Evolution Of Velocity Dispersion Research Articles

Tully-Fisher relation of late-type galaxies at 0.6 ≤ z ≤ 2.5

We present a study of the stellar and baryonic Tully-Fisher relation within the redshift range of 0.6 ≤ z ≤ 2.5, utilizing observations of star-forming galaxies. This dataset comprises of disk-like galaxies spanning a stellar mass range of 8.89 ≤ log(Mstar [M⊙]) ≤ 11.5, a baryonic mass range of 9.0 ≤ log(Mbar [M⊙]) ≤ 11.5, and a circular velocity range of 1.65 ≤ log(Vc [km/s]) ≤ 2.85. We estimated the stellar masses of these objects using spectral energy distribution fitting techniques, while the gas masses were determined via scaling relations. Circular velocities were directly derived from the rotation curves (RCs), after meticulously correcting for beam smearing and pressure support. Our analysis confirms that our sample adheres to the fundamental mass-size relations of galaxies and reflects the evolution of velocity dispersion in galaxies, in line with previous findings. This reaffirms the reliability of our photometric and kinematic parameters (i.e., Mstar and Vc), thereby enabling a comprehensive examination of the Tully-Fisher relation. To attain robust results, we employed a novel orthogonal likelihood fitting technique designed to minimize intrinsic scatter around the best-fit line, as required at high redshifts. For the stellar Tully-Fisher relation, we obtained a slope of α = 3.03 ± 0.25, an offset of β = 3.34 ± 0.53, and an intrinsic scatter of ζint = 0.08 dex. Correspondingly, the baryonic Tully-Fisher relation yielded α = 3.21 ± 0.28, β = 3.16 ± 0.61, and ζint = 0.09 dex. Our findings indicate a subtle deviation in the stellar and baryonic Tully-Fisher relation with respect to local studies, which is most likely due to the evolutionary processes governing disk formation.

The scaling relations of galaxies back in time: The road toward virialization

Context. The structural scaling relations (SSRs) of galaxies, that is, the observed correlations between effective radius, effective surface intensity, and velocity dispersion, are important tools for understanding how evolution proceeds. Aims. In this paper, we aim to demonstrate that the evolution of the SSRs back in time is governed by a combination of virial theorem (VT) and the relation L = L0′(t)σβ(t), where the parameters β and L0′ vary with time and from galaxy to galaxy. Methods. Using the WINGS database for the galaxies at redshift z = 0 and the Illustris-1 and IllustrisTNG databases of artificial galaxies, for the galaxies up to redshift z = 4, we analyse the SSRs back in time and, by means of simple algebraic expressions for L0′ and β (functions of time and other physical quantities), we derive the expected paths followed by galaxies in the various SSRs toward the distributions observed at z = 0. Results. The distribution of galaxies in the SSRs is ultimately related to the evolution in luminosity and velocity dispersion, which are empirically mirrored by the L = L0′(t)σβ(t) law. Furthermore, the β parameter works as a thermometer of the virialization of a galaxy. This parameter can assume either positive or negative values, and its absolute value attains high values when the galaxy is close to the virial condition, while it tends to zero when the galaxy is far from this condition. Conclusions. As the SSRs change with time, the method proposed in this paper allows us to decipher the temporal evolution of galaxies.

Dark matter relic abundance beyond kinetic equilibrium

We introduce DRAKE, a numerical precision tool for predicting the dark matter relic abundance also in situations where the standard assumption of kinetic equilibrium during the freeze-out process may not be satisfied. DRAKE comes with a set of three dedicated Boltzmann equation solvers that implement, respectively, the traditionally adopted equation for the dark matter number density, fluid-like equations that couple the evolution of number density and velocity dispersion, and a full numerical evolution of the phase-space distribution. We review the general motivation for these approaches and, for illustration, highlight three concrete classes of models where kinetic and chemical decoupling are intertwined in a way that quantitatively impacts the relic density: (i) dark matter annihilation via a narrow resonance, (ii) Sommerfeld-enhanced annihilation and (iii) ‘forbidden’ annihilation to final states that are kinematically inaccessible at threshold. We discuss all these cases in some detail, demonstrating that the commonly adopted, traditional treatment can result in an estimate of the relic density that is wrong by up to an order of magnitude. The public release of DRAKE, along with several examples of how to calculate the relic density in concrete models, is provided at drake.hepforge.org

X-shooter Spectroscopy and HST Imaging of 15 Massive Quiescent Galaxies at z ≳ 2

Abstract We present a detailed analysis of a large sample of spectroscopically confirmed massive quiescent galaxies (MQGs; log(M */M ⊙) ∼ 11.5) at z ≳ 2. This sample comprises 15 galaxies selected in the COSMOS and UDS fields by their bright K-band magnitudes and followed up with Very Large Telescope (VLT) X-shooter spectroscopy and Hubble Space Telescope (HST)/WFC3 H F160W imaging. These observations allow us to unambiguously confirm their redshifts, ascertain their quiescent nature and stellar ages, and reliably assess their internal kinematics and effective radii. We find that these galaxies are compact, consistent with the high-mass end of the stellar mass–size relation for quiescent galaxies at z = 2. Moreover, the distribution of the measured stellar velocity dispersions of the sample is consistent with the most massive local early-type galaxies from the MASSIVE Survey, showing that evolution in these galaxies is dominated by changes in size. The HST images reveal, as surprisingly high, that 40% of the sample has tidal features suggestive of mergers and companions in close proximity, including three galaxies experiencing ongoing major mergers. The absence of velocity dispersion evolution from z = 2 to 0, coupled with a doubling of the stellar mass, with a factor of 4 size increase and the observed disturbed stellar morphologies, supports dry minor mergers as the primary drivers of the evolution of the MQGs over the last 10 billion yr.

Stellar Velocity Dispersion of a Massive Quenching Galaxy at z = 4.01

Abstract We present the first stellar velocity dispersion measurement of a massive quenching galaxy at z = 4. The galaxy is first identified as a massive z ≥ 4 galaxy with suppressed star formation from photometric redshifts based on deep multiband data. A follow-up spectroscopic observation with MOSFIRE on Keck revealed strong multiple absorption features, which are identified as Balmer lines, giving a secure redshift of z = 4.01. This is the most distant quiescent galaxy known to date. Thanks to the high S/N of the spectrum, we are able to estimate the stellar velocity dispersion, , making a significant leap from the previous highest redshift measurement at z = 2.8. Interestingly, we find that the velocity dispersion is consistent with that of massive galaxies today, implying no significant evolution in velocity dispersion over the last 12 Gyr. Based on a stringent upper limit on its physical size from deep optical images (r eff < 1.3 kpc), we find that its dynamical mass is consistent with the stellar mass inferred from photometry. Furthermore, the galaxy is located on the mass fundamental plane extrapolated from lower redshift galaxies. The observed no strong evolution in σ suggests that the mass in the core of massive galaxies does not evolve significantly, while most of the mass growth occurs in the outskirts of the galaxies, which also increases the size. This picture is consistent with a two-phase formation scenario in which mass and size growth is due to accretion in the outskirts of galaxies via mergers. Our results imply that the first phase may be completed as early as z ∼ 4.

Merger History of Central Galaxies in Semi-analytic Models of Galaxy Formation

Abstract We investigate the dynamical evolution of galaxies in groups with different formation epochs. Galaxy groups have been selected to be in different dynamical states, namely dynamically old and dynamically young, which reflect their early and late formation times, respectively, based on their halo mass assembly. The brightest galaxies in dynamically young groups have suffered their last major galaxy merger typically ∼2 Gyr more recently than their counterparts in dynamically old groups. Furthermore, we study the evolution of velocity dispersion in these two classes and compare them with the analytic models of isolated halos. The velocity dispersion of dwarf galaxies in high-mass, dynamically young groups increases slowly in time, while the analogous dispersion in dynamically old, high-mass groups is constant. In contrast, the velocity dispersion of giant galaxies in low-mass groups decreases rapidly at late times. This increasing velocity bias is caused by dynamical friction, and starts much earlier in the dynamically old groups. The recent Radio-SAGE model of galaxy formation suggests that radio luminosities of central galaxies, considered to be tracers of AGN activity, are enhanced in halos that assembled more recently, independent of the time since the last major merger.

KMOS LENsing Survey (KLENS): Morpho-kinematic analysis of star-forming galaxies at z ~ 2

We present results from the KMOS LENsing Survey (KLENS), which is exploiting gravitational lensing to study the kinematics of 24 star-forming galaxies at 1.4 < z < 3.5 with a median mass of log(M⋆∕M⊙) = 9.6 and a median star formation rate (SFR) of 7.5 M⊙ yr−1. We find that 25% of these low mass/low SFR galaxies are rotation-dominated, while the majority of our sample shows no velocity gradient. When combining our data with other surveys, we find that the fraction of rotation-dominated galaxies increases with the stellar mass, and decreases for galaxies with a positive offset from the main sequence (higher specific star formation rate). We also investigate the evolution of the intrinsic velocity dispersion, σ0, as a function of the redshift, z, and stellar mass, M⋆, assuming galaxies in quasi-equilibrium (Toomre Q parameter equal to 1). From the z − σ0 relation, we find that the redshift evolution of the velocity dispersion is mostly expected for massive galaxies (log(M⋆∕M⊙) > 10). We derive a M⋆ − σ0 relation, using the Tully–Fisher relation, which highlights that a different evolution of the velocity dispersion is expected depending on the stellar mass, with lower velocity dispersions for lower masses, and an increase for higher masses, stronger at higher redshift. The observed velocity dispersions from this work and from comparison samples spanning 0 < z < 3.5 appear to follow this relation, except at higher redshift (z > 2), where we observe higher velocity dispersions for low masses (log(M⋆∕M⊙) ~ 9.6) and lower velocity dispersions for high masses (log(M⋆∕M⊙) ~ 10.9) than expected. This discrepancy could, for instance, suggest that galaxies at high redshift do not satisfy the stability criterion, or that the adopted parametrization of the specific star formation rate and molecular properties fail at high redshift.

The KMOS Cluster Survey (KCS). III. Fundamental Plane of Cluster Galaxies at z ≃ 1.80 in JKCS 041*

Abstract We present data for 16 galaxies in the overdensity JKCS 041 at as part of the K-band Multi-Object Spectrograph (KMOS) Cluster Survey (KCS). With 20 hr integrations, we have obtained deep absorption-line spectra from which we derived velocity dispersions for seven quiescent galaxies. We combined photometric parameters derived from Hubble Space Telescope images with the dispersions to construct a fundamental plane (FP) for quiescent galaxies in JKCS 041. From the zero-point evolution of the FP, we derived a formation redshift for the galaxies of , corresponding to a mean age of 1.4 ± 0.2 Gyr. We tested the effect of structural and velocity dispersion evolution on our FP zero-point and found a negligible contribution when using dynamical mass-normalized parameters ( ) but a significant contribution from stellar-mass-normalized parameters ( ). From the relative velocities of the galaxies, we probed the 3D structure of these 16 confirmed members of JKCS 041 and found that a group of galaxies in the southwest of the overdensity had systematically higher velocities. We derived ages for the galaxies in the different groups from the FP. We found that the east-extending group had typically older galaxies ( Gyr) than those in the southwest group (0.3 ± 0.2 Gyr). Although based on small numbers, the overdensity dynamics, morphology, and age results could indicate that JKCS 041 is in formation and may comprise two merging groups of galaxies. This result could link large-scale structure to ages of galaxies for the first time at this redshift.

Exploring SMBH assembly with semi-analytic modelling

We develop a semi-analytic model to explore different prescriptions of supermassive black hole (SMBH) fuelling. This model utilises a merger-triggered burst mode in concert with two possible implementations of a long-lived steady mode for assembling the mass of the black hole in a galactic nucleus. We improve modelling of the galaxy-halo connection in order to more realistically determine the evolution of a halo's velocity dispersion. We use four model variants to explore a suite of observables: the M-sigma relation, mass functions of both the overall and broad-line quasar population, and luminosity functions as a function of redshift. We find that "downsizing" is a natural consequence of our improved velocity dispersion mappings, and that high-mass SMBHs assemble earlier than low-mass SMBHs. The burst mode of fuelling is sufficient to explain the assembly of SMBHs to z=2, but an additional steady mode is required to both assemble low-mass SMBHs and reproduce the low-redshift luminosity function. We discuss in detail the trade-offs in matching various observables and the interconnected modelling components that govern them. As a result, we demonstrate the utility as well as the limitations of these semi-analytic techniques.

The Evolution of Bulge-dominated Field Galaxies from z ≈ 1 to the Present

Abstract We analyze the stellar populations and evolutionary history of bulge-dominated field galaxies at redshifts as part of the Gemini/Hubble Space Telescope (HST) Galaxy Cluster Project (GCP). High signal-to-noise optical spectroscopy from the Gemini Observatory and imaging from the HST are used to analyze a total of 43 galaxies, focusing on the 30 passive galaxies in the sample. Using the size–mass and velocity dispersion–mass relations for the passive field galaxies we find no significant evolution of sizes or velocity dispersions at a given dynamical mass between and the present. We establish the Fundamental Plane and study mass-to-light (M/L) ratios. The M/L versus dynamical mass relation shows that the passive field galaxies follow a relation with a steeper slope than the local comparison sample, consistent with cluster galaxies in the GCP at z = 0.86. This steeper slope indicates that the formation redshift is mass dependent, in agreement with “downsizing,” meaning that the low-mass galaxies formed their stars more recently while the high-mass galaxies formed theirs at higher redshift. The zero-point differences of the scaling relations for the M/L ratios imply a formation redshift of for the passive field galaxies. This is consistent with the line index which implies a formation redshift of .

The KMOS Cluster Survey (KCS). I. The Fundamental Plane and the Formation Ages of Cluster Galaxies at Redshift 1.4 < Z < 1.6*

Abstract We present the analysis of the fundamental plane (FP) for a sample of 19 massive red-sequence galaxies ( ) in three known overdensities at from the K-band Multi-object Spectrograph (KMOS) Cluster Survey, a guaranteed-time program with spectroscopy from the KMOS at the VLT and imaging from the Hubble Space Telescope. As expected, we find that the FP zero-point in B band evolves with redshift, from the value 0.443 of Coma to −0.10 ± 0.09, −0.19 ± 0.05, and −0.29 ± 0.12 for our clusters at z = 1.39, z = 1.46, and z = 1.61, respectively. For the most massive galaxies ( ) in our sample, we translate the FP zero-point evolution into a mass-to-light-ratio M/L evolution, finding , , to , respectively. We assess the potential contribution of the galaxy structural and stellar velocity dispersion evolution to the evolution of the FP zero-point and find it to be ∼6%–35% of the FP zero-point evolution. The rate of M/L evolution is consistent with galaxies evolving passively. Using single stellar population models, we find an average age of Gyr for the galaxies in our massive and virialized cluster at z = 1.39, Gyr in a massive but not virialized cluster at z = 1.46, and Gyr in a protocluster at z = 1.61. After accounting for the difference in the age of the universe between redshifts, the ages of the galaxies in the three overdensities are consistent within the errors, with possibly a weak suggestion that galaxies in the most evolved structure are older.

Velocity Dispersion, Size, Sérsic Index, and Dn4000: The Scaling of Stellar Mass with Dynamical Mass for Quiescent Galaxies

Abstract We examine the relation between stellar mass, velocity dispersion, size, Sérsic index, and for a volume-limited sample of ∼40,000 quiescent galaxies in the SDSS. At a fixed stellar mass, galaxies with higher have larger velocity dispersions and smaller sizes. is a proxy for stellar population age, thus these trends suggest that older galaxies typically have larger velocity dispersions and smaller sizes. We combine velocity dispersion and size into a dynamical mass estimator, . At a fixed stellar mass, depends on . The Sérsic index is also correlated with . The dependence of and Sérsic index on suggests that quiescent galaxies are not structurally homologous systems. We derive an empirical correction for non-homology which is consistent with the analytical correction derived from the virial theorem. After accounting for non-homologous galactic structure, we measure , where M * is the stellar mass and M d is the dynamical mass derived from the velocity dispersion and size; stellar mass is directly proportional to dynamical mass. Quiescent galaxies appear to be in approximate virial equilibrium, and deviations of the fundamental plane parameters from the expected virial relation may result from mass-to-light ratio variations, selection effects, and the non-homology of quiescent galaxies. We infer the redshift evolution of velocity dispersion and size for galaxies in our sample assuming purely passive evolution. The inferred evolution is inconsistent with direct measurements at higher redshifts. Thus quiescent galaxies do not passively evolve. Quiescent galaxies have properties that are consistent with standard galaxy formation in ΛCDM. They form at different epochs and evolve modestly, increasing their size, velocity dispersion, and Sérsic index after they cease star formation.

Star formation inside a galactic outflow.

Recent observations have revealed massive galactic molecular outflows that may have the physical conditions (high gas densities) required to form stars. Indeed, several recent models predict that such massive outflows may ignite star formation within the outflow itself. This star-formation mode, in which stars form with high radial velocities, could contribute to the morphological evolution of galaxies, to the evolution in size and velocity dispersion of the spheroidal component of galaxies, and would contribute to the population of high-velocity stars, which could even escape the galaxy. Such star formation could provide in situ chemical enrichment of the circumgalactic and intergalactic medium (through supernova explosions of young stars on large orbits), and some models also predict it to contribute substantially to the star-formation rate observed in distant galaxies. Although there exists observational evidence for star formation triggered by outflows or jets into their host galaxy, as a consequence of gas compression, evidence for star formation occurring within galactic outflows is still missing. Here we report spectroscopic observations that unambiguously reveal star formation occurring in a galactic outflow at a redshift of 0.0448. The inferred star-formation rate in the outflow is larger than 15 solar masses per year. Star formation may also be occurring in other galactic outflows, but may have been missed by previous observations owing to the lack of adequate diagnostics.

Formation and settling of a disc galaxy during the last 8 billion years in a cosmological simulation

We present results of a high-resolution zoom cosmological simulation of the evolution of a low-mass galaxy with a maximum velocity of V=100 km/s at z=0, using the initial conditions from the AGORA project (Kim et al. 2014). The final disc-dominated galaxy is consistent with local disc scaling relations, such as the stellar-to-halo mass relation and the baryonic Tully-Fisher. The galaxy evolves from a compact, dispersion-dominated galaxy into a rotation-dominated but dynamically hot disc in about 0.5 Gyr (from z=1.4 to z=1.2). The disc dynamically cools down for the following 7 Gyr, as the gas velocity dispersion decreases over time, in agreement with observations. The primary cause of this slow evolution of velocity dispersion in this low-mass galaxy is stellar feedback. It is related to the decline in gas fraction, and to the associated gravitational disk instability, as the disc slowly settles from a global Toomre Q>1 turbulent disc to a marginally unstable disc (Q=1).

The fundamental plane of EDisCS galaxies (Corrigendum)

We discovered a mistake in Eqs. (7) and (10) of Saglia et al. (2010, A&A, 524, A6), which propagates to Tables 8 and 9 and Fig. 25. We revise the tables, the figure and the affected statements in the paper. As a result, the reduction in the luminosity evolution due to the effects of the size and velocity dispersion evolution is smaller than claimed in Saglia et al. (2010).

The most massive galaxies in clusters are already fully grown atz ∼ 0.5

By constructing scaling relations for galaxies in the massive cluster MACSJ0717.5 at $z=0.545$ and comparing with those of Coma, we model the luminosity evolution of the stellar populations and the structural evolution of the galaxies. We calculate magnitudes, surface brightnesses and effective radii using HST/ACS images and velocity dispersions using Gemini/GMOS spectra, and present a catalogue of our measurements for 17 galaxies. We also generate photometric catalogues for $\sim 3000$ galaxies from the HST imaging. With these, we construct the colour-magnitude relation, the fundamental plane, the mass-to-light versus mass relation, the mass-size relation and the mass-velocity dispersion relation for both clusters. We present a new, coherent way of modelling these scaling relations simultaneously using a simple physical model in order to infer the evolution in luminosity, size and velocity dispersion as a function of redshift, and show that the data can be fully accounted for with this model. We find that (a) the evolution in size and velocity dispersion undergone by these galaxies between $z \sim 0.5$ and $z \sim 0$ is mild, with $R_e(z) \sim (1+z)^{-0.40\pm0.32}$ and $\sigma(z) \sim (1+z)^{0.09 \pm 0.27}$, and (b) the stellar populations are old, $\sim 10$ Gyr, with a $\sim 3$ Gyr dispersion in age, and are consistent with evolving purely passively since $z \sim 0.5$ with $\Delta \log M/L_B = -0.55_{-0.07}^{+0.15} z$. The implication is that these galaxies formed their stars early and subsequently grew dissipationlessly so as to have their mass already in place by $z \sim 0.5$, and suggests a dominant role for dry mergers, which may have accelerated the growth in these high-density cluster environments.

Evolution of velocity dispersion along cold collisionless flows

The infall of cold dark matter onto a galaxy produces cold collisionless flows and caustics in its halo. If a signal is found in the cavity detector of dark matter axions, the flows will be readily apparent as peaks in the energy spectrum of photons from axion conversion, allowing the densities, velocity vectors and velocity dispersions of the flows to be determined. We discuss the evolution of velocity dispersion along cold collisionless flows in one and two dimensions. A technique is presented for obtaining the leading behaviour of the velocity dispersion near caustics. The results are used to derive an upper limit on the energy dispersion of the Big Flow from the sharpness of its nearby caustic, and a prediction for the dispersions in its velocity components.

QUIESCENT COMPACT GALAXIES AT INTERMEDIATE REDSHIFT IN THE COSMOS FIELD. II. THE FUNDAMENTAL PLANE OF MASSIVE GALAXIES

We examine the relation between surface brightness, velocity dispersion, and size?the fundamental plane (FP)?for quiescent galaxies at intermediate redshifts in the COSMOS field. The COSMOS sample consists of ?150 massive quiescent galaxies with an average velocity dispersion of?? ? 250 km s?1 and redshifts between 0.2 < z < 0.8. More than half of the galaxies in the sample are compact. The COSMOS galaxies exhibit a tight relation (?0.1 dex scatter) between surface brightness, velocity dispersion, and size. At a fixed combination of velocity dispersion and size, the COSMOS galaxies are brighter than galaxies in the local universe. These surface brightness offsets are correlated with the rest-frame g ? z color and Dn4000 index; bluer galaxies and those with smaller Dn4000 indices have larger offsets. Stellar population synthesis models indicate that the massive COSMOS galaxies are younger and therefore brighter than similarly massive quiescent galaxies in the local universe. Passive evolution alone brings the massive compact quiescent (MCQ) COSMOS galaxies onto the local FP at z = 0. Therefore, evolution in size or velocity dispersion for MCQ galaxies since z ? 1 is constrained by the small scatter observed in the FP. We conclude that MCQ galaxies at z ? 1 are not a special class of objects but rather the tail of the mass and size distribution of the normal quiescent galaxy population.

REDSHIFT EVOLUTION OF THE DYNAMICAL PROPERTIES OF MASSIVE GALAXIES FROM SDSS-III/BOSS

We study the redshift evolution of the dynamical properties of ~180,000 massive galaxies from SDSS-III/BOSS combined with a local early-type galaxy sample from SDSS-II in the redshift range 0.1<z< 0.6. The typical stellar mass of this sample is Mstar~2x10^{11} Msun. We analyze the evolution of the galaxy parameters effective radius, stellar velocity dispersion, and the dynamical to stellar mass ratio with redshift. As the effective radii of BOSS galaxies at these redshifts are not well resolved in the SDSS imaging we calibrate the SDSS size measurements with HST/COSMOS photometry for a sub-sample of galaxies. We further apply a correction for progenitor bias to build a sample which consists of a coeval, passively evolving population. Systematic errors due to size correction and the calculation of dynamical mass, are assessed through Monte Carlo simulations. At fixed stellar or dynamical mass, we find moderate evolution in galaxy size and stellar velocity dispersion, in agreement with previous studies. We show that this results in a decrease of the dynamical to stellar mass ratio with redshift at >2sigma significance. By combining our sample with high-redshift literature data we find that this evolution of the dynamical to stellar mass ratio continues beyond z~0.7 up to z>2 as Mdyn/Mstar~ (1+z)^{-0.30+/- 0.12} further strengthening the evidence for an increase of Mdyn/Mstar with cosmic time. This result is in line with recent predictions from galaxy formation simulations based on minor merger driven mass growth, in which the dark matter fraction within the half-light radius increases with cosmic time.

Analytical model for non-thermal pressure in galaxy clusters

Non-thermal pressure in the intracluster gas has been found ubiquitously in numerical simulations, and observed indirectly. In this paper we develop an analytical model for intracluster non-thermal pressure in the virial region of relaxed clusters. We write down and solve a first-order differential equation describing the evolution of non-thermal velocity dispersion. This equation is based on insights gained from observations, numerical simulations, and theory of turbulence. The non-thermal energy is sourced, in a self-similar fashion, by the mass growth of clusters via mergers and accretion, and dissipates with a time-scale determined by the turnover time of the largest turbulence eddies. Our model predicts a radial profile of non-thermal pressure for relaxed clusters. The non-thermal fraction increases with radius, redshift, and cluster mass, in agreement with numerical simulations. The radial dependence is due to a rapid increase of the dissipation time-scale with radii, and the mass and redshift dependence comes from the mass growth history. Combing our model for the non-thermal fraction with the Komatsu-Seljak model for the total pressure, we obtain thermal pressure profiles, and compute the hydrostatic mass bias. We find typically 10% bias for the hydrostatic mass enclosed within $r_{500}$.