SF2A 2014 J. Ballet, F. Bournaud, F. Martins, R. Monier and C. Reyl´ e (eds) ON THE STABILITY OF SELF-GRAVITATING FILAMENTS J. Freundlich 1 , C. J. Jog 2 , F. Combes 1 and S. Anathpindika 2 Abstract. Filamentary structures are very common in astrophysical environments and are observed at various scales. On a cosmological scale, matter is usually distributed along filaments, and filaments are also typical features of the interstellar medium. Within a cosmic filament, matter can possibly contract and form galaxies, whereas an interstellar gas filament can clump into a series of bead-like structures which can then turn into stars. To investigate the growth of such instabilities and the properties of the resulting substruc- tures, we consider idealized self-gravitating filaments and derive the dispersion relation for perturbations within them. We assume no specific density distribution, treat matter as a fluid, and use hydrodynamics to derive the linearized equations that govern the growth of perturbations. Assuming small local perturbations leads to a dispersion relation analogous to the spherical Jeans case: perturbations of size higher than the Jeans length collapse and asymmetries regarding their growth rates arise only because of rotation. For perturbations of arbitrary size, the dispersion relation retains its complex terms: all modes are potentially unstable, but elongated perturbations near the axis of the cylinder grow faster. Prolate substructures and global collapse are favored, which is corroborated by most observations of interstellar filaments. Keywords: gravitation, hydrodynamics, instabilities, large-scale structure of the Universe, ISM: structure 1 Introduction Although filaments have been observed since decades within molecular clouds (e.g., Schneider & Elmegreen 1979), cosmological simulations and high-resolution observations of the interstellar medium only recently showed the key role played by filamentary structures at various scales in astrophysics. Filamentary structures are indeed ubiquitous and involved in processes as varied as gas accretion onto galaxies and the formation of stars in the interstellar medium. On cosmological scales, matter is usually distributed along filaments, forming a cosmic web that connects galaxies to one another (e.g., Bond et al. 1996) and provides a gas reservoir from which galaxies grow and accrete (e.g., Kereˇ s et al. 2005; Dekel et al. 2009). The inner core of many of these filaments may be predominantly made of gas, as notably shown by simulations by Harford et al. (2008), motivating models which treat them as self-gravitating, isothermal or barotropic cylinders in hydrostatic equilibrium. In the interstellar medium, observations show filamentary structures on much smaller scales (e.g., Andr´ e et al. 2010; Arzoumanian et al. 2011). Motivated by Herschel observations of star-forming environments, Andr´ e et al. (2010) suggest a scenario in which the formation of turbulence-driven filaments in the interstellar medium represents the first step towards core and star formation. The densest filaments would then fragment into pre-stellar cores owing to gravitational instability. Simulations reveal filamentary features arising either from turbulence (e.g., Padoan et al. 2001) or from intermediate stages of gravitational collapse (e.g., Gomez & Vazquez-Semadeni 2014). 2 Studying the growth of instabilities through linearized equations The standard Jeans instability describes the collapse of a spherical gas cloud when the inner pressure is not strong enough to support the self-gravitating gas. The cylindrical case is more complicated and has not been fully investigated yet. Our goal is to obtain a dispersion relation for small perturbations arising in an idealized 1 LERMA, Observatoire de Paris, CNRS, 61 av. de l’Observatoire, 75014 Paris, France 2 Department of Physics, Indian Institute of Science, Bangalore 560012, India c  Soci´ et´ e Francaise d’Astronomie et d’Astrophysique (SF2A) 2014