\magnification=\magstep1


$$
N_p \quad = \quad \left( {{4 \pi}\over{3}} \thinspace R^3 \right) \ n_p
$$

$$
N_{\rm recomb} \quad = \quad {{N_p}\over{t_r}} \quad = \quad 
        {{\left( {{4 \pi}\over{3}} \thinspace R^3 \right) \ n_p}\over{t_r}}
$$

$$
N_{\rm UV} \quad = \quad N_{\rm recomb} \quad = \quad
        {{\left( {{4 \pi}\over{3}} \thinspace R^3 \right) \ n_p}\over{t_r}}
$$

$$
R_{eq} \quad = \quad {\left( {{3 \thinspace N_{\rm UV}}\over{4 \pi n_p}} \right)}^{1/3}
             \ {\left( t_r \right)}^{1/3}
$$

$$
{\rm volume\ swept} \quad = \quad \sigma v_e t_r 
$$

$$
{\rm volume\ per\ electron} \quad = \quad {{1}\over{n_e}}
$$

$$
\sigma v_e t_r \quad = \quad {{1}\over{n_e}}
$$

$$
t_r \quad = \quad {{1}\over{\sigma v_e n_e}}
$$

$$
v_e \quad = \quad \sqrt{ {{3 k T}\over{m_e}} }
$$

$$
\sigma \quad \simeq \quad 5.9 \times 10^{-25} \ {\rm m}^2
$$

$$
\sigma_{\rm ionize} \quad \simeq \quad 6 \times 10^{-22} \ {\rm m}^2
$$

$$
{\rm mfp\ of\ ionizing\ photon} \quad = \quad
           {{1}\over{\sigma_{\rm ionize} \ n_H}}
$$


$$
R_{\rm Stromgren} \quad = \quad {\left( {{3 \thinspace N_{\rm UV}}\over{4 \pi }} \right)}^{1/3}
             \ {\left( {{1}\over{\sigma v_e}}  \right)}^{1/3}  \ n_p^{-2/3}
$$



\bye

