Stellar Distribution Function Research Articles

The capture and escape of stars

The shape of galaxies depends on their orbital populations. These populations change through capture into and escape from resonance. Capture problems fall into distinct cases depending upon the shape of the potential well. To visualise the effective potential well for orbital capture, a diagrammatic approach to the resonant perturbation theory of Born is presented. These diagrams we call equiaction sections. To illustrate their use, we present examples drawn from both galactic and Solar System dynamics. The probability of capture for generic shapes of the potential well is calculated. A number of predictions are made. First, there are barred galaxies that possess two outer rings of gas and stars (type R1R2). We show how to relate changes in the pattern speed and amplitude of the bar to the strength of the two rings. Secondly, under certain conditions, small disturbances can lead to dramatic changes in orbital shape. This can be exploited as a mechanism to pump counter-rotating stars and gas into the nuclei of disk galaxies. Tidal resonant forcing of highly inclined orbits around a central mass causes a substantial increase in the likelihood of collision. Thirdly, the angular momentum of a potential well is changed by the passage of stars across or capture into the well. This can lead to the creation of holes, notches and high velocity tails in the stellar distribution function, whose form we explicitly calculate.

Bloated stars as AGN broad-line clouds: the emission-line profiles

The Bloated Stars Scenario proposes that AGN broad line emission originates in the winds or envelopes of bloated stars (BS). Alexander and Netzer (1994) established that ~ 5e4 BSs with dense, decelerating winds can reproduce the observed emission line spectrum and avoid rapid collisional destruction. Here, we use the observed properties of AGN line profiles to further constrain the model parameters. In the BS model, the origin of the broad profiles is the stellar velocity field in the vicinity of the central black hole. We use a detailed photoionization code and a model of the stellar distribution function to calculate the BS emission line profiles and compare them to a large sample of AGN CIV, CIII] and MgII profiles. We find that the BSs can reproduce the general shape and width of typical AGN profiles as well as the line ratios if (i) The ionizing luminosity to black hole mass ratio is low enough. (ii) The broad line region size is limited by some cutoff mechanism. (iii) The fraction of the BSs in the stellar population falls off roughly as r^-2. (iv) The wind density and/or velocity are correlated with the black hole mass and ionizing luminosity. Under these conditions the strong tidal forces near the black hole play an important role in determining the line emission properties of the BSs. Some discrepancies remain: the calculated BS profiles tend to have weaker wings than the observed ones, and the differences between the profiles of different lines are somewhat smaller than those observed.

The Mass of the Milky Way

view Abstract Citations (241) References (51) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Mass of the Milky Way Kochanek, Christopher S. Abstract We use the Jaffe model as a global mass distribution for the Galaxy and determine the circular velocity υc and the Jaffe radius rj using the satellites of the Galaxy, estimates of the local escape velocity of stars, the constraints imposed by the known rotation curve of the disk, and the Local Group timing model. The models include the systematic uncertainties in the isotropy of the satellite orbits, the form of the stellar distribution function near the escape velocity, and the ellipticity of the M31/Galaxy orbit. If we include the Local Group timing constraint, then Leo I is bound, υc = 220±30km s-1, and rj = 200 kpc (115 kpc ≲ rj ≲ 360 kpc) at 90% confidence. The satellite orbits are nearly isotropic with β = 1 -σ2/θ/σ2r = 0.2 (-0.5 ≲ β ≲ 0.6), and the stellar distribution function near the escape velocity is f(ɛ) ∝ ɛk with kr = 4.4 (1.6 ≲ kr ≲ 8.5), where kr = k + 5/2. While not an accurate measurement of k, it is consistent with models of violent relaxation (k = 3/2). The mass inside 50 kpc is (4.9±1.1) × 1011 Msun. Higher mass models require that M31 is on its second orbit and that the halo is larger than the classical tidal limit of the Galaxy/M31 binary system. Such models must have a significant fraction of the Local Group mass in an extended Local Group halo. Lower mass models require that both M31 and Leo I are unbound, but there is no plausible mechanism to produce observed deviations of M31 and Leo I from their expected velocities in an unbound system. If we do not use the Local Group timing model, the median mass of the Galaxy increases significantly, and the error bars broaden. Using only the satellite, escape velocity, and disk rotation curve constraints, the median mass interior to 50 kpc is 3.9 (5.1), and the 90% confidence interval is 3.2-5.5 (4.0-6.4) without (with) Leo I in units of 1011 Msun. The lower bound without Leo I is 65% of the mass expected for a continuation of a flat rotation curve. Publication: The Astrophysical Journal Pub Date: January 1996 DOI: 10.1086/176724 arXiv: arXiv:astro-ph/9505068 Bibcode: 1996ApJ...457..228K Keywords: CELESTIAL MECHANICS; STELLAR DYNAMICS; GALAXY: FUNDAMENTAL PARAMETERS; GALAXIES: LOCAL GROUP; GALAXY: KINEMATICS AND DYNAMICS; Astrophysics E-Print: 31 pages, uuencoded compressed postscript without figures, complete file including figures available at ftp://cfa0.harvard.edu/outgoing/kochanek/milkyway.ps.Z (0.5 MByte, compressed), submitted to ApJ full text sources arXiv | ADS | data products SIMBAD (26)

Bloated Stars as BLR Clouds: Numeric Results

The ‘Bloated Stars Scenario’ proposes that AGN broad line emission originates in the winds or envelopes of bloated stars (BS) (see e.g. Kazanas 1989 and references therein). Its main advantage over BLR cloud models is the gravitational confinement of the gas and its major difficulty the large estimated number of BSs and the resulting high collisional and evolutionary mass loss rates (see e.g. Begelman & Sikura 1991). Previous work on this model did not include detailed calculations of the line spectrum, modeled solar neighborhood super giants (SG) and used very simplified stellar distribution functions for the nucleus. Here (Alexander & Netzer, 1993) we calculate the emission line ratios by applying a detailed numerical photoionization code (Rees, Netzer & Ferland, 1989) to the wind and by assuming a detailed nucleus model (Murphy, Cohn & Durisen, 1990). Allowing for the yet unknown effects of the AGN's extreme conditions on stars and stellar evolution, we study a wide range of simplified wind structures rather than confine ourselves to normal SGs. Our model consists of a spherically symmetric outflowing wind that emanates from the surface of the BS (R∗ = 1013 cm, M∗ = 0.8M⊙, M = 10−6M⊙/yr) whose size and edge density are determined by various processes: Comptonization by the central continuum source (calculated self consistently for our Lion = 1046 erg/s model continuum by the photoionization code), tidal disruption by the black hole (Mbh = 8 × 107M⊙) and the limit set by the assumption that the wind's mass ≤ 0.2M⊙. This results in a large range of wind sizes, from 1013 to 1016 cm. We find that the line emission spectrum is mainly determined by the conditions at the edge of the wind rather than by its internal structure. Comptonization results in a very high ionization parameter at the edge which produces an excess of unobserved broad high excitation forbidden lines. The finite mass constraint limits the wind's size, increases the edge density and thus improves the results. Studying power-law wind structures (v(R) = v∗(R/R∗)−α where v∗ is the wind's base velocity at the BS's surface), we find that slow, decelerating, mass-constrained flows (v∗ = 50 m/s, α = 0.5) with high gas densities (108 to 1012 cm−3) are as successful as cloud models in reproducing the overall observed line spectrum. The Mg II λ2798 and N V λ1240 lines are however under-produced in our models. The denser the winds, the more efficient they are as BLR clouds. By calculating the Lα emission from the wind we adjust the number of BSs so as to obtain the BLR's observed EW(Lα). We find that only ∼ 5 × 104 BSs with dense winds (v∗ = 50 m/s, α = 0.5) are required in the inner 1/3 pc (∼ 0.005 of the total stellar population). This small fraction approaches that of SGs in the solar neighborhood. The calculated mass loss from such a small number of BSs is consistent with the observational constraints. We find that the required number of BSs, and consequently their mass loss rate, are a very sensitive functions of the wind's density structure (a ∼ 104 factor between the slow v∗ = 50 m/s, α = 0.5 model and the fast v∗ = 50 km/s, α = −2 model). In particular, high mass loss rules out SG-like BSs (v∗ = 10 km/s, α = 0). We conclude that BSs with dense winds can reproduce the BLR line spectrum and be supported by the stellar population without excessive mass loss and collisional destruction rates. The question whether such hitherto unobserved stars actually exist in the BLR remains open.

Inhomogeneous chemical evolution of the Galactic disk

We present analytical models for inhomogeneous chemical evolution (ICE) of systems in which the star formation history resembles a series of bursts, localized in space and/or time, with intermittent periods of remixing. The additional parameter of this model is the metallicity increment of bursting subsystems, but this parameter is constrained by the spread in the age-metallicity relation. We apply this model to the solar annulus in the Galactic disk and show that ICE models yield an improved fit to the observed shape of the stellar abundance distribution function (ADF). The G-dwarf problem can be alleviated with ICE models, but infall of metal poor gas and/or some preenrichment of the disk during the epoch of protogalactic evolution is still required to explain the paucity of low-metallicity dwarfs. ICE models also suggest an explanation of the reduced frequency of metal-rich G-dwarfs relative to the predictions of the simple model. It does not seem likely that chemical evolution of the solar annulus proceeded in a medium that was well-mixed at all times.

Central moments of the velocity distribution from a superposition of two stellar systems

From an analytical viewpoint, the superposition of two stellar velocity distribution functions and the set of central velocity moments up to fourth order are studied. For the Galaxy, any specific type of symmetry or differential movement has not been assumed. General expressions of the total central moments starting from the partial ones, which are associated with both distribution function components, are obtained and, when each component is quadratic in peculiar velocities and of Schwarzschild type, the constraints between the total central moments are studied

The local velocity distribution from a stellar system with two Gaussian components

The superposition of two stellar velocity distribution functions of Gaussian type is applied in order to study the velocity distribution of a stellar sample in the galactic plane. Assuming that the Galaxy is symmetric respect to this plane, the relationships to be fulfilled by the central velocity moments of the total stellar sample are obtained. If the total velocity distribution is consistent with this type of superposition, the partial moments of the subsamples, which are associated with the Gaussian components, can be calculated and, also, the population percentage. Our results show that first component corresponds to young disk population and the second one to intermediate disk population.

Box-shaped galactic bulges. I - Equilibrium models

view Abstract Citations (50) References (19) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Box-shaped Galactic Bulges. I. Equilibrium Models Rowley, Gregg Abstract Simple, self-consistent stellar distribution function models are presented which have a box-shaped appearance and are cylindrically rotating. The distribution function is a function only of the two classical isolating integrals of an axisymmetric system, energy (E) and the component of angular momentum parallel to the axis of symmetry (h_z_). The major difference between these models and previous ones is the mode of truncation of the distribution function. The truncation contour has the same functional form as all of the other contours of the distribution function. This ensures that the mean rotation in the meridional plane of the models is identically cylindrical. It is argued that this mode of truncation is similar to what would be found in a system that has suffered significant dissipation. A second paper in this series presents comparisons of the models to photometric and kinematic observations of a number of box-shaped galactic bulges. Publication: The Astrophysical Journal Pub Date: August 1988 DOI: 10.1086/166542 Bibcode: 1988ApJ...331..124R Keywords: Astronomical Models; Distribution Functions; Galactic Structure; Milky Way Galaxy; Star Distribution; Angular Momentum; Angular Velocity; Galactic Evolution; Velocity Distribution; Astrophysics; GALAXIES: INTERNAL MOTIONS; GALAXIES: STRUCTURE full text sources ADS | data products SIMBAD (5) NED (4)

Massive Black Holes in Galactic Halos?

We consider the idea that galaxy halos are composed of massive black holes, as a possible resolution of two problems: the composition of dark halos, and the heating of stellar disks. Scattering of disk stars by halo black holes with mass MH, velocity dispersion σH and number density nH causes the stellar velocity dispersion to increase with time t as σ≈(Dt)1/2 for t large, where D α nHM2H in Λ/σH, and in Λ is the Coulomb logarithm. This time-dependence is in good agreement with observations, as is the prediction for the axial ratios of the velocity ellipsoid σu: σv: σw. To account for the magnitude of the disk velocity dispersion in the solar neighbourhood, we require MH≈2 × 106M⊙. The stellar distribution function is predicted to be approximately isothermal at low epicyclic energies, in the Fokker-Planck regime in which the effect of the many distant, weak encounters dominates, but with a power-law tail at high energies produced by the relatively rare close encounters. This tail has the form N(E)αE−2, where E is the horizontal or vertical epicyclic energy, and N(E) is the number of stars per unit area of the disk, per unit E. The fraction of stars in this power-law tail depends only on the value of in Λ, and is about 1% for typical values. This provides a possible explanation for the high velocity A stars found in the solar neighbourhood. This disk heating mechanism can also account for the approximate constancy of the disk scaleheight with radius that is observed in other spiral galaxies, although this does not result as naturally as the other properties.

Stellar distribution functions for galactic disks

La fonction de distribution dans l'espace des phases pour un disque stellaire galactique a symetrie axiale est determinee en fonction de trois integrales du mouvement particulieres. On a calcule un ensemble de 2000 orbites resultant des simulations numeriques du chauffage d'une population stellaire initialement froide par des nuages moleculaires geants. Les integrales d'action J R et Jθ ne sont pas correlees de sorte que les amplitudes des mouvements radiaux et verticaux des etoiles sont decouplees

The problem of maximizing functionals in Newtonian stellar dynamics, and its relation to thermodynamic and dynamical stability

view Abstract Citations (43) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The problem of maximizing functionals in Newtonian stellar dynamics, and its relation to thermodynamic and dynamical stability. Ipser, J. R. ; Horwitz, G. Abstract In an earlier paper, one of us studied the general problem of determining which Newtonian stellar systems maximize (or minimize) locally, at fixed energy and stellar masses, the functional given by the phase-space integral of a chosen function of the stellar distribution function. The criterion derived there for a maximum is incorrect. In this paper, a correct criterion is derived. Examples of the functionals associated with various types of equilibrium stellar systems are presented. In many cases, members of an appropriately constructed one-parameter sequence of spherical equilibria cease to maximize their associated functionals at the first maximum of the binding energy per unit mass along the sequence. Further, it is shown that an isotropic stellar system in collisionless equilibrium is dynamically stable to collisionless perturbations if it does maximize the unique (up to sign and constants) functional which it extremizes. For isothermal clusters, this implies a connection between dynamical stability and so-called thermodynamic stability: such a cluster is dynamically stable if it is thermodynamically stable in the sense that it locally maximizes the Boltzmann entropy. In many cases, it also implies a sufficient, bindingenergy criterion for dynamical stability: members of an appropriate spherical equilibrium sequence are dynamically stable to spherical collisionless perturbations at least up to the first maximum of the binding energy per unit mass along the sequence. Subject headings: instabilities - stars: stellar dynamics Publication: The Astrophysical Journal Pub Date: September 1979 DOI: 10.1086/157347 Bibcode: 1979ApJ...232..863I Keywords: Dynamic Stability; Functionals; Newton Theory; Stellar Motions; Integral Equations; Isotropic Media; Perturbation Theory; Relativity; Star Clusters; Thermodynamic Properties; Astrophysics; Stellar Dynamics; Stellar Systems:Stability full text sources ADS |

Tidal Interaction of Galaxies

view Abstract Citations (13) References (20) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Tidal Interaction of Galaxies Knobloch, Edgar Abstract In this paper we investigate the effect of weak tidal interactions among galaxies in clusters. We obtain the Fokker-Planck coefficients for this process by making use of the impulse approximation. We assume that the outer regions of a galaxy are well described by isotropically distributed stellar orbits with small angnlar momenta. In this case the stellar distribution function is a function of energy and time only. We are able to solve the resulting Fokker-Planck equation for power-law potentials, and find that there are no globally self-consistent solutions of this type. Our calculation indicates that stars are preferentially stripped from the outer regions rather than forming a relaxed extended envelope. We conclude that the effect of the fluctuating term in the tidal force is qualitatively similar to that of the usual secular term. Subject headings: galaxies: clusters of - galaxies: strncture - stars: stellar dynamics Publication: The Astrophysical Journal Supplement Series Pub Date: November 1978 DOI: 10.1086/190556 Bibcode: 1978ApJS...38..253K full text sources ADS |

The equilibrium of cool stellar disks

view Abstract Citations (4) References (15) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The equilibrium of cool stellar disks. Zweibel, E. Abstract A system of finite infinitesimally thin self-gravitating disks is constructed. The sequence begins with the uniformly rotating disk and approaches the isothermal disk, and the latter members of the sequence resemble real disk galaxies. A mathematical description of cool stellar disks is presented for application to model galaxies in which the ratio of random to mean velocity is small, except near the center. Phase-space moments of the modified Schwarzschild stellar distribution function are shown to have an expansion in powers of the ratio of the star's random kinetic energy to gravitational potential energy Publication: The Astrophysical Journal Pub Date: May 1978 DOI: 10.1086/156126 Bibcode: 1978ApJ...222..103Z Keywords: Galactic Structure; Gravitational Fields; Star Distribution; Stellar Motions; Astronomical Models; Distribution Functions; Kinetic Energy; Potential Energy; Rotating Disks; Astrophysics; Galaxies:Stellar Dynamics; Stellar Disks: Models full text sources ADS |