Prolate Rotation Research Articles

A stream come true: Connecting tidal tails, shells, streams, and planes with galaxy kinematics and formation history

Context. The rapidly improving quality and resolution of both low surface brightness observations and cosmological simulations of galaxies enable us to address the important question of how the formation history is imprinted in the outer unrelaxed regions of galaxies, and to inspect the correlations of these imprints with another tracer of galaxy formation, the internal kinematics. Aims. Using the hydrodynamical cosmological simulation called Magneticum Pathfinder, we identified tidal tails, shells, streams, and satellite planes, and connected them to the amount of rotational support and the formation histories of the host galaxies. This presents the first combined statistical census considering all these four types of features in hydrodynamical cosmological simulations. Methods. Tidal features were visually classified from a three-dimensional rendering of the simulated galaxies by several scientists independently. Only features that were identified by at least half of the participating individuals were considered to be existing features. The data on satellite planes and kinematic properties of the simulated galaxies were taken from previous work. The results were compared to observations, especially from the MATLAS survey. Results. Generally, prominent features are much more common around elliptical than around disk galaxies. Shells are preferentially found around kinematically slowly rotating galaxies in both simulations and observations, while streams can be found around all types of galaxies, with a slightly higher probability to be present around less rotationally supported galaxies. Tails and satellite planes, however, appear independently of the internal kinematics of the central galaxy, indicating that they are formed through processes that have not (yet) affected the internal kinematics. Prolate rotators have the overall highest probability to exhibit tidal features, but the highest likelihood for a specific type of feature is found for galaxies with kinematically distinct cores (KDCs), nearly 20% of which exhibit streams. Conclusions. As shells are formed through radial merger events while streams are remnants of circular merger infall, this suggests that the orbital angular momentum of the merger event plays a more crucial role in transforming the host galaxy than previously anticipated. The existence of a shell around a given slow rotator furthermore is a sign of a radial merger formation for this particular slow rotator because one-third of the galaxies with a shell were transformed into slow rotators by the merger event that also caused the shells. The appearance of a stream around a KDC is a direct indicator for the multiple merger formation pathway of that KDC as opposed to the major merger pathway.

Reorientation Rates of Structural and Kinematic Axes in Simulated Massive Galaxies and the Origins of Prolate Rotation

In this work, we analyze a sample of ∼4000 massive (M * ≥ 1011 M ⊙ at z = 0) galaxies in TNG300, the (300Mpc)3 box of the IllustrisTNG simulation suite. We characterize the shape and kinematics of these galaxies with a focus on the kinematic misalignment (Ψint) between the angular momentum (AM) and morphological major axis. We find that the traditional purely shape- or kinematics-based classifications are insufficient to characterize the diversity of our sample and define a new set of classes based on the rates of change of the galaxies’ morphological and kinematic axes. We show that these classes are mostly stable over time and correspond to six distinct populations of galaxies: the rapid AM reorienters (58% of our sample), unsettled galaxies (10%), spinning disks (10%), twirling cigars (16%), misaligned slow reorienters (3%), and regular prolate rotators (galaxies that display major axis rotation; 2%). We demonstrate that the most-recent significant (mass-ratio μ > 1/10) mergers of these galaxies are the primary cause for their present-day properties and find that these mergers are best characterized at the point of the satellite’s final infall—that is, much closer to the final coalescence than has been previously thought. We show that regular prolate rotators evolve from spinning disk progenitors that experience a radial merger along their internal AM direction. Finally, we argue that these regular prolate rotators are distinct from the similarly sized population of rapid AM reorienters with large Ψint, implying that a large Ψint is not a sufficient condition for major axis rotation.

Origin of stellar prolate rotation in a cosmologically simulated faint dwarf galaxy

ABSTRACT Stellar prolate rotation in dwarf galaxies is rather uncommon, with only two known galaxies in the Local Group showing such feature (Phoenix and And II). Cosmological simulations show that in massive early-type galaxies prolate rotation likely arises from major mergers. However, the origin of such kinematics in the dwarf galaxies regime has only been explored using idealized simulations. Here, we made use of hydrodynamical cosmological simulations of dwarfs galaxies with stellar mass between 3 × 105 and 5 × 108 M⊙ to explore the formation of prolate rotators. Out of 27 dwarfs, only one system showed clear rotation around the major axis, whose culprit is a major merger at $z$ = 1.64, which caused the transition from an oblate to a prolate configuration. Interestingly, this galaxy displays a steep metallicity gradient, reminiscent of the one measured in Phoenix and And II: this is the outcome of the merger event that dynamically heats old, metal-poor stars, and of the centrally concentrated residual star formation. Major mergers in dwarf galaxies offer a viable explanation for the formation of such peculiar systems, characterized by steep metallicity gradients and prolate rotation.

Ubiquitous signs of interactions in early-type galaxies with prolate rotation

Context. A small fraction of early-type galaxies (ETGs) show prolate rotation; that is, they rotate around their long photometric axis. In simulations, certain configurations of galaxy mergers are known to produce this type of rotation. Aims. We investigate the association of prolate rotation and signs of galaxy interactions among the observed galaxies. Methods. We collected a sample of 19 nearby ETGs with distinct prolate rotation from the literature and inspected their ground-based deep optical images for interaction signs – 18 in archival images and 1 in a new image obtained with the Milanković telescope. Results. Tidal tails, shells, disturbed asymmetric stellar halos, or ongoing interactions are present in all the 19 prolate rotators. Comparing this with the frequency of tidal disturbance among the general sample of ETGs of a roughly similar mass range and surface-brightness limit, we estimate that the chance probability of such an observation is only 0.00087. We also find a significant overabundance of prolate rotators that are hosting multiple stellar shells. The visible tidal features imply a relatively recent galaxy interaction. That agrees with the Illustris large-scale cosmological hydrodynamical simulation, where prolate rotators are predominantly formed in major mergers during the last 6 Gyr. In the appendix, we present the properties of an additional galaxy, NGC 7052, a prolate rotator for which no deep images are available, but for which an HST image revealed the presence of a prominent shell, which had not been reported before.

Multiple chemodynamic stellar populations of the Ursa Minor dwarf spheroidal galaxy

ABSTRACT We present a Bayesian method to identify multiple (chemodynamic) stellar populations in dwarf spheroidal galaxies (dSphs) using velocity, metallicity, and positional stellar data without the assumption of spherical symmetry. We apply this method to a new Keck/Deep Imaging Multi-Object Spectrograph (DEIMOS) spectroscopic survey of the Ursa Minor (UMi) dSph. We identify 892 likely members, making this the largest UMi sample with line-of-sight velocity and metallicity measurements. Our Bayesian method detects two distinct chemodynamic populations with high significance (in logarithmic Bayes factor, ln B ∼ 33). The metal-rich ([Fe/H] = −2.05 ± 0.03) population is kinematically colder (radial velocity dispersion of $\sigma _v=4.9_{-1.0}^{+0.8} \, \mathrm{km} \, \mathrm{s}^{-1}$) and more centrally concentrated than the metal-poor ($[{\rm Fe/H}]=-2.29_{-0.06}^{+0.05}$) and kinematically hotter population ($\sigma _v =11.5_{-0.8}^{+0.9}\, \mathrm{km} \, \mathrm{s}^{-1}$). Furthermore, we apply the same analysis to an independent Multiple Mirror Telescope (MMT)/Hectochelle data set and confirm the existence of two chemodynamic populations in UMi. In both data sets, the metal-rich population is significantly flattened (ϵ = 0.75 ± 0.03) and the metal-poor population is closer to spherical ($\epsilon =0.33_{-0.09}^{+0.12}$). Despite the presence of two populations, we are able to robustly estimate the slope of the dynamical mass profile. We found hints for prolate rotation of order ${\sim}2 \, \mathrm{km} \, \mathrm{s}^{-1}$ in the MMT data set, but further observations are required to verify this. The flattened metal-rich population invalidates assumptions built into simple dynamical mass estimators, so we computed new astrophysical dark matter annihilation (J) and decay profiles based on the rounder, hotter metal-poor population and inferred $\log _{10}{(J(0{^{\circ}_{.}}5)/{\rm GeV^{2} \, cm^{-5}})}\approx 19.1$ for the Keck data set. Our results paint a more complex picture of the evolution of UMi than previously discussed.

Understanding nuclear dynamics in the SD-pair shell model: From pre-vibration to collective rotation

Influences of some important residual interactions considered in the shell model on nuclear dynamics and the shape phase evolution have been investigated in the SD-pair shell model for an identical nucleon system. The results indicate that the collective modes including the spherical vibration, prolate rotation, and X(5)-like critical dynamics are mainly driven by the competition between the monopole-pairing interaction and the quadrupole-quadrupole interaction, and the main features of the U(5)-SU(3) phase transition can be well reproduced within the SD-pair model. In addition, it is also shown that the quadrupole-pairing interaction cannot be ignored in reproducing an ideal rotational spectrum with a realistic single-particle configuration.

Prospects for recovering galaxy intrinsic shapes from projected quantities

The distribution of three dimensional intrinsic galaxy shapes has been a longstanding open question. The difficulty stems from projection effects meaning one must rely on statistical methods applied to galaxy samples to infer intrinsic shape distributions. Theoretical work using analytical galaxy potentials suggests a relationship between galaxy intrinsic shape (as defined by its "triaxiality", in practice a proxy for how prolate a galaxy is) and the intrinsic misalignment angle between kinematic and morphological axes ($\Psi_{\rm int}$). This relationship reduces the number of unknowns, providing more reliable inferred intrinsic shape distributions than methods using photometry alone. Here we explore the connection between intrinsic shape and stellar kinematics using cosmological hydrodynamical simulations from the Illustris project. The strongest relationship we find is that galaxy intrinsic flattening is correlated with specific angular momentum (j) with high j galaxies being flatter than galaxies with low specific angular momentum. Our analysis shows that, although the majority of kinematically misaligned galaxies exhibit prolate shapes, examples of kinematically aligned prolate galaxies are also present. Clearly a direct correspondence between prolate shape and minor-axis rotation (often referred to as "prolate rotation") is not present in Illustris. Thus, we demonstrate that the assumption of a simple relationship between $\Psi_{\rm int}$ and intrinsic shape commonly employed in shape recovery studies is not valid for Illustris galaxies. We suggest improvements on the method as well as some alternative methods for future work in this area.

Dark matter distribution and dynamics of dwarf spheroidal galaxies

AbstractI review the current status of dynamical modelling of dwarf spheroidal galaxies focusing on estimates of their dark matter content. Starting with the simplest methods using the velocity dispersion profiles I discuss the inherent issues of mass-anisotropy degeneracy and contamination by unbound stars. I then move on to methods of increasing complexity, aiming to break the degeneracy, up to recent applications of the Schwarzschild orbit superposition method. The dynamical modelling is placed in the context of possible scenarios for the formation of dwarf spheroidal galaxies, including the tidal stirring model and mergers of dwarf galaxies. The two scenarios are illustrated with examples from simulations: a comparison between the tidal evolution of dwarfs with cuspy and cored dark matter profiles and the formation of a dwarf spheroidal with prolate rotation.

The intricate link between galaxy dynamics and intrinsic shape (or why so-called prolate rotation is a misnomer)

AbstractMany recent integral field spectroscopy (IFS) survey teams have used stellar kinematic maps combined with imaging to statistically infer the underlying distributions of galaxy intrinsic shapes. With now several IFS samples at our disposal, the method, which was originally proposed by M. Franx and collaborators in 1991, is gaining in popularity, having been so far applied to ATLAS3D, SAMI, MANGA and MASSIVE. We present results showing that a commonly assumed relationship between dynamical and intrinsic shape alignment does not hold in Illustris, affecting our ability to recover accurate intrinsic shape distributions. A further implication is that so-called “prolate rotation”, where the bulk of stars in prolate galaxies are thought to rotate around the projected major axis, is a misnomer.

SDSS-IV MaNGA: The Intrinsic Shape of Slow Rotator Early-type Galaxies

Abstract By inverting the distributions of galaxies’ apparent ellipticities and misalignment angles (measured around the projected half-light radius R e) between their photometric and kinematic axes, we study the intrinsic shape distribution of 189 slow rotator early-type galaxies with stellar masses 2 × 1011 M ⊙ < M * < 2 × 1012 M ⊙, extracted from a sample of about 2200 galaxies with integral-field stellar kinematics from the data release 14 (DR14) of the fourth-generation Sloan Digital Sky Survey IV (SDSS-IV) Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) integral field unit (IFU) survey. Thanks to the large sample of slow rotators, Graham et al. showed that there is clear structure in the misalignment angle distribution, with two peaks at both 0° and 90° misalignment (characteristic of oblate and prolate rotation, respectively). Here we invert the observed distribution from Graham et al. The large sample allows us to go beyond the known fact that slow rotators are weakly triaxial and to place useful constraints on their intrinsic triaxiality distribution (around 1 R e) for the first time. The shape inversion is generally non-unique. However, we find that, for a wide set of model assumptions, the observed distribution clearly requires a dominant triaxial-oblate population. For some of our models, the data suggest a minor triaxial-prolate population, but a dominant prolate population is ruled out.

Kinematics of simulated galaxies – I. Connecting dynamical and morphological properties of early-type galaxies at different redshifts

State-of-the-art integral field surveys like $\mathrm{ATLAS^{3D}}$, SLUGGS, CALIFA, SAMI, and MaNGA provide large data sets of kinematical observations of early-type galaxies (ETGs), yielding constraints on the formation of ETGs. Using the cosmological hydrodynamical \textit{Magneticum Pathfinder} simulations, we investigate the paradigm of fast and slow rotating ETGs in a fully cosmological context. We show that the ETGs within the \textit{Magneticum} simulation are in remarkable agreement with the observations, revealing fast and slow rotators quantified by the angular momentum proxy $\lambda_{\mathrm{R}}$ and the flattening $\epsilon$ with the observed prevalence. Taking full advantage of the three-dimensional data, we demonstrate that the dichotomy between fast and slow rotating galaxies gets enhanced, showing an upper and lower population separated by an underpopulated region in the edge-on $\lambda_{\mathrm{R}}$-$\epsilon$ plane. Following the evolution of the $\lambda_{\mathrm{R}}$-$\epsilon$ plane through cosmic time, we find that, while the upper population is already in place at $z=2$, the lower population gets statistically significant below $z=1$ with a gradual increase. At least $50\%$ of the galaxies transition from fast to slow rotators on a short timescale, in most cases associated to a significant merger event. Furthermore, we connect the $M_{*}$-$j_{*}$ plane, quantified by the $b$-value, with the $\lambda_{\mathrm{R}}$-$\epsilon$ plane, revealing a strong correlation between the position of a galaxy in the $\lambda_{\mathrm{R}}$-$\epsilon$ plane and the $b$-value. Going one step further, we classify our sample based on features in their velocity map, finding all five common kinematic groups, also including the recently observed group of prolate rotators, populating distinct regions in the $\lambda_{\mathrm{R}}$-$b$ plane.

The major merger origin of the Andromeda II kinematics

AbstractProlate rotation (i.e. rotation around the long axis) has been reported for two Local Group dwarf galaxies: Andromeda II, a dwarf spheroidal satellite of M31, and Phoenix, a transition type dwarf galaxy. The prolate rotation may be an exceptional indicator of a past major merger between dwarf galaxies. We showed that this type of rotation cannot be obtained in the tidal stirring scenario, in which the satellite is transformed from disky to spheroidal by tidal forces of the host galaxy. However, we successfully reproduced the observed Andromeda II kinematics in controlled, self-consistent simulations of mergers between equal-mass disky dwarf galaxies on a radial or close-to-radial orbit. In simulations including gas dynamics, star formation and ram pressure stripping, we are able to reproduce more of the observed properties of Andromeda II: the unusual rotation, the bimodal star formation history and the spatial distribution of the two stellar populations, as well as the lack of gas. We support this scenario by demonstrating the merger origin of prolate rotation in the cosmological context for sufficiently resolved galaxies in the Illustris large-scale cosmological hydrodynamical simulation.

Galaxies with Prolate Rotation in Illustris

Abstract Tens of early-type galaxies have been recently reported to possess prolate rotation of the stellar component, i.e., a significant amount of rotation around the major axis, including two cases in the Local Group. Although expected theoretically, this phenomenon is rarely observed and remains elusive. We study its origin using the population of well-resolved galaxies in the Illustris cosmological simulation. We identify 59 convincing examples of prolate rotators at the present time, more frequently among more massive galaxies, with the number varying very little with redshift. We follow their evolution back in time using the main progenitor branch galaxies of the Illustris merger trees. We find that the emergence of prolate rotation is strongly correlated with the time of the last significant merger that the galaxy experienced, although other evolutionary paths leading to prolate rotation are also possible. The transition to prolate rotation most often happens around the same time as the transition to prolate shape of the stellar component. The mergers leading to prolate rotation have slightly more radial orbits and higher mass ratios, and they occur at more recent times than mergers in the reference sample of twin galaxies we construct for comparison. However, they cover a wide range of initial conditions in terms of the mass ratio, merger time, radiality of the progenitor orbits, and relative orientations of progenitor spins with respect to the orbital angular momenta. About half of our sample of prolate rotators was created during gas-rich mergers, and the newly formed stars usually support prolate rotation.

Elliptical galaxies spinning on end

Galaxy Structure Most elliptical galaxies have a flattened shape and rotate around their shortest axis, like a wheel. However, some galaxies, known as prolate rotators, are elongated and spin around their longest axis, like a spindle. Tsatsi et al. searched a large survey of galaxy kinematics to discover eight previously unknown examples. They estimate that around 10% of elliptical galaxies are prolate rotators, but the fraction is higher at higher masses. They use simulations to show that prolate rotators can form if two massive spiral galaxies merge in a polar orientation, potentially explaining the evolutionary history of these systems. Astron. Astrophys. 606 , A62 (2017).

CALIFA reveals prolate rotation in massive early-type galaxies: A polar galaxy merger origin?

We present new evidence for eight early-type galaxies (ETGs) from the CALIFA Survey that show clear rotation around their major photometric axis ("prolate rotation"). These are LSBCF560-04, NGC 0647, NGC 0810, NGC 2484, NGC 4874, NGC 5216, NGC 6173 and NGC 6338. Including NGC 5485, a known case of an ETG with stellar prolate rotation, as well as UGC 10695, a further possible candidate for prolate rotation, we report ten CALIFA galaxies in total that show evidence for such a feature in their stellar kinematics. Prolate rotators correspond to ~9% of the volume-corrected sample of CALIFA ETGs, a fraction much higher than previously reported. We find that prolate rotation is more common among the most massive ETGs. We investigate the implications of these findings by studying N-body merger simulations, and show that a prolate ETG with rotation around its major axis could be the result of a major polar merger, with the amplitude of prolate rotation depending on the initial bulge-to-total stellar mass ratio of its progenitor galaxies. Additionally, we find that prolate ETGs resulting from this formation scenario show a correlation between their stellar line-of-sight velocity and higher order moment h_3, opposite to typical oblate ETGs, as well as a double peak of their stellar velocity dispersion along their minor axis. Finally, we investigate the origin of prolate rotation in polar galaxy merger remnants. Our findings suggest that prolate rotation in massive ETGs might be more common than previously expected, and can help towards a better understanding of their dynamical structure and formation origin.

Prolate rotation and metallicity gradient in the transforming dwarf galaxy Phoenix

Transition type dwarf galaxies are thought to be systems undergoing the process of transformation from a star-forming into a passively evolving dwarf, which makes them particularly suitable to study evolutionary processes driving the existence of different dwarf morphological types. Here we present results from a spectroscopic survey of ~200 individual red giant branch stars in the Phoenix dwarf, the closest transition type with a comparable luminosity to "classical" dwarf galaxies. We measure a systemic heliocentric velocity V = -21.2 km/s. Our survey reveals the clear presence of prolate rotation, which is aligned with the peculiar spatial distribution of the youngest stars in Phoenix. We speculate that both features might have arisen from the same event, possibly an accretion of a smaller system. The evolved stellar population of Phoenix is relatively metal-poor (<[Fe/H]> = -1.49+/-0.04 dex) and shows a large metallicity spread ($\sigma_{\rm [Fe/H]} = 0.51\pm0.04$\,dex), with a pronounced metallicity gradient of -0.13+/-0.01 dex per arcmin similar to luminous, passive dwarf galaxies. We also report a discovery of an extremely metal-poor star candidate in Phoenix and discuss the importance of correcting for spatial sampling when interpreting the chemical properties of galaxies with metallicity gradients. This study presents a major leap forward in our knowledge of the internal kinematics of the Phoenix transition type dwarf galaxy, and the first wide area spectroscopic survey of its metallicity properties.

THE ORIGIN OF PROLATE ROTATION IN DWARF SPHEROIDAL GALAXIES FORMED BY MERGERS OF DISKY DWARFS

Motivated by the discovery of prolate rotation of stars in Andromeda II, a dwarf spheroidal companion of M31, we study its origin via mergers of disky dwarf galaxies. We simulate merger events between two identical dwarfs changing the initial inclination of their disks with respect to the orbit and the amount of orbital angular momentum. On radial orbits the amount of prolate rotation in the merger remnants correlates strongly with the inclination of the disks and is well understood as due to the conservation of the angular momentum component of the disks along the merger axis. For non-radial orbits prolate rotation may still be produced if the orbital angular momentum is initially not much larger than the intrinsic angular momentum of the disks. The orbital structure of the remnants with significant rotation is dominated by box orbits in the center and long-axis tubes in the outer parts. The frequency analysis of stellar orbits in the plane perpendicular to the major axis reveals the presence of two families roughly corresponding to inner and outer long-axis tubes. The fraction of inner tubes is largest in the remnant forming from disks oriented most vertically initially and is responsible for the boxy shape of the galaxy. We conclude that prolate rotation results from mergers with a variety of initial conditions and no fine tuning is necessary to reproduce this feature. We compare the properties of our merger remnants to those of dwarfs resulting from the tidal stirring scenario and the data for Andromeda II.

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.

STELLAR KINEMATICS OF THE ANDROMEDA II DWARF SPHEROIDAL GALAXY

We present kinematical profiles and metallicity for the M31 dwarf spheroidal (dSph) satellite galaxy Andromeda II (And II) based on Keck DEIMOS spectroscopy of 531 red giant branch stars. Our kinematical sample is among the largest for any M31 satellite and extends out to two effective radii (r_eff = 5.3' = 1.1 kpc). We find a mean systemic velocity of -192.4+-0.5 km/s and an average velocity dispersion of sigma_v = 7.8+-1.1 km/s. While the rotation velocity along the major axis of And II is nearly zero (<1 km/s), the rotation along the minor axis is significant with a maximum rotational velocity of v_max=8.6+-1.8 km/s. We find a kinematical major axis, with a maximum rotational velocity of v_max=10.9+-2.4 km/s, misaligned by 67 degrees to the isophotal major axis. And II is thus the first dwarf galaxy with evidence for nearly prolate rotation with a v_max/sigma_v = 1.1, although given its ellipticity of epsilon = 0.10, this object may be triaxial. We measured metallicities for a subsample of our data, finding a mean metallicity of [Fe/H] = -1.39+- 0.03 dex and an internal metallicity dispersion of 0.72+-0.03 dex. We find a radial metallicity gradient with metal-rich stars more centrally concentrated, but do not observe a significant difference in the dynamics of two metallicity populations. And II is the only known dwarf galaxy to show minor axis rotation making it a unique system whose existence offers important clues on the processes responsible for the formation of dSphs.

Theory study of shapes in 187,189Tl nuclei

The nuclear shapes of 187,189Tl were investigated theoretically in this work. The total routhian surface (TRS) calculations were performed for N=106 and 108 isotopes of thallium (Z=81). The single-particle energies were obtained from the deformed Woods-Saxon potential, with the Lipkin-Nogami (LN) treatment of pairing. It is found that the collective oblate rotation coexists with the high-K prolate rotation. Superdeformed prolate rotation is included in this calculation.