Stellar population models of Lick indices with variable element abundance ratios

We provide the whole set of Lick indices from CN1 to TiO2 of Simple Stellar Population models with, for the first time, variable element abundance ratios, [alpha/Fe]=0.0, 0.3, 0.5, [alpha/Ca]=-0.1, 0.0, 0.2, 0.5, and [alpha/N]=-0.5, 0.0. The models cover ages between 1 and 15 Gyr, metallicities between 1/200 and 3.5 solar. Our models are free from the intrinsic alpha/Fe bias that was imposed by the Milky Way template stars up to now, hence they reflect well-defined alpha/Fe ratios at all metallicities. The models are calibrated with Milky Way globular clusters for which metallicities and alpha/Fe ratios are known from independent spectroscopy of individual stars. The metallicities that we derive from the Lick indices Mgb and Fe5270 are in excellent agreement with the metallicity scale by Zinn & West (1984), and we show that the latter provides total metallicity rather than iron abundance. We can reproduce the relatively strong CN-absorption features CN1 and CN2 of galactic globular clusters with models in which nitrogen is enhanced by a factor three. An enhancement of carbon, instead, would lead to serious inconsistencies with the indices Mg1 and C24668. The calcium sensitive index Ca4227 of globular clusters is well matched by our models with [Ca/Fe]= 0.3, including the metal-rich Bulge clusters NGC 6528 and NGC 6553. From our alpha/Fe enhanced models we infer that the index [MgFe] defined by Gonzalez (1993) is quite independent of alpha/Fe, but still slightly decreases with increasing alpha/Fe. We define a slight modification of this index that is completely independent of alpha/Fe and serves best as a tracer of total metallicity. Searching for blue indices that give similar information as Mgb and Fe, we find that CN1 and Fe4383 may be best suited to estimate alpha/Fe ratios of objects at redshifts z~1. (Abridged)Comment: 17 pages, 7 figures, plus 8 pages model tables. Accepted by MNRAS. Models are also available in at ftp://ftp.mpe.mpg.de/people/dthomas/SSP

Spontaneous Breaking of Classical PT Symmetry

The classical trajectories of the family of complex PT-symmetric Hamiltonians $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) form closed orbits. All such complex orbits that have been studied in the past are PT symmetric (left-right symmetric). The periods of these orbits exhibit an unusual dependence on the parameter $\epsilon$. There are regions in $\epsilon$ of smooth behavior interspersed with regions of rapid variation. It is demonstrated that the onset of rapid variation is associated with strange new kinds of classical trajectories that have never been seen previously. These rare kinds of trajectories are not PT symmetric and occur only for special rational values of $\epsilon$.Comment: 16 pages, 17 figures, accepted for publication in the Journal of Mathematical Physic

New Clues on the Calcium Underabundance in Early-Type Galaxies

We use our new stellar population models, which include effects from variable element abundance ratios, to model the Ca4227 absorption line indices of early-type galaxies (Trager et al.), and to derive calcium element abundances. We find that calcium, although being an alpha-element, is depressed with respect to the other alpha-elements by up to a factor 2. This confirms quantitatively earlier speculations that early-type galaxies are calcium underabundant. We find a clear correlation between alpha/Ca ratio and central velocity dispersion, which implies that more massive galaxies are more calcium underabundant. Interestingly this correlation extends down to the dwarf spheroidal galaxies of the Local Group for which alpha/Ca ratios have been measured from high-resolution spectroscopy of individual stars (Shetrone et al.). The increase of the calcium underabundance with galaxy mass balances the higher total metallicities of more massive galaxies, so that calcium abundance in early-type galaxies is fairly constant and in particular does not increase with increasing galaxy mass. This result may be the key to understand why the CaII triplet absorption of early-type galaxies at 8600 A is constant to within 5 per cent over a large range of velocity dispersions (Saglia et al.; Cenarro et al.). The origin of the calcium underabundance in early-type galaxies remains yet to be understood. We argue that formation timescales are disfavoured to produce calcium underabundance, and that the option of metallicity dependent supernova yields may be the most promising track to follow.Comment: 5 pages, 2 figures. Accepted by MNRA

Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: Complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im(E) -> 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as ReE increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure