uniformity of initial states: The calculations show that the final state is very sensitive to the initial conditions. Collisionless stellar dynamics does not result in a relaxation process in which different initial states all approach the same characteristic final state. Nor would this be expected, considering the nature of the evolutionary mechanism (macroscopic mass flow resulting from incomplete mixing of the relative phases of individual orbits). The only reason to postulate a relaxation process is the remarkable observation that all ellipticals have the same relative luminosity distribution. However, the dynamical calculations, together with this observation, allow only one interpretation: the initial states of all spherical galaxies were the same except for scale. The uniformity of all initial states simply means that the gaseous protogalaxies had all evolved to an equilibrium configuration before fragmentation began to develop. The only alternative, star formation in nonequilibrium configurations, would necessarily result in many different initial states. The thermal, U,, kinetic, Ti, and potential, Qi, energies for an initial state in hydrostatic equilibrium must obey a relation analogous to the virial theorem: Ti + Us=, (2) 2 (2) Adjustment to dynamical equilibrium: A stellar system in the final steady state must obey the virial theorem,,2 = H, (3) 2 where the total mechanical energy determined by the initial state, Ti + Qt = H = Tf + Of, is invariant. It is clear that the relative amounts of potential and kinetic energy in the initial state will influence the subsequent evolution. The potential energy, together with the total mass, serves to define a mean radius 7, Q M2 r which provides a measure of the dilution or concentration of mass in the system. If the initial stellar state is formed with comparatively little kinetic energy, 0 < T < 0/2, evolution results in a net concentration of the mass distribution, since the potential energy must decrease in order to satisfy the virial theorem (3); for relatively large kinetic energy, Q/2 < T < Q, evolution will result in a net dilution of the system. When the virial theorem (3) is obeyed in the initial state, only a small readjustment of mass is necessary to establish equilibrium. These considerations are verified by dynamical calculations as shown in Figure 3. Galaxies are considerably more concentrated than gaseous systems, as revealed

Q2Q2Ok ' 6 ASTRONOMY: P. M. CAMPBELL PROC. N. A. S. 1. I 0.8 0 z 4 ~~~~~~INITIAL STATE FINAL STATES 0 01 02Si / =O.825 Ti =0.1 1 ~~~~~~~~~~~~x Ti z 0.4125 a Ti =0.8 0 ~~~~~~~23 RADIUS, r FIG. 3.Final steady states corresponding to isothermal initial states with different amounts of kinetic energy, Ti. The energies are scaled to the magnitude of the potential energy of a sphere of unit radius (distance scale of initial state) with p = const. 1.0 0.8 o ' * P., EMPIRICAL MASS DISTRIBUTION cr C6 2 * ISOTHERMAL INITIALLY I.4 _ 7yi4/3 POLYTROPE INITIALLY n* y3/2 POLYTROPE INITIALLY c02 ' 4 0.6 s8 1.0 RADIUS, r FIG. 4.Comparison of final states corresponding to the initial states of Fig. 2 with the cumulative mass distribution in spherical galaxies. The initial kinetic and potential energies are: isothermal (T = 0.005, Q = 0.825); y = 1.5 polytrope (T = 0.06, R = 1.00); oy = 1.33 polytrope (T = 0.096, Q = 1.51). The energies are scaled to the magnitude of the potential energy of a sphere of unit radius (distance scale of initial state) with p = const. in Figure 2. Therefore, it follows that stars formed in the initial gaseous state had relatively small random motions in order for the dynamical evolution to produce the necessary concentration of the initial mass distribution. Interpreted dynamically, the effect of small initial velocities is to produce nearly radial orbits, which result in congestion of the central regions. (3) Comparison with observation: The closest fit that could be obtained for each of the initial mass distributions of Figure 2 is illustrated in Figure 4 on the same scale as the observed final state. Although the polytrope for My = 4/3 gives the best overall fit, the truncated isothermal case gives a better fit in the inner regions, where the empirical luminosity relation is well established. In the outer regions (r > 5re), where de Vaucouleurs' law has not been verified by accurate observations, the maximum deviation is only a few per cent. (4) Scale of initial states: A further result of the calculations is that the scale of the final state is known in relation to that of the initial state. For the isothermal where r, is the effective radius of case, the radius of the initial configuration is 4 r,,

VOL. 55, 1966 ASTRONOMY: P. M. CAMPBELL 7 the observed distribution; for the y = 4/3 polytrope, the radius of the initial state is approximately 7 re,. Assuming an isothermal initial condition, the protogalaxy of a typical giant elliptical was characterized by the following properties: mass, M 8.1044 gmi; radius, R 4.1022 cm; average density, + = 3.1024 gm/cm3; temperature, T 5.1060K. The temperature necessary to support the initial state in equilibrium is obtained directly from equation (2), the kinetic energy of mass motion being negligible. Hoyle's Theory of Star Formation.Hoyle's theory of star formation predicts anl isothermal initial state and is based rather critically on narrow ranges of temperature.11 In Hoyle's theory, fragmentation is triggered by turbulent compression which develops as a direct result of the isothermal condition. The dynamical calculations indicate that a truncated isothermal distribution does indeed result in a final state consistent with observation (under the assumption of a uniform mass spectrum). However, the temperature obtained for the initial state is an order of magnitude greater than Hoyle's upper limit of 3.105'K. Although the masses and radii of galaxies are not known with precision, the need for some revision of the star formation theory is indicated. * Work performed under the auspices of the U.S. Atomic Energy Commission, AEC contract no. W7405Eng48. 1 Campbell, P. M., these PROCEEDINGS, 48, 1993 (1962). 2 de Vaucouleurs, G., in Handbuch der Physik (Berlin: SpringerVerlag, 1959), vol. 53, p. 320. 3 Miller, R. H., and K. H. Prendergast, Astrophys. J., 136, 713 (1962). 4 Burbidge, E. M., G. R. Burbidge, and R. A. Fish, Astrophys. J., 134, 251 (1961). 6 Page, T., Astron. J., 64, 53 (1959). 6 Fish, R. A., Astrophys. J., 139, 284 (1964). 7 Ralston, A., in Mathematical Methods for Digital Computers (New York: John Wiley and Sons, 1960), p. 95. 8 Romanelli, M. J., in Mathematical Methods for Digital Computers (New York: John Wiley and Sons, 1960), p. 110. 9Milne, W. E., Numerical Calculus (Princeton: University Press, 1949), p. 275. "Eddington, A. S., The Internal Constitution of the Stars (New York: Dover, 1926), p. 79. "Hoyle, F., Astrophys. J., 118, 513 (1953).