First Core

After $\rho \mbox{\raisebox{0.3ex}{$>$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} \rho_A$, gas is no more isothermal and becomes adiabatic. Between $10^{-13}{\rm g\,cm^{-3}}\mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-... ...{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 10^{-9}{\rm g\,cm^{-3}}$, gas obeys the $\gamma =5/3$ polytropes. Above the density $\rho \mbox{\raisebox{0.3ex}{$>$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} \rho_{\rm A}$, the optical depth for the thermal radiation exceeds unity $\tau \mbox{\raisebox{0.3ex}{$>$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 1$ and radiative cooling can not compensate the compressional heating. As long as the temperature is low as $T\mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 100{\rm K}$, neither rotation nor vibration is excited for H$_2$ molecule. Even H$_2$ gas behaves like single-atom molecule. Thus $\gamma\simeq 5/3$.

Between $10^{-9}{\rm g\,cm^{-3}}\mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0... ....3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 10^{-7.5}{\rm g\,cm^{-3}}$, the exponent becomes $\gamma=7/5$, which characterizes that the gas consists of two-atom molecule ${\rm H}_2$.

In this phase, relatively large gas pressure supports against the gravity and the cloud becomes hydrostatic (points number 4-6 of Figure 4.14). This is called as ``first core'' made by the molecular hydrogen. The density structure of the first core is well represented by a polytrope sphere with the specific heat ratio of $\gamma=7/5$ or the polytropic index $n=2\frac{1}{2}$. From equation (C.11) in Appendix C.1, such a polytrope has a mass-density relation as

$$M_{c1}\propto \rho_c^{1/10},$$ (4.100)

where $M_{c1}$ and $\rho _c$ represent, respectively, the mass of the first core and the central density. At the beginning, the core mass is equal to $M_{c1}\simeq 0.01M_\odot$. As long as the mass increases a factor 3, the central density increases 5 orders of magnitude.