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Walter Dehnen

Publications and source records attributed to Walter Dehnen.

At least 73 records · Page 4Linked to original sources
arXiv2005

Weighing the young stellar discs around Sgr A*

It is believed that young massive stars orbiting Sgr A* in two stellar discs on scales of 0.1-0.2 parsecs were formed either farther out in the Galaxy and then quickly migrated inward, or in situ in a massive self-gravitating disc. Comparing N-body evolution of stellar orbits with observational constraints, we set upper limits on the masses of the two stellar systems. These masses turn out to be few times lower than the expected total stellar mass estimated from the observed young high-mass stellar population and the standard galactic IMF. If these stars were formed in situ, in a massive self-gravitating disc, our results suggest that the formation of low-mass stars was suppressed by a factor of at least a few, requiring a top-heavy initial mass function (IMF) for stars formed near sgr A*.

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arXiv2005

Halo evolution in the presence of a disc bar

Angular momentum transfer from a rotating stellar bar has been proposed by Weinberg & Katz (2002) as a mechanism to destroy dark-matter cusps in a few rotation periods. The N-body simulations performed by these authors in support of their claim employed spherical harmonics for the force computation and were, as shown by Sellwood (2003), very sensitive to inclusion of asymmetric terms (odd l,m). In order to disentangle possible numerical artifacts due to the usage of spherical harmonics from genuine stellar dynamical effects, we performed similar experiments using a tree code and find that significant cusp destruction requires substantially more angular momentum than is realistically available. However, we find that the simplified model (a N-body halo torqued by a rotating bar pinned to the origin) undergoes an instability in which the cusp moves away from the origin. In presence of this off-centring, spherical density profiles centred on the origin display an apparent cusp-removal. We strongly suspect that it is this effect which Weinberg & Katz observed. When suppressing the artificial instability, cusp removal is very slow and requires much more angular momentum to be transferred to the halo than a realistic stellar bar possibly possesses.

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arXiv2005

Dynamical insight into dark-matter haloes

We investigate, using the spherical Jeans equation, self-gravitating dynamical equilibria satisfying a relation rho/sigma_r^3 propto r^-alpha, which holds for simulated dark-matter haloes over their whole resolved radial range. Considering first the case of velocity isotropy, we find that this problem has only one solution with realistic density profile, which occurs only for a critical value of alpha_crit = 35/18 ~= 1.94, which is consistent with the empirical value of 1.9+/-0.05. We extend our analysis in two ways: first we introduce a parameter epsilon to allow for a more general relation rho/σ_r^epsilon propto r^-alpha; second we consider velocity anisotropy. If we assume beta(r) := 1- sigma_theta^2 / sigma_r^2 to be linearly related to the logarithmic density slope gamma(r) := -dln(rho)/dln(r), which is in agreement with simulations, the problem remains analytically tractable and is equivalent to the simpler isotropic case: there exists only one physical solution, which occurs at a critical alpha value. Remarkably, this value of alpha and the density and velocity-dispersion profiles depend only on epsilon and the value beta_0 := beta(r=0), but not on the slope of the linear beta-gamma relation. For epsilon=3, alpha_crit = 35/18 - 2beta_0/9 and the resulting density profile is fully analytic (as are the velocity dispersion and circular speed) with an inner cusp rho propto r^{-(7+10beta_0)/9} and a very smooth transition to a steeper outer power-law asymptote. These models are in excellent agreement with the density, velocity-dispersion and anisotropy profiles of simulated dark-matter haloes over their full resolved radial range. If epsilon=3 is a universal constant, some scatter in beta_0 ~= 0 may account for some diversity in the density profiles. (ABRIDGED)

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arXiv2005

Phase-space mixing and the merging of cusps

Collisionless stellar systems are driven towards equilibrium by mixing of phase-space elements. I show that the excess-mass function D(f)= int_{F(x,v)>f} (F(x,v)-f) d^3x d^3v (with F(x,v) the coarse-grained distribution function) always decreases on mixing. D(f) gives the excess mass from values of F(x,v))>f. This novel form of the mixing theorem extends the maximum phase-space density argument to all values of f. The excess-mass function can be computed from N-body simulations and is additive: the excess mass of a combination of non-overlapping systems is the sum of their individual D(f). I propose a novel interpretation for the coarse-grained distribution function, which avoids conceptual problems with the mixing theorem. As an example application, I show that for self-gravitating cusps (rho propto r^{-gamma} as r->0) the excess mass D propto f^{-2(3-gamma)/(6-gamma)} as f->oo, i.e. steeper cusps are less mixed than shallower ones, independent of the shape of surfaces of constant density or details of the distribution function (e.g. anisotropy). This property, together with the additivity of D(f) and the mixing theorem, implies that a merger remnant cannot have a cusp steeper than the steepest of its progenitors. Furthermore, I argue that the remnant's cusp should not be shallower either, implying that the steepest cusp always survives.

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arXiv2004

Hierarchical Merging, Ultraluminous and Hyperluminous X-ray Sources

Various arguments strongly suggest that the population of ultraluminous X-ray sources (ULXs: apparent X-ray luminosity > Eddington limit for 10 Msun ~10^39 erg/s) in nearby galaxies are mostly stellar-mass X-ray binaries in unusual evolutionary stages. However there are indications that the very brightest systems may be difficult to explain this way. Accordingly we consider the class of hyperluminous X-ray sources (HLXs) (i.e. those with apparent bolometric luminosities >~10^41 erg/s). Because this class is small (currently only the M82 object is a secure member) we do not need to invoke a new formation mechanism for its black holes. We explore instead the idea that HLXs may be the nuclei of satellite galaxies captured during hierarchical merging. The observed correlation between AGN and tidal interactions implies that HLX activity would switch on during passage through the host galaxy, close to pericentre. This suggests that HLXs should appear near the host galaxy, be associated with star formation, and thus possibly with ULXs.

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arXiv2003

Kinematic response of the outer stellar disk to a central bar

We study, using direct orbit integrations, the kinematic response of the outer stellar disk to the presence of a central bar, as in the Milky-Way. We find that the bar's outer Lindblad resonance (OLR) causes significant perturbations of the velocity moments. With increasing velocity dispersion, the radius of these perturbations is shifted outwards, beyond the nominal position of the OLR, but also the disk becomes less responsive. If we follow Dehnen (2000) in assuming that the OLR occurs just inside the Solar circle and that the Sun lags the bar major axis by ~20 degrees, we find (1) no significant radial motion of the local standard of rest (LSR), (2) a vertex deviation of \~10 degrees and (3) a lower ratio sigma_2/sigma_1 of the principal components of the velocity- dispersion tensor than for an unperturbed disk. All of these are actually consistent with the observations of the Solar-neighbourhood kinematics. Thus it seems that at least the lowest-order deviations of the local-disk kinematics from simple expectations based on axisymmetric equilibrium can be attributed entirely to the influence of the Galactic bar.

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arXiv2003

The Oort Constants Measured from Proper Motions

The Oort constants describe the local variations of the stellar streaming field. Classically, they are determined from stellar proper motions. We discuss problems arising in this procedure. A large, hitherto overlooked, source of systematic error arises from longitudinal variations of the mean stellar parallax. Together with the solar reflex motion, these variations create contributions to the longitudinal proper motions mu_l*(l) that are indistinguishable from the Oort Constants at the 20% level. Fortunately, we can correct for this "mode mixing" using the latitudinal proper motions mu_b(l). We use ~10^6 stars from the ACT/Tycho-2 catalogs with proper motion errors of ~ 3 mas/yr. We find significant deviations from expectations based on a smooth axisymmetric equilibrium disk, in particular non-zero C for old red giant stars. We also find variations of the Oort constants with the asymmetric drift of the sub-sample. These correlations are different in nature than those expected for an axisymmetric Galaxy. The most reliable tracers for the ``true'' Oort constants are red giants, which are old enough to be in equilibrium and distant enough to be unaffected by possible local anomalies. For these stars we find, A ~ 16, B ~- 17, A - B ~ 33, and C ~- 10 km/s/kpc with internal errors of about 1-2 and external error of perhaps the same order. These values are consistent with our knowledge of the Milky Way (flat rotation curve and Omega=A - B ~ 28 +/- 2). Based on observations made with the ESA Hipparcos astrometry satellite. (Abridged)

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arXiv2002

Kinematic study of the disrupting globular cluster Palomar 5 using VLT spectra

Wide-field photometric data from the Sloan Digital Sky Survey have recently revealed that the Galactic globular cluster Palomar 5 is in the process of being tidally disrupted (Odenkirchen et al. 2001). Here we investigate the kinematics of this sparse remote star cluster using high resolution spectra from the Very Large Telescope (VLT). Twenty candidate cluster giants located within 6 arcmin of the cluster center have been observed with the UV-Visual Echelle Spectrograph (UVES) on VLT-UT2. The spectra provide radial velocities with a typical accuracy of 0.15 km/s. We find that the sample contains 17 certain cluster members with very coherent kinematics, two unrelated field dwarfs, and one giant with a deviant velocity, which is most likely a cluster binary showing fast orbital motion. From the confirmed members we determine the heliocentric velocity of the cluster as -58.7 +- 0.2 km/s. The total line-of-sight velocity dispersion of the cluster stars is 1.1 +- 0.2 km/s (all members) or 0.9 +- 0.2 km/s (stars on the red giant branch only). This is the lowest velocity dispersion that has so far been measured for a stellar system classified as a globular cluster. The shape of the velocity distribution suggests that there is a significant contribution from orbital motions of binaries and that the dynamical part of the velocity dispersion is therefore still substantially smaller than the total dispersion. ... (abridged)

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arXiv2002

A Hierarchical O(N) Force Calculation Algorithm

A novel code for the approximate computation of long-range forces between N mutually interacting bodies is presented. The code is based on a hierarchical tree of cubic cells and features mutual cell-cell interactions which are calculated via a Cartesian Taylor expansion in a symmetric way, such that total momentum is conserved. The code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort. For N>10^4, the computational costs are found empirically to rise sub-linear with N. For applications in stellar dynamics, this is the first competitive code with complexity O(N), it is faster than the standard tree code by a factor ten or more.

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arXiv2001

New insights on the Draco dwarf spheroidal galaxy from SDSS: a larger radius and no tidal tails

We have investigated the spatial extent and structure of the Draco dwarf spheroidal galaxy using deep wide-field multicolor CCD photometry from the Sloan Digital Sky Survey (SDSS). Our study covers an area of 27 square degrees around the center of the Draco dwarf. We show that the spatial distribution of Draco's red giants, red horizontal branch stars and subgiants down to i=21.7 mag does not provide evidence for the existence of tidally induced tails or a halo of unbound stars. The radial profile can be fit by King models as well as by a generalized exponential. The core radius and the limiting (or tidal) radius along the major axis are 7.7' and 40.1', respectively, making Draco 40% larger than previously measured. Down to our magnitude limit tidal effects can only exist at a level of 10^-3 of the central surface density of Draco or below. The regular structure of Draco found from the new data argues against it being a portion of an unbound tidal stream and lends support to the assumption of dynamical equilibrium. We estimate Draco's total mass to be 2.2 - 3.5 times 10^7 solar masses. We obtain an overall mass-to-light ratio of 146+-42 or 92+-28 solar masses depending on the details of the mass and luminosity estimates. In summary, our results strengthen the case for a strongly dark matter dominated, bound stellar system. (Abstract strongly abridged).

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arXiv2000

Towards optimal softening in 3D N-body codes: I. Minimizing the force error

In N-body simulations of collisionless stellar systems, the forces are softened to reduce the shot noise. Softening modifies gravity at r=|x-y| smaller than softening length epsilon and the softened forces are increasingly biased for ever larger epsilon. There is, thus, some optimum between reducing the fluctuations and introducing a bias. Here, analytical relations are derived for the amplitudes of the bias and the fluctuations in the limit of small epsilon and large N. It is shown that the fluctuations of the force are generated locally, in contrast to the variations of the potential, which originate from noise in the whole system. Based on the asymptotic relations and using numerical experiments, I study the dependence of the resulting force error on N, epsilon, and on the functional form by which Newtonian gravity is replaced. The Plummer softening, where each body is replaced by a Plummer sphere of scale radius epsilon, yields significantly larger force errors than do methods in which the bodies are replaced by density kernels of finite extent. I also give special kernels, which reduce the errors even further. These kernels largely compensate the errors made with too small inter-particle forces at r<epsilon by exceeding Newtonian forces at r epsilon. Additionally, the possibilities of locally adapting epsilon and of using unequal weights for the bodies are investigated. These various techniques allow, without increasing N, to reduce the rms force error by a factor 2 compared to Plummer softening with constant epsilon. The results of this study are directly relevant to N-body simulations using direct summation techniques or the tree code. (abridged)

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arXiv2000

The Age of the Solar Neighbourhood

High-quality Hipparcos data for a complete sample of nearly 12000 main-sequence and subgiant stars, together with Padua isochrones, are used to constrain the star-formation history of the solar neigbourhood and the processes that stochastically accelerate disk stars. The velocity dispersion of a coeval group of stars is found to increase with time from ~8 kms at birth as t^{0.33}. In the fits, the slope of the IMF near 1 Msun proves to be degenerate with the rate at which the star-formation rate declines. If the slope of the IMF is to lie near Salpeter's value, -2.35, the star-formation rate has to be very nearly constant. The age of the solar neighbourhood is found to be 11.2+/-0.75 Gyr with remarkably little sensitivity to variations in the assumed metallicity distribution of old disk stars. This age is only a Gyr younger than the age of the oldest globular clusters when the same isochrones and distance scale are employed. It is compatible with current indications of the redshift of luminous galaxy formation only if there is a large cosmological constant. A younger age is formally excluded because it provides a poor fit to the number density of red stars. Since this density is subject to a significantly uncertain selection function, ages as low as 9 Gyr are plausible even though they lie outside our formal error bars.

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arXiv2000

A Very Fast and Momentum-Conserving Tree Code

The tree code for the approximate evaluation of gravitational forces is extended and substantially accelerated by including mutual cell-cell interactions. These are computed by a Taylor series in Cartesian coordinates and in a completely symmetric fashion, such that Newton's third law is satisfied by construction and hence momentum exactly conserved. The computational effort is further reduced by exploiting the mutual symmetry of the interactions. For typical astrophysical problems with N=10^5 and at the same level of accuracy, the new code is about four times faster than the tree code. For large N, the computational costs are found to scale almost linearly with N, which can also be supported by a theoretical argument, and the advantage over the tree code increases with ever larger N.

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arXiv1999

The Effect of the Outer Lindblad Resonance of the Galactic Bar on the Local Stellar Velocity Distribution

Hydro-dynamical modeling of the inner Galaxy suggest that the radius of the outer Lindblad resonance (OLR) of the Galactic bar lies in the vicinity of the Sun. How does this resonance affect the distribution function in the outer parts of a barred disk, and can we identify any effect of the resonance in the velocity distribution f(v) actually observed in the solar neighborhood? To answer these questions, detailed simulations of f(v) in the outer parts of an exponential stellar disks with nearly flat rotation curves and a rotating central bar have been performed. For a model resembling the old stellar disk, the OLR causes a distinct feature in f(v) over a significant fraction of the outer disk. For positions <2kpc outside the OLR radius and at bar angles of \~10-70 degrees, f(v) inhibits a bi-modality between the low-velocity stars moving like the local standard of rest (LSR) and a secondary mode of stars predominantly moving outward and rotating more slowly than the LSR. Such a bi-modality is indeed present in f(v) inferred from the Hipparcos data for late-type stars in the solar neighborhood. If one interpretes this observed bi-modality as induced by the OLR -- and there are hardly any viable alternatives -- then one is forced to deduce that the OLR radius is slightly smaller than Ro. Moreover, by a quantitative comparison of the observed with the simulated distributions one finds that the pattern speed of the bar is 1.85+/-0.15 times the local circular frequency, where the error is dominated by the uncertainty in bar angle and local circular speed. Also other, less prominent but still significant, features in the observed f(v) resemble properties of the simulated velocity distributions, in particular a ripple caused by orbits trapped in the outer 1:1 resonance.

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arXiv1999

The Pattern Speed of the Galactic Bar

Most late-type stars in the solar neighborhood have velocities similar to the local standard of rest (LSR), but there is a clearly separated secondary component corresponding to a slower rotation and a mean outward motion. Detailed simulations of the response of a stellar disk to a central bar show that such a bi-modality is expected from outer-Lindblad resonant scattering. When constraining the run of the rotation curve by the proper motion of Sgr A* and the terminal gas velocities, the value observed for the rotation velocity separating the two components results in a value of (53+/-3)km/s/kpc for the pattern speed of the bar, only weakly dependent on the precise values for Ro and bar angle phi.

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arXiv1999

Approximating Stellar Orbits: Improving on Epicycle Theory

Already slightly eccentric orbits, such as those occupied by many old stars in the Galactic disk, are not well approximated by Lindblad's epicycle theory. Here, alternative approximations for flat orbits in axisymmetric stellar systems are derived and compared to results from numeric integrations. All of these approximations are more accurate than Lindblad's classical theory. I also present approximate, but canonical, maps from ordinary phase-space coordinates to a set of action-angle variables. Unfortunately, the most accurate orbit approximation leads to non-analytical R(t). However, from this approximation simple and yet very accurate estimates can be derived for the peri- and apo-centers, frequencies, and actions integrals of galactic orbits, even for high eccentricities. Moreover, further approximating this approximation allows for an analytical R(t) and still an accurate approximation to galactic orbits, even with high eccentricities.

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arXiv1999

Simple Distribution Functions for Stellar Disks

Distribution functions (DFs) for dynamically warm thin stellar disks residing in arbitrary axisymmetric potentials are presented which approximately reproduce pre-described surface-density and velocity-dispersion profiles. The functional form of the DFs is obtained by `warming-up' a model made entirely of circular orbits. This can be done in various ways giving different functional forms for the DF. In the best case, the DF reproduces the pre-described profiles to within a few per cent for a typical case reminiscent to the old stellar disk in the Milky Way. This match may be improved to about one per cent or better by a simple iterative method. An algorithm is given to draw phase-space points randomly from the DFs for the purpose of, e.g., N-body simulations. All the relevant computer programs are available from the author.

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