### First Core

After , gas is no more isothermal and becomes adiabatic. Between , gas obeys the polytropes. Above the density , the optical depth for the thermal radiation exceeds unity and radiative cooling can not compensate the compressional heating. As long as the temperature is low as , neither rotation nor vibration is excited for H molecule. Even H gas behaves like single-atom molecule. Thus .

Between , the exponent becomes , which characterizes that the gas consists of two-atom molecule .

In this phase, relatively large gas pressure supports against the gravity and the cloud becomes hydrostatic (points number 4-6 of Figure 4.16). 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 or the polytropic index . From equation (C.11) in Appendix C.1, such a polytrope has a mass-density relation as

 (4.110)

where and represent, respectively, the mass of the first core and the central density. At the beginning, the core mass is equal to . As long as the mass increases a factor 3, the central density increases 5 orders of magnitude.

Kohji Tomisaka 2009-12-10