Virial Relation Research Articles

Quasilocal energy and thermodynamic equilibrium conditions

Equilibrium thermodynamic laws are typically applied to horizons in general relativity without stating the conditions that bring them into equilibrium. We fill this gap by applying a new thermodynamic interpretation to a generalized Raychaudhuri equation for a worldsheet orthogonal to a closed spacelike two-surface, the ‘screen’, which encompasses a system of arbitrary size in nonequilibrium with its surroundings in general. In the case of spherical symmetry this enables us to identify quasilocal thermodynamic potentials directly related to standard quasilocal energy definitions. Quasilocal thermodynamic equilibrium is defined by minimizing the mean extrinsic curvature of the screen. Moreover, without any direct reference to surface gravity, we find that the system comes into quasilocal thermodynamic equilibrium when the screen is located at a generalized apparent horizon. Examples of the Schwarzschild, Friedmann–Lemaître and Lemaître–Tolman geometries are investigated and compared. Conditions for the quasilocal thermodynamic and hydrodynamic equilibrium states to coincide are also discussed, and a quasilocal virial relation is suggested as a potential application of this approach.

THE PHYSICAL NATURE OF THE COSMIC ACCRETION OF BARYONS AND DARK MATTER INTO HALOS AND THEIR GALAXIES

The cosmic accretion of both dark matter and baryons into halos typically is measured using some evolving virial relation, but recent work suggests that most halo growth at late cosmic time (z < 2) is not physical but is rather the by-product of an evolving virial radius ("pseudo-evolution"). Using Omega25, a suite of cosmological simulations that incorporate both dark matter and gas dynamics with differing treatments of gas cooling, star formation, and thermal feedback, we systematically explore the physics that governs cosmic accretion into halos and their galaxies. Physically meaningful cosmic accretion of both dark matter and baryons occurs at z > 1 across our halo mass range: M_200m = 10^{11-14} M_sun. However, dark matter, because it is dissipationless, is deposited (in a time-average sense) at > R_200m(z) in a shell-like manner, such that dark-matter mass and density experience little-to-no physical growth at any radius within a halo at z < 1. In contrast, gas, because it is able to cool radiatively, experiences significant accretion at all radii, at a rate that roughly tracks the accretion rate at R_200m, at all redshifts. Infalling gas starts to decouple from dark matter at ~2 R_200m and continues to accrete to smaller radii until the onset of strong angular-momentum support at ~ 0.1 R_200m. Thus, while the growth of dark matter is subject to pseudo-evolution, the growth of baryons is not. The fact that the accretion rate of gas on galactic scales tracks the accretion rate near R_200m provides insight into the tight relations between the masses/sizes of galaxies and those of their host halos across cosmic time.

Anisotropic stellar structure equations for magnetized strange stars

The fact that a fermion system in an external magnetic field breaks the spherical symmetry suggests that its intrinsic geometry is axisymmetric rather than spherical. In this work we analyze the impact of anisotropic pressures, due to the presence of a magnetic field, in the structure equations of a magnetized quark star. We assume a cylindrical metric and an anisotropic energy momentum tensor for the source. We found that there is a maximum magnetic field that a strange star can sustain, closely related to the violation of the virial relations.

A NEW BLACK HOLE MASS ESTIMATE FOR OBSCURED ACTIVE GALACTIC NUCLEI

We propose a new method for estimating the mass of a supermassive black hole, applicable to obscured AGNs. This method estimates the black hole mass using the width of the narrow core of the neutral FeKa emission line in X-rays and the distance of its emitting region from the black hole based on the isotropic luminosity indicator via the luminosity scaling relation. We collect the line width data of the neutral FeKa line core for seven type-1 AGNs and seven type-2 AGNs obtained by the Chandra HETGS. Assuming the virial relation between the locations and the velocity widths of the neutral FeKa line core and the broad Hb emission line, the luminosity scaling relation of the neutral FeKa line core emitting region is estimated. We find that the FWHM of the neutral FeKa line core falls between that of the broad Balmer emission lines and the corresponding value at the dust reverberation radius for most of the type-1 AGNs and for all of the type-2 AGNs. This suggests that significant fraction of photons of the neutral FeKa line core originates between the outer BLR and the inner dust torus in most cases. The black hole mass M_FeKa estimated with this method is then compared with other black hole mass estimates, such as the broad emission-line reverberation mass M_rev for the type-1 AGNs, the mass M_H2O based on the H2O maser and the single-epoch mass estimate M_pol based on the polarized broad Balmer lines for the type-2 AGNs. We find that M_FeKa is consistent with M_rev for the most of the type-1 AGNs and with M_pol for all of the type-2 AGNs. We also find that M_FeKa is correlated well with M_H2O for the type-2 AGNs. These results suggest that M_FeKa is a potential indicator of the black hole mass especially for obscured AGNs. In contrast, M_FeKa for which the same virial factor as for M_rev and M_pol is adopted is systematically larger than M_H2O by about a factor of about 5. (abridged)

Virial tests for post-Newtonian stationary black-hole—disk systems

We investigated hydrodynamical post-Newtonian models of selfgravitating stationary black-hole-disk systems. The post-Newtonian scheme presented here and also in our recent paper is a continuation of previous, purely Newtonian studies of selfgravitating hydrodynamical disks rotating according to the Keplerian rotation law. The post-Newtonian relativistic corrections are significant even at the 1PN level. The 1PN correction to the angular velocity can be of the order of 10% of its Newtonian value. It can be expressed as a combination of geometric and hydrodynamical terms. Moreover, in contrast to the Newtonian Poincaré-Wavre theorem, it depends both on the distance from the rotation axis and the distance from the equatorial plane.In the technical part of this note we derive virial relations valid up to 1PN order. We show that they are indeed satisfied by our numerical solutions.

The virial relation for the Q-balls in the thermal logarithmic potential revisited analytically

We study the properties of Q-balls dominated by the thermal logarithmic potential analytically instead of estimating the characters with only some specific values of model variables numerically. In particular, the analytical expressions for radius and energy of this kind of Q-ball are obtained. According to these explicit expressions we demonstrate strictly that the large Q-balls enlarge and the small ones become smaller in the background with lower temperature. The energy per unit charge will not be divergent if the charge is enormous. We find that the lower temperature will lead the energy per unit charge of Q-ball smaller. We also prove rigorously the necessary conditions that the model parameters should satisfy to keep the stability of the Q-balls. When one of model parameters of Q-balls, K, is positive, the Q-balls will not form or survive unless the temperature is high enough. In the case of negative K, the Q-balls are stable no matter the temperature is high or low.

Magnetic moment of the positronium ion and virial relations

We derive a hypervirial relation for the positronium ion, a three-body bound state of two electrons and a positron. It relates expectation values of three operators and reconciles two recently published, seemingly inequivalent formulas for the magnetic moment of the ion. As a by-product, the precision of the leading binding correction is improved.

ONE PLANE FOR ALL: MASSIVE STAR-FORMING AND QUIESCENT GALAXIES LIE ON THE SAME MASS FUNDAMENTAL PLANE ATz∼ 0 ANDz∼ 0.7

Scaling relations between galaxy structures and dynamics have been studied extensively for early and late-type galaxies, both in the local universe and at high redshifts. The abundant differences between the properties of disky and elliptical, or star-forming and quiescent, galaxies seem to be characteristic of the local Universe; such clear distinctions begin to disintegrate as observations of massive galaxies probe higher redshifts. In this Paper, we investigate the existence the mass fundamental plane of all massive galaxies ($\sigma\gtrsim$ 100 km/s). This work includes local galaxies (0.05<z<0.07) from the SDSS, in addition to 31 star-forming and 72 quiescent massive galaxies at intermediate redshift (z~0.7) with absorption line kinematics from deep Keck-DEIMOS spectra and structural parameters from HST imaging. In two parameter scaling relations, star-forming and quiescent galaxies differ structurally and dynamically. However, we show that massive star-forming and quiescent galaxies lie on nearly the same mass fundamental plane, or the relationship between stellar mass surface density, stellar velocity dispersion, and effective radius. The scatter in this relation (measured about $\log\sigma$) is low: 0.072 dex (0.055 dex intrinsic) at z~0 and 0.10 dex (0.08 dex intrinsic) at z~0.7. This three dimensional surface is not unique: virial relations, with or without a dependence on luminosity profile shapes, can connect galaxy structures and stellar dynamics with similar scatter. This result builds on the recent finding that mass fundamental plane has been stable for early-type galaxies since z~2 (Bezanson et al. 2013). As we now find this also holds for star-forming galaxies to z~0.7, this implies that these scaling relations of galaxies will be minimally susceptible to progenitor biases due to the evolving stellar populations, structures, and dynamics of galaxies through cosmic time.

A novel formulation by Lagrangian variational principle for rotational equilibria: towards multidimensional stellar evolutions

Abstract We have developed a new formulation to obtain self-gravitating, axisymmetric configurations in permanent rotation. The formulation is based on the Lagrangian variational principle, and treats not only barotropic but also baroclinic equations of state, for which angular momentum distributions are not necessarily cylindrical. We adopt a Monte Carlo technique, which is analogous to those employed in other fields, e.g. nuclear physics, in minimizing the energy functional, which is evaluated on a triangulated mesh. This Letter is a proof of principle and detailed comparisons with existing results will be reported in the sequel, but some test calculations are presented, in which we have achieved an error of O(10−4) in the virial relation. We have in mind the application of this method to two-dimensional calculations of the evolutions of rotating stars, for which the Lagrangian formulation is best suited.

EARLY-TYPE GALAXIES IN THECHANDRACOSMOS SURVEY

We study a sample of 69 X-ray detected Early Type Galaxies (ETGs), selected from the Chandra COSMOS survey, to explore the relation between the X-ray luminosity of hot gaseous halos (L_X, gas) and the integrated stellar luminosity (L_K) of the galaxies, in a range of redshift extending out to z=1.5. In the local universe a tight steep relationship has been stablished between these two quantities (L_X,gas~ L_K^4.5) suggesting the presence of largely virialized halos in X-ray luminous systems. We use well established relations from the study of local universe ETGs, together with the expected evolution of the X-ray emission, to subtract the contribution of low mass X-ray binary populations (LMXBs) from the X-ray luminosity of our sample. Our selection minimizes the presence of active galactic nuclei (AGN), yielding a sample representative of normal passive COSMOS ETGs; therefore the resulting luminosity should be representative of gaseous halos, although we cannot exclude other sources such as obscured AGN, or enhanced X-ray emission connected with embedded star formation in the higher z galaxies. We find that most of the galaxies with estimated L_X<10^42 erg/s and z<0.55 follow the L_X,gas- L_K relation of local universe ETGs. For these galaxies, the gravitational mass can be estimated with a certain degree of confidence from the local virial relation. However, the more luminous (10^42<L_X<10^43.5 erg/s) and distant galaxies present significantly larger scatter; these galaxies also tend to have younger stellar ages. The divergence from the local L_X,gas - L_K relation in these galaxies implies significantly enhanced X-ray emission, up to a factor of 100 larger than predicted from the local relation. We discuss the implications of this result for the presence of hidden AGN, and the evolution of hot halos, in the presence of nuclear and star formation feedback.

The discrepancy between dynamical and stellar masses in massive compact galaxies traces non-homology

For many massive compact galaxies, their dynamical masses ($M_\mathrm{dyn} \propto \sigma^2 r_\mathrm{e}$) are lower than their stellar masses ($M_\star$). We analyse the unphysical mass discrepancy $M_\star / M_\mathrm{dyn} > 1$ on a stellar-mass-selected sample of early-type galaxies ($M_\star \gtrsim 10^{11} \ \mathrm{M_\odot}$) at redshifts $z \sim 0.2$ to $z \sim 1.1$. We build stacked spectra for bins of redshift, size and stellar mass, obtain velocity dispersions, and infer dynamical masses using the virial relation $M_\mathrm{dyn} \equiv K \ \sigma_\mathrm{e}^2 r_\mathrm{e} / G$ with $K = 5.0$; this assumes homology between our galaxies and nearby massive ellipticals. Our sample is completed using literature data, including individual objects up to $z \sim 2.5$ and a large local reference sample from the Sloan Digital Sky Survey (SDSS). We find that, at all redshifts, the discrepancy between $M_\star$ and $M_\mathrm{dyn}$ grows as galaxies depart from the present-day relation between stellar mass and size: the more compact a galaxy, the larger its $M_\star / M_\mathrm{dyn}$. Current uncertainties in stellar masses cannot account for values of $M_\star / M_\mathrm{dyn}$ above 1. Our results suggest that the homology hypothesis contained in the $M_\mathrm{dyn}$ formula above breaks down for compact galaxies. We provide an approximation to the virial coefficient $K \sim 6.0 \left[ r_\mathrm{e} / (3.185 \ \mathrm{kpc}) \right]^{-0.81} \left[ M_\star / (10^{11} \ \mathrm{M_\odot}) \right]^{0.45}$, which solves the mass discrepancy problem. A rough approximation to the dynamical mass is given by $M_\mathrm{dyn} \sim \left[ \sigma_\mathrm{e} / (200 \ \mathrm{km \ s^{-1}}) \right]^{3.6} \left[ r_\mathrm{e} / (3 \ \mathrm{kpc}) \right]^{0.35} 2.1 \times 10^{11} \ \mathrm{M_\odot}$.

The one-dimensional Schrödinger–Coulomb operator with definite parity

As a model for one-dimensional hydrogenic systems, the Schrödinger operator with the singular potential −|x|−1 is studied by analyzing it in its reducing subspaces corresponding to even and odd parity. We derive spectral properties common to all self-adjoint extensions of the reduced operators and extract the boundary conditions at x = 0 specific to the different self-adjoint extensions. A transcendental equation for the eigenvalues is derived which leads to a construct that we call the ‘spectral helix’. For the distinguished self-adjoint extension with Dirichlet boundary conditions, virial and supersymmetry relations are established.

General virial theorem for modified-gravity MOND

An important and useful relation is known to hold in two specific MOND theories. It pertains to low-acceleration, isolated systems of pointlike masses, m_p, at positions r_p, subject to gravitational forces F_p. It reads sum_p r_p.F_p=-(2/3)(Ga0)^{1/2}[(\sum_p m_p)^{3/2}-\sum_p m_p^{3/2}]; a0 is the MOND acceleration constant. Here I show that this relation holds in the nonrelativistic limit of any modified-gravity MOND theory. It follows from only the basic tenets of MOND (as applied to such theories): departure from standard dynamics at accelerations below a0, and space-time scale invariance in the nonrelativistic, low-acceleration limit. This implies space-dilatation invariance of the static, gravitational-field equations, which, in turn, leads to the above point-mass virial relation. Thus, the various MOND predictions and tests based on this relation hold in any modified-gravity MOND theory. Since we do not know that any of the existing MOND theories point in the right direction, it is important to identify such predictions that hold in a much larger class of theories. Among these predictions are the MOND two-body force for arbitrary masses, and a general mass-velocity-dispersion relation of the form sigma^2=(2/3)(MGa0)^{1/2}[1-\sum_p (m_p/M)^{3/2}], where M is the total mass.

Black-hole mass estimates for a homogeneous sample of bright flat-spectrum radio quasars

We have selected a complete sample of flat-spectrum radio quasars (FSRQs) from the WMAP 7-yr catalog within the SDSS area, all with measured redshift, and have compared the black hole mass estimates based on fitting a standard accretion disk model to the `blue bump' with those obtained from the commonly used single epoch virial method. The sample comprises 79 objects with a flux density limit of 1 Jy at 23 GHz, 54 of which (68%) have a clearly detected `blue bump'. Thirty-four of the latter have, in the literature, black hole mass estimates obtained with the virial method. The mass estimates obtained from the two methods are well correlated. If the calibration factor of the virial relation is set to $f=4.5$, well within the range of recent estimates, the mean logarithmic ratio of the two mass estimates is equal to zero with a dispersion close to the estimated uncertainty of the virial method. The fact that the two independent methods agree so closely in spite of the potentially large uncertainties associated with each lends strong support to both of them. The distribution of black-hole masses for the 54 FSRQs in our sample with a well detected blue bump has a median value of $7.4\times 10^{8}\,M_\odot$. It declines at the low mass end, consistent with other indications that radio loud AGNs are generally associated with the most massive black holes, although the decline may be, at least partly, due to the source selection. The distribution drops above $\log(M_\bullet/M_\odot) = 9.4$, implying that ultra-massive black holes associated with FSRQs must be rare.

A SYSTEMATIC ANALYSIS OF CAUSTIC METHODS FOR GALAXY CLUSTER MASSES

We quantify the expected observed statistical and systematic uncertainties of the escape velocity as a measure of the gravitational potential and total mass of galaxy clusters. We focus our attention on low redshift (z < 0.15) clusters, where large and deep spectroscopic datasets currently exist. Utilizing a suite of Millennium Simulation semi-analytic galaxy catalogs, we find that the dynamical mass, as traced by either the virial relation or the escape velocity, is robust to variations in how dynamical friction is applied to "orphan" galaxies in the mock catalogs (i.e., those galaxies whose dark matter halos have fallen below the resolution limit). We find that the caustic technique recovers the known halo masses (M_200) with a third less scatter compared to the virial masses. The bias we measure increases quickly as the number of galaxies used decreases. For N_gal > 25, the scatter in the escape velocity mass is dominated by projections along the line-of-sight. Algorithmic uncertainties from the determination of the projected escape velocity profile are negligible. We quantify how target selection based on magnitude, color, and projected radial separation can induce small additional biases into the escape velocity masses. Using N_gal = 150 (25), the caustic technique has a per cluster scatter in ln(M|M_200) of 0.3 (0.5) and bias 1+/-3% (16+/-5%) for clusters with masses > 10^14M_solar at z<0.15.

Virial relation and first law of thermodynamics in scalar-tensor theories of gravity

Virial relations are satisfied for equilibria and quasiequilibria of self-gravitating systems in general relativity and Newtonian gravity. First-law relations are also satisfied along sequences of equilibria and quasiequilibria of self-gravitating systems. These relations are useful for analyzing numerical solutions of equilibria and quasiequilibria. We derive these relations in scalar-tensor theories of gravity in an explicit form. The implication of these relations to compact binary evolution is also discussed.

The ATLAS3D project – XV. Benchmark for early-type galaxies scaling relations from 260 dynamical models: mass-to-light ratio, dark matter, Fundamental Plane and Mass Plane

We study the volume-limited and nearly mass-selected (stellar mass Mstars ≳ 6 × 109 M⊙) ATLAS3D sample of 260 early-type galaxies (ETGs, ellipticals Es and lenticulars S0s). We construct detailed axisymmetric dynamical models (Jeans Anisotropic MGE), which allow for orbital anisotropy, include a dark matter halo and reproduce in detail both the galaxy images and the high-quality integral-field stellar kinematics out to about 1Re, the projected half-light radius. We derive accurate total mass-to-light ratios (M/L)e and dark matter fractions fDM, within a sphere of radius |$r={R_{\rm e}}$| centred on the galaxies. We also measure the stellar (M/L)stars and derive a median dark matter fraction fDM = 13 per cent in our sample. We infer masses MJAM ≡ L × (M/L)e ≈ 2 × M1/2, where M1/2 is the total mass within a sphere enclosing half of the galaxy light. We find that the thin two-dimensional subset spanned by galaxies in the |$(M_{\rm JAM},\sigma _e,R_{\rm e}^{\rm maj})$| coordinates system, which we call the Mass Plane (MP) has an observed rms scatter of 19 per cent, which implies an intrinsic one of 11 per cent. Here, |$R_{\rm e}^{\rm maj}$| is the major axis of an isophote enclosing half of the observed galaxy light, while σe is measured within that isophote. The MP satisfies the scalar virial relation |$M_{\rm JAM}\propto \sigma _e^2 R_{\rm e}^{\rm maj}$| within our tight errors. This show that the larger scatter in the Fundamental Plane (FP) (L, σe, Re) is due to stellar population effects [including trends in the stellar initial mass function (IMF)]. It confirms that the FP deviation from the virial exponents is due to a genuine (M/L)e variation. However, the details of how both Re and σe are determined are critical in defining the precise deviation from the virial exponents. The main uncertainty in masses or M/L estimates using the scalar virial relation is in the measurement of Re. This problem is already relevant for nearby galaxies and may cause significant biases in virial mass and size determinations at high redshift. Dynamical models can eliminate these problems. We revisit the (M/L)e-σe relation, which describes most of the deviations between the MP and the FP. The best-fitting relation is |$({\rm M/L})_e\propto \sigma _e^{0.72}$| (r band). It provides an upper limit to any systematic increase of the IMF mass normalization with σe. The correlation is more shallow and has smaller scatter for slow rotating systems or for galaxies in Virgo. For the latter, when using the best distance estimates, we observe a scatter in (M/L)e of 11 per cent, and infer an intrinsic one of 8 per cent. We perform an accurate empirical study of the link between σe and the galaxies circular velocity Vcirc within 1Re (where stars dominate) and find the relation max (Vcirc) ≈ 1.76 × σe, which has an observed scatter of 7 per cent. The accurate parameters described in this paper are used in the companion Paper XX (Cappellari et al.) of this series to explore the variation of global galaxy properties, including the IMF, on the projections of the MP.

DEEP NEAR-INFRARED SPECTROSCOPY OF PASSIVELY EVOLVING GALAXIES ATz≳ 1.4

[Abridged] We present the results of new near-IR spectroscopic observations of passive galaxies at z>1.4 in a concentration of BzK-selected galaxies in the COSMOS field. The observations have been conducted with Subaru/MOIRCS, and have resulted in absorption lines and/or continuum detection for 18 out of 34 objects. This allows us to measure spectroscopic redshifts for a sample almost complete to K(AB)=21. COSMOS photometric redshifts are found in fair agreement overall with the spectroscopic redshifts, with a standard deviation of ~0.05; however, ~30% of objects have photometric redshifts systematically underestimated by up to ~25%. We show that these systematic offsets in photometric redshifts can be removed by using these objects as a training set. All galaxies fall in four distinct redshift spikes at z=1.43, 1.53, 1.67 and 1.82, with this latter one including 7 galaxies. SED fits to broad-band fluxes indicate stellar masses in the range of ~4-40x10^10Msun and that star formation was quenched ~1 Gyr before the cosmic epoch at which they are observed. The spectra of several individual galaxies have allowed us to measure their Hdelta_F and Dn4000 indices, which confirms their identification as passive galaxies, as does a composite spectrum resulting from the coaddition of 17 individual spectra. The effective radii of the galaxies have been measured on the HST/ACS F814W image, confirming the coexistence at these redshifts of passive galaxies which are substantially more compact than their local counterparts with others that follow the local size-stellar mass relation. For the galaxy with best S/N spectrum we were able to measure a velocity dispersion of 270+/-105 km/s, indicating that this galaxy lies closely on the virial relation given its stellar mass and effective radius.

Computing the local pressure in molecular dynamics simulations

Computer simulations of inhomogeneous soft matter systems often require accurate methods for computing the local pressure. We present a simple derivation, based on the virial relation, of two equivalent expressions for the local (atomistic) pressure in a molecular dynamics simulation. One of these expressions, previously derived by other authors via a different route, involves summation over interactions between particles within the region of interest; the other involves summation over interactions across the boundary of the region of interest. We illustrate our derivation using simulations of a simple osmotic system; both expressions produce accurate results even when the region of interest over which the pressure is measured is very small.

The virial theorem and the ground state problem in polaron theory

The virial theorem for the translation-invariant theory of a polaron [3] is discussed. It is shown that, in [3], Tulub made a nonoptimal choice of variational parameters in the strong-coupling limit, which led to the violation of the virial relations. The introduction of an additional variational parameter to the test function reduces the polaron energy and makes it possible to satisfy the relations of the virial theorem for a strong-coupling polaron (the Pekar 1 : 2 : 3 : 4 theorem).