Collapse of a massive pre-stellar core with hybrid RT and hydrodynamics

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Towards higher masses: when radiation comes into play

In this section I set the seeds for this thesis by presenting the historical difficulty encountered when trying to explain massive star formation with the previous scenario.
Let us consider a protostar accreting from a spherically-symmetric envelope (1D view, see the top panel of Fig. 1.5). On the one hand, gas is free-falling onto the protostar, accelerated at agrav = GM?/r2, where r is the distance to the protostar of mass M?, and G the gravitational constant. On the other hand, the protostar’s radiation exerts a pressure on the inflow, corre-sponding to a radiative acceleration arad = kL?/(4pr2c) where L? is the protostar’s luminosity and c is the speed of light. The medium is composed of gas and dust ( 1% in mass) and its opacity k is dominated by the dust continuum opacity. At the protostellar densities, gas and dust are well-coupled so the force acting on dust is efficiently transmitted to the gas. Above 1500 K, hence close to the protostar, dust grains sublimate and the medium is optically-thin so the protostar radiation propagates freely. The radiation is progressively absorbed beyond this region. Interstellar dust opacities globally increase with a decreasing wavelength, thus the UV photons are more quickly absorbed than the others. The gas-dust mixture heats up and emits radiation in the infrared (IR), corresponding to a temperature of a few hundreds Kelvins. For a star to become massive, the inflow must resist the pressure from this re-processed radi-ation. The ratio of the radiative acceleration to the gravitational acceleration is given by the dimensionless number called the Eddington factor kL /4pr2c ’8 105 k M? 2.5 G = ? , (1.1) GM?/r2 1 cm2 g 1 M tostar. Bottom: schematic view of the accretion via a disk onto a protostar. Dust re-emission escapes preferentially via the poles, where the medium is transparent. Credits: Kuiper et al. (2010b).
where we have used a simplified mass-luminosity relation: L? _ M?3.5 (valid between 2 and 55 M ). The dust opacity k varies by several orders of magnitude with the photon frequency so there is no typical value. We can however take k’5 cm2g 1 (Krumholz, 2017) as a lower limit adapted to protostellar disks in the Milky Way. Thus, the mass above which G > 1 is equal to M’23 M . Reasoning on the dynamical versus radiative pressure leads to a similar result and requires to compute the dust sublimation radius. The uncertainty we had on the opacity is replaced by the uncertainties on the various prescriptions used to compute this radius (density profile, temperature profile), which can end up as a factor 1.5 on the stellar mass (Larson & Starrfield, 1971). Above this mass, radiative acceleration is sufficient to reverse the flow. Numerical simulations performed by Kuiper et al. (2010b) under the hypothesis of spherical.
This calculation shows a main difference between low- and high-mass star formation. Any model must explain how to overcome this radiation pressure barrier, since the existence of stars above this limit is clear (e.g., Crowther et al. 2010). A substancial part of the answer came from the 2D frequency-dependent radiation-hydrodynamical simulations of Yorke & Sonnhal-ter (2002) (confirmed later by Kuiper et al. 2010b, see bottom panel of Fig. 1.5). Their work has shown the emergence of the accretion via a disk. Thanks to their careful treatment of radia-tion, they were able to capture the strong anisotropy of the radiation field. They showed that in a protostar-disk system radiation tends to leave through the poles (the so-called, flashlight effect), which lowers the radiation pressure onto the accretion flow and hence lets the star gain-ing more mass. They obtained a 42.9 M star from the collapse of a 120 M pre-stellar core. Therefore, disk accretion, as in low-mass star formation, is a viable and sufficient accretion mode for massive stars. As I will show now, several models have passed this radiation barrier test and attempted to formulate a complete theory for massive star formation.

Current models of massive star formation

At small scales, disk accretion has been shown to be effective (Yorke & Sonnhalter, 2002) to overcome the radiation pressure barrier. The global models have attempted to explain the origin of the mass of massive stars, whether it has been pre-assembled (and how) or brought by environmental interactions. Two classes of models exist, between which the main difference is whether massive stars form in isolation or not. I will first focus on the latter class, where the process that leads to a massive star includes many dynamical interactions between a protostar and its environment.
Accretion-induced collisions are among the mechanisms able to circumvent this radiation pressure barrier problem (Bonnell et al. 1998). Since it occurs above 20 M , Bonnell et al. (1998) have proposed that intermediate-mass stars ( 10 M ) form through the classical star formation scenario, accreting gas from their surrounding, until they come close enough to each-other to finally merge and lead to a higher-mass object. This model predicts that massive stars would form at the center of stellar clusters. The isolated massive stars we observe would have been ejected from their birth site by gravitational interactions, like the runaway stars we have mentioned above. It would not, however, explain how massive stars with low proper motions would have formed.
In a larger frame, the competitive accretion model has emerged (Bonnell & Davies, 1998), based on the observation of mass-segregation in the Trapezium cluster in Orion, i.e. the more massive stars being located in the center. This has been further studied in numerical simu-lations, without radiative transfer (Bonnell & Davies 1998, Bonnell et al. 2001, Bonnell et al. 2004). They have simulated the dynamics of a stellar cluster forming from a turbulent molecu-lar cloud and compared the final mass of stars to the masses of their clumps (from which stars form), their envelopes and their subsequent accretion events. They find no correlation between a star mass and its clump mass and a good correlation between a star mass and its envelope mass (for low-mass stars but not for high-mass stars). The envelope is defined as the spher-ical mass reservoir where at least 99% of the gas ends up in the star, i.e. which is not shared by other stars. Furthermore, they find a clear correlation between a star final mass and the mass subsequently accreted apart from its parent clump. Their results show that the mass of a massive star is governed by its ability to accrete the common gas and is independent of the mass of the clump from which it formed, while its envelope mass is not sufficient. They also find a linear relation between the mass of a massive star and the number of companions, which is consistent with multiple systems being more frequent for massive stars than for low-mass stars. However, these studies suffer the absence of radiative transfer and magnetic fields. In particular, pure hydrodynamical simulations would favor fragmentation by ignoring stabiliz-ing effects from magnetic fields (Commerc¸on et al. 2011a, Myers et al. 2013) and/or turbulence.
At molecular cloud scales, the Global Hierarchical Collapse model (GHC,Vazquez´-Semadeni et al. 2016, Vazquez´-Semadeni et al. 2019) naturally includes massive star formation as a part of a hierarchical fragmentation model, in which small-scale collapses occur within large-scale collapses. In this frame, filaments consist in flows of infalling gas (”ridges”) coming from the larger scales and directed towards the central zones, called ”hubs”, where massive stars form. As in the competitive accretion model, it relies on a ”clump-fed” accretion (Smith et al., 2009) in contrast to ”core-fed” accretion of isolated models (McKee & Tan, 2003) (see below). This model is supported by the ubiquiticity of filaments, some of them containing large-scale infall motions (Schneider et al. 2010, Csengeri et al. 2011). Observations with the Atacama Large Mil-limeter Array (ALMA) of individually collapsing prestellar clumps and high mass infall rates onto massive cores (e.g, Neupane et al. 2020) are consistent with this picture, in contrast with isolated cores in the core-fed accretion models. In this frame, the accretion rate is supposed to increase with time in the area of the molecular cloud under global collapse.As a by-product, it also favors interactions and therefore collisions (Bonnell et al., 2003).
The inertial-inflow model (Padoan et al., 2019) aims at linking low- and high-mass star for-mation to large-scale turbulence driven by supernovæ explosions. This supersonic turbulence leads to density peaks which can collapse gravitationally (so-called turbulent fragmentation). The power-law end of the IMF is a natural consequence of the turbulence being scale-free, whose exponent is fixed by the temperature, the mean density and the velocity dispersion (Padoan et al., 1997). In this model, a low- to intermediate-mass star can form from a pre-stellar core before it accretes enough material from large-scale converging inertial flows to become a massive star. Consistently, the timescale to form stars in this scenario scales with the mass: the time it takes to gather 95% of the final stellar mass Mf is 0.51Myr (Mf/1 M )0.54, which gives 2 Myr for a 10 M star. This is the most recent model of its class.
The turbulent core accretion model (McKee & Tan, 2003) is in the second class of models. It is a somehow scaled-up version of the low-mass star formation model. It aims at explaining the mass of massive stars by inheritance from their parent core mass. In this model, the core undergoes Jeans fragmentation at higher masses (than in the low-mass star formation theory) because turbulence and magnetic fields help stabilizing it. Turbulence has a second role: it generates large gas velocities which can therefore circumvent the radiation pressure barrier, where the accretion rate goes as c3s/G in the isothermal case (Shu, 1977), as cs is the isothermal sound speed and would be larger when accounting for non-thermal motions (McKee & Tan, 2002). Reaching the star final mass takes one to several free-fall times, typically 100 kyr. This model suffers a lack of consistency with observations. Indeed, it predicts the existence of starless massive pre-stellar cores, these massive clumps with no stellar activity, on the verge to collapse. However, to this date there only exist few candidates (e.g., Nony et al. 2018). Hence, whether it occurs or not, this pathway should not be the most frequent to form massive stars. In spite of this apparent tension, this model is often used in massive star formation simulations as it permits to probe the small-scale dynamics while not having to deal with the global dynamics of the molecular cloud.
The need for observational constraints is striking in the case of the turbulent core accretion model since it relies on the existence of a particular structure, the massive pre-stellar core. Re-cent technological advances in the millimiter and infrared regime, together with polarization measurements to deduce the magnetic field properties, have not ruled-out any of the models above yet. Nonetheless, they have made the multi-physical aspect of massive star formation clear. Although theoretically-challenging, this richness also provides complementary informa-tions on the birth and evolutionary path of massive protostars, as I will show.

The legacy of Spitzer and Herschel, the advent of ALMA

High-mass stars are embedded in very dense clouds at more than 1 kpc (except Orion, right panel of Fig. 1.9). Their observation requires far-IR (Spitzer and Herschel space observatories), (sub)millimeter imaging (SubMillimeter Array, SMA, and ALMA) at high angular resolution, and spectroscopy to identify gas motion (infall, outflows) and masers. Here I present the ob-servational constraints regarding the conditions in which high-mass stars form.
A possible tool to derive an empirical evolutionary sequence has been provided by proto-stellar radiation. Ionizing radiation from young massive stars can form HII regions that expand and eventually destroy the parent cloud. The size of this region has been considered as a clock to measure its evolution (see Churchwell 2002 for a review), from hyper-compact HII region (HCHII, see also Keto 2007), to ultra-compact HII regions (UCHII), compact HII regions and finally standard HII regions (see Fig. 1.6). These sources are strong emitters of free-free radia-tion (also called Bremsstrahlung), i.e. when free electrons are scattered by protons without being captured. They mainly radiate at centimeter wavelengths for their typical thermal spectrum in HII regions (Te 104 K), as they are being deflected (Wynn-Williams & Becklin, 1974). In addition, the changes in density (traced in, e.g., CO then 13CS or N2H+ for denser gas), tem-perature, molecular abundances and masers (e.g, methanol and OH) in the massive protostar vicinity provide an evolutionary path as well. This gives a temporal axis to understand massive star formation and a more precise view of particular epochs, often linked to particular spatial scales, is required, together with large statistics.

Absorption, emission, scattering

Let us go further in our description of radiation transport. The extinction accounts for the radiation losses, hence a mean free path of photons is defined as lp,n 1/cn. Let us note that the absorption, emission and scattering cross sections of various materials are isotropic in the fluid rest frame. At each frequency corresponds a cross section of interaction (cm2) which can vary significantly with the frequency. Once multiplied by the particle density, this gives the absorption and scattering coefficients (and therefore the extinction coefficient) above. In the laboratory frame, these would be affected by Doppler shift and aberration effects, due to the fluid motion.
We have defined c and h as absorption and emission coefficients, respectively. It would be useful, however, to distinguish between thermal absorption/emission and scattering. As an example of thermal absorption, an atom is excited by an incoming photon and de-excited by the collision with another particle which inherit from the photon energy in the form of kinetic energy. Then, the material internal energy is increased by the photon radiative energy it has absorbed. Similarly, thermal emission converts thermal energy of a hot gas into radiative energy.
On the opposite, when a photon is scattered by a charged particle (Compton scattering, called Thompson scattering in the case of a low-energy photon), an atom or a molecule (Rayleigh scattering if elastic, i.e. no exchange of energy, Raman scattering otherwise), it is deviated from its original direction with a possible small change of frequency, hence the gas thermal energy is (close to) unchanged. Indeed, there can be a tiny change of frequency which means that some energy has been gained by the atom/particule. For simplicity, we will consider it to be zero.
Therefore, we split the extinction coefficient into the sum of an absorption coefficient k and scattering coefficient s, i.e. c(x, t; n, n) = k(x, t; n, n) + s(x, t; n, n). (2.20).
Likewise, we decompose the emission coefficient as h(x, t; n, n) = ht(x, t; n, n) + hs(x, t; n, n), (2.21).

Hybrid radiative transfer

As can be seen in Sects. 2.2.2 and 2.2.3, the FLD and M1 methods present complementarities. The FLD method is well-suited in optically-thick media, and includes one equation. The M1 method is more advanced and adapted for describing unidirectional radiation beams, but its closure relation is more complex.
Motivated by the question of massive star formation, we have undertaken the coupling of the two methods, both already implemented in RAMSES (Commerc¸on et al. 2011b and Rosdahl et al. 2013). Indeed, their definition of the radiative acceleration differs, which is of main inter-est for us (as mentioned in Sect. 1.7). Our aim is to follow the protostellar radiation propagation and absorption with the M1 method. Meanwhile, the heating of the surrounding gas-and-dust mixture and its reemitted radiation is handled with the FLD method. In that view, we have chosen a gray formalism for both modules, but multigroup approaches are available (Rosdahl et al. 2013, Gonzalez´ et al. 2015 for the FLD).
The equations governing the M1 module of the hybrid approach are therefore 8 ¶E r F = ¶t + > < > ¶F + c2r P = : ¶t . kPcE + Er?, (2.46) .
where Er? is the radiative energy injection rate by stellar sources. The gas thermal emission term in Eq. 2.42 has also disappeared, since the M1 module only deals with stellar radiation here. In this work, we have focused on a single source for simplicity and because of the problems aris-ing with multiple sources within the M1 method, as mentioned in Sect. 2.2.3. We recall that this model is gray, so the opacity and extinction coefficients are frequency-averaged coefficients. More than that, we have first considered isotropic scattering so cF = kF + sF, where kF and sF would be the flux-averaged absorption and scattering opacities, respectively. The assumption that the spectral shapes of E and F are similar is made, so that sF and kF are approximated by the Planck mean scattering and absorption opacities, respectively, i.e. sF ’ sP and kF ’ kP. Doing so, all the averages are weighted by the Planck function, hence they only depend on the temperature. The mean opacity and extinction coefficients are evaluated at the stellar temper-ature. We consider the spectral distribution of stellar radiation to be that of a blackbody. After showing that scattering does not contribute significantly in our pure radiative transfer tests (Sect. 3.5), we have neglected it in collapse calculations, with cF = kP. Since the averaged opac-ity only depends on the temperature, we note kP,? = kP(T?) the Planck mean opacity computed at the stellar effective temperature T?.
Finally, the set of equations we obtain is 8 ¶t + r FM1 = kP,? cEM1 + EM1, > ¶EM1 . ? ¶FM1 + c 2 PM1 = kP,? cFM1, >.

Table of contents :

List of Figures
List of Tables
1 Introduction 
1.1 Preamble and definitions
1.2 The central role of massive stars
1.3 A first step : Low-mass star formation
1.3.1 Theory
1.3.2 Empirical evolutionary sequence
1.4 Towards higher masses: when radiation comes into play
1.5 Current models of massive star formation
1.6 The legacy of Spitzer and Herschel, the advent of ALMA
1.7 Recent numerical advances
1.8 This work
1.9 Essentials
1.9.1 Stability of a cloud: order of magnitude consideration
1.9.2 Pertubative analysis
1.9.3 Virial theorem applied to a collapsing cloud
1.9.4 Radiative or magnetic outflows?
2 Radiative Transfer 
2.1 Fundamental quantities and equation of transfer
2.1.1 Definitions
2.1.2 The radiative transfer equation
2.1.3 Absorption, emission, scattering
2.2 Moment models
2.2.1 Gray radiative transfer
2.2.2 Flux-limited diffusion
2.2.3 M1 model
2.2.4 Hybrid radiative transfer
2.3 Radiation Hydrodynamics equations
2.3.1 Why Radiation Hydrodynamics in star formation?
2.3.2 Equations in the non-relativistic regime
3 RAMSES and the Hybrid Radiative Transfer method 
3.1 The RAMSES code
3.1.1 The AMR structure
3.1.2 Solving the Euler equations on a Cartesian grid
3.1.3 Magneto-hydrodynamics with ambipolar diffusion
3.2 The Flux-Limited Diffusion implementation
3.2.1 The explicit step for the conservative part
3.2.2 The implicit step for gas-radiation coupling and diffusion
3.3 RAMSES-RT
3.3.1 Radiation transport
3.3.2 Radiation injection
3.3.3 Gas-radiation coupling
3.4 A hybrid implementation for stellar irradiation
3.5 Pure radiative transfer tests
3.5.1 Optically-thin and moderately optically-thick cases
3.5.2 Very optically-thick case
3.5.3 Temperature structure with isotropic scattering
3.5.4 Performance test
3.5.5 Perspectives
4 Massive Star Formation with RadiationHydrodynamics 
4.1 Collapse of a massive pre-stellar core with hybrid RT and hydrodynamics
4.1.1 Included physics
4.1.2 Setup
4.1.3 Results – overview
4.1.4 Disk properties
4.1.5 Radiative cavities – outflows
4.1.6 Accretion via Rayleigh-Taylor instabilities?
4.1.7 Physical outcomes
4.1.8 Performance
4.2 Modelling disk fragmentation in numerical codes
4.2.1 Context
4.2.2 Initial conditions
4.2.3 Disk properties
4.2.4 Stellar properties
4.2.5 Run with secondary sink particles
4.2.6 Extension of the comparison study and perspectives
5 Collapse of turbulent cores with radiation-magneto-hydrodynamics 
5.1 Context
5.2 Methods
5.2.1 Radiation magneto-hydrodynamical model
5.2.2 Physical setup
5.2.3 Resolution and sink particles
5.2.4 Analysis: disk and outflow identification
5.3 Temporal evolution
5.3.1 Overview
5.3.2 Alignment between the angular momentum and the magnetic field
5.3.3 Interchange instability
5.3.4 Sink mass history
5.3.5 Disk properties
5.4 Outflows
5.4.1 Origin
5.4.2 A channel for radiation?
5.4.3 Outflow velocity, mass, dynamical time, ejection rate
5.4.4 Outlow momentum rate
5.4.5 Opening angles
5.4.6 Alignment with the magnetic field
5.5 Conclusions
6 Conclusions and Perspectives
Appendices
A Basics of Virial theorem
B Core gravitational and rotational energy
C Luminosity injection in the sink particle volume
Bibliography

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