arXiv subjects

Duncan A. Forbes

Publications and source records attributed to Duncan A. Forbes.

At least 55 records · Page 3Linked to original sources

arXiv2024

Analysis of Galaxies at the Extremes: A Kinematic Analysis of the Virgo Cluster Dwarfs VCC 9 and VCC 1448 using the Keck Cosmic Web Imager

We present spatially resolved Keck Cosmic Web Imager stellar spectroscopy of the Virgo cluster dwarf galaxies VCC 9 and VCC 1448. These galaxies have similar stellar masses and large half-light radii but very different globular cluster (GC) system richness ($\sim$25 vs. $\sim$99 GCs). Using the KCWI data, we spectroscopically confirm 10 GCs associated with VCC 1448 and one GC associated with VCC 9. We make two measurements of dynamical mass for VCC 1448 based on the stellar and GC velocities respectively. VCC 1448's mass measurements suggest that it resides in a halo in better agreement with the expectation of the stellar mass -- halo mass relationship than the expectation from its large GC counts. For VCC 9, the dynamical mass we measure agrees with the expected halo mass from both relationships. We compare VCC 1448 and VCC 9 to the GC-rich galaxy Dragonfly 44 ($\sim74$ GCs), which is similar in size but has $\sim 1$ dex less stellar mass than either Virgo galaxy. In dynamical mass -- GC number space, Dragonfly 44 and VCC 1448 exhibit richer GC systems given their dynamical mass than that of VCC 9 and other `normal' galaxies. We also place the galaxies in kinematics -- ellipticity space finding evidence of an anticorrelation between rotational support and the fraction of a galaxy's stellar mass in its GC system. i.e., VCC 9 is more rotationally supported than VCC 1448, which is more rotationally supported than Dragonfly 44. This trend may be expected if a galaxy's GC content depends on its natal gas properties at formation.

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arXiv2024

A Catalogue and Analysis of Ultra-Diffuse Galaxy Spectroscopic Properties

In order to help facilitate the future study of ultra-diffuse galaxies (UDGs) we compile a catalogue of their spectroscopic properties. Using it, we investigate some of the biases inherent in the current UDG sample that have been targeted for spectroscopy. In comparison to a larger sample of UDGs studied via their spectral energy distributions (SED), current spectroscopic targets are intrinsically brighter, have higher stellar mass, are larger, more globular cluster-rich, older, and have a wider spread in their metallicities. In particular, many spectroscopically studied UDGs have a significant fraction of their stellar mass contained within their globular cluster (GC) system. We also search for correlations between parameters in the catalogue. Of note is a correlation between alpha element abundance and metallicity as may be expected for a `failed galaxy' scenario. However, the expected correlations of metallicity with age are not found and it is unclear if this is evidence against a `failed galaxy' scenario or simply due to the low number statistics and the presence of outliers. Finally, we attempt to segment our catalogue into different classes using a machine learning K-means method. We find that the clustering is very weak and that it is currently not warranted to split the catalogue into multiple, distinct sub-populations. Our catalogue is available online and we aim to maintain it beyond the publication of this work.

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arXiv2024

Origin of the correlation between stellar kinematics and globular cluster system richness in ultra-diffuse galaxies

Observational surveys have found that the dynamical masses of ultra-diffuse galaxies (UDGs) correlate with the richness of their globular cluster (GC) system. This could be explained if GC-rich galaxies formed in more massive dark matter haloes. We use simulations of galaxies and their GC systems from the E-MOSAICS project to test whether the simulations reproduce such a trend. We find that GC-rich simulated galaxies in galaxy groups have enclosed masses that are consistent with the dynamical masses of observed GC-rich UDGs. However, simulated GC-poor galaxies in galaxy groups have higher enclosed masses than those observed. We argue that GC-poor UDGs with low stellar velocity dispersions are discs observed nearly face on, such that their true mass is underestimated by observations. Using the simulations, we show that galactic star-formation conditions resulting in dispersion-supported stellar systems also leads to efficient GC formation. Conversely, conditions leading to rotationally-supported discs leads to inefficient GC formation. This result may explain why early-type galaxies typically have richer GC systems than late-type galaxies. This is also supported by comparisons of stellar axis ratios and GC specific frequencies in observed dwarf galaxy samples, which show GC-rich systems are consistent with being spheroidal, while GC-poor systems are consistent with being discs. Therefore, particularly for GC-poor galaxies, rotation should be included in dynamical mass measurements from stellar dynamics.

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arXiv2024

Discovery of Globular Cluster Candidates in the Dwarf Irregular Galaxy IC 2574 Using HST/ACS Imaging

We report the discovery of 23 globular cluster (GC) candidates around the relatively isolated dwarf galaxy IC 2574 within the Messier 81 (M81) group, at a distance of 3.86 Mpc. We use observations from the HST Advanced Camera for Surveys (ACS) to analyse the imaging in the F814W and F555W broadband filters. Our GC candidates have luminosities ranging from $-5.9 \geq M_V \geq -10.4$ and half-light radii of $1.4 \leq r_h \leq 11.5$ pc. We find the total number of GCs ($N_{\mathrm{GC}})=27\pm5$ after applying completeness corrections, which implies a specific frequency of $S_N = 4.0\pm0.8$, consistent with expectations based on its luminosity. The GC system appears to have a bimodal colour distribution, with 30% of the GC candidates having redder colours. We also find 5 objects with extremely blue colours that could be young star clusters linked to an intense star formation episode that occurred in IC 2574 $\sim$1 Gyr ago. We make an independent measurement of the halo mass of IC 2574 from its kinematic data, which is rare for low mass galaxies, and find log $M_{200} = 10.93 \pm 0.08$. We place the galaxy on the well-known GC system mass-halo mass relation and find that it agrees well with the observed near-linear relation. IC 2574 has a rich GC population for a dwarf galaxy, which includes an unusually bright $ω$ Cen-like GC, making it an exciting nearby laboratory for probing the peculiar efficiency of forming massive GCs in dwarf galaxies.

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arXiv2024

Constraining the stellar populations of ultra-diffuse galaxies in the MATLAS survey using spectral energy distribution fitting

We use spectral energy distribution (SED) fitting to place constraints on the stellar populations of 59 ultra-diffuse galaxies (UDGs) in the low-to-moderate density fields of the MATLAS survey. We use the routine PROSPECTOR, coupled with archival data in the optical from DECaLS, and near- and mid-infrared imaging from WISE, to recover the stellar masses, ages, metallicities and star formation timescales of the UDGs. We find that a subsample of the UDGs lies within the scatter of the mass-metallicity relation (MZR) for local classical dwarfs. However, another subsample is more metal-poor, being consistent with the evolving MZR at high-redshift. We investigate UDG positioning trends in the mass-metallicity plane as a function of surface brightness, effective radius, axis ratio, local volume density, mass-weighted age, star formation timescale, globular cluster (GC) counts and GC specific frequency. We find that our sample of UDGs can be separated into two main classes. Class A: Comprised of UDGs with lower stellar masses, prolonged star formation histories (SFHs), more elongated, inhabiting less dense environments, hosting fewer GCs, younger, consistent with the classical dwarf MZR, and fainter. Class B: UDGs with higher stellar masses, rapid SFHs, rounder, inhabiting the densest of our probed environments, hosting on average the most numerous GC systems, older, consistent with the high-redshift MZR (i.e., consistent with early-quenching), and brighter. The combination of these properties suggests that UDGs of Class A are consistent with a `puffed-up dwarf' formation scenario, while UDGs of Class B seem to be better explained by `failed galaxy' scenarios.

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arXiv2024

Do Ultra Diffuse Galaxies with Rich Globular Clusters Systems have Overly Massive Halos?

Some Ultra Diffuse Galaxies (UDGs) appear to host exceptionally rich globular cluster (GC) systems compared to normal galaxies of the same stellar mass. After re-examing these claims, we focus on a small sample of UDGs from the literature that have {\it both} rich GC systems (N$_{GC}$ $> 20$) and a measured galaxy velocity dispersion. We find that UDGs with more GCs have higher dynamical masses and that GC-rich UDGs are dark matter dominated within their half-light radii. We extrapolate these dynamical masses to derive total halo masses assuming cuspy and cored mass profiles. We find reasonable agreement between halo masses derived from GC numbers (assuming the GC number - halo mass relation) and from cored halo profiles. This suggests that GC-rich UDGs do {\it not} follow the standard stellar mass - halo mass relation, occupying overly massive cored halos for their stellar mass. A similar process to that invoked for some Local Group dwarfs, of early quenching, may result in GC-rich UDGs that have failed to form the expected mass of stars in a given halo (and thus giving the appearance of overly an massive halo). Simulations that correctly reproduce the known properties of GC systems associated with UDGs are needed.

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arXiv2023

J-PLUS: Detecting and studying extragalactic globular clusters -- the case of NGC 1023

Extragalactic globular clusters (GCs) are key objects for studying the history of galaxies. The arrival of wide-field surveys such as the Javalambre Photometric Local Universe Survey (J-PLUS) offers new possibilities for the study of these systems. We perform the first study of GCs in J-PLUS to recover information about the history of NGC 1023 taking advantage of wide-field images and 12 filters. We develop the semiautomatic pipeline GCFinder that detects GC candidates in J-PLUS images and can also be adapted to similar surveys. We study the stellar population properties of a sub-sample of GC candidates using spectral energy distribution (SED) fitting. We find 523 GC candidates in NGC 1023, of which $\sim$300 are new. We identify subpopulations of GC candidates, where age and metallicity distributions have multiple peaks. By comparing our results with simulations, we report a possible broad age-metallicity relation, evidence that NGC 1023 experienced accretion events in the past. The dominating age peak is at $10^{10}$ yr. We report a correlation between masses and ages that suggests that massive GC candidates are more likely to survive the turbulent history of the host galaxy. Modeling the light of NGC 1023, we find two spiral-like arms and detect a displacement of the galaxy's photometric center with respect to the outer isophotes and center of GC distribution ($\sim$700 pc and $\sim$1600 pc, respectively), which could be the result of ongoing interaction between NGC 1023 and NGC 1023A. By studying the GC system of NGC 1023 with J-PLUS we showcase the power of multi-band surveys for this kind of study and find evidence of a complex accretion history of the host galaxy.

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arXiv2023

The star formation histories of quiescent ultra-diffuse galaxies and their dependence on environment and globular cluster richness

We derive the stellar population parameters of 11 quiescent ultra-diffuse galaxies (UDGs) from Keck/KCWI data. We supplement these with 14 literature UDGs, creating the largest spectroscopic sample of UDGs to date (25). We find a strong relationship between their $α$-enhancement and their star formation histories: UDGs that formed on very short timescales have elevated [Mg/Fe] abundance ratios, whereas those forming over extended periods present lower values. Those forming earlier and faster are overall found in high-density environments, being mostly early infalls into the cluster. No other strong trends are found with infall times. We analyze the stellar mass-metallicity, age-metallicity and [Mg/Fe]-metallicity relations of the UDGs, comparing them to other types of low mass galaxies. Overall, UDGs scatter around the established stellar mass--metallicity relations of classical dwarfs. We find that GC-rich UDGs have intermediate-to-old ages, but previously reported trends of galaxy metallicity and GC richness are not reproduced with this spectroscopic sample due to the existence of GC-rich UDGs with elevated metallicities. In addition, we also find that a small fraction of UDGs could be 'failed-galaxies', supported by their GC richness, high $α$-abundances, fast formation timescales and that they follow the mass-metallicity relation of z~2 galaxies. Finally, we also compare our observations to simulated UDGs. We caution that there is not a single simulation that can produce the diverse UDG properties simultaneously, in particular the low metallicity failed-galaxy like UDGs.

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arXiv2023

Looking into the faintEst WIth MUSE (LEWIS): on the nature of ultra-diffuse galaxies in the Hydra-I cluster.I. Project description and preliminary results

Looking into the faintEst WIth MUSE (LEWIS) is an ESO large observing programme aimed at obtaining the first homogeneous integral-field spectroscopic survey of 30 extremely low-surface brightness (LSB) galaxies in the Hydra I cluster of galaxies, with MUSE at ESO-VLT. The majority of LSB galaxies in the sample (22 in total) are ultra-diffuse galaxies (UDGs). The distribution of systemic velocities Vsys ranges between 2317 km/s and 5198 km/s and is centred on the mean velocity of Hydra I (Vsys = 3683 $\pm$ 46 km/s). Considering the mean velocity and the velocity dispersion of the cluster, 17 out of 20 targets are confirmed cluster members. To assess the quality of the data and demonstrate the feasibility of the science goals, we report the preliminary results obtained for one of the sample galaxies, UDG11. For this target, we derived the stellar kinematics, including the 2-dimensional maps of line-of-sight velocity and velocity dispersion, constrained age and metallicity, and studied the globular cluster (GC) population hosted by the UDG. Results are compared with the available measurements for UDGs and dwarf galaxies in literature. By fitting the stacked spectrum inside one effective radius, we find that UDG11 has a velocity dispersion $σ= 20 \pm 8$ km/s, it is old ($10\pm1$ Gyr), metal-poor ([M/H]=-1.17$\pm$0.11 dex) and has a total dynamical mass-to-light ratio M$/L_V\sim 14$, comparable to those observed for classical dwarf galaxies. The spatially resolved stellar kinematics maps suggest that UDG11 does not show a significant velocity gradient along either major or minor photometric axes. We find two GCs kinematically associated with UDG11. The estimated total number of GCs in UDG11, corrected for the spectroscopic completeness limit, is $N_{GC}= 5.9^{+2.2}_ {-1.8}$, which corresponds to a GC specific frequency of $S_N = 8.4^{+3.2}_{-2.7}$.

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arXiv2023

Investigating the Dark Matter Halo of NGC 5128 using a Discrete Dynamical Model

As the nearest accessible massive early-type galaxy, NGC 5128 presents an exceptional opportunity to measure dark matter halo parameters for a representative elliptical galaxy. Here we take advantage of rich new observational datasets of large-radius tracers to perform dynamical modeling of NGC 5128, using a discrete axisymmetric anisotropic Jeans approach with a total tracer population of nearly 1800 planetary nebulae, globular clusters, and dwarf satellite galaxies extending to a projected distance of $\sim250$ kpc from the galaxy center. We find that a standard NFW halo provides an excellent fit to nearly all the data, excepting a subset of the planetary nebulae that appear to be out of virial equilibrium. The best-fit dark matter halo has a virial mass of ${\rm M}_{vir}=4.4^{+2.4}_{-1.4}\times10^{12} {\rm M}_{\odot}$, and NGC 5128 appears to sit below the mean stellar mass--halo mass and globular cluster mass--halo mass relations, which both predict a halo virial mass closer to ${\rm M}_{vir} \sim 10^{13} {\rm M}_{\odot}$. The inferred NFW virial concentration is $c_{vir}=5.6^{+2.4}_{-1.6}$, nominally lower than $c_{vir} \sim 9$ predicted from published $c_{vir}$--${\rm M}_{vir}$ relations, but within the $\sim 30\%$ scatter found in simulations. The best-fit dark matter halo constitutes only $\sim10\%$ of the total mass at 1 effective radius but $\sim50\%$ at 5 effective radii. The derived halo parameters are relatively insensitive to reasonable variations in the tracer population considered, tracer anisotropies, and system inclination. Our analysis highlights the value of comprehensive dynamical modeling of nearby galaxies, and the importance of using multiple tracers to allow cross-checks for model robustness.

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arXiv2023

Keck Spectroscopy of NGC 1052-DF9: Stellar Populations in the Context of the NGC 1052 Group

In this study, we use Keck/KCWI spectroscopy to measure the age, metallicity and recessional velocity of NGC~1052-DF9 (DF9), a dwarf galaxy in the NGC~1052 group. We compare these properties to those of two other galaxies in the group, NGC~1052-DF2 and NGC~1052-DF4, which have low dark matter content. The three galaxies are proposed constituents of a trail of galaxies recently hypothesised to have formed as part of a ``bullet dwarf'' collision. We show that the ages and total metallicities of the three galaxies are within uncertainties of one another which may be expected if they share a related formation pathway. However, the recessional velocity we recover for DF9 (1680 $\pm$ 10 km s$^{-1}$) is higher than predicted for a linearly projected interpretation of the ``bullet dwarf'' trail. DF9 is then either not part of the trail or the correlation of galaxy velocities along the trail is not linear in 2D projection due to their 3D geometry. After examining other proposed formation pathways for the galaxies, none provide a wholly satisfactory explanation for all of their known properties. We conclude further work is required to understand the formation of this interesting group of galaxies.

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arXiv2023

The large-scale structure of globular clusters in the NGC 1052 group

Prompted by the many controversial claims involving the NGC 1052 group, including that it hosts two dark matter-free galaxies with overluminous and monochromatic globular cluster (GC) systems, here we map out the large-scale structure (LSS) of GCs over the entire group. To recover the LSS, we use archival optical CFHT imaging data. We recover two GC density maps, one based on universal photometric properties of GCs from simple stellar population models, and one based on the properties of spectroscopically confirmed GCs in DF2 and DF4 (the two dwarf galaxies with overluminous GC populations). Both selection methods reveal overdensities around the massive galaxies in the group, as well as around NGC 1052 itself, that are coincident with the positions of previously identified stellar streams and tidal features. No intragroup GCs are found connecting these structures to any of the dwarf galaxies. We find, however, two other dwarfs in the group hosting GC systems. These include RCP32 with 2 GCs with ages equivalent to the GCs around NGC 1052, and DF9 with 3 GCs with ages similar to the GCs around DF2 and DF4. We conclude that the GC distribution in the group does not strongly support any formation scenario in particular. It favours, nonetheless, scenarios relying on galaxy-galaxy interactions and on the coeval formation of GCs around the DM-free dwarf galaxies. These may include the recently proposed bullet-dwarf formation, as well as high-redshift tidal dwarf galaxy models.

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arXiv2023

A Trail of the Invisible: Blue Globular Clusters Trace the Radial Density Distribution of the Dark Matter -- Case Study of NGC 4278

We present new, deep optical observations of the early-type galaxy NGC 4278, which is located in a small loose group. We find that the galaxy lacks fine substructure, i.e., it appears relaxed, out to a radius of $\sim$70 kpc. Our $g$- and $i$-band surface brightness profiles are uniform down to our deepest levels of $\sim$28 mag arcsec$^{-2}$. This spans an extremely large radial range of more than 14 half-mass radii. Combined with archival globular cluster (GC) number density maps and a new analysis of the total mass distribution obtained from archival Chandra X-ray data, we find that the red GC subpopulation traces well the stellar mass density profile from 2.4 out to even 14 half-mass radii, while the blue GC subpopulation traces the total mass density profile of the galaxy over a large radial range. Our results reinforce the scenario that red GCs form mostly in-situ along with the stellar component of the galaxy, while the blue GCs are more closely aligned with the total mass distribution in the halo and were accreted along with halo matter. We conclude that for galaxies where the X-ray emission from the hot halo is too faint to be properly observable and as such is not available to measure the dark matter profile, the blue GC population can be used to trace this dark matter component out to large radii.

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arXiv2022

The galaxy morphology-density relation in the EAGLE simulation

The optical morphology of galaxies is strongly related to galactic environment, with the fraction of early-type galaxies increasing with local galaxy density. In this work we present the first analysis of the galaxy morphology-density relation in a cosmological hydrodynamical simulation. We use a convolutional neural network, trained on observed galaxies, to perform visual morphological classification of galaxies with stellar masses $M_\ast > 10^{10} \, \mathrm{M}_\odot$ in the EAGLE simulation into elliptical, lenticular and late-type (spiral/irregular) classes. We find that EAGLE reproduces both the galaxy morphology-density and morphology-mass relations. Using the simulations, we find three key processes that result in the observed morphology-density relation: (i) transformation of disc-dominated galaxies from late-type (spiral) to lenticular galaxies through gas stripping in high-density environments, (ii) formation of lenticular galaxies by merger-induced black hole feedback in low-density environments, and (iii) an increasing fraction of high-mass galaxies, which are more often elliptical galaxies, at higher galactic densities.

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arXiv2022

Does the virial mass drive the intra-cluster light? The relationship between the ICL and M$_{vir}$ from VEGAS

In this Letter we revisit the relationship between the fraction of the intra-cluster light (ICL) and both the virial mass and the fraction of Early Type Galaxies in the host halo. This is based on a statistically significant and homogeneous sample of 22 groups and clusters of galaxies in the local Universe ($z \leq 0.05$), obtained with the VST Early-type GAlaxy Survey (VEGAS). Taking advantage of the long integration time and large area of the VEGAS images, we are able to map the galaxy outskirts and ICL down to $μ_g$ $\geq$ 29-30 mag/arcsec$^2$ and out to hundreds of kpc. With this data-set, we have expanded the sample of ICL measurements, doubling the previous measures available from the literature for z $\leq$ 0.05. The main result of this work is the lack of any significant trend between the fraction of ICL and the virial mass of the host environment, covering a wide range of virial masses ( $\sim$ $10^{12.5} \leq M_{vir} \leq 10^{15.5} M_{\odot}$), in agreement with some theoretical studies. Since the new data points are all derived with the same methodology and from the same observational setup, and all have comparable depth, the large observed scatter indicates an intrinsic variation in the ICL fraction.On the other hand, there is a weak relation between the fraction of ICL and the fraction of Early Type Galaxies in the host halo, where a larger fraction of ICL is found in groups and clusters of galaxies dominated by earlier morphological types, indicating a connection between the ICL and the dynamical state of the host system.

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arXiv2022

Keck Spectroscopy of the Coma Cluster Ultra-Diffuse Galaxy Y358: Dynamical Mass in a Wider Context

We examine ultra-diffuse galaxies (UDGs) and their relation to non-UDGs in mass-radius-luminosity space. We begin by publishing Keck/KCWI spectroscopy for the Coma cluster UDG Y358, for which we measure both a recessional velocity and velocity dispersion. Our recessional velocity confirms association with the Coma cluster and Y358's status as a UDG. From our velocity dispersion (19 $\pm$ 3 km s$^{-1}$) we calculate a dynamical mass within the half-light radius which provides evidence for a core in Y358's dark matter halo. We compare this dynamical mass, along with those for globular cluster (GC)-rich/-poor UDGs in the literature, to mass profiles for isolated, gas-rich UDGs and UDGs in the NIHAO/FIRE simulations. We find GC-poor UDGs have dynamical masses similar to isolated, gas-rich UDGs, suggesting an evolutionary pathway may exist between the two. Conversely, GC-rich UDGs have dynamical masses too massive to be easily explained as the evolution of the isolated, gas-rich UDGs. The simulated UDGs match the dynamical masses of the GC-rich UDGs. However, once compared in stellar mass -- halo mass space, the FIRE/NIHAO simulated UDGs do not match the halo masses of either the isolated, gas-rich UDGs or the GC-rich UDGs at the same stellar mass. Finally, we supplement our data for Y358 with other UDGs that have measured velocity dispersions in the literature. We compare this sample to a wide range of non-UDGs in mass-radius-luminosity space, finding UDGs have a similar locus to non-UDGs of similar luminosity with the primary difference being their larger half-light radii.

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arXiv2022

The stellar populations of quiescent ultra-diffuse galaxies from optical to mid-infrared spectral energy distribution fitting

We use spectral energy distribution (SED) fitting to place constraints on the stellar population properties of 29 quiescent ultra-diffuse galaxies (UDGs) across different environments. We use the fully Bayesian routine PROSPECTOR coupled with archival data in the optical, near, and mid-infrared from Spitzer and WISE under the assumption of an exponentially declining star formation history. We recover the stellar mass, age, metallicity, dust content, star formation time scales and photometric redshifts (photo-zs) of the UDGs studied. Using the mid-infrared data, we probe the existence of dust in UDGs. Although its presence cannot be confirmed, we find that the inclusion of small amounts of dust in the models brings the stellar populations closer to those reported with spectroscopy. Additionally, we fit the redshifts of all galaxies. We find a high accuracy in recovering photo-zs compared to spectroscopy, allowing us to provide new photo-z estimates for three field UDGs with unknown distances. We find evidence of a stellar population dependence on the environment, with quiescent field UDGs being systematically younger than their cluster counterparts. Lastly, we find that all UDGs lie below the mass--metallicity relation for normal dwarf galaxies. Particularly, the globular cluster (GC)-poor UDGs are consistently more metal-rich than GC-rich ones, suggesting that GC-poor UDGs may be puffed-up dwarfs, while most GC-rich UDGs are better explained by a failed galaxy scenario. As a byproduct, we show that two galaxies in our sample, NGC 1052-DF2 and NGC 1052-DF4, share equivalent stellar population properties, with ages consistent with 8 Gyr. This finding supports formation scenarios where the galaxies were formed together.

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arXiv2022

Finite shock model of density in supersonic turbulence

The probability distribution of density in isothermal, supersonic, turbulent gas is approximately lognormal. This behaviour can be traced back to the shock waves travelling through the medium, which randomly adjust the density by a random factor of the local sonic Mach number squared. Provided a certain parcel of gas experiences a large number of shocks, due to the central limit theorem, the resulting distribution for density is lognormal. We explore a model in which parcels of gas undergo finite number of shocks before relaxing to the ambient density, causing the distribution for density to deviate from a lognormal. We confront this model with numerical simulations with various r.m.s. Mach numbers ranging from subsonic as low as 0.1 to supersonic at 25. We find that the fits to the finite formula are an order of magnitude better than a lognormal. The model naturally extends even to subsonic flows, where no shocks exist.

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