Isothermal Phase

Below $\rho \mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 10^{-13}{\rm g cm^{-3}}$, gas is essentially isothermal $T\propto \rho^0$. This corresponds to points number 1-3 of Figure 4.16. Since internal energy of gas is transferred to the thermal energy of dusts by collisions, the main coolant in this regime is the dust thermal radiation. The cooling rate per mass is

$$\Lambda=4\kappa_p\sigma T^4,$$ (4.107)

where $\kappa_p(10{\rm K})\sim 0.01{\rm cm^2 g^{-1}}$ and $\sigma$ represent Planck mean absorption coefficient and the Stephan-Boltzman constant. Main heating process for $\rho \mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 10^{-14}{\rm g cm^{-3}}$ is cosmic-ray heating: $\epsilon_{\rm CR} \sim 2\times 10^{-4} {\rm erg s^{-1} g^{-1}}$ (Goldsmith & Langer 1978). Balance between these two majors asks that temperature is constant as

$$T\sim 3{\rm K}\left(\frac{\epsilon_{\rm CR}}{2\times 10^{-4... ... \left(\frac{\kappa_p}{0.01{\rm cm^2 g^{-1}}})\right)^{1/2}.$$ (4.108)

Heating rate due to dynamical compression,

$$\Gamma=-p\frac{d (1/\rho)}{d t}\simeq c_s^2 (4\pi G \rho),$$ (4.109)

increases according to the contraction and it balances with the above cooling at the density $\rho_{\rm A}\sim 10^{-14}{\rm g cm^{-3}}$ (Masunaga, Miyama, & Inutsuka 1998).

The cloud in this phase experiences the dynamical contraction as described in section 4.5. Structure of $\rho (r)$ and $v_r(r)$ is well represented by the Larson-Penston self-similar solution.