Triaxial Equilibrium Research Articles

Some Remarks on the Equilibrium of Granular Materials Described by Constitutive Relations That Depend on the Gradients of the Density or Volume Fraction

Abstract One of the ways of determining the properties of granular materials is to subject the material to a tri-axial state of stress with three distinct eigenvalues, in a cubical tri-axial testing equipment, the material tested being in a static state. A constitutive relation for granular materials that does not allow the body to be in tri-axial static equilibrium would not be a reasonable model as experiments clearly show that granular materials can exist in such tri-axial states. In this short note, we observe that a large class of models that have been developed and used to describe the response of granular materials will not allow them to be in a static tri-axial state of stress with three different principal stresses.

The Trans-Neptunian Object (84922) 2003 VS2 through Stellar Occultations

Abstract We present results from three world-wide campaigns that resulted in the detections of two single-chord and one multi-chord stellar occultations by the plutino object (84922) 2003 VS2. From the single-chord occultations in 2013 and 2014 we obtained accurate astrometric positions for the object, while from the multi-chord occultation on 2014 November 7, we obtained the parameters of the best-fitting ellipse to the limb of the body at the time of occultation. We also obtained short-term photometry data for the body in order to derive its rotational phase during the occultation. The rotational light curve present a peak-to-peak amplitude of 0.141 ± 0.009 mag. This allows us to reconstruct the 3D shape of the body, with principal semi-axes of a = 313.8 ± 7.1 km, km, and km, which is not consistent with a Jacobi triaxial equilibrium figure. The derived spherical volume equivalent diameter of km is about 5% larger than the radiometric diameter of 2003 VS2 derived from Herschel data of 523 ± 35 km, but still compatible with it within error bars. From those results we can also derive the geometric albedo ( ) and, under the assumption that the object is a Maclaurin spheroid, the density for the plutino. The disappearances and reappearances of the star during the occultations do not show any compelling evidence for a global atmosphere considering a pressure upper limit of about 1 microbar for a pure nitrogen atmosphere, nor secondary features (e.g., rings or satellite) around the main body.

Numerical Studies of Astrophysical Objects at Fesenkov Astrophysical Institute (FAI)

Numerical studies of astrophysical objects are a relatively new direction in Fesenkov Astrophysical Institute (FAI) and is mainly represented by the Laboratory of Cosmology, Stellar Dynamics and Computational Astrophysics. The lab seeks to understand the evolution of gravitating systems at various scales – from star clusters to galaxies to large-scale structure of the universe as a whole, and tackles these problems both through analytical methods and through numerical simulations. The particular focus is on numerical simulations of star clusters, especially those found in active galactic nuclei – this is a topic of oldestablished collaboration with colleagues from Astronomisches Rechen-Institut (Heidelberg) and National Astronomical Observatories of China (Beijing). The prominent example is STARDISK project dedicated to the numerical research of active galactic nuclei as multicomponent systems composed of compact stellar cluster, gaseous accretion disk and a supermassive black hole. It is demonstrated that an accretion disk can noticeably decelerate stars and thus enhance the accretion rate onto the black hole. In 2013 FAI hosted the MODEST-13 International Workshop dedicated to modeling of star clusters. Recently a new project has been approved aimed at construction of triaxial equilibrium N-body systems that can be of great help in various numerical experiments with disk galaxies. There are also long standing plans to perform cosmological simulations of large scale structures to test a new approach to dark matter and energy actively developed at FAI. For numerical calculations, FAI has a small, but growing computer cluster consisting of several high-performance computing servers equipped with computational GPU cards.

Lopsidedness of Self-consistent Galaxies Caused by the External Field Effect of Clusters

Abstract Adopting Schwarzschild’s orbit-superposition technique, we construct a series of self-consistent galaxy models, embedded in the external field of galaxy clusters in the framework of Milgrom’s MOdified Newtonian Dynamics (MOND). These models represent relatively massive ellipticals with a Hernquist radial profile at various distances from the cluster center. Using N-body simulations, we perform a first analysis of these models and their evolution. We find that self-gravitating axisymmetric density models, even under a weak external field, lose their symmetry by instability and generally evolve to triaxial configurations. A kinematic analysis suggests that the instability originates from both box and nonclassified orbits with low angular momentum. We also consider a self-consistent isolated system that is then placed in a strong external field and allowed to evolve freely. This model, just like the corresponding equilibrium model in the same external field, eventually settles to a triaxial equilibrium as well, but has a higher velocity radial anisotropy and is rounder. The presence of an external field in the MOND universe generically predicts some lopsidedness of galaxy shapes.

WARPS AND BARS FROM THE EXTERNAL TIDAL TORQUES OF TUMBLING DARK HALOS

The dark matter halos in $\Lambda$CDM cosmological simulations are triaxial and highly flattened. In many cases, these triaxial equilibria are also tumbling slowly, typically about their short axes, with periods of order a Hubble time. Halos may therefore exert a slowly changing external torque on spiral galaxies that can affect their dynamical evolution in interesting ways. We examine the effect of the external torques exerted by a tumbling quadrupolar tidal field on the evolution of spiral galaxies using N-body simulations with realistic, disk galaxy models. We measure the amplitude of the external quadrupole moments of dark halos in cosmological simulations and use these to force disk galaxy models in a series of N-body experiments for a range of pattern speeds. We find that the torques are strong enough to induce long lived transient warps in disks similar to those observed in real spirals and also induce the bar instability at later times in some galaxy models that are otherwise stable for long periods of time in isolation. We also observe forced spiral structure near the edge of the disk where normally self-gravity is too weak to be responsible for such structure. This overlooked influence of dark halos may well be responsible for many of the peculiar aspects of disk galaxy dynamics.

Self‐consistent Models of Triaxial Galaxies in MOND Gravity

The Bekenstein-Milgrom gravity theory with a modified Poisson equation is tested here for the existence of triaxial equilibrium solutions. Using the nonnegative least-square method, we show that self-consistent triaxial galaxies exist for baryonic models with a mild density cusp, ρ ∼ Σ/r. Self-consistency is achieved for a wide range of central concentrations, Σ ∼ 10-1000 M☉ pc−2, representing low to high surface brightness galaxies. Our results demonstrate for the first time that the orbit superposition technique is fruitful for constructing galaxy models beyond Newtonian gravity, and triaxial cuspy galaxies might exist without the help of cold dark matter.

Chaos and secular evolution of triaxial N-body galactic models due to an imposed central mass

We investigate the response of triaxial non-rotating N-body models of elliptical galaxies with smooth centers, initially in equilibrium, under the presence of a central mass assumed to be due mainly to a massive central black hole. We examine the fraction of mass in chaotic motion and the resulting secular evolution of the models. Four cases of the size of the central mass are investigated, namely m = 0.0005, 0.0010, 0.0050, 0.0100 in units of the total mass of the galaxy. We find that a central mass with value m < 0.005 increases the mass fraction in chaotic motion from the level of 25-35% (that appears in the case of smooth centers) to the level of 50-80% depending on the value of m and on the initial maximum ellipticity of the system. However, most of this mass moves in chaotic orbits with Lyapunov numbers too small to develop chaotic diffusion in a Hubble time. Thus their secular evolution is so slow that it can be neglected in a Hubble time. Larger central masses (m ≥ 0.005) give initially about the same fractions of mass in chaotic motion as for smaller m, but the Lyapunov numbers are concentrated to larger values, so that a secular evolution of the self-consistent models is prominent. These systems evolve in time tending to a new equilibrium. During their evolution they become self-organized by converting chaotic orbits to ordered orbits of the Short Axis Tube type. The mechanism of such a self-organization is investigated. The rate of this evolution depends on m and on the value of the initial maximum ellipticity of the system. For m = 0.01 and a large initial maximum ellipticity E max 7, equilibrium can be achieved in one Hubble time, forming an oblate spheroidal configuration. For the same value of m and initial maximum elipticity E max 3.5, or for E max 7, but m = 0.005, oblate equilibrium configurations can also be achieved, but in much longer times. Furthermore, we find that, form = 0.005 and E max 3.5, triaxial equilibrium configurations can be formed. The fraction of mass in chaotic motion in the equilibrium configurations is in the range of 12-25%.

Triaxial bifurcations of rapidly rotating spheroids

A rotating system, such as a star, liquid drop, or atomic nucleus, may rotate as an oblate spheroid about its symmetry axis or, if the angular velocity is greater than a critical value, as a triaxial ellipsoid about a principal axis. The oblate and triaxial equilibrium configurations minimize the total energy, the sum of the rotational kinetic energy plus the potential energy. For a star or galaxy the potential is the self-gravitating potential, for a liquid drop, the surface tension energy, and for a nucleus, the potential is the sum of the repulsive Coulomb energy plus the attractive surface energy. A simple, but accurate, Padé approximation to the potential function is used for the energy minimization problem that permits closed analytic expressions to be derived. In particular, the critical deformation and angular velocity for bifurcation from MacLaurin spheroids to Jacobi ellipsoids is determined analytically in the approximation.

The triaxial rotation vibration model in the XeBa region

Collective quadrupole degrees of freedom give rise to vibrations and rotations in nuclei. The axial Rotation Vibration Model (RVM) is here extended to describe also triaxial equilibrium shapes with β and γ vibrations allowing for the interaction between vibrations and rotations. This Triaxial Rotation Vibration Model (TRVM( is applied to Ba and Xe isotopes with A ≈ 120 to 130. This are has recently been pointed out to be a region for the O(6) limit of the Interacting Boson Approximation (IBA). The present work shows that the TRVM can equally well describe these nuclei concerning their excitation energies and E2 branching ratios.

Lifetimes of a shears band in 139Sm

Lifetimes have been measured in 139Sm with the DSAM method, using the reaction 62Ni( 81Br,p3n) 139Sm at 350 MeV. In the negative-parity regular dipole band based on the 25 2 − state at 3325 keV values of 0.60(21), 0.40(14) and 0.25(8) ps were found for the 337, 409 and 473 keV transitions, respectively. The deduced B(M1) and B(E2) are in agreement with the predictions of the tilted axis cranking model, allowing a triaxial equilibrium shape for the assumed νh 11 2 ⊗ πh 11 2 2 configuration.

Chaos and the Shapes of Elliptical Galaxies

Hubble Space Telescope observations reveal that the density of stars in most elliptical galaxies rises toward the center in a power-law cusp. Many of these galaxies also contain central dark objects, possibly supermassive black holes. The gravitational force from a steep cusp or black hole will destroy most of the box orbits that constitute the “backbone” of a triaxial stellar system. Detailed modeling demonstrates that the resulting chaos can preclude a self-consistent, strongly triaxial equilibrium. Most elliptical galaxies may therefore be nearly axisymmetric, either oblate or prolate.

Dynamics of Elliptical Galaxies

Elliptical galaxies were once thought to be similar in their structure and dynamics to rotationally flattened bodies like stars. The discovery that elliptical galaxies rotate much more slowly than a fluid body with the same shape has led to a qualitative change in our understanding of the dynamics of these systems. It is now believed that elliptical galaxies are fully triaxial in shape. Self-consistent triaxial equilibria have been constructed and appear to be long-lived; they are made possible by the existence of conserved quantities, or integrals of motion, for galactic potentials without rotational symmetry. Many self-consistent equilibria are unstable; the nonexistence of elliptical galaxies with axis ratios more extreme than 3:1 is probably the result of such an instability. There is evidence for strong central mass concentrations, perhaps massive black holes, at the centers of some nearby galaxies. Recent observations suggest that many elliptical galaxies formed through the merger of two or more spiral galaxies.

Orbital structure of triaxial equilibrium models of various shapes

To bring out the link between the observed shapes of triaxial dynamical systems (ellipticals, bulges, bars) and their orbital populations inferred from models.

The shape and internal structure of Mimas

Limb profiles from the six best Voyager images of Mimas have been used to determine the shape of the satellite. Correction of image distortions allows coordinates on the limbs to be located with an accuracy of approximately one-half picture element: about 0.5 km for the two best images and between 1 and 2 km for the other images. Ellipses fit to the limbs show that the shape of Mimas is well represented by a triaxial ellipsoid; it is the smallest satellite observed for which this is the case. The ratio of the differences of the axes, ( b − c)/( a − c), is 0.27 ± 0.04, indicating that the satellite is close to hydrostatic equilibrium. This is the first bservation of a satellite in the solar system with a triaxial equilibrium figure. Using the satellite mass determined by Kozai (Y. Kozai 1957, Ann. Tokyo Astron. Obs. 2nd Ser. 5, 73–106) from observations of the libration period and the libration amplitude of the Mimas-Tethys resonance, and a second-order theory for the ellipsoidal figure of equilibrium, we deduce that the satellite has a mean radius 〈 R〉 (after allowing for limb topography) of 198.8 ± 0.6 km and a mean density of 1.137 ± 0.018 g/cm 3 and that the difference between the long and short axes, a − c, is 16.9 ± 0.7 km. The expected value of a − c for a comparable, but homogeneous, satellite in hydrostatic equilibrium is 20.3 ± 0.3 km. We conclude that Mimas is probably differentiated. The satellite may have a rocky core of radius (0.44 ± 0.09) 〈 R〉, in which case the material outside the core probably has a mean density of 0.96 ± 0.08 g/cm 3, consistent with that of uncompressed, but moderately contaminated, water-ice. If the matrix of the mantle material is water-ice, then the silicate mass fraction of Mimas is 0.27 ± 0.04; Mimas is markedly deficient in rock. An alternative interpretation of our data is that the material of Mimas has a cosmic composition of ice and rock and is undifferentiated, but is highly porous down to a depth determined by the crushing strength of cold ice. However, this interpretation requires that the surface porosity of the satellite is very high, between 20 and 60%. Further conclusions depend on assumptions about the rheology of the mantle material and on the thermal history of the satellite. If the rheology of the mantle is determined by the properties of ice I, and the satellite does not have a deep, porous megaregolith and has not been heated since the time of its formation, then (a) it is likely that Mimas was formed close to its present orbital radius in a state of near, if not exact, resonance with Tethys and (b) it is unlikely that Mimas has been disrupted since the time of its formation. We consider, however, that it is more likely that either the thermal history of Mimas is complex and the satellite has been heated since the time of its formation, possibly due to tidal damping of its orbital eccentricity on temporary capture into resonance, or that the satellite does have a deep, porous megaregolith.

Microscopic Description of Anharmonic Gamma-Vibrations by Means of the Selfconsistent-Collective-Coordinate Method. II

We investigate, in a fully microscopic way, the double gamma·vibrational states in l68Er by means of the ·selfconsistent-collective-coordinate (See) method proposed by Marumori, Maskawa, Sakata and Kuriyama. In applying the see method, we propose a new prescription to derive a quantized collective Hamiltonian. The collective potential is defined in terms of the Wigner transform of the quantized collective Hamiltonian. It is different from the potential defined by the energy expectation values with respect to the time-even Slater determinants, and we find that it acquires a much stronger tendency toward the triaxial equilibrium shape than the latter. Numerical calculations show that 168Er is situated just in the transitional region toward the triaxial equilibrium shape. The nuclear anharmonicities associated with low-frequency vibrational modes have been an outstanding subject in nuclear many-body problems. We encounter genuine non-linear vibrations, for instance, in transitional nuclei lying in the intermediate region between the spherical and deformed equilibrium shapes. For the aim of constructing a fully microscopic theory capable of describing the large-amplitude collective motions such as the non-linear nuclear vibrations, Marumori, Maskawa, Sakata and Kuriyama 1 ) have proposed a new theory called the selfconsistent-collective-coordinate (See) method. The quadrupole collective modes associated with the y-degree of freedom have been thought to constitute another typical example of large-amplitude, nuclear non-linear phenomena. 2 ) But, it is only recently that significant anharmonic character of the y vibrations is demonstrated in a clear-cut way.3) In this paper, we shall analyse the microscopic structure of the anharmonicities associated with the y-vibrational modes by means ofthe see method. ll For this aim, we shall propose a new prescription to quantize the classical collective Hamiltonian derived by the see method. We also propose a new procedure to relate the resulting quantized Hamiltonian with the phenomenological Bohr Hamiltonian. 4

Microscopic Description of Anharmonic Gamma-Vibrations by Means of the Selfconsistent-Collective-Coordinate Method. I

We investigate, in a fully microscopic way, the double gamma·vibrational states in l68Er by means of the ·selfconsistent-collective-coordinate (See) method proposed by Marumori, Maskawa, Sakata and Kuriyama. In applying the see method, we propose a new prescription to derive a quantized collective Hamiltonian. The collective potential is defined in terms of the Wigner transform of the quantized collective Hamiltonian. It is different from the potential defined by the energy expectation values with respect to the time-even Slater determinants, and we find that it acquires a much stronger tendency toward the triaxial equilibrium shape than the latter. Numerical calculations show that 168Er is situated just in the transitional region toward the triaxial equilibrium shape. The nuclear anharmonicities associated with low-frequency vibrational modes have been an outstanding subject in nuclear many-body problems. We encounter genuine non-linear vibrations, for instance, in transitional nuclei lying in the intermediate region between the spherical and deformed equilibrium shapes. For the aim of constructing a fully microscopic theory capable of describing the large-amplitude collective motions such as the non-linear nuclear vibrations, Marumori, Maskawa, Sakata and Kuriyama 1 ) have proposed a new theory called the selfconsistent-collective-coordinate (See) method. The quadrupole collective modes associated with the y-degree of freedom have been thought to constitute another typical example of large-amplitude, nuclear non-linear phenomena. 2 ) But, it is only recently that significant anharmonic character of the y vibrations is demonstrated in a clear-cut way.3) In this paper, we shall analyse the microscopic structure of the anharmonicities associated with the y-vibrational modes by means ofthe see method. ll For this aim, we shall propose a new prescription to quantize the classical collective Hamiltonian derived by the see method. We also propose a new procedure to relate the resulting quantized Hamiltonian with the phenomenological Bohr Hamiltonian. 4

Interplay of Gamma-Vibrations and Aligned-Quasiparticles at High-Spin Yrast Region

On the basis of the self-consistently determined diabatic quasiparticle representation, RP A calculations in the rotating frame are performed for the r-vibrations around the triaxial equilibrium shapes. The doubly stretched quadrupole forces are used as residual interactions. It is found that the negative-signa­ ture r-vibrational modes created on the s-band well retain their collective characters, while the positive­ signature modes tend to lose their identities with increasing rotational frequency.

The asteroids as outcomes of catastrophic collisions

The role of catastrophic collisions in the evolution of the asteroids is discussed in detail, employing extrapolations of experimental results on the outcomrs of high-velocity impacts. We determine the range of the probable largest collision for target asteroids of different sizes during the solar system's lifetime, and we conclude that all the asteroids have undergone collisional events capable of overcoming the material's solid-state cohesion. Such events do not lead inescapably to complete disruption of the targets, because (i) for a previously unfractured target, experiments show that fragments of significant size can survive breakup, depending on the energy and geometry of the collision; (ii) self-gravitation can easily cause a reaccumulation of fragments for targets exceeding a critical size, which seems to be of the order of 100 km. In the intermediate diameter range 100⪅ D ⪅300 km, where formation of gravitationally bound “rubble piles” is frequent, the transfer of angular momentum can be large enough to produce objects with triaxial equilibrium shapes (Jacobi ellipsoids) or to cause fission into binary systems. In the same size range, low-velocity escape of collisional fragments can also occur, leading to the formation of dynamical families. Asteroids smaller than ∼100 km are mostly multigeneration fragments, while for D⪆300 km the collisional process produces nearly spheroidal objects covered by megaregoliths; whether their rotation is “primordial” or collisionally generated depends critically on the past flux of colliders. The complex and size-dependent phenomenology predicted by the theory compares satisfactorily with the observational evidence, as derived both by a classification of asteroids in terms of their size, spin rate, and lightcurve amplitude, and by a comparison between the rotational properties of family and nonfamily asteroids. The fundamental result of this investigation is that almost all asteroids are outcomes of catastrophic collisions, and that these events cause either complete fragmentation of the target bodies or, at least, drastic readjustments of their internal structure, shape, and spin rate.

Triaxial equilibrium models for elliptical galaxies with slow figure rotation

view Abstract Citations (157) References (13) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Triaxial equilibrium models for elliptical galaxies with slow figure rotation Schwarzschild, M. Abstract An earlier numerical construction of a triaxial model for a stellar system in dynamical equilibrium has been extended, first by adding a slow figure rotation, and second by adding rotational streaming on Z-tube orbits. The figure rotation was kept low so that the main resonances, including the outer edge of Binney's instability strip, fall outside the model. This figure rotation induces weak rotational streaming, prograde on box orbits and retrograde on X-tube orbits. The resulting net streaming has a very low velocity (although substantially higher than the figure rotation by itself) and a complicated pattern, prograde in the inner portion and retrograde in the outer portion. The addition of prograde Z-tube orbits, circling around the short (Z) rotation axis, introduces strong rotational streaming on these orbits. The net effect of this streaming is, however, substantially diluted by the box orbits, which are needed for the support of the triaxial figure. The maximum net rotational streaming velocity thus permitted for the figure here used (1:1.25:2) appears sufficient to model the observed rotation of most giant ellipticals. Publication: The Astrophysical Journal Pub Date: December 1982 DOI: 10.1086/160531 Bibcode: 1982ApJ...263..599S Keywords: Astronomical Models; Elliptical Galaxies; Galactic Rotation; Orbit Calculation; Stellar Motions; Angular Velocity; Dynamic Characteristics; Mathematical Models; Velocity Distribution; Astrophysics full text sources ADS |

Triaxial equilibrium ellipsoids among the asteroids?

We present a physical model to explain the existence of a class of large-lightcurve-amplitude, rapidly rotating asteroids found most commonly among objects in the size range 100–300 km diameter. A significant correlation between rotation period and lightcurve amplitude exists for asteroids in this size range in the sense that those with larger amplitudes spin more rapidly and hence these objects have high rotational angular momenta. Since this is a property of Jacobi ellipsoids, we have investigated whether these asteriods might be examples of triaxial equilibrium ellipsoids. We find that objects rotating with periods of 6 hr must have densities between 1.1 and 1.4 g cm −3, while those rotating in 4 hr would have densities between 2.4 and 3.2 g cm −3. If this model is valid then at least some of these asteroids have rather low mean densities. The reality of this result and its interpretation in terms of collisional evolution of the asteroids is discussed.