Experiments have been conducted with a two-dimensional model of a galaxy, consisting of 2000 rods of mass. Varying amounts of differential rotation are given to a bar and a cylindrical galaxy. Spiral and loop structures are formed and in one case an S-shaped density wave is seen to grow from a symmetric cylinder. This wave is expected to be a long-lived feature in a real galaxy with a negligible collision rate.
In a previous paper (Hockney 1967) the writer described computer experiments with a galactic model of 2000 rod-like stars. These experiments showed the evolution of a cylindrical galaxy of stars when given different initial amounts of solid-body rotation. When the rotation was insufficient to balance gravitational attraction short-lived spiral and barred spiral structures were observed which showed a similarity to some real galaxies. These structures existed for only a fraction of a galactic rotation and after five rotations a structureless equilibrium distribution, similar to an elliptical galaxy, was reached. In one case a spiral changed to a barred spiral in one-fifth of a rotation. Very similar computer experiments have been conducted by Hohl 1967a, Hohl 1967b) using 500 and 4000 stars. In this communication we describe further experiments concerned with the influence of differential rotation on the evolution of both a bar and cylindrical galaxy.
The tendency of rotating star systems to form filamentary condensations has previously been observed (Hockney 1967) plate 5 at t = 0.5; Hohl 1961a, Hohl 1961b). These condensations can equally well be regarded as embryo galaxies condensing from the protogalactic medium. It is realistic therefore to study the evolution of rotating elongated star systems.
The stars were initially placed at random within an ellipse with a ratio of major to minor axes of ten. The average density was constant although there were small deviations due to the use of only 2000 stars. The initial velocities were chosen to approximately balance gravitational attraction. The 2000 stars within the ellipse were used to simulate the bar of the galaxy. The nucleus of the galaxy was represented by a single fixed mass rod at the center of the system. Three values (0, 0.5, 2.0) were taken for the ratio of the mass of the nucleus to the mass of the bar.
Figure 1 shows the development of the bar in the absence of a nucleus. The dominant feature is the formation of a binary system which lasts for three rotations. After three rotations the two systems merge and form a structureless galaxy.
Figure 2 shows the case of a small nucleus equal to half the total mass of the bar stars. During the first two rotations the bar contracts and is wrapped up into a spiral structure by the differential rotation. Comparisons between real galaxies with an r2 force law between stars and the results from the computer model of rod-like stars with an r-1 force must be viewed with considerable caution. However, one cannot help but note the striking similarity between the computed structure after one rotation and the whirlpool galaxy (M 51). After 2.5 rotations the two spiral arms have merged to form two intersecting elliptical rings. Such structures are not observed frequently in the sky, but there is a similarity between NCC 2523 (Sandage 1961, plate 48) and the computed galaxy at 2.2 rotations. By five rotations all structure has disappeared. Similar results are obtained with a nucleus equal to twice the mass of the bar, only now the differential rotation is greater and the ring pattern appears sooner. Again all structure has disappeared after five rotations.
In these experiments there are two effects tending to wash out spiral structure in the galaxy. The first effect is real and is the differential rotation which, after producing spirals from the bar in the first place, changes the spiral arms into an interlocking ring system by further wrapping up and finally merging the arms. The second effect is due to deficiencies in the computer model. The use of 2000 stars instead of 1011 stars in a real galaxy vastly increases the collision rate and reduces the relaxation time. The effect of these collisions is to increase the velocity dispersion of the star system. The collision time in the model has been estimated to be about sixteen rotations (Hockney 1967). In this estimate the collision time is the time required for the velocity dispersion to become equivalent to a Jeans length equal to the diameter of the galaxy. Clearly no structure can remain after this time unless there is an active gravitational instability to support it. Similarly a spiral arm with a diameter less than one-fifth of the system cannot be maintained longer than one-fifth of a collision time - that is to say longer than about three rotations. The loss of structure toward the end of the computer runs can be attributed to the exaggerated collision rate, although the main spiral features have been turned into ring structures by the differential rotation before this. However, to be confident about how long an imposed spiral structure will last, one must await experiments with 50,000 or 100,000 stars. In this case collisions will not become important until after about 50 rotations. Such experiments are now feasible using the techniques of this model on such computers as the CDC 6600 and IBM 360/91. Also an entirely new type of model has been reported recently by Miller and Prendergast (1967) which can move 120,000 stars.
Experiments with a constant density cylinder in the absence of differential rotation have shown an instability which leads to a fluting or scalloping of the edge of the cylinder (Hockney 1967, Plate 2; Hohl 1967a, Hohl 1967b). Further experiments were therefore carried out to see if these flutes would be drawn out into spiral structures by differential rotation. Two thousand stars were used to represent the disk of the galaxy and differential rotation was introduced, as in the bar experiments, by means of a fixed mass or nucleus at the center. The ratio of mass of the nucleus to mass of the disk was taken as 0.5, 2.0 and 10.0. We find that the flutes are not drawn out into spirals and that the amount of fluting decreases as the amount of differential rotation increases. For a nucleus with ten times the mass of the disk there is no evidence of flutes.
The fluting of a uniform cylinder occurs where the density gradient is highest - namely at the edge. This suggested that a gradual density gradient throughout the galaxy might produce a structure in the interior of the galaxy. To test this possibility we have conducted two experiments with a balanced rotating cylinder with no nucleus. The density, however, varies inversely as the radius, and in the balanced condition this results in differential rotation in which angular frequency varies inversely as the square root of the radius. A combination of density gradient and differential rotation occurs in real galaxies and also in the theory of Lin and Shu (1965) which predicts the presence of density waves.
Figure 3 shows the first case. Multiple spiral features are visible at half a rotation after which an S-shaped density wave appears. The wave is faintly visible at 1.5 rotations and grows for the next three rotations, being clearly visible at 4.5 rotations. Another calculation for a slightly smaller galaxy has been made which again showed the S-shaped density wave. The wave reached a maximum amplitude at five rotations and subsequently decayed due to the exaggerated effect of collisions. The wave was no longer visible after seven rotations. It is to be expected that the wave would have a longer lifetime in a real galaxy with a neglible collision rate.
A 16-mm movie film has been made from these computer experiments together with those described earlier (Hockney 1967). The star distribution at each time-step of the calculation is plotted on 35-mm microfilm using a Stromberg Carlson 4020 microfilm recorder. This film, when reduced photographically to 16 mm and projected at 16 frames per second, gives a dramatic visual display of the experiments. Figures 1-3 are still frames taken at intervals from this movie. An early version of the film was shown to the meeting of the Astronomical Society of the Pacific at Pasadena (June 1967), and the final version to the International Astronomical Union colloquium on the Gravitational Problem of N Bodies at Paris, France (August 1967).
The writer would like to thank Profs. P. A. Sturrock and O. Buneman for encouraging the application of computer models to the galactic problem. The work was sponsored by the Air Force under contract AF 49(638)1321. The calculations were performed at the Stanford Computation Center and the film was made on the Lockheed Stromberg-Carlson 4020 facility.
Hockney, R W, . 1967, Gravitational Experiments with a Cylindrical Galaxy, The Astrophysical Journal, Volume 150, 797.
Hohl, F, 1967a, One- and Two-Dimensional Models To Study the Evolution of Stellas Systems, Symposium on Computer Simulation of Plasma and Many-Body Problems, NASA SP-153, 1967, pp323-336.
Hohl, F, 1967b, Computer Solutions of the Gravitational N-Body Problem, International Astronomical Union Colloquium on the Gravitational Problems of N Bodies, Paris, France, August 1967.
Lin, C C, Shu, F H, 1966, On the spiral structure of disk galaxies, ii. Outline of a theory of density waves, Proc Natl Acad Sci, 55, 229
Miller, R H, Prendergast, K H, 1968, Stellar Dynamics in a Discrete Phase Space, Astrophysical Journal, vol. 151, p.699.
Sandage, A, 1961, The Hubble Atlas of Galaxies, Carnegie Institute of Washington Pub, 618. Washington, D.C.