Triaxial Galaxy Research Articles

The MAGPI survey: orbital distributions, intrinsic shapes, and mass profiles for MAGPI-like eagle galaxies using Schwarzschild dynamical models

ABSTRACT Schwarzschild dynamical models are now regularly employed in large surveys of galaxies in the local and distant Universe to derive information on galaxies’ intrinsic properties such as their orbital structure and their (dark matter and stellar) mass distribution. Comparing the internal orbital structures and mass distributions of galaxies in the distant Universe with simulations is key to understanding what physical processes are responsible for shaping galaxy properties. However, it is first crucial to understand whether observationally derived properties are directly comparable with intrinsic ones in simulations. To assess this, we build Schwarzschild dynamical models for MUSE-like IFS (integral field spectroscopy) cubes (constructed to be like those obtained by the Middle Ages Galaxy Properties with Integral Field Spectroscopy, MAGPI survey) of 75 galaxies at $z \sim$ 0.3 from the eagle simulations. We compare the true particle-derived properties with the galaxies’ model-derived properties. In general, we find that the models can recover the true galaxy properties qualitatively well, with the exception of the enclosed dark matter, where we find a median offset of 48 per cent, which is due to the assumed Navarro–Frenk–White profile not being able to reproduce the dark matter distribution in the inner region of the galaxies. We then compare our model-derived properties with Schwarzschild models-derived properties of observed MAGPI galaxies and find good agreement between MAGPI and eagle: the majority of our galaxies (57 per cent) have non-oblate shapes within 1 effective radius. More triaxial galaxies show higher fractions of hot orbits in their inner regions and tend to be more radially anisotropic.

Stellar Dynamical Modeling—Counting Conserved Quantities

Knowing the conserved quantities that a galaxy’s stellar orbits conform to is important in helping us understand the stellar distribution and structures within the galaxy. Isolating integrals of motion and resonances are particularly important, non-isolating integrals less so. We compare the behavior and results of two methods for counting the number of conserved quantities, one based on the correlation integral approach and the other a more recent method using machine learning. Both methods use stellar orbit trajectories in phase space as their only input, and we create such trajectories from theoretical spherical, axisymmetric, and triaxial model galaxies. The orbits have known isolating integrals and resonances. We find that neither method is fully effective in recovering the numbers of these quantities, nor in determining the number of non-isolating integrals. From a computer performance perspective, we find the correlation integral approach to be the faster. Determining the algebraic formulae of (multiple) conserved quantities from the trajectories has not been possible due to the lack of an appropriate symbolic regression capability. Notwithstanding the shortcomings we have noted, it may be that the methods are usable as part of a trajectory analysis tool kit.

Accuracy and precision of triaxial orbit models I: SMBH mass, stellar mass, and dark-matter halo

ABSTRACT We investigate the accuracy and precision of triaxial dynamical orbit models by fitting 2D mock observations of a realistic N-body merger simulation resembling a massive early-type galaxy with a supermassive black hole. We show that we can reproduce the triaxial N-body merger remnant’s correct black hole mass, stellar mass-to-light ratio and total enclosed mass (inside the half-light radius) for several different tested orientations with an unprecedented accuracy of 5–10 per cent. Our dynamical models use the entire non-parametric line-of-sight velocity distribution (LOSVD) rather than parametric LOSVDs or velocity moments as constraints. Our results strongly suggest that state-of-the-art integral-field projected kinematic data contain only minor degeneracies with respect to the mass and anisotropy recovery. Moreover, this also demonstrates the strength of the Schwarzschild method in general. We achieve the proven high recovery accuracy and precision with our newly developed modelling machinery by combining several advancements: (i) our new semiparametric deprojection code probes degeneracies and allows us to constrain the viewing angles of a triaxial galaxy; (ii) our new orbit modelling code SMART uses a 5-dim orbital starting space to representatively sample in particular near-Keplerian orbits in galaxy centres; (iii) we use a generalized information criterion AICp to optimize the smoothing and to compare different mass models to avoid biases that occur in χ2-based models with varying model flexibilities.

Testing the robustness of DYNAMITE triaxial Schwarzschild modelling: The effects of correcting the orbit mirroring

In the past 15 yr, the triaxial Schwarzschild orbit-superposition code developed by van den Bosch and van de Ven in Leiden has been widely applied to study the dynamics of galaxies. Recently, a bug was reported in the orbit calculation of this code, specifically in the mirroring procedure that is used to speed up the computation. We have fixed the incorrect mirroring in the DYNAMITE code, which is the publicly-released successor of the Leiden triaxial Schwarzschild code. In this study, we provide a thorough quantification of how this bug has affected the results of dynamical analyses performed with this code. We compare results obtained with the original and corrected versions of DYNAMITE, and discuss the differences in the phase-space distribution of a single orbit and in the global stellar orbit distribution, in the mass estimate of the central black hole in the highly triaxial galaxy PGC 46832, and in the measurement of intrinsic shape and enclosed mass for more than 50 galaxies. Focusing on the typical scientific applications of the Schwarzschild method, in all our tests we find that differences are negligible with respect to the statistical and systematic uncertainties. We conclude that previous results with the Leiden triaxial Schwarzschild code are not significantly affected by the incorrect mirroring.

Accuracy and precision of triaxial orbit models – II. Viewing angles, shape, and orbital structure

ABSTRACT We explore the potential of our novel triaxial modelling machinery in recovering the viewing angles, the shape, and the orbit distribution of galaxies by using a high-resolution N-body merger simulation. Our modelling technique includes several recent advancements. (i) Our new triaxial deprojection algorithm shape3d is able to significantly shrink the range of possible orientations of a triaxial galaxy and therefore to constrain its shape relying only on photometric information. It also allows to probe degeneracies, i.e. to recover different deprojections at the same assumed orientation. With this method we can constrain the intrinsic shape of the N-body simulation, i.e. the axis ratios p = b/a and q = c/a, with Δp and Δq ≲ 0.1 using only photometric information. The typical accuracy of the viewing angles reconstruction is 15°–20°. (ii) Our new triaxial Schwarzschild code smart exploits the full kinematic information contained in the entire non-parametric line-of-sight velocity distributions along with a 5D orbital sampling in phase space. (iii) We use a new generalized Akaike information criterion AICp to optimize the smoothing and to select the best-fitting model, avoiding potential biases in purely χ2-based approaches. With our deprojected densities, we recover the correct orbital structure and anisotropy parameter β with Δβ ≲ 0.1. These results are valid regardless of the tested orientation of the simulation and suggest that even despite the known intrinsic photometric and kinematic degeneracies the above described advanced methods make it possible to recover the shape and the orbital structure of triaxial bodies with unprecedented accuracy.

Triaxial Orbit-based Dynamical Modeling of Galaxies with Supermassive Black Holes and an Application to Massive Elliptical Galaxy NGC 1453

Abstract Most stellar-dynamical determinations of the masses of nearby supermassive black holes (SMBHs) have been obtained with the orbit superposition technique under the assumption of axisymmetry. However, few galaxies—in particular massive early-type galaxies—obey exact axisymmetry. Here we present a revised orbit superposition code and a new approach for dynamically determining the intrinsic shapes and mass parameters of triaxial galaxies based on spatially resolved stellar kinematic data. The triaxial TriOS code described here corrects an error in the original van den Bosch et al. code that gives rise to incorrect projections for most orbits in triaxial models and can significantly impact parameter search results. The revised code also contains significant improvements in orbit sampling, mass constraints, and run time. Furthermore, we introduce two new parameter-searching strategies—a new set of triaxial shape parameters and a novel grid-free sampling technique—that together lead to a remarkable gain in efficiency in locating the best-fit model. We apply the updated code and search method to NGC 1453, a fast-rotating massive elliptical galaxy. A full 6D parameter search finds p = b / a = 0.933 − 0.015 + 0.014 and q = c/a = 0.779 ± 0.012 for the intrinsic axis ratios and T = 0.33 ± 0.06 for the triaxiality parameter. Despite the deviations from axisymmetry, the best-fit SMBH mass, stellar mass-to-light ratio, and dark matter enclosed mass for NGC 1453 are consistent with the axisymmetric results. More comparisons between axisymmetric and triaxial modeling are needed before drawing general conclusions.

SMART: A new implementation of Schwarzschild’s Orbit Superposition technique for triaxial galaxies and its application to an N-body merger simulation

Abstract We present SMART, a new 3D implementation of the Schwarzschild Method and its application to a triaxial N-body merger simulation. SMART fits full line-of-sight velocity distributions (LOSVDs) to determine the viewing angles, black hole, stellar and dark matter (DM) masses and the stellar orbit distribution of galaxies. Our model uses a 5D orbital starting space to ensure a representative set of stellar trajectories adaptable to the integrals-of-motion space and it is designed to deal with non-parametric stellar and DM densities. SMART’s efficiency is demonstrated by application to a realistic N-body merger simulation including supermassive black holes which we model from five different projections. When providing the true viewing angles, 3D stellar luminosity profile and normalized DM halo, we can (i) reproduce the intrinsic velocity moments and anisotropy profile with a precision of $\sim 1\%$ and (ii) recover the black hole mass, stellar mass-to-light ratio and DM normalization to better than a few percent accuracy. This precision is smaller than the currently discussed differences between initial-stellar-mass functions and scatter in black hole scaling relations. Further tests with toy models suggest that the recovery of the anisotropy in triaxial galaxies is almost unique when the potential is known and full LOSVDs are fitted. We show that orbit models even allow the reconstruction of full intrinsic velocity distributions, which contain more information than the classical anisotropy parameter. Surprisingly, the orbit library for the analysed N-body simulation’s gravitational potential contains orbits with net rotation around the intermediate axis that is stable over some Gyrs.

Cosmic Distributions of Stellar Tidal Disruptions by Massive Black Holes at Galactic Centers

Abstract Stars can be consumed (either tidally disrupted or swallowed whole) by massive black holes (MBHs) at galactic centers when they move into the vicinity of the MBHs. In this study, we investigate the rates of stellar consumption by central MBHs and their cosmic distributions, including the effects of triaxial galaxy shapes in enhancing the reservoir of low-angular-momentum stars and incorporating realistic galaxy distributions. We find that the enhancement in the stellar consumption rates due to triaxial galaxy shapes can be significant, by a factor of ∼3 for MBH mass – and up to more than one order of magnitude for . Only for are the stellar consumption rates significantly higher in galaxies with steeper inner surface brightness profiles. The average (per galaxy) stellar consumption rates correlate with central MBH masses positively for and negatively for . The volumetric stellar tidal disruption rates are for MBHs in the mass range 105– at z = 0; and the volumetric stellar consumption rates by MBHs with higher masses are , which can be the stellar tidal disruption rate if the high-mass BHs are extremely spinning Kerr BHs or the rate of being swallowed if those BHs are Schwarzschild ones. The volumetric stellar consumption rates decrease with increasing redshift, and the decrease is relatively mild for – and stronger for higher . Most of the stellar tidal disruption events (TDEs) at z = 0 occur in galaxies with mass , and about 1%–2% of the TDEs can occur in high-mass galaxies with .

Numerical modelling of X-shaped radio galaxies using back-flow model

AbstractThe focus of this work is to comprehensively understand hydro-dynamical back-flows and their role in dynamics and non-thermal spectral signatures particularly during the initial phase of X-shaped radio galaxies. In this regard, we have performed axisymmetric (2D) and three dimensional (3D) simulations of relativistic magneto-hydrodynamic jet propagation from tri-axial galaxies. High-resolution dynamical modelling of axisymmetric jets has demonstrated the effect of magnetic field strengths on lobe and wing formation. Distinct X-shape formation due to back-flow and pressure gradient of ambient is also observed in our 3D dynamical run. Furthermore, the effect of radiative losses and diffusive shock acceleration on the particle spectral evolution is demonstrated, which particularly highlights how crucial their contributions are in the emission signature of these galaxies. This imparts a significant effect on the galaxy’s equipartition condition, indicating that one must be careful in extending its use in estimating other parameters, as the criterion evolves with time.

How to break the density-anisotropy degeneracy in spherical stellar systems

We present a new non-parametric Jeans code, GravSphere, that recovers the density $\rho(r)$ and velocity anisotropy $\beta(r)$ of spherical stellar systems, assuming only that they are in a steady-state. Using a large suite of mock data, we confirm that with only line-of-sight velocity data, GravSphere provides a good estimate of the density at the projected stellar half mass radius, $\rho(R_{1/2})$, but is not able to measure $\rho(r)$ or $\beta(r)$, even with 10,000 tracer stars. We then test three popular methods for breaking this $\rho-\beta$ degeneracy: using multiple populations with different $R_{1/2}$; using higher order `Virial Shape Parameters' (VSPs); and including proper motion data. We find that two populations provide an excellent recovery of $\rho(r)$ in-between their respective $R_{1/2}$. However, even with a total of $\sim 7,000$ tracers, we are not able to well-constrain $\beta(r)$ for either population. By contrast, using 1000 tracers with higher order VSPs we are able to measure $\rho(r)$ over the range $0.5 < r/R_{1/2} < 2$ and broadly constrain $\beta(r)$. Including proper motion data for all stars gives an even better performance, with $\rho$ and $\beta$ well-measured over the range $0.25 < r/R_{1/2} < 4$. Finally, we test GravSphere on a triaxial mock galaxy that has axis ratios typical of a merger remnant, $[1:0.8:0.6]$. In this case, GravSphere can become slightly biased. However, we find that when this occurs the data are poorly fit, allowing us to detect when such departures from spherical symmetry become problematic.

THE FINAL-PARSEC PROBLEM IN THE COLLISIONLESS LIMIT

A binary supermassive black hole loses energy via ejection of stars in a galactic nucleus, until emission of gravitational waves becomes strong enough to induce rapid coalescence. Evolution via the gravitational slingshot requires that stars be continuously supplied to the binary, and it is known that in spherical galaxies the reservoir of such stars is quickly depleted, leading to stalling of the binary at parsec-scale separations. Recent N-body simulations of galaxy mergers and isolated nonspherical galaxies suggest that this stalling may not occur in less idealized systems. However, it remains unclear to what degree these conclusions are affected by collisional relaxation, which is much stronger in the numerical simulations than in real galaxies. In this study, we present a novel Monte Carlo method that can efficiently deal with both collisional and collisionless dynamics, and with galaxy models having arbitrary shapes. We show that without relaxation, the final-parsec problem may be overcome only in triaxial galaxies. Axisymmetry is not enough, but even a moderate departure from axisymmetry is sufficient to keep the binary shrinking. We find that the binary hardening rate is always substantially lower than the maximum possible, "full-loss-cone" rate, and that it decreases with time, but that stellar-dynamical interactions are nevertheless able to drive the binary to coalescence on a timescale <=1 Gyr in any triaxial galaxy.

Formation of elongated galaxies with low masses at high redshift

We report the identification of elongated (triaxial or prolate) galaxies in cosmological simulations at $z\simeq2$. These are preferentially low-mass galaxies ($M_s \le 10^{9.5} \ M_\odot$), residing in dark-matter (DM) haloes with strongly elongated inner parts, a common feature of high-redshift DM haloes in the $\Lambda$CDM cosmology. Feedback slows formation of stars at the centres of these halos, so that a dominant and prolate DM distribution gives rise to galaxies elongated along the DM major axis. As galaxies grow in stellar mass, stars dominate the total mass within the galaxy half-mass radius, making stars and DM rounder and more oblate. A large population of elongated galaxies produces a very asymmetric distribution of projected axis ratios, as observed in high-z galaxy surveys. This indicates that the majority of the galaxies at high redshifts are not discs or spheroids but rather galaxies with elongated morphologies.

Applying Schwarzschild's orbit superposition method to barred or non-barred disc galaxies

We present an implementation of the Schwarzschild orbit superposition method which can be used for constructing self-consistent equilibrium models of barred or non-barred disc galaxies, or of elliptical galaxies with figure rotation. This is a further development of the publicly available code SMILE; its main improvements include a new efficient representation of an arbitrary gravitational potential using two-dimensional spline interpolation of Fourier coefficients in the meridional plane, as well as the ability to deal with rotation of the density profile and with multicomponent mass models. We compare several published methods for constructing composite axisymmetric disc--bulge--halo models and demonstrate that our code produces the models that are closest to equilibrium. We also apply it to create models of triaxial elliptical galaxies with cuspy density profiles and figure rotation, and find that such models can be found and are stable over many dynamical times in a wide range of pattern speeds and angular momenta, covering both slow- and fast-rotator classes. We then attempt to create models of strongly barred disc galaxies, using an analytic three-component potential, and find that it is not possible to make a stable dynamically self-consistent model for this density profile. Finally, we take snapshots of two N-body simulations of barred disc galaxies embedded in nearly-spherical haloes, and construct equilibrium models using only information on the density profile of the snapshots. We demonstrate that such reconstructed models are in near-stationary state, in contrast with the original N-body simulations, one of which displayed significant secular evolution.

Time-Frequency Analysis of Asymmetric Triaxial Galaxy Model Including Effect of Spherical Dark Halo Component

A method of time-frequency analysis (TFA) based on wavelets is applied to study the phase space structure of three-dimensional asymmetric triaxial galaxy enclosed by spherical dark halo component. The investigation is carried out in the presence and absence of dark halo component. Time-frequency analysis is based on the extraction of instantaneous frequency from the phase of the continuous wavelet transform. This method is comparatively fast and reliable. This method can differentiate periodic from quasi-periodic, chaotic sticky from chaotic non-sticky, ordered from chaotic and also, it can accurately determine the time interval of the resonance trapping and transitions too. Apart from that, the phenomenon of transient chaos can be explained with the help of time-frequency analysis. Comparison with the method of total angular momentum (denoted as Ltot) proposed recently is also presented.

Andromeda II as a merger remnant

Abstract Using N-body simulations, we study the origin of prolate rotation observed in the kinematic data for Andromeda II (And II), a dwarf spheroidal satellite of M31. We propose an evolutionary model for the origin of And II involving a merger between two discy dwarf galaxies with different disc scalelengths. The dwarfs are placed on a radial orbit towards each other with their angular momenta inclined by 90 deg. The merger remnant forms a stable triaxial galaxy with rotation only around the longest axis whose origin is naturally explained as due to the symmetry of the initial configuration and the conservation of angular momentum components along the direction of the merger. The stars originating from the two dwarfs show significantly different surface density profiles while having very similar kinematics as required by the data. We also study an alternative scenario for the formation of And II, via tidal stirring of a discy dwarf. While intrinsic rotation occurs naturally in this model as a remnant of the initial rotation of the disc, it is mostly around the shortest axis of the stellar component. We conclude that the velocity distribution in And II is much more naturally explained by a scenario involving a past merger.

Evolution of binary supermassive black holes and the final-parsec problem

AbstractI review the evolution of binary supermassive black holes and focus on the stellar-dynamical mechanisms that may help to overcome the final-parsec problem – the possible stalling of the binary at a separation much larger than is required for an efficient gravitational wave emission. RecentN-body simulations have suggested that a departure from spherical symmetry in the nucleus of the galaxy may keep the rate of interaction of stars with the binary at a high enough level so that the binary continues to shrink rather rapidly. However, a major problem of all these simulations is that they do not probe the regime where collisionless effects are dominant – in other words, the number of particles in the simulation is still not sufficient to reach the asymptotic behavior of the system. I present a novel Monte Carlo method for simulating both collisional and collisionless evolution of non-spherical stellar systems, and apply it for the problem of binary supermassive black hole evolution. I show that in triaxial galaxies the final-parsec problem is largely non-existent, while in the axisymmetric case it seems to still exist in the limit of purely collisionless regime relevant for real galaxies, but disappears in theN-body simulations where the feasible values ofNare still too low to get rid of collisional effects.

THE FINAL-PARSEC PROBLEM IN NONSPHERICAL GALAXIES REVISITED

We consider the evolution of supermassive black hole binaries at the center of spherical, axisymmetric, and triaxial galaxies, using direct N-body integrations as well as analytic estimates. We find that the rates of binary hardening exhibit a significant N-dependence in all the models, at least for N in the investigated range of 10^5<=N<=10^6. Binary hardening rates are also substantially lower than would be expected if the binary loss cone remained full, as it would be if the orbits supplying stars to the binary were being efficiently replenished. The difference in binary hardening rates between the spherical and nonspherical models is less than a factor of two even in the simulations with the largest N. By studying the orbital populations of our models, we conclude that the rate of supply of stars to the binary via draining of centrophilic orbits is indeed expected to be much lower than the full-loss-cone rate, consistent with our simulations. We argue that the binary's evolution in the simulations is driven in roughly equal amounts by collisional and collisionless effects, even at the highest N-values currently accessible. While binary hardening rates would probably reach a limiting value for large N, our results suggest that we cannot approach that rate with currently available algorithms and computing hardware. The extrapolation of results from N-body simulations to real galaxies is therefore not straightforward, casting doubt on recent claims that triaxiality or axisymmetry alone are capable of solving the final-parsec problem in gas-free galaxies.

Loss-cone dynamics

Supermassive black holes can capture or disrupt stars that come sufficiently close. This paper reviews the dynamical processes by which stars or stellar remnants are placed onto loss-cone orbits and the implications for feeding rates. The capture rate is well defined for spherical galaxies with nuclear relaxation times that are shorter than the galaxy's age. However, even the dense nucleus of the Milky Way may be less than one relaxation time old, and this is certainly the case for more massive galaxies; the capture rate in such galaxies is an initial-value problem with poorly-known initial conditions and the rate can be much higher, or much lower, than the rate in a collisionally-relaxed nucleus. In nonspherical (axisymmetric, triaxial) galaxies, torquing of orbits by the mean field can dominate perturbations due to random encounters, leading to much higher capture rates than in the spherical geometry, particularly in (massive) galaxies with long central relaxation times. Relativistic precession plays a crucial role in mediating the capture of compact remnants from regions very near to the black hole, by destroying the orbital correlations that would otherwise dominate the torques. The complex dynamics of relativistic loss cones are not yet well enough understood for accurate estimates of compact-object (‘EMRI’) capture rates to be made.

Orbital orientation evolution of massive binary black holes at the centres of non-spherical galaxies

At the centre of a spherical and kinematically isotropic galaxy, the orientation of a massive binary black hole (BBH) orbit (i.e. the direction of the BBH orbital angular momentum) undergoes a random walk. If the stars in a spherical system have non-zero total angular momentum, the BBH orbital orientation evolves towards aligning with the total stellar angular momentum direction. In this article, we show that a triaxial galaxy has an alignment-erasing effect, i.e. the alignment of the BBH orientation towards the galaxy rotation axis can be decreased significantly or erased. We also show that in a non-rotating axisymmetric galaxy the BBH orbital orientation evolves towards the axisymmetric axis and precesses about it in a retrograde direction. Our results provide a step towards understanding the spin orientation of the final merged black hole (and hence the probable orientation of any jet produced) within its host galaxy and may help to constrain the recoil velocity of the merged black hole arising from gravitational wave radiation as well.

STRUCTURAL EVOLUTION OF EARLY-TYPE GALAXIES TOz= 2.5 IN CANDELS

Projected axis ratio measurements of 880 early-type galaxies at redshifts 1<z<2.5 selected from CANDELS are used to reconstruct and model their intrinsic shapes. The sample is selected on the basis of multiple rest-frame colors to reflect low star-formation activity. We demonstrate that these galaxies as an ensemble are dust-poor and transparent and therefore likely have smooth light profiles, similar to visually classified early-type galaxies. Similar to their present-day counterparts, the z>1 early-type galaxies show a variety of intrinsic shapes; even at a fixed mass, the projected axis ratio distributions cannot be explained by the random projection of a set of galaxies with very similar intrinsic shapes. However, a two-population model for the intrinsic shapes, consisting of a triaxial, fairly round population, combined with a flat (c/a~0.3) oblate population, adequately describes the projected axis ratio distributions of both present-day and z>1 early-type galaxies. We find that the proportion of oblate versus triaxial galaxies depends both on the galaxies' stellar mass, and - at a given mass - on redshift. For present-day and z<1 early-type galaxies the oblate fraction strongly depends on galaxy mass. At z>1 this trend is much weaker over the mass range explored here (10^10<M*/M_sun<10^11), because the oblate fraction among massive (M*~10^11 M_sun) was much higher in the past: 0.59+-0.10 at z>1, compared to 0.20+-0.02 at z~0.1. In contrast, the oblate fraction among low-mass early-type galaxies (log(M*/M_sun)<10.5) increased toward the present, from 0.38+-0.11 at z>1 to 0.72+-0.06 at z=0. [Abridged]