Cappellari (2008) presented a flexible and efficient method to model the stellar kinematics of anisotropic axisymmetric and spherical stellar systems. The spherical formalism could be used to model the line-of-sight velocity second moments allowing f
or essentially arbitrary radial variation in the anisotropy and general luminous and total density profiles. Here we generalize the spherical formalism by providing the expressions for all three components of the projected second moments, including the two proper motion components. A reference implementation is now included in the public JAM package available at http://purl.org/cappellari/software
I review our understanding of classic dynamical scaling relations, relating luminosity, size and kinematics of early-type galaxies. Using unbiased determinations of galaxy mass profiles from stellar dynamical models, a simple picture has emerged in w
hich scaling relations are driven by virial equilibrium, accompanied by a trend in the stellar mass-to-light ratio (M/L). This picture confirms the earliest insights. The trend is mainly due to the combined variation of age, metallicity and the stellar initial mass function (IMF). The systematic variations best correlate with the galaxy velocity dispersion, which traces the bulge mass fraction. This indicates a link between bulge growth and quenching of star formation. Dark matter is unimportant within the half-light radius, where the total mass profile is close to isothermal ($rhopropto r^{-2}$).
The distribution of galaxies on the mass-size plane as a function of redshift or environment is a powerful test for galaxy formation models. Here we use integral-field stellar kinematics to interpret the variation of the mass-size distribution in two
galaxy samples spanning extreme environmental densities. The samples are both identically and nearly mass-selected (stellar mass M*>6e9 Msun) and volume-limited. The first consists of nearby field galaxies from the Atlas3D parent sample. The second consists of galaxies in the Coma Cluster (Abell 1656), one of densest environments for which good resolved spectroscopy can be obtained. The mass-size distribution in the dense environment differs from the field one in two ways: (i) spiral galaxies are replaced by bulge-dominated disk-like fast-rotator early-type galaxies (ETGs), which follow the SAME mass-size relation and have the SAME mass distribution as in the field sample; (ii) the slow rotator ETGs are segregated in mass from the fast rotators, with their size increasing proportionally to their mass. A transition between the two processes appears around the stellar mass M_crit=2e11 Msun. We interpret this as evidence for bulge growth (outside-in evolution) and bulge-related environmental quenching dominating at low masses, with little influence from merging, while significant dry mergers (inside-out evolution) and halo-related quenching driving the mass and size growth at the high-mass end. The existence of these two processes naturally explains the diverse size evolution of galaxies of different masses and the separability of mass and environmental quenching.
This is an addendum to the paper by Cappellari (2008, MNRAS, 390, 71), which presented a simple and efficient method to model the stellar kinematics of axisymmetric stellar systems. The technique reproduces well the integral-field kinematics of real
galaxies. It allows for orbital anisotropy (three-integral distribution function), multiple kinematic components, supermassive black holes and dark matter. The paper described the derivation of the projected second moments and we provided a reference software implementation. However only the line-of-sight component was given in the paper. For completeness we provide here all the six projected second moments, including radial velocities and proper motions. We present a test against realistic N-body galaxy simulations.
Since Edwin Hubble introduced his famous tuning fork diagram more than 70 years ago, spiral galaxies and early-type galaxies (ETGs) have been regarded as two distinct families. The spirals are characterized by the presence of disks of stars and gas i
n rapid rotation, while the early-types are gas poor and described as spheroidal systems, with less rotation and often non-axisymmetric shapes. The separation is physically relevant as it implies a distinct path of formation for the two classes of objects. I will give an overview of recent findings, from independent teams, that motivated a radical revision to Hubbles classic view of ETGs. These results imply a much closer link between spiral galaxies and ETGs than generally assumed.
The evolution of masses and sizes of passive (early-type) galaxies with redshift provides ideal constraints to galaxy formation models. These parameters can in principle be obtained for large galaxy samples from multi-band photometry alone. However t
he accuracy of photometric masses is limited by the non-universality of the IMF. Galaxy sizes can be biased at high redshift due to the inferior quality of the imaging data. Both problems can be avoided using galaxy dynamics, and in particular by measuring the galaxies stellar velocity dispersion. Here we provide an overview of the efforts in this direction.
We study the particle energy distribution in the cocoon surrounding Cygnus A, using radio images between 151 MHz and 15 GHz and a 200 ks Chandra ACIS-I image. We show that the excess low frequency emission in the the lobe further from Earth cannot be
explained by absorption or excess adiabatic expansion of the lobe or a combination of both. We show that this excess emission is consistent with emission from a relic counterlobe and a relic counterjet that are being re-energized by compression from the current lobe. We detect hints of a relic hotspot at the end of the relic X-ray jet in the more distant lobe. We do not detect relic emission in the lobe nearer to Earth as expected from light travel-time effects assuming intrinsic symmetry. We determine that the duration of the previous jet activity phase was slightly less than that of the current jet-active phase. Further, we explain some features observed at 5 and 15 GHz as due to the presence of a relic jet.
(Abridged) We present mass models of a sample of 14 spiral and 14 S0 galaxies that constrain their stellar and dark matter content. For each galaxy we derive the stellar mass distribution from near-infrared photometry under the assumptions of axisymm
etry and a constant Ks-band stellar mass-to-light ratio, (M/L)_Ks. To this we add a dark halo assumed to follow a spherically symmetric NFW profile and a correlation between concentration and dark mass within the virial radius, M_DM. We solve the Jeans equations for the corresponding potential under the assumption of constant anisotropy in the meridional plane, beta_z. By comparing the predicted second velocity moment to observed long-slit stellar kinematics, we determine the three best-fitting parameters of the model: (M/L)_Ks, M_DM and beta_z. These simple axisymmetric Jeans models are able to accurately reproduce the wide range of observed stellar kinematics, which typically extend to ~2-3 Re or, equivalently, ~0.5-1 R_25. We find a median stellar mass-to-light ratio at Ks-band of 1.09 (solar units) with an rms scatter of 0.31. We present preliminary comparisons between this large sample of dynamically determined stellar mass-to-light ratios and the predictions of stellar population models. The stellar population models predict slightly lower mass-to-light ratios than we measure. The mass models contain a median of 15 per cent dark matter by mass within an effective radius Re, and 49 per cent within the optical radius R_25. Dark and stellar matter contribute equally to the mass within a sphere of radius 4.1 Re or 1.0 R_25. There is no evidence of any significant difference in the dark matter content of the spirals and S0s in our sample.
We consider the problem of approximating the empirical Shannon entropy of a high-frequency data stream under the relaxed strict-turnstile model, when space limitations make exact computation infeasible. An equivalent measure of entropy is the Renyi e
ntropy that depends on a constant alpha. This quantity can be estimated efficiently and unbiasedly from a low-dimensional synopsis called an alpha-stable data sketch via the method of compressed counting. An approximation to the Shannon entropy can be obtained from the Renyi entropy by taking alpha sufficiently close to 1. However, practical guidelines for parameter calibration with respect to alpha are lacking. We avoid this problem by showing that the random variables used in estimating the Renyi entropy can be transformed to have a proper distributional limit as alpha approaches 1: the maximally skewed, strictly stable distribution with alpha = 1 defined on the entire real line. We propose a family of asymptotically unbiased log-mean estimators of the Shannon entropy, indexed by a constant zeta > 0, that can be computed in a single-pass algorithm to provide an additive approximation. We recommend the log-mean estimator with zeta = 1 that has exponentially decreasing tail bounds on the error probability, asymptotic relative efficiency of 0.932, and near-optimal computational complexity.
We present a simple and efficient anisotropic generalization of the semi-isotropic (two-integral) axisymmetric Jeans formalism which is used to model the stellar kinematics of galaxies. The following is assumed: (i) a constant mass-to-light ratio M/L
and (ii) a velocity ellipsoid that is aligned with cylindrical coordinates (R,z) and characterized by the classic anisotropy parameter beta_z=1-sigma_z^2/sigma_R^2. Our simple models are fit to SAURON integral-field observations of the stellar kinematics for a set of fast-rotator early-type galaxies. With only two free parameters (beta_z and the inclination) the models generally provide remarkably good descriptions of the shape of the first (V) and second (V_rms=sqrt{V^2+sigma^2}) velocity moments, once a detailed description of the surface brightness is given. This is consistent with previous findings on the simple dynamical structure of these objects. With the observationally-motivated assumption that beta_z>0, the method is able to recover the inclination. The technique can be used to determine the dynamical mass-to-light ratios and angular momenta of early-type fast-rotators and spiral galaxies, especially when the quality of the data does not justify more sophisticated modeling approaches. This formalism allows for the inclusion of dark matter, supermassive black holes, spatially varying anisotropy, and multiple kinematic components.
