Why haven't loose globular clusters collapsed yet?

We report on the discovery of a surprising observed correlation between the slope of the low-mass stellar global mass function (GMF) of globular clusters (GCs) and their central concentration parameter c=log(r_t/r_c), i.e. the logarithmic ratio of tidal and core radii. This result is based on the analysis of a sample of twenty Galactic GCs with solid GMF measurements from deep HST or VLT data. All the high-concentration clusters in the sample have a steep GMF, most likely reflecting their initial mass function. Conversely, low-concentration clusters tend to have a flatter GMF implying that they have lost many stars via evaporation or tidal stripping. No GCs are found with a flat GMF and high central concentration. This finding appears counter-intuitive, since the same two-body relaxation mechanism that causes stars to evaporate and the cluster to eventually dissolve should also lead to higher central density and possibly core-collapse. Therefore, more concentrated clusters should have lost proportionately more stars and have a shallower GMF than low concentration clusters, contrary to what is observed. It is possible that severely depleted GCs have also undergone core collapse and have already recovered a normal radial density profile. It is, however, more likely that GCs with a flat GMF have a much denser and smaller core than suggested by their surface brightness profile and may well be undergoing collapse at present. In either case, we may have so far seriously underestimated the number of post core-collapse clusters and many may be lurking in the Milky Way.Comment: Four pages, one figure, accepted for publication in ApJ Letter

Greedy kernel methods for accelerating implicit integrators for parametric ODEs

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features

WIYN Open Cluster Study 1: Deep Photometry of NGC 188

We have employed precise V and I photometry of NGC 188 at WIYN to explore the cluster luminosity function (LF) and study the cluster white dwarfs (WDs). Our photometry is offset by V = 0.052 (fainter) from Sandage (1962) and Eggen & Sandage (1969). All published photometry for the past three decades have been based on these two calibrations, which are in error by 0.05 +- 0.01. We employ the Pinsonneault etal (1998) fiducial main sequence to derive a cluster distance modulus of 11.43 +- 0.08. We report observations that are >= 50% complete to V = 24.6 and find that the cluster central-field LF peaks at M_I ~ 3 to 4. This is unlike the solar neighborhood LF and unlike the LFs of dynamically unevolved portions of open and globular clusters, which rise continuously until M_I ~ 9.5. Although we find that >= 50% of the unresolved cluster objects are multiple systems, their presence cannot account for the shape of the NGC 188 LF. For theoretical reasons (Terlevich 1987; Vesperini & Heggie 1997) having to do with the survivability of NGC 188 we believe the cluster is highly dynamically evolved and that the missing low luminosity stars are either in the cluster outskirts or have left the cluster altogether. We identify nine candidate WDs, of which we expect three to six are bona fide cluster WDs. The luminosities of the faintest likely WD indicates an age (Bergeron, Wesemael, & Beauchamp 1995) of 1.14 +- 0.09 Gyrs. This is a lower limit to the cluster age and observations probing to V = 27 or 28 will be necessary to find the faintest cluster WDs and independently determine the cluster age. While our age limit is not surprising for this ~6 Gyr old cluster, our result demonstrates the value of the WD age technique with its very low internal errors. (abridged)Comment: 26 pages, uuencoded gunzip'ed latex + 16 postscrip figures, to be published in A

Contribution of White Dwarfs to Cluster Masses

I present a literature search through 31 July 1997 of white dwarfs (WDs) in open and globular clusters. There are 36 single WDs and 5 WDs in binaries known among 13 open clusters, and 340 single WDs and 11 WDs in binaries known among 11 globular clusters. From these data I have calculated WD mass fractions for four open clusters (the Pleiades, NGC 2168, NGC 3532, and the Hyades) and one globular cluster (NGC 6121). I develop a simple model of cluster evolution that incorporates stellar evolution but not dynamical evolution to interpret the WD mass fractions. I augment the results of my simple model with N-body simulations incorporating stellar evolution (Terlevich 1987; de la Feunte Marcos 1996; Vesperini & Heggie 1997). I find that even though these clusters undergo moderate to strong kinematical evolution the WD mass fraction is relatively insensitive to kinematical evolution. By comparing the cluster mass functions to that of the Galactic disk, and incorporating plausibility arguments for the mass function of the Galactic halo, I estimate the WD mass fraction in these two populations. I assume the Galactic disk is ~10 Gyrs old (Winget et al. 1987; Liebert, Dahn, & Monet 1988; Oswalt et al. 1996) and that the Galactic halo is ~12 Gyrs old (Reid 1997b; Gratton et al. 1997; Chaboyer et al. 1998), although the WD mass fraction is insensitive to age in this range. I find that the Galactic halo should contain 8 to 9% (alpha = -2.35) or perhaps as much as 15 to 17% (alpha = -2.0) of its stellar mass in the form of WDs. The Galactic disk WD mass fraction should be 6 to 7% (alpha = -2.35), consistent with the empirical estimates of 3 to 7% (Liebert, Dahn, & Monet 1988; Oswalt et al. 1996). (abridged)Comment: 20 pages, uuencoded gunzip'ed latex + 3 postscrip figures, to be published in AJ, April, 199

The Evolution of Globular Clusters in the Galaxy

We investigate the evolution of globular clusters using N-body calculations and anisotropic Fokker-Planck (FP) calculations. The models include a mass spectrum, mass loss due to stellar evolution, and the tidal field of the parent galaxy. Recent N-body calculations have revealed a serious discrepancy between the results of N-body calculations and isotropic FP calculations. The main reason for the discrepancy is an oversimplified treatment of the tidal field employed in the isotropic FP models. In this paper we perform a series of calculations with anisotropic FP models with a better treatment of the tidal boundary and compare these with N-body calculations. The new tidal boundary condition in our FP model includes one free parameter. We find that a single value of this parameter gives satisfactory agreement between the N-body and FP models over a wide range of initial conditions. Using the improved FP model, we carry out an extensive survey of the evolution of globular clusters over a wide range of initial conditions varying the slope of the mass function, the central concentration, and the relaxation time. The evolution of clusters is followed up to the moment of core collapse or the disruption of the clusters in the tidal field of the parent galaxy. In general, our model clusters, calculated with the anisotropic FP model with the improved treatment for the tidal boundary, live longer than isotropic models. The difference in the lifetime between the isotropic and anisotropic models is particularly large when the effect of mass loss via stellar evolution is rather significant. On the other hand the difference is small for relaxation- dominated clusters which initially have steep mass functions and high central concentrations.Comment: 36 pages, 11 figures, LaTeX; added figures and tables; accepted by Ap

Cataclysmic Variables and a New Class of Faint UV Stars in the Globular Cluster NGC 6397

We present evidence that the globular cluster NGC 6397 contains two distinct classes of centrally-concentrated UV-bright stars. Color-magnitude diagrams constructed from U, B, V, and I data obtained with the HST/WFPC2 reveal seven UV-bright stars fainter than the main-sequence turnoff, three of which had previously been identified as cataclysmic variables (CVs). Lightcurves of these stars show the characteristic ``flicker'' of CVs, as well as longer-term variability. A fourth star is identified as a CV candidate on the basis of its variability and UV excess. Three additional UV-bright stars show no photometric variability and have broad-band colors characteristic of B stars. These non-flickering UV stars are too faint to be extended horizontal branch stars. We suggest that they could be low-mass helium white dwarfs, formed when the evolution of a red giant is interrupted, due either to Roche-lobe overflow onto a binary companion, or to envelope ejection following a common-envelope phase in a tidal-capture binary. Alternatively, they could be very-low-mass core-He-burning stars. Both the CVs and the new class of faint UV stars are strongly concentrated toward the cluster center, to the extent that mass segregation from 2-body relaxation alone may be unable to explain their distribution.Comment: 11 pages plus 3 eps figures; LaTeX using aaspp4.sty; to appear in The Astrophysical Journal Letter

Preface

This is the preface of the special volume of Discrete Mathematics dedicated to Combinatorics 2008 we curated as guest editors

Linking the proximal tibiofibular joint to hominid locomotion: A morphometric study of extant species

Objectives: We perform a comparative assessment of shape variation of the proximal fibula in extant humans and great apes, intending to investigate the possible link between proximal fibular shape and locomotor patterns.Methods: Our sample includes 94 fibulae of 37 Homo sapiens, 15 Gorilla, 17 Pongo, and 25 Pan. Fibular morphology was investigated through three-dimensional (semi)landmark-based geometric morphometric methods.Results: We found unique features of the human fibular head compared to that of great apes (i.e., oblique articular surface, the presence of the styloid process, specific morphology of muscle attachment sites), supporting the functional role of this bone in relation to human obligate bipedalism. Great apes also showed distinctive traits in their proximal fibula morphology, in agreement with differences in locomotor behavior.Conclusion: The morphology of the proximal fibula in extant humans and great apes is indicative of locomotor behavior, offering the potential for the comparative analysis of fossil hominin remains