arXiv subjects
Publications and source records attributed to M. van den Berg.
We consider the torsion function for the Dirichlet Laplacian $-Δ$, and for the Schrödinger operator $- Δ+ V$ on an open set $Ω\subset \R^m$ of finite Lebesgue measure $0<|Ω|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.
The dynamical production of low-mass X-ray binaries and brighter cataclysmic variables (CVs) in dense globular clusters is well-established. We investigate how the X-ray emissivity of fainter X-ray binaries (principally CVs and coronally active binaries) varies between different environments. We compile calculations (largely from the literature) of the X-ray emissivity of old stellar populations, including open and globular clusters and several galaxies. We investigate three literature claims of unusual X-ray sources in low-density stellar populations. We show that a suggested quiescent neutron star in the open cluster NGC 6819 is a foreground M dwarf. We show that the suggested diffuse X-ray emission from an old nova shell in the globular cluster NGC 6366 is actually a background galaxy cluster. And we show that a suggested population of quiescent X-ray binaries in the Sculptor Dwarf Galaxy is mostly (perhaps entirely) background galaxies. We find that above densities of $10^4$ M$_{\odot}$/pc$^3$, the X-ray emissivity of globular clusters increases, due to dynamical production of X-ray emitting systems. Below this density, globular clusters have lower X-ray emissivity than the other populations, and we do not see a strong dependence of X-ray emissivity due to density effects. We find significant correlations between X-ray emissivity and binary fraction, metallicity, and density. Sampling these fits via bootstrap techniques gives less significant correlations, but confirms the effect of metallicity on low-density populations, and that of density on the full globular cluster sample.
Let $Ω$ be an open convex set in ${\mathbb R}^m$ with finite width, and let $v_Ω$ be the torsion function for $Ω$, i.e. the solution of $-Δv=1, v\in H_0^1(Ω)$. An upper bound is obtained for the product of $\Vert v_Ω\Vert_{L^{\infty}(Ω)}λ(Ω)$, where $λ(Ω)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(Ω)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_Ω\Vert_{L^{1}(Ω)}λ(Ω)\ge \frac{π^2}{24}$, and that this bound is sharp.
We report mid-UV (MUV) observations taken with HST/WFC3, Swift/UVOT and GALEX/NUV of the transitional millisecond pulsars XSS J12270-4859 and PSR J1023+0038 during their radio pulsar states. Both systems were detected in our images and showed MUV variability. At similar orbital phases, the MUV luminosities of both pulsars are comparable. This suggests that the emission processes involved in both objects are similar. We estimated limits on the mass ratio, companion's temperature, inclination, and distance to XSS J12270-4859 by using a Markov Chain Monte Carlo algorithm to fit published folded optical light curves. Using the resulting parameters, we modeled MUV light curves in our HST filters. The resulting models failed to fit our MUV observations. Fixing the mass ratio of XSS J12270-4859 to the value reported in other studies, we obtained a distance of ~3.2 kpc. This is larger than the one derived from radio measurements (~1.4 kpc). Assuming a uniform prior for the mass ratio, the distance is similar to that from radio measurements. However, it requires an undermassive companion (~0.01 M_sun). We conclude that a direct heating model alone can not fully explain the observations in optical and MUV. Therefore, an additional radiation source is needed. The source could be an intrabinary shock which contributes to the MUV flux and likely to the optical one as well. During the radio pulsar state, the MUV orbital variations of PSR J1023+0038 detected with GALEX, suggests the presence of an asymmetric intrabinary shock.
To confirm the nature of the donor star in the ultra-compact X-ray binary candidate 47 Tuc X9, we obtained optical spectra (3,000$-$10,000 Å) with the Hubble Space Telescope / Space Telescope Imaging Spectrograph. We find no strong emission or absorption features in the spectrum of X9. In particular, we place $3σ$ upper limits on the H$α$ and HeII $λ4686$ emission line equivalent widths $-$EW$_{\mathrm{Hα}} \lesssim 14$ Å and $-$EW$_{\mathrm{HeII}} \lesssim 9$ Å, respectively. This is much lower than seen for typical X-ray binaries at a similar X-ray luminosity (which, for $L_{\mathrm{2-10 keV}} \approx 10^{33}-10^{34}$ erg s$^{-1}$ is typically $-$EW$_{\mathrm{Hα}} \sim 50$ Å). This supports our previous suggestion (by Bahramian et al.) of an H-poor donor in X9. We perform timing analysis on archival far-ultraviolet, $V$ and $I$-band data to search for periodicities. In the optical bands we recover the seven-day superorbital period initially discovered in X-rays, but we do not recover the orbital period. In the far-ultraviolet we find evidence for a 27.2 min period (shorter than the 28.2 min period seen in X-rays). We find that either a neutron star or black hole could explain the observed properties of X9. We also perform binary evolution calculations, showing that the formation of an initial black hole / He-star binary early in the life of a globular cluster could evolve into a present-day system such as X9 (should the compact object in this system indeed be a black hole) via mass-transfer driven by gravitational wave radiation.
We present 22 new (+3 confirmed) cataclysmic variables (CVs) in the non core-collapsed globular cluster 47 Tucanae (47 Tuc). The total number of CVs in the cluster is now 43, the largest sample in any globular cluster so far. For the identifications we used near-ultraviolet (NUV) and optical images from the Hubble Space Telescope, in combination with X-ray results from the Chandra X-ray Observatory. This allowed us to build the deepest NUV CV luminosity function of the cluster to date. We found that the CVs in 47 Tuc are more concentrated towards the cluster center than the main sequence turnoff stars. We compared our results to the CV populations of the core-collapsed globular clusters NGC 6397 and NGC 6752. We found that 47 Tuc has fewer bright CVs per unit mass than those two other clusters. That suggests that dynamical interactions in core-collapsed clusters play a major role creating new CVs. In 47 Tuc, the CV population is probably dominated by primordial and old dynamically formed systems. We estimated that the CVs in 47 Tuc have total masses of approx. 1.4 M_sun. We also found that the X-ray luminosity function of the CVs in the three clusters is bimodal. Additionally, we discuss a possible double degenerate system and an intriguing/unclassified object. Finally, we present four systems that could be millisecond pulsar companions given their X-ray and NUV/optical colors. For one of them we present very strong evidence for being an ablated companion. The other three could be CO- or He-WDs.
We obtain bounds for the expected loss of torsional rigidity of a cylinder $Ω_L=(-L/2,L/2) \times Ω\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $Ω_L,$ and runs until the first time it exits $Ω_L$. These bounds are expressed in terms of the geometry of the cross-section $Ω\subset \R^2$. It is shown that if $Ω$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in $\R^3$ with radius $1,$ and runs until the first time it exits this ball.
Multi-decade observing campaigns of the globular clusters 47 Tucanae and M15 have led to an outstanding number of discoveries. Here, we report on the latest results of the long-term observations of the pulsars in these two clusters. For most of the pulsars in 47 Tucanae we have measured, among other things, their higher-order spin period derivatives, which have in turn provided stringent constraints on the physical parameters of the cluster, such as its distance and gravitational potential. For M15, we have studied the relativistic spin precession effect in PSR B2127+11C. We have used full-Stokes observations to model the precession effect, and to constrain the system geometry. We find that the visible beam of the pulsar is swiftly moving away from our line of sight and may very soon become undetectable. On the other hand, we expect to see the opposite emission beam sometime between 2041 and 2053.
Let $Ω$ be an open set in Euclidean space with finite Lebesgue measure $|Ω|$. We obtain some properties of the set function $F:Ω\mapsto \R^+$ defined by $$ F(Ω)=\frac{T(Ω)λ_1(Ω)}{|Ω|} ,$$ where $T(Ω)$ and $λ_1(Ω)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound $F(Ω)\le 1,$ and show that $$F(Ω)\le 1- ν_m T(Ω)|Ω|^{-1-\frac2m},$$ where $ν_m$ depends only on $m$. For any $m=2,3,\dots$ and $ε\in (0,1)$ we construct an open set $Ω_ε\subset \R^m$ such that $F(Ω_ε)\ge 1-ε$.
For the past couple of decades, the Parkes radio telescope has been regularly observing the millisecond pulsars in 47 Tucanae (47 Tuc). This long-term timing program was designed to address a wide range of scientific issues related to these pulsars and the globular cluster where they are located. In this paper, the first of a series, we address one of these objectives: the characterization of four previously known binary pulsars for which no precise orbital parameters were known, namely 47 Tuc P, V, W and X (pulsars 47 Tuc R and Y are discussed elsewhere). We determined the previously unknown orbital parameters of 47 Tuc V and X and greatly improved those of 47 Tuc P and W. For pulsars W and X we obtained, for the first time, full coherent timing solutions across the whole data span, which allowed a much more detailed characterization of these systems. 47 Tuc W, a well-known tight eclipsing binary pulsar, exhibits a large orbital period variability, as expected for a system of its class. 47 Tuc X turns out to be in a wide, extremely circular, 10.9-day long binary orbit and its position is ~3.8 arcmin away from the cluster center, more than three times the distance of any other pulsar in 47 Tuc. These characteristics make 47 Tuc X a very different object with respect to the other pulsars of the cluster.
We report the detection of steady radio emission from the known X-ray source X9 in the globular cluster 47 Tuc. With a double-peaked C IV emission line in its ultraviolet spectrum providing a clear signature of accretion, this source had been previously classified as a cataclysmic variable. In deep ATCA imaging from 2010 and 2013, we identified a steady radio source at both 5.5 and 9.0 GHz, with a radio spectral index (defined as $S_ν\proptoν^α$) of $α=-0.4\pm0.4$. Our measured flux density of $42\pm4$ microJy/beam at 5.5 GHz implies a radio luminosity ($νL_ν$) of 5.8e27 erg/s, significantly higher than any previous radio detection of an accreting white dwarf. Transitional millisecond pulsars, which have the highest radio-to-X-ray flux ratios among accreting neutron stars (still a factor of a few below accreting black holes at the same X-ray luminosity), show distinctly different patterns of X-ray and radio variability than X9. When combined with archival X-ray measurements, our radio detection places 47 Tuc X9 very close to the radio/X-ray correlation for accreting black holes, and we explore the possibility that this source is instead a quiescent stellar-mass black hole X-ray binary. The nature of the donor star is uncertain; although the luminosity of the optical counterpart is consistent with a low-mass main sequence donor star, the mass transfer rate required to produce the high quiescent X-ray luminosity of 1e33 erg/s suggests the system may instead be ultracompact, with an orbital period of order 25 minutes. This is the fourth quiescent black hole candidate discovered to date in a Galactic globular cluster, and the only one with a confirmed accretion signature from its optical/ultraviolet spectrum.
We report the discovery of the likely white dwarf companions to radio millisecond pulsars 47 Tuc Q and 47 Tuc S in the globular cluster 47 Tucanae. These blue stars were found in near-ultraviolet images from the Hubble Space Telescope for which we derived accurate absolute astrometry, and are located at positions consistent with the radio coordinates to within 0.016 arcsec (0.2sigma). We present near-ultraviolet and optical colours for the previously identified companion to millisecond pulsar 47 Tuc U, and we unambiguously confirm the tentative prior identifications of the optical counterparts to 47 Tuc T and 47 Tuc Y. For the latter, we present its radio-timing solution for the first time. We find that all five near-ultraviolet counterparts have U300-B390 colours that are consistent with He white dwarf cooling models for masses ~0.16-0.3 Msun and cooling ages within ~0.1-6 Gyr. The Ha-R625 colours of 47 Tuc U and 47 Tuc T indicate the presence of a strong Ha absorption line, as expected for white dwarfs with an H envelope.
Results are obtained for two minimization problems: $$I_k(c)=\inf \{λ_k(Ω): Ω \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(Ω)= c \},$$ and $$J_k(c)=\inf\{λ_k(Ω): Ω \textup{quasi-open in}\ \mathbb{R}^m, |Ω|\le 1, \mathcal {P}(Ω)\le c \},$$ where $c>0$, $λ_k(Ω)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(Ω)$, $|Ω|$ denotes the Lebesgue measure of $Ω$, $\mathcal{P}(Ω)$ denotes the perimeter of $Ω$, and where $\mathcal{T}$ is in a suitable class set functions. The latter include for example the perimeter of $Ω$, and the moment of inertia of $Ω$ with respect to its center of mass.
We present the results of a photometric search for variable stars in the core of the Galactic globular cluster M4. The input data are a large and unprecedented set of deep Hubble Space Telescope WFC3 images (large program GO-12911; 120 orbits allocated), primarily aimed at probing binaries with massive companions by detecting their astrometric wobbles. Though these data were not optimised to carry out a time-resolved photometric survey, their exquisite precision, spatial resolution and dynamic range enabled us to firmly detect 38 variable stars, of which 20 were previously unpublished. They include 19 cluster-member eclipsing binaries (confirming the large binary fraction of M4), RR Lyrae, and objects with known X-ray counterparts. We improved and revised the parameters of some among published variables.
We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schrödinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.
Let (M,g) be a compact Riemannian manifold without boundary. Let D be a compact subdomain of M with smooth boundary. We examine the heat content asymptotics for the heat flow from D into M where both the initial temperature and the specific heat are permitted to have controlled singularities on the boundary of D. The operator driving the heat process is assumed to be an operator of Laplace type
The M4 Core Project with HST is designed to exploit the Hubble Space Telescope to investigate the central regions of M4, the Globular Cluster closest to the Sun. In this paper we combine optical and near-infrared photometry to study multiple stellar populations in M4. We detected two sequences of M-dwarfs containing ~38% (MS_I) and ~62% (MS_II) of MS stars below the main-sequence (MS) knee. We compare our observations with those of NGC2808, which is the only other GCs where multiple MSs of very low-mass stars have been studied to date. We calculate synthetic spectra for M-dwarfs, assuming the chemical composition mixture inferred from spectroscopic studies of stellar populations along the red giant branch, and different Helium abundances, and we compare predicted and observed colors. Observations are consistent with two populations, one with primordial abundance and another with enhanced nitrogen and depleted oxygen.
We study the variational problem $$\inf \{λ_k(Ω): Ω \textup{open in}\ \R^m,\ |Ω| < \infty, \ \h(\partial Ω) \le 1 \},$$ where $λ_k(Ω)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(Ω)$, $\h(\partial Ω)$ is the $(m-1)$- dimensional Hausdorff measure of the boundary of $Ω$, and $|Ω|$ is the Lebesgue measure of $Ω$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $Ω_{2,k}$. If $m \ge 2$, and if $Ω_{m,k}$ is a minimiser, then $Ω_{m,k}^*:= \textup{int}(\overline{Ω_{m,k}})$ is also a minimiser, and $\R^m\setminus Ω_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $Ω_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $Ω_{m,k}$ has at most $4$ components. Furthermore $Ω_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.