I came across several papers (e.g. Looking for outflow and infall signatures in high mass star forming regions P. D. Klaassen, L. Testi and H. Beuther, 2011) talking about infall signatures in star formation. Can you please explain the physical meaning of it?
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The paper you linked is specifically trying to understand massive $(M\gtrsim 8\:\mathrm{M_\odot})$ star formation. It is still an open question as to exactly how very massive stars form. Do they form all at once in a giant accretion event ("top-down" formation) or do they form by having a bunch of low mass stars merge ("bottom-up" formation) or by some other process altogether? The authors of the paper state that it is particularly tricky to study high mass star formation.
Evolution in low mass systems is often characterized by changes in the spectral energy distributions (SEDs). This is not as easily done in high mass systems (Molinari et al. 2008)1, which is why studying the gas dynamics is very important.
Basically, they can't easily figure out what's going on by looking at the SED of the high mass star forming region, so instead, they're going to look at how the gas is flowing and moving during the formation. Klassen et al. break their observations into observing "infall" and "outflow" (via two different molecular transitions). What they mean by this is that one molecular transition (that of the $J=4-3$ transition in $\mathrm{HCO^+}$ and $\mathrm{H^{13}CO^+}$ molecules) will show evidence of gas flowing into the core, i.e. infall, while another molecular transition (that of the $J=8-7$ transition in the $\mathrm{SiO}$ molecule) will show evidence of gas flowing out of the core, i.e., outflow.
During star formation, you tend to get both types of flows. Infall will occur because the gas is simply accreting onto the core due to gravity. Outflow will occur because the accreting matter and central core get very hot, causing an energy release outwards.
You can see precisely how they track infall flows in Section 3.2. They have a paragraph discussing precisely how they determine that the gas is falling into the core from the observations of the $\mathrm{HCO^+}$ and $\mathrm{H^{13}CO^+}$ molecules.
Double peaked or asymmetric H$\mathrm{HCO^+}$ line profiles in which the brightest emission is blueward of the source rest velocity can be interpreted as due to infall if the $\mathrm{H^{13}CO^+}$ is single peaked at the rest velocity. If the optically thin $\mathrm{H^{13}CO^+}$ also has a double peaked profile, the double peak in both lines is likely due to multiple components along the line of sight. Churchwell et al. (2010)2 describe the different mechanisms that could produce a double peak in an $\mathrm{HCO^+}$ line profile towards massive star forming regions, and conclude that self absorption best fits observed line profiles. The blue or red skew of the absorption profile can then be used to distinguish infall (blue) or outflow (red).
I won't go into detail to precisely explain the spectroscopic interpretation of the above quote, but suffice it to say, that is their method of detecting matter infalling towards the core of the high mass star.
1 Molinari, S., Pezzuto, S., Cesaroni, R., et al. 2008, Astronomy and Astrophysics, 481, 345
2 Churchwell, E., Sievers, A., & Thum, C. 2010, Astronomy and Astrophysics, 513, 9
