In all our simulations independent of the inflow Mach number, the radial velocity dispersion at the filament boundary amounts to about 0.85 times the total equilibrium velocity dispersion of the non-self-gravitational case for which a functional form can be found in the appendix. Thus, we can calculate the theoretical radius and central density of the filament at every line-mass and therefore we can make predictions on the fragmentation length and time-scales of cores forming in an accreting filament using the gravitational fragmentation model. This model was successfully applied to explain several observed core distances (Jackson et al. 2010; Miettinen 2012; Busquet et al. 2013; Beuther et al. 2015; Contreras et al. 2016; Heigl et al. 2016; Kainulainen et al. 2016) however it is not able to explain all observations (André et al. 2010; Kainulainen et al. 2013; Takahashi et al. 2013; Lu et al. 2014; Wang et al. 2014; Henshaw et al. 2016; Teixeira et al. 2016; Kainulainen et al. 2017; Lu et al. 2018; Palau et al. 2018; Williams et al. 2018; Zhou et al. 2019). It predicts that small density perturbations in the linear regime along the filament axis of the form:
(32)$$\begin{eqnarray*}
\rho (r, x, t) = \rho _0(r) \left(1 + \epsilon \exp (ikx -i\omega t)\right)
\end{eqnarray*}$$will grow for values of k where the dispersion relation ω2(k) is negative. Here ρ0 is the unperturbed initial density, k = 2π/λ is the wave vector with λ being the perturbation length, x is the filament axis, ω = 1/τ is the growth rate with τ being the growth time-scale, t the time variable, and ϵ the perturbation strength. The fastest growing, or dominant, fragmentation length scale λdom as well as the growth time-scale of the dominant mode τdom depend on the current line-mass as well as the current central density of the filament and are given by the pre-calculated (Nagasawa 1987) and interpolated values in Fischera & Martin (2012), shown by their Table E.1. We use these values to determine the length scale of the fastest growing mode at every line-mass for the same mass accretion rate but for different inflow Mach numbers as shown in Fig. 10. As one can see, the dominant fragmentation length changes over the evolution of the line-mass. At the boundary values it vanishes to zero and it has a maximum at about fcyl = 0.4. The figure is self-similar for different mass accretion rates, with a lower rate leading to a larger dominant fragmentation length. For a constant accretion rate, the fragmentation length does not vary much for different inflow Mach numbers. Only for large and for very low inflow Mach numbers, the fragmentation length is slightly larger. As the dominant fragmentation length constantly changes as fcyl grows, it is hard to make predictions of what will be the final distance between forming cores. But the curves have a maximum which allows us to make a prediction about the minimum number of cores that will form. For instance, a filament with an inflow Mach number of 4.0 and a length of 0.2 pc will form at least one core. As soon as the first core forms, the further evolution of the filament is also influenced by the gravitational attraction of the core. This makes the formation of additional cores even more unpredictable.
Citation: Wang et al. 2014
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