arXiv subjects

Thomas Peters

Publications and source records attributed to Thomas Peters.

At least 19 recordsLinked to original sources

arXiv2025

ALMA-QUARKS: Few-Thousand-Year Hatching out of "Egg": The Supersonic Breakout of a Hypercompact H II Region from Its Parental Hot Core

The kinematic evolution of hypercompact H II (HC H II) regions around young high-mass stars remains poorly understood due to complex interactions with parental environs. We present ALMA QUARKS/ATOMS 1.3 mm/3 mm observations (the highest resolution $\sim0.01$ pc) of a deeply embedded HC H II region (diameter $\sim0.015$ pc, electron density $\sim2\times10^{5}$ cm$^{-3}$) exhibiting a striking $\gtrsim20$ km s$^{-1}$ global redshift seen in optically thin H30$\alpha$/H40$\alpha$ recombination lines relative to its parental hot molecular core within a hub-filament system. The 1.3 mm continuum data reveal a distinct 0.1-pc arc and a perpendicular 0.04-pc tail. We propose that this morphology arises from a dynamic champagne flow: the slow expansion of HC H II region into a pre-existing filament forms the arc and associated low-velocity (few km s$^{-1}$) SiO shocks. Meanwhile, in the opposite direction ionized gas escapes along a steep density gradient traced by the tail and high-velocity (20 km s$^{-1}$) SiO emission. We reject the bow shock scenario in which ionized gas co-moves with a runaway high-mass star because shocked gas in the arc aligns with the hub velocity, contradicting the bow shock prediction. Non-LTE radiative transfer modeling further rules out infall of ionized gas as the velocity shift origin. We conclude that this exceptional HC H II region is undergoing a few-thousand-year transition phase of "hatching out of the egg": the ionized gas of HC H II region has just broken out of its parental hot core and now is flowing outward supersonically. This work highlights how anisotropic density distributions induce supersonically anisotropic ionized flows that govern HC H II region evolution.

astro-ph.GA↗

arXiv2025

ngVLA Synthetic Observations of Ionized Gas in Massive Protostars

Massive star formation involves significant ionization in the innermost regions near the central object, such as gravitationally trapped H II regions, jets, ionized disks, or winds. Resolved observations of the associated continuum and recombination line emission are crucial for guiding theory. The next-generation Very Large Array (ngVLA) will enable unprecedented observations of thermal emission with 1 mas resolution, providing a new perspective on massive star formation at scales down to a few astronomical units at kiloparsec distances. This work presents synthetic interferometric ngVLA observations of the free-free continuum (93-GHz band), $\mathrm{H41\alpha}$, and $\mathrm{H38\alpha}$ recombination lines from ionized jets and disks around massive protostars. Using the sf3dmodels Python package, we generate gas distributions based on analytical models, which are then processed through the RADMC-3D radiative transfer code. Our results indicate that the ngVLA can easily resolve, both spatially and spectrally, the ionized jet from a 15 $\mathrm{M_\odot}$ protostar at 700 pc, distinguishing between collimated jets and wide-angle winds, and resolving their launching radii, widths, and any substructure down to a few astronomical units. Detailed studies of radio jets launched by massive protostars will be feasible up to distances of $\sim 2$ kpc. Furthermore, ngVLA will be able to study in detail the ionized disks around massive ($> 10~\mathrm{M_\odot}$) protostars up to distances from 4 to 12 kpc, resolving their kinematics and enabling the measurement of their central masses across the Galaxy. These observations can be conducted with on-source integrations of only a few hours.

astro-ph.GA↗

arXiv2024

Gravitational collapse at low to moderate Mach numbers: The relationship between star formation efficiency and the fraction of mass in the massive object

The formation of massive objects via gravitational collapse is relevant both for explaining the origin of the first supermassive black holes and in the context of massive star formation. Here, we analyze simulations of the formation of massive objects pursued by different groups and in various environments, concerning the formation of supermassive black holes, primordial stars, as well as present-day massive stars. We focus particularly on the regime of small virial parameters, i.e., low ratios of the initial kinetic to gravitational energy, low to moderate Mach numbers, and the phase before feedback is very efficient. We compare the outcomes of collapse under different conditions using dimensionless parameters, particularly the star formation efficiency \epsilon_*, the fraction f_* of mass in the most massive object relative to the total stellar mass, and the fraction f_{\rm tot} of mass of the most massive object as a function of the total mass. We find that in all simulations analyzed here, f_{\rm tot} increases as a function of \epsilon_*, although the steepness of the increase depends on the environment. The relation between f_* and \epsilon_* is found to be more complex and also strongly depends on the number of protostars present at the beginning of the simulations. We show that a collision parameter, estimated as the ratio of the system size divided by the typical collision length, allows us to approximately characterize whether collisions will be important. We analyze the statistical correlation between the dimensionless quantities using the Spearman coefficient and confirm via a machine learning analysis that good predictions of f_* can be obtained from \epsilon_* together with a rough estimate of the collision parameter. This suggests that a good estimate of the mass of the most massive object can be obtained once the maximum efficiency for a given environment is known.

astro-ph.GA↗

arXiv2022

Clustered Formation of Massive Stars within an Ionized Rotating Disk

We present ALMA observations with a 800 au resolution and radiative-transfer modelling of the inner part ($r\approx6000$ au) of the ionized accretion flow around a compact star cluster in formation at the center of the luminous ultra-compact (UC) HII region G10.6-0.4. We modeled the flow with an ionized Keplerian disk with and without radial motions in its outer part, or with an external Ulrich envelope. The MCMC fits to the data give total stellar masses $M_\star$ from 120 to $200~M_\odot$, with much smaller ionized-gas masses $M_\mathrm{ion-gas} = 0.2$ to $0.25~M_\odot$. The stellar mass is distributed within the gravitational radius $R_g\approx 1000$ to 1500 au, where the ionized gas is bound. The viewing inclination angle from the face-on orientation is $i = 49$ to $56~\deg$. Radial motions at radii $r > R_g$ converge to $v_{r,0} \approx 8.7$ km/s, or about the speed of sound of ionized gas, indicating that this gas is marginally unbound at most. From additional constraints on the ionizing-photon rate and far-IR luminosity of the region, we conclude that the stellar cluster consists of a few massive stars with $M_\mathrm{star} = 32$ to $60~M_\odot$, or one star in this range of masses accompanied by a population of lower-mass stars. Any active accretion of ionized gas onto the massive (proto)stars is residual. The inferred cluster density is very large, comparable to that reported at similar scales in the Galactic Center. Stellar interactions are likely to occur within the next Myr.

astro-ph.GA↗

arXiv2020

Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres $\textit{increases}$ when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximised though an optimal $\textit{lowering}$ of the voltage on the larger sphere at an optimal sphere separation.

physics.class-ph↗

arXiv2018

The SILCC project - V. The impact of magnetic fields on the chemistry and the formation of molecular clouds

Magnetic fields are ubiquitously observed in the interstellar medium (ISM) of present-day star-forming galaxies with dynamically relevant energy densities. Using three-dimensional magneto-hydrodynamic (MHD) simulations of the supernova (SN) driven ISM in the flux-freezing approximation (ideal MHD) we investigate the impact of the magnetic field on the chemical and dynamical evolution of the gas, fragmentation and the formation of molecular clouds. We follow the chemistry with a network of six species (H$^{+}$, H, H$_2$, C$^+$, CO, free electrons) including local shielding effects. We find that magnetic fields thicken the disc by a factor of a few to a scale height of $\sim100\,\mathrm{pc}$, delay the formation of dense (and molecular) gas by $\sim25\,\mathrm{Myr}$ and result in differently shaped gas structures. The magnetised gas fragments into fewer clumps, which are initially at subcritical mass-to-flux ratios, $M/\Phi\approx0.3(M/\Phi)_\mathrm{crit}$, and accrete gas preferentially parallel to the magnetic field lines until supercritial mass-to-flux ratios of up to order 10 are reached. The accretion rates onto molecular clouds scale with $\dot{M}\propto M^{1.5}$. The median of the inter-cloud velocity dispersion is $\sim2-5\,\mathrm{km\,s}^{-1}$ and lower than the internal velocity dispersion in the clouds ($\sim3-7\,\mathrm{km\,s}^{-1}$). However, individual cloud-cloud collisions occur at speeds of a few $10\,\mathrm{km\,s}^{-1}$.

astro-ph.GA↗

arXiv2018

Ionised gas kinematics in bipolar H II regions

Stellar feedback plays a fundamental role in shaping the evolution of galaxies. Here we explore the use of ionised gas kinematics in young, bipolar H II regions as a probe of early feedback in these star-forming environments. We have undertaken a multiwavelength study of a young, bipolar H II region in the Galactic disc, G$316.81-0.06$, which lies at the centre of a massive ($\sim10^3$ M$_{\odot}$) infrared-dark cloud filament. It is still accreting molecular gas as well as driving a $\sim 0.2$ pc ionised gas outflow perpendicular to the filament. Intriguingly, we observe a large velocity gradient ($47.81 \pm 3.21$ km s$^{-1}$ pc$^{-1}$) across the ionised gas in a direction perpendicular to the outflow. This kinematic signature of the ionised gas shows a reasonable correspondence with the simulations of young H II regions. Based on a qualitative comparison between our observations and these simulations, we put forward a possible explanation for the velocity gradients observed in G$316.81-0.06$. If the velocity gradient perpendicular to the outflow is caused by rotation of the ionised gas, then we infer that this rotation is a direct result of the initial net angular momentum in the natal molecular cloud. If this explanation is correct, this kinematic signature should be common in other young (bipolar) H II regions. We suggest that further quantitative analysis of the ionised gas kinematics of young H II regions, combined with additional simulations, should improve our understanding of feedback at these early stages.

astro-ph.SR↗

arXiv2018

Radiative transfer calculations of the diffuse ionised gas in disc galaxies with cosmic ray feedback

The large vertical scale heights of the diffuse ionised gas (DIG) in disc galaxies are challenging to model, as hydrodynamical models including only thermal feedback seem to be unable to support gas at these heights. In this paper, we use a three dimensional Monte Carlo radiation transfer code to post-process disc simulations of the Simulating the Life-Cycle of Molecular Clouds (SILCC) project that include feedback by cosmic rays. We show that the more extended discs in simulations including cosmic ray feedback naturally lead to larger scale heights for the DIG which are more in line with observed scale heights. We also show that including a fiducial cosmic ray heating term in our model can help to increase the temperature as a function of disc scale height, but fails to reproduce observed DIG nitrogen and sulphur forbidden line intensities. We show that, to reproduce these line emissions, we require a heating mechanism that affects gas over a larger density range than is achieved by cosmic ray heating, which can be achieved by fine tuning the total luminosity of ionising sources to get an appropriate ionising spectrum as a function of scale height. This result sheds a new light on the relation between forbidden line emissions and temperature profiles for realistic DIG gas distributions.

astro-ph.GA↗

arXiv2016

The impact of magnetic fields on the chemical evolution of the supernova-driven ISM

We present three-dimensional magneto-hydrodynamical simulations of the self-gravitating interstellar medium (ISM) in a periodic (256 pc)$^3$ box with a mean number density of 0.5 cm$^{-3}$. At a fixed supernova rate we investigate the multi-phase ISM structure, H$_{2}$ molecule formation and density-magnetic field scaling for varying initial magnetic field strengths (0, $6\times 10^{-3}$, 0.3, 3 $μ$G). All magnetic runs saturate at mass weighted field strengths of $\sim$ 1 $-$ 3 $μ$G but the ISM structure is notably different. With increasing initial field strengths (from $6\times 10^{-3}$ to 3 $μ$G) the simulations develop an ISM with a more homogeneous density and temperature structure, with increasing mass (from 5% to 85%) and volume filling fractions (from 4% to 85%) of warm (300 K $<$ T $<$ 8000 K) gas, with decreasing volume filling fractions (VFF) from $\sim$ 35% to $\sim$ 12% of hot gas (T $> 10^5$ K) and with a decreasing H$_{2}$ mass fraction (from 70% to $<$ 1%). Meanwhile the mass fraction of gas in which the magnetic pressure dominates over the thermal pressure increases by a factor of 10, from 0.07 for an initial field of $6\times 10^{-3}$ $μ$G to 0.7 for a 3 $μ$G initial field. In all but the simulations with the highest initial field strength self-gravity promotes the formation of dense gas and H$_{2}$, but does not change any other trends. We conclude that magnetic fields have a significant impact on the multi-phase, chemical and thermal structure of the ISM and discuss potential implications and limitations of the model.

astro-ph.GA↗

arXiv2016

The SILCC project --- IV. Impact of dissociating and ionising radiation on the interstellar medium and Halpha emission as a tracer of the star formation rate

We present three-dimensional radiation-hydrodynamical simulations of the impact of stellar winds, photoelectric heating, photodissociating and photoionising radiation, and supernovae on the chemical composition and star formation in a stratified disc model. This is followed with a sink-based model for star clusters with populations of individual massive stars. Stellar winds and ionising radiation regulate the star formation rate at a factor of ~10 below the simulation with only supernova feedback due to their immediate impact on the ambient interstellar medium after star formation. Ionising radiation (with winds and supernovae) significantly reduces the ambient densities for most supernova explosions to rho < 10^-25 g cm^-3, compared to 10^-23 g cm^-3 for the model with only winds and supernovae. Radiation from massive stars reduces the amount of molecular hydrogen and increases the neutral hydrogen mass and volume filling fraction. Only this model results in a molecular gas depletion time scale of 2 Gyr and shows the best agreement with observations. In the radiative models, the Halpha emission is dominated by radiative recombination as opposed to collisional excitation (the dominant emission in non-radiative models), which only contributes ~1-10 % to the total Halpha emission. Individual massive stars (M >= 30 M_sun) with short lifetimes are responsible for significant fluctuations in the Halpha luminosities. The corresponding inferred star formation rates can underestimate the true instantaneous star formation rate by factors of ~10.

astro-ph.GA↗

arXiv2016

The turbulent life of dust grains in the supernova-driven, multi-phase interstellar medium

Dust grains are an important component of the interstellar medium (ISM) of galaxies. We present the first direct measurement of the residence times of interstellar dust in the different ISM phases, and of the transition rates between these phases, in realistic hydrodynamical simulations of the multi-phase ISM. Our simulations include a time-dependent chemical network that follows the abundances of H^+, H, H_2, C^+ and CO and take into account self-shielding by gas and dust using a tree-based radiation transfer method. Supernova explosions are injected either at random locations, at density peaks, or as a mixture of the two. For each simulation, we investigate how matter circulates between the ISM phases and find more sizeable transitions than considered in simple mass exchange schemes in the literature. The derived residence times in the ISM phases are characterised by broad distributions, in particular for the molecular, warm and hot medium. The most realistic simulations with random and mixed driving have median residence times in the molecular, cold, warm and hot phase around 17, 7, 44 and 1 Myr, respectively. The transition rates measured in the random driving run are in good agreement with observations of Ti gas-phase depletion in the warm and cold phases in a simple depletion model, although the depletion in the molecular phase is under-predicted. ISM phase definitions based on chemical abundance rather than temperature cuts are physically more meaningful, but lead to significantly different transition rates and residence times because there is no direct correspondence between the two definitions.

astro-ph.GA↗

arXiv2016

Simulating the Formation of Massive Protostars: I. Radiative Feedback and Accretion Disks

We present radiation hydrodynamic simulations of collapsing protostellar cores with initial masses of 30, 100, and 200 M$_{\odot}$. We follow their gravitational collapse and the formation of a massive protostar and protostellar accretion disk. We employ a new hybrid radiative feedback method blending raytracing techniques with flux-limited diffusion for a more accurate treatment of the temperature and radiative force. In each case, the disk that forms becomes Toomre-unstable and develops spiral arms. This occurs between 0.35 and 0.55 freefall times and is accompanied by an increase in the accretion rate by a factor of 2-10. Although the disk becomes unstable, no other stars are formed. In the case of our 100 and 200 M$_{\odot}$ simulation, the star becomes highly super-Eddington and begins to drive bipolar outflow cavities that expand outwards. These radiatively-driven bubbles appear stable, and appear to be channeling gas back onto the protostellar accretion disk. Accretion proceeds strongly through the disk. After 81.4 kyr of evolution, our 30 M$_{\odot}$ simulation shows a star with a mass of 5.48 M$_{\odot}$ and a disk of mass 3.3 M$_{\odot}$, while our 100 M$_{\odot}$ simulation forms a 28.8 M$_{\odot}$ mass star with a 15.8 M$_{\odot}$ disk over the course of 41.6 kyr, and our 200 M$_{\odot}$ simulation forms a 43.7 M$_{\odot}$ star with an 18 M$_{\odot}$ disk in 21.9 kyr. In the absence of magnetic fields or other forms of feedback, the masses of the stars in our simulation do not appear limited by their own luminosities.

astro-ph.GA↗

arXiv2016

Impact of initial models and variable accretion rates on the pre-main-sequence evolution of massive and intermediate-mass stars and the early evolution of HII regions

Massive star formation requires the accretion of gas at high rate while the star is already bright. Its actual luminosity depends sensitively on the stellar structure. We compute pre-main-sequence tracks for massive and intermediate-mass stars with variable accretion rates and study the evolution of stellar radius, effective temperature and ionizing luminosity, starting at $2\,M_\odot$ with convective or radiative structures. The radiative case shows a much stronger swelling of the protostar for high accretion rates than the convective case. For radiative structures, the star is very sensitive to the accretion rate and reacts quickly to accretion bursts, leading to considerable changes in photospheric properties on timescales as short as 100 - 1000 yr. The evolution for convective structures is much less influenced by the instantaneous accretion rate, and produces a monotonically increasing ionizing flux that can be many orders of magnitude smaller than in the radiative case. For massive stars, it results in a delay of the HII region expansion by up to 10,000 yr. In the radiative case, the HII region can potentially be engulfed by the star during the swelling, which never happens in the convective case. We conclude that the early stellar structure has a large impact on the radiative feedback during the pre-main-sequence evolution of massive protostars and introduces an important uncertainty that should be taken into account. Because of their lower effective temperatures, our convective models may hint at a solution to an observed discrepancy between the luminosity distribution functions of massive young stellar objects and compact HII regions.

astro-ph.SR↗

arXiv2016

Direct detection of precursors of gas giants formed by gravitational instability with the Atacama Large Millimetre/sub-millimetre Array

Phases of gravitational instability are expected in the early phases of disk evolution, when the disk mass is still a substantial fraction of the mass of the star. Disk fragmentation into sub-stellar objects could occur in the cold exterior part of the disk. Direct detection of massive gaseous clumps on their way to collapse into gas giant planets would offer an unprecedented test of the disk instability model. Here we use state-of-the-art 3D radiation-hydro simulations of disks undergoing fragmentation into massive gas giants, post-processed with the RADMC-3D ray-tracing code to produce dust continuum emission maps. These are then fed into the Common Astronomy Software Applications (CASA) ALMA simulator. The synthetic maps show that both overdense spiral arms and actual clumps at different stages of collapse can be detected with the Atacama Large Millimetre/sub-millimetre Array (ALMA) in the full configuration at the distance of the Ophiuchus star forming region (125 pc). The detection of clumps is particularly effective at shorter wavelengths (690 GHz) combining two resolutions with multi-scale clean. Furthermore, we show that a flux-based estimate of the mass of a protoplanetary clump can be from comparable to a factor of 3 higher than the gravitationally bound clump mass. The estimated mass depends on the assumed opacity, and on the gas temperature, which should be set using the input of radiation-hydro simulations. We conclude that ALMA has the capability to detect "smoking gun" systems that are a signpost of the disk instability model for gas giant planet formation.

astro-ph.EP↗

arXiv2016

Launching cosmic-ray-driven outflows from the magnetized interstellar medium

We present a hydrodynamical simulation of the turbulent, magnetized, supernova (SN)-driven interstellar medium (ISM) in a stratified box that dynamically couples the injection and evolution of cosmic rays (CRs) and a self-consistent evolution of the chemical composition. CRs are treated as a relativistic fluid in the advection-diffusion approximation. The thermodynamic evolution of the gas is computed using a chemical network that follows the abundances of H+, H, H2, CO, C+, and free electrons and includes (self-)shielding of the gas and dust. We find that CRs perceptibly thicken the disk with the heights of 90% (70%) enclosed mass reaching ~1.5 kpc (~0.2 kpc). The simulations indicate that CRs alone can launch and sustain strong outflows of atomic and ionized gas with mass loading factors of order unity, even in solar neighborhood conditions and with a CR energy injection per SN of 10^50 erg, 10% of the fiducial thermal energy of an SN. The CR-driven outflows have moderate launching velocities close to the midplane (~100 km/s) and are denser (ρ~1e-24 - 1e-26 g/cm^3), smoother, and colder than the (thermal) SN-driven winds. The simulations support the importance of CRs for setting the vertical structure of the disk as well as the driving of winds.

astro-ph.GA↗

arXiv2015

The SILCC (SImulating the LifeCycle of molecular Clouds) project - II. Dynamical evolution of the supernova-driven ISM and the launching of outflows

The SILCC project (SImulating the Life-Cycle of molecular Clouds) aims at a more self-consistent understanding of the interstellar medium (ISM) on small scales and its link to galaxy evolution. We present three-dimensional (magneto)hydrodynamic simulations of the ISM in a vertically stratified box including self-gravity, an external potential due to the stellar component of the galactic disc, and stellar feedback in the form of an interstellar radiation field and supernovae (SNe). The cooling of the gas is based on a chemical network that follows the abundances of H+, H, H2, C+, and CO and takes shielding into account consistently. We vary the SN feedback by comparing different SN rates, clustering and different positioning, in particular SNe in density peaks and at random positions, which has a major impact on the dynamics. Only for random SN positions the energy is injected in sufficiently low-density environments to reduce energy losses and enhance the effective kinetic coupling of the SNe with the gas. This leads to more realistic velocity dispersions (σ_HI ~ 0.8σ_(300-8000K) ~ 10-20km/s, σ_Hα~ 0.6σ_(8000-3e5K) ~ 20-30km/s), and strong outflows with mass loading factors of up to 10 even for solar neighbourhood conditions. Clustered SNe abet the onset of outflows compared to individual SNe but do not influence the net outflow rate. The outflows do not contain any molecular gas and are mainly composed of atomic hydrogen. The bulk of the outflowing mass is dense (ρ~ 1e-25-1e-24g/cc) and slow (v ~ 20-40km/s) but there is a high-velocity tail of up to v ~ 500km/s with ρ~ 1e-28-1e-27g/cc.

astro-ph.GA↗

arXiv2015

Impact of supernova and cosmic-ray driving on the surface brightness of the galactic halo in soft X-rays

The halo of the Milky Way contains a hot plasma with a surface brightness in soft X-rays of the order $10^{-12}$erg cm$^{-2}$ s$^{-1}$ deg$^{-2}$. The origin of this gas is unclear, but so far numerical models of galactic star formation have failed to reproduce such a large surface brightness by several orders of magnitude. In this paper, we analyze simulations of the turbulent, magnetized, multi-phase interstellar medium including thermal feedback by supernova explosions as well as cosmic-ray feedback. We include a time-dependent chemical network, self-shielding by gas and dust, and self-gravity. Pure thermal feedback alone is sufficient to produce the observed surface brightness, although it is very sensitive to the supernova rate. Cosmic rays suppress this sensitivity and reduce the surface brightness because they drive cooler outflows. Self-gravity has by far the largest effect because it accumulates the diffuse gas in the disk in dense clumps and filaments, so that supernovae exploding in voids can eject a large amount of hot gas into the halo. This can boost the surface brightness by several orders of magnitude. Although our simulations do not reach a steady state, all simulations produce surface brightness values of the same order of magnitude as the observations, with the exact value depending sensitively on the simulation parameters. We conclude that star formation feedback alone is sufficient to explain the origin of the hot halo gas, but measurements of the surface brightness alone do not provide useful diagnostics for the study of galactic star formation.

astro-ph.GA↗

arXiv2015

Mach number study of supersonic turbulence: The properties of the density field

We model driven, compressible, isothermal, turbulence with Mach numbers ranging from the subsonic ($\mathcal{M} \approx 0.65$) to the highly supersonic regime ($\mathcal{M}\approx 16 $). The forcing scheme consists both solenoidal (transverse) and compressive (longitudinal) modes in equal parts. We find a relation $σ_{s}^2 = \mathrm{b}\log{(1+\mathrm{b}^2\mathcal{M}^2)}$ between the Mach number and the standard deviation of the logarithmic density with $\mathrm{b} = 0.457 \pm 0.007$. The density spectra follow $\mathcal{D}(k,\,\mathcal{M}) \propto k^{ζ(\mathcal{M})}$ with scaling exponents depending on the Mach number. We find $ζ(\mathcal{M}) = α\mathcal{M}^β$ with a coefficient $α$ that varies slightly with resolution, whereas $β$ changes systematically. We extrapolate to the limit of infinite resolution and find $α= -1.91 \pm 0.01,\, β=-0.30\pm 0.03$. The dependence of the scaling exponent on the Mach number implies a fractal dimension $D=2+0.96 \mathcal{M}^{-0.30}$. We determine how the scaling parameters depend on the wavenumber and find that the density spectra are slightly curved. This curvature gets more pronounced with increasing Mach number. We propose a physically motivated fitting formula $\mathcal{D}(k) = \mathcal{D}_0 k^{ζk^η}$ by using simple scaling arguments. The fit reproduces the spectral behaviour down to scales $k\approx 80$. The density spectrum follows a single power-law $η= -0.005 \pm 0.01$ in the low Mach number regime and the strongest curvature $η= -0.04 \pm 0.02$ for the highest Mach number. These values of $η$ represent a lower limit, as the curvature increases with resolution.

astro-ph.SR↗