We explore large-scale hydrodynamics of H \Rmnum 2 regions for various self-similar shock flows of a polytropic gas cloud under self-gravity and with quasi-spherical symmetry .
We formulate cloud dynamics by invoking specific entropy conservation along streamlines and obtain global self-similar “ champagne flows ” for a conventional polytropic gas with shocks as a subclass .
Molecular cloud cores are ionized and heated to high temperatures after the onset of nuclear burning of a central protostar .
We model subsequent evolutionary processes in several ways and construct possible self-similar shock flow solutions .
We may neglect the mass and gravity of the central protostar .
The ionization and heating of the surrounding medium drive outflows in the inner cloud core and a shock travels outwards , leading to the so-called “ champagne phase ” with an expanding outer cloud envelope .
Complementarily , we also consider the expansion of a central cavity around the centre .
As the inner cloud expands plausibly due to powerful stellar winds , a cavity ( i.e. , ‘ void ’ or ‘ bubble ’ ) can be created around the centre , and when the cavity becomes sufficiently large , one may neglect the gravity of the central protostar .
We thus present self-similar shock solutions for “ champagne flows ” with an expanding central void .
We compare our solutions with isothermal solutions and find that the generalization to the polytropic regime brings about significant differences of the gas dynamics , especially for cases of n < 1 , where n is a key scaling index in the self-similar transformation .
We also compare our global polytropic self-similar solutions with numerical simulations on the expansion of H \Rmnum 2 regions .
We further explore other possible dynamic evolutions of H \Rmnum 2 regions after the initiation of nuclear burning of the central protostar , for example asymptotic inflows or contractions far from the cloud centre and the ongoing infall around a central protostar .
In particular , it is possible to use the downstream free-fall solution with shocks to describe the dynamic evolution of H \Rmnum 2 regions shortly after the nascence of the central protostar .
We also give an analysis on the invariant form of self-similar polytropic flows by ignoring self-gravity .