Abstract
Star formation is a complex multi-scale phenomenon that is of significant importance for astrophysics in general. Stars and star formation are key pillars in observational astronomy from local star forming regions in the Milky Way up to high-redshift galaxies. From a theoretical perspective, star formation and feedback processes (radiation, winds, and supernovae) play a pivotal role in advancing our understanding of the physical processes at work, both individually and of their interactions. In this review we will give an overview of the main processes that are important for the understanding of star formation. We start with an observationally motivated view on star formation from a global perspective and outline the general paradigm of the life-cycle of molecular clouds, in which star formation is the key process to close the cycle. After that we focus on the thermal and chemical aspects in star forming regions, discuss turbulence and magnetic fields as well as gravitational forces. Finally, we review the most important stellar feedback mechanisms.
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Notes
In fluid dynamics, an eddy is understood as the swirling current of a fluid or a ‘blob of vorticity’, but there is a good tradition in hydrodynamic turbulence of avoiding any formal definition of a turbulent eddy, see page 52 in Davidson (2004).
In realistic situations, both assumptions are usually not satisfied at large scales. However, if they are valid at small scales and far from boundaries of the flow or its other special regions, this general theoretical framework still remains useful.
Note that the Fourier transforms of homogeneous random functions are, generally, random distributions, i.e. not ordinary functions of their argument, \(\boldsymbol{k}\). One way to deal with this mathematical difficulty is to replace the ordinary integrals with generalized stochastic Fourier-Stieltjes integrals (Batchelor 1953); another way is to use low- or high-pass filtering (e.g. coarse-graining), which allows one to deal lusively with ordinary functions (Frisch 1995).
Envisioned by Lewis F. Richardson in 1922 (Richardson 1965).
A similar relation, \(\langle \delta u_{\parallel }(r) [\delta \theta (r)]^{2}\rangle =- \frac{4}{3}\epsilon _{\theta } r\), was obtained by Yaglom (1949) for temperature fluctuations in turbulent flows. Here, \(\delta \theta (r)\) is the temperature increment and \(\epsilon _{\theta }\) is the mean dissipation rate of temperature fluctuations. An anisotropic generalization of this relation exploited in Galtier and Banerjee (2011), \(\boldsymbol{\nabla }\boldsymbol{\cdot }\langle |\delta \boldsymbol{u}(\boldsymbol{r})|^{2}\delta \boldsymbol{u}( \boldsymbol{r})\rangle =-4\epsilon \), avoids projection onto the ∥ direction and is sometimes called the von Kármán-Howarth-Monin relation (Frisch 1995; Antonia et al. 1997).
The same splitting based on \(\boldsymbol{u}_{\mathrm{s}}\) and \(\boldsymbol{u}_{\mathrm{d}}\) does not work for the Reynolds stress \(R_{ij}\equiv \langle \rho u_{i}u_{j}\rangle \), since in addition to solenoidal and dilatational stresses, there is also non-zero cross Reynolds stress (Lele 1994).
There are hydrodynamic and acoustic pressure fluctuations. The pseudo-sound represents vortical pressure fluctuations advected with the fluid velocity, while acoustic waves propagate at the speed of sound. In the pseudo-sound component, the dilatational velocity field is in equilibrium with the solenoidal pressure. Both types of fluctuations can be measured by the observer.
The sonic scale, \(k_{\mathrm{s}}\), is defined as a scale at which the root mean squared velocity fluctuations are equal to the mean sound speed, \(\int _{k_{\mathrm{s}}}^{\infty }E(k)dk=\langle c_{\mathrm{s}}\rangle ^{2}\).
This type of modeling is usually called an implicit large-eddy simulation (ILES) (Sytine et al. 2000) or a coarse DNS since the dissipation is of purely numerical origin and depends on the numerical method used.
A similar slope was also obtained in simulations by W.-C. Müller (2005, private communication).
In this context, point-splitting regularization refers to the use of two-point statistics (such as correlation functions, structure functions, and power spectra) when the products of fields at the same spatial (temporal) location are not mathematically well-defined.
The coarse-graining operation is a simple convolution \(\langle \boldsymbol{a}\rangle _{r}(\boldsymbol{x},t)=\int \phi _{r}(\boldsymbol{y})\boldsymbol{a}(\boldsymbol{x}+ \boldsymbol{y},t)d\boldsymbol{y}\) with a smooth mollifier \(\phi _{r}(\boldsymbol{y})=\phi (\boldsymbol{y}/r)/r^{3}\) such that \(\int \phi _{r}(\boldsymbol{y})d\boldsymbol{y}=1\). This type of smooth filtering with compact support in space is used to single out the large scale component of a field variable \(\boldsymbol{a}(\boldsymbol{x},t)\) corresponding to length scales \(>r\).
Here, ∗ denotes the complex conjugate and we used the convolution theorem to cast the Fourier transform of the correlation function \(R_{\boldsymbol{j}\boldsymbol{u}}(\boldsymbol{r})\) using the Fourier transforms of \(\boldsymbol{j}\) and \(\boldsymbol{u}\). Using the symmetric cross-covariance makes sure that the spectral kinetic energy density, \(K(\boldsymbol{k})\), is real. Finally, it follows from Parseval’s theorem that \(K=\int K(\boldsymbol{k})d\boldsymbol{k}=R_{\boldsymbol{j}\boldsymbol{u}}(0)/2\).
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Acknowledgements
We thank the staff of the International Space Science Institute (ISSI) for their generous hospitality and for creating a stimulating collaborative environment. P.G. acknowledges funding from the European Research Council under ERC-CoG grant CRAGSMAN-646955. S.S.R.O. acknowledges funding from NSF Career grant AST-1650486. J.M.D.K. gratefully acknowledges funding from the German Research Foundation (DFG) in the form of an Emmy Noether Research Group (grant number KR4801/1-1) and a DFG Sachbeihilfe Grant (grant number KR4801/2-1), from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme via the ERC Starting Grant MUSTANG (grant agreement number 714907), and from Sonderforschungsbereich SFB 881 “The Milky Way System” (subproject B2) of the DFG. R.S.K. acknowledges support from the Deutsche Forschungsgemeinschaft via the SFB 881 “The Milky Way System” (subprojects B1, B2, and B8) as well as funding from the Heidelberg Cluster of Excellence STRUCTURES in the framework of Germany’s Excellence Strategy (grant EXC-2181/1 - 390900948). A.G.K. acknowledges support from the NASA ATP Grant No. 80NSSC18K0561 and NASA TCAN Grant No. NNH17ZDA001N. M.P. acknowledges funding from the INAF PRIN-SKA 2017 program 1.05.01.88.04.
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Star Formation
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Girichidis, P., Offner, S.S.R., Kritsuk, A.G. et al. Physical Processes in Star Formation. Space Sci Rev 216, 68 (2020). https://doi.org/10.1007/s11214-020-00693-8
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DOI: https://doi.org/10.1007/s11214-020-00693-8