Scattering, Migration, Re-circularization and Relaxation to Build Out Galaxy Disks with Exponential Profiles

Abstract

Scattering of stars by interstellar clouds or massive clumps increases the stellar velocity dispersion and promotes a radial disk profile that is exponential. Here we show that such scattering reaches a steady-state distribution function of stellar eccentricity, after which eccentricity increases and decreases occur at equal rates. The implication is that clump/cloud scattering recircularizes eccentric stellar orbits, keeping the stellar velocity dispersion in a limited range. This re-circularization regulates disk heating and maintains kinematic coherence, contributing to the longevity of disk structures. The eccentricity distribution function and the presence of recircularizing cloud-star interactions are independent of cloud mass but the timescale to reach equilibrium decreases with increasing mass. The calculations are made in the simplest possible disk system to highlight the effects of scattering without contamination from spiral waves, star formation, and other processes. The calculations also reveal a bifurcation in the disk evolutions whereby in a minority of cases temporary asymmetries in the clump spatial distribution drive the disks to an end state of increased velocity dispersion and orbital eccentricity corresponding to early type disks. Overall the models emphasize an important physical process that can make and maintain an exponential stellar disk in all galaxies with a cloudy interstellar medium.

Figures (5)

• Figure 1: Evolution of the fiducial (Normal) model. Each row shows four snapshots at times (in typical disk units described above) shown in the second row. The top row shows x-y (disk plane) views of the star particles and the massive clumps (asterisks). The star particles are color coded and binned by their initial radii, and only half of those used in the model are plotted. Extensive mixing is evident at late times. The second row shows x-z views of the disk, showing the effects of vertical scattering. The third row shows the disk plane again, but with color coding of the eccentricity; blue colors are the lowest eccentricity, while red are the highest (blue for $e<0.3$, cyan for $0.3<e<0.5$, yellow for $0.5<e<0.65$, and red for $e>0.65$.) The fourth row shows two quantities. The red dotted curves give the surface density profiles; refer to the base ten log scale on the far right panel of the plots, and which applies to all of the panels. The evolution from the initial flat profile to near exponential profiles is evident. The dots show the eccentricities of the star particles up to a maximum of 0.95. The eccentricity evolution is also shown by the blue, solid histograms of the mean eccentricity versus radius. The global mean eccentricity is given at the bottom.
• Figure 2: Evolution in a (Anomalous) model that produced a fat disk. The model parameters and evolutionary graph sequence are the same as in Fig. \ref{['fig:f1']}, but with slightly different stochastic initial conditions (see text). The result is similar surface profile and eccentricity distribution evolution, but much more vertical thickening. Many particles with high eccentricities leave the disk. A substantial population with low to moderate eccentricity remains at all radii in the disk, even at late times.
• Figure 3: Sample particle trajectories in dimensionless units. Each panel shows two trajectories beginning from very similar initial conditions. In Panel a) the two trajectories both show episodes of recircularization. The lower, bold, red one is an example of one that after several scattering and recircularization events ends in a nearly circular orbit at close to half its initial radius. In Panel b) the trajectories become moderately eccentric and remain so throughout the run. In Panel c) one trajectory evolves to significant eccentricity, while the second partially recircularizes at a somewhat larger than initial radius. In Panel d) one trajectory becomes highly eccentric, while the second partially recircularizes sporadically at a smaller than initial radius.
• Figure 4: Evolution of various quantities describing the orbit structure in six model discs, all with individual clump masses of 0.010 units. Time is in dimensionless units, except for panel b), multiply by 40 Myr for the typical disk time units above. Panel a) The evolution of mean orbital eccentricity for selected particles in 6 different runs of the simulation (see text for particle-inclusion criterion). Thin black curves show the results of the 6 runs. The thick blue line shows the mean, and the dashed cyan lines show one standard deviation from the mean. The vertical lines show estimates of the rapid increase timescale of the eccentricity (leftmost) and the onset of the steady state of the eccentricity and its mean (rightmost). These lines are replicated in the other panels. We note also that at a representative radius of 4.0 units the circular orbital period is about 25 units or about 220 Myr in the typical disk scaling, and roughly scales with radius. Panel b) The mean specific radial velocity dispersion for all particles in the model runs (with arbitrary colors). This panel contains an additional black curve with points marked by x symbols, showing the averaged 10times model. This model has clump masses reduced by a factor of 10 ($m_{cl} = 0.001$ units), and the number of clumps increased by the same factor. The number of stars is reduced in each 10times model run, so the curve here represents the average of four such runs. The vertical scale corresponds to the example units for disk galaxies in km s$^{-1}$ from Sec. 2.2 above. The velocity dispersion scale for the dwarf galaxies example (not shown) is nearly the same. Panel c) Specific angular momentum evolution for each run averaged over all particles. Panel d) Average relative change as a function of time in the semi-major axes of all particles. This quantity is the average of the ratios of the difference between the semi-major axis of all particles at the time plotted and its semi-major axis near the beginning of the run, at $t = 0.1$ units, divided by the latter quantity. This is equivalent to the mean relative gyro-center evolution of the particles.
• Figure 5: Panel a) shows end-of-run eccentricities for models with a range of clump masses in dimensionless units, but constant clump number. The solid line shows a least squares fit to all of the models except that represented by the upper leftmost cross (+) and the averaged 10times model (x). Panel b) shows estimates of the time to form an exponential profile (in dimensionless units multiply by 40 Myr for typical disk units above) in runs with a range of clump masses. Crosses show estimates estimated visually from run outputs, while the arrow shows the end-of-run time for a runs that did not completely establish an exponential profile. The criterion for establishing an exponential was a straight line running over a factor of about 3 in radius in a log(surface density) versus radius plot, and a clear change in surface density within that radial range. The point in the lower left corresponds to that in the upper left of panel a) and the averaged 10times model is shown by the x symbol.