Ionization Rate

Figure 4.4: Ionization fractions $X/n_{\rm H}$ are plotted against the number density $n_{\rm H}$. The ionization rate of an H$_2$ molecule by cosmic rays outside the cloud is taken $\zeta_0 = 1 \times 10^{-17} {\rm s^{-1}}$. 20% of C and O and 2% of metallic elements are assumed to remain in the gas phase and the rest in grains.

In the dense clouds, the ionization fraction is low. Since the uv/optical radiations from stars can not reach the cloud center, potential ionization comes from the cosmic ray particles. In this case the rate of ionization per volume is given as $\zeta n_n$, where $\zeta\sim 10^{-17}{\rm s^{-1}}$. In Figure 4.4, the ionization fraction for various density is shown (Nakano, Nishi, & Umebayashi 2002). This clearly shows that the fraction of ions decrease approximately in proportion to $n_{\rm H}^{-1/2}$ for $n_{\rm H}\mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 10^8{\rm cm^{-3}}$. This is understood as follows: Equilibrium balance between one kind of ion $m^+$ and its neutral $m^0$

$$m^0 \leftrightarrow m^+ + e^{-},$$ (4.35)

is considered. The recombination (reaction from the right to left) rate per unit volume is expressed $\alpha n_e n_{m^+}$, while the ionization rate (left to right) per unit volume is $\zeta n_{m^0}$, where $\alpha$ means the recombination rate coefficient. Nakano (1984) obtained in the range of $10^2{\rm cm^{-3}}\mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{... ...isebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 10^8{\rm cm^{-3}}$

$$\rho_i = C \rho_n^{1/2},$$ (4.36)

where the numerical factor $C=4.46\times 10^{-16}{\rm g^{1/2}\,cm^{-3/2}}$.