Self-Similar Graphs

For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every vertex of the previous graph by a copy of $G$ and every edge of the previous graph by a copy of $J$ the new graph is constructed. We call these graphs {\em self-similar} graphs. We are interested in delineating those pairs $(G,J)$ for which the chromatic numbers of the graphs in the sequence are bounded. Here we have some partial results. When $G$ is a complete graph and $J$ is a special matching we show that every graph in the resulting sequence is an {\em expander} graph.Comment: 13 pages, 1 tabl

Curvature and Acoustic Instabilities in Rotating Fluid Disks

The stability of a rotating fluid disk to the formation of spiral arms is studied in the tightwinding approximation in the linear regime. The dispersion relation for spirals that was derived by Bertin et al. is shown to contain a new, acoustic instability beyond the Lindblad resonances that depends only on pressure and rotation. In this regime, pressure and gravity exchange roles as drivers and inhibitors of spiral wave structures. Other instabilities that are enhanced by pressure are also found in the general dispersion relation by including higher order terms in the small parameter 1/kr for wavenumber k and radius r. These instabilities are present even for large values of Toomre's parameter Q. Unstable growth rates are determined in four cases: a self-gravitating disk with a flat rotation curve, a self-gravitating disk with solid body rotation, a non-self-gravitating disk with solid body rotation, and a non-self-gravitating disk with Keplerian rotation. The most important application appears to be as a source of spiral structure, possibly leading to accretion in non-self-gravitating disks, such as some galactic nuclear disks, disks around black holes, and proto-planetary disks. All of these examples have short orbital times so the unstable growth time can be small.Comment: 30 pages, 5 figures, scheduled for ApJ 520, August 1, 199

The Formation of the First Stars II. Radiative Feedback Processes and Implications for the Initial Mass Function

We consider the radiative feedback processes that operate during the formation of the first stars, including the photodissociation of H_2, Ly-alpha radiation pressure, formation and expansion of an HII region, and disk photoevaporation. These processes may inhibit continued accretion once the stellar mass has reached a critical value, and we evaluate this mass separately for each process. Photodissociation of H_2 in the local dark matter minihalo occurs relatively early in the growth of the protostar, but we argue this does not affect subsequent accretion since by this time the depth of the potential is large enough for accretion to be mediated by atomic cooling. However, neighboring starless minihalos can be affected. Ionization creates an HII region in the infalling envelope above and below the accretion disk. Ly-alpha radiation pressure acting at the boundary of the HII region is effective at reversing infall from narrow polar directions when the star reaches ~20-30Msun, but cannot prevent infall from other directions. Expansion of the HII region beyond the gravitational escape radius for ionized gas occurs at masses ~50-100Msun, depending on the accretion rate and angular momentum of the inflow. However, again, accretion from the equatorial regions can continue since the neutral accretion disk has a finite thickness and shields a substantial fraction of the accretion envelope from direct ionizing flux. At higher stellar masses, ~140Msun in the fiducial case, the combination of declining accretion rates and increasing photoevaporation-driven mass loss from the disk act to effectively halt the increase in the protostellar mass. We identify this process as the mechanism that terminates the growth of Population III stars... (abridged)Comment: 31 pages, including 10 figures, accepted to Ap

Flows, Fragmentation, and Star Formation. I. Low-mass Stars in Taurus

The remarkably filamentary spatial distribution of young stars in the Taurus molecular cloud has significant implications for understanding low-mass star formation in relatively quiescent conditions. The large scale and regular spacing of the filaments suggests that small-scale turbulence is of limited importance, which could be consistent with driving on large scales by flows which produced the cloud. The small spatial dispersion of stars from gaseous filaments indicates that the low-mass stars are generally born with small velocity dispersions relative to their natal gas, of order the sound speed or less. The spatial distribution of the stars exhibits a mean separation of about 0.25 pc, comparable to the estimated Jeans length in the densest gaseous filaments, and is consistent with roughly uniform density along the filaments. The efficiency of star formation in filaments is much higher than elsewhere, with an associated higher frequency of protostars and accreting T Tauri stars. The protostellar cores generally are aligned with the filaments, suggesting that they are produced by gravitational fragmentation, resulting in initially quasi-prolate cores. Given the absence of massive stars which could strongly dominate cloud dynamics, Taurus provides important tests of theories of dispersed low-mass star formation and numerical simulations of molecular cloud structure and evolution.Comment: 32 pages, 9 figures: to appear in Ap

Statistics of Core Lifetimes in Numerical Simulations of Turbulent, Magnetically Supercritical Molecular Clouds

We present measurements of the mean dense core lifetimes in numerical simulations of magnetically supercritical, turbulent, isothermal molecular clouds, in order to compare with observational determinations. "Prestellar" lifetimes (given as a function of the mean density within the cores, which in turn is determined by the density threshold n_thr used to define them) are consistent with observationally reported values, ranging from a few to several free-fall times. We also present estimates of the fraction of cores in the "prestellar", "stellar'', and "failed" (those cores that redisperse back into the environment) stages as a function of n_thr. The number ratios are measured indirectly in the simulations due to their resolution limitations. Our approach contains one free parameter, the lifetime of a protostellar object t_yso (Class 0 + Class I stages), which is outside the realm of the simulations. Assuming a value t_yso = 0.46 Myr, we obtain number ratios of starless to stellar cores ranging from 4-5 at n_thr = 1.5 x 10^4 cm^-3 to 1 at n_thr = 1.2 x 10^5 cm^-3, again in good agreement with observational determinations. We also find that the mass in the failed cores is comparable to that in stellar cores at n_thr = 1.5 x 10^4 cm^-3, but becomes negligible at n_thr = 1.2 x 10^5 cm^-3, in agreement with recent observational suggestions that at the latter densities the cores are in general gravitationally dominated. We conclude by noting that the timescale for core contraction and collapse is virtually the same in the subcritical, ambipolar diffusion-mediated model of star formation, in the model of star formation in turbulent supercritical clouds, and in a model intermediate between the previous two, for currently accepted values of the clouds' magnetic criticality.Comment: 25 pages, 8 figures, ApJ accepted. Fig.1 animation is at http://www.astrosmo.unam.mx/~e.vazquez/turbulence/movies/Galvan_etal07/Galvan_etal07.htm

Dilute gas of ultracold two-level atoms inside a cavity; generalized Dicke model

We consider a gas of ultracold two-level atoms confined in a cavity, taking into account for atomic center-of-mass motion and cavity mode variations. We use the generalized Dicke model, and analyze separately the cases of a Gaussian, and a standing wave mode shape. Owing to the interplay between external motional energies of the atoms and internal atomic and field energies, the phase-diagrams exhibit novel features not encountered in the standard Dicke model, such as the existence of first and second order phase transitions between normal and superradiant phases. Due to the quantum description of atomic motion, internal and external atomic degrees of freedom are highly correlated leading to modified normal and superradiant phases.Comment: 10 pages, 7 figure

Turbulent Cooling Flows in Molecular Clouds

We propose that inward, subsonic flows arise from the local dissipation of turbulent motions in molecular clouds. Such "turbulent cooling flows" may account for recent observations of spatially extended inward motions towards dense cores. These pressure-driven flows may arise from various types of turbulence and dissipation mechanisms. For the example of MHD waves and turbulence damped by ion-neutral friction, sustained cooling flow requires that the outer gas be sufficiently turbulent, that the inner gas have marginal field-neutral coupling, and that this coupling decrease sufficiently rapidly with increasing density. These conditions are most likely met at the transition between outer regions ionized primarily by UV photons and inner regions ionized primarily by cosmic rays. If so, turbulent cooling flows can help form dense cores, with speeds faster than expected for ambipolar diffusion. Such motions could reduce the time needed for dense core formation and could precede and enhance the motions of star-forming gravitational infall.Comment: To appear ApJL, Nov.10, 4 ApJ style pages, Postscrip

A Prediction of Brown Dwarfs in Ultracold Molecular Gas

A recent model for the stellar initial mass function (IMF), in which the stellar masses are randomly sampled down to the thermal Jeans mass from hierarchically structured pre-stellar clouds, predicts that regions of ultra-cold CO gas, such as those recently found in nearby galaxies by Allen and collaborators, should make an abundance of Brown Dwarfs with relatively few normal stars. This result comes from the low value of the thermal Jeans mass, considering that the hierarchical cloud model always gives the Salpeter IMF slope above this lower mass limit. The ultracold CO clouds in the inner disk of M31 have T~3K and pressures that are probably 10 times higher than in the solar neighborhood. This gives a mass at the peak of the IMF equal to 0.01 Msun, well below the Brown Dwarf limit of 0.08 Msun. Using a functional approximation to the IMF, the ultracold clouds would have 50% of the star-like mass and 90% of the objects below the Brown Dwarf limit. The brightest of the Brown Dwarfs in M31 should have an apparent, extinction-corrected K-band magnitude of ~21 mag in their pre-main sequence phase.Comment: 13 pages, 2 figures, to be published in Astrophysical Journal, Vol 522, September 10, 199

On the rotating wave approximation in the adiabatic limit

I revisit a longstanding question in quantum optics; When is the rotating wave approximation justified? In terms of the Jaynes-Cummings and Rabi models I demonstrate that the approximation in general breaks down in the adiabatic limit regardless of system parameters. This is explicitly shown by comparing Berry phases of the two models, where it is found that this geometrical phase is strictly zero in the Rabi model contrary to the non-trivial Berry phase of the Jaynes-Cummings model. The source of this surprising result is traced back to different topologies in the two models.Comment: 8 pages, 3 figure