Strömgren sphere

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In astrophysics, a Strömgren sphere is the sphere of ionized hydrogen (H II) around a young star of the spectral classes O or B. The most prominent example is the Rosette Nebula. It was derived by and later named after Bengt Strömgren^[1].

An idealized calculation is simple, let's suppose the region is exactly spherical and fully ionized (x=1) and composed only of hydrogen so the numerical density of protons equals the density of electrons (Failed to parse (Missing texvc executable; please see math/README to configure.): n_e = n_p ), then the Strömgren radius will be the region where the recombination rate equals the ionization rate. We will consider the recombination rate Failed to parse (Missing texvc executable; please see math/README to configure.): N_R

of all energy levels which is

Failed to parse (Missing texvc executable; please see math/README to configure.): N_R = \sum_{n=2}^{\infty}N_n

Failed to parse (Missing texvc executable; please see math/README to configure.): N_n

is the recombiantion rate of the n-th energy level. The reason we have excluded n=1 is that if a photon with enough energy recombines in to the ground level the hydrogen atom will release another photon capable of ionizing up to the ground level. This is important as electric dipole mechanism always make the ionization up to the ground level so we exclude n=1 to add these ionizing field effect. Now, the recombination rate of a particular energy level Failed to parse (Missing texvc executable; please see math/README to configure.): N_n
is (with Failed to parse (Missing texvc executable; please see math/README to configure.): n_e=n_p

Failed to parse (Missing texvc executable; please see math/README to configure.): N_n=n_e n_p \beta_{n}(T_e)=n_e^2 \beta_{n}(T_e)

where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_{n}(T_e)

is the recombination coefficient of the nth energy level in a unitary volume at a temperature Failed to parse (Missing texvc executable; please see math/README to configure.): T_e
which is the temperature of the electrons and is usually the same of the sphere. So after doing the sum we arrive to:

Failed to parse (Missing texvc executable; please see math/README to configure.): N_R=n_e^2 \beta_2(T_e)

Where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_2(T_e)

is the total recombination rate and has an approximate value of:

Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_2(T_e) \approx 2 \times 10^{-22} T_e^{3/4} \ \mathrm{[m^{3} s^{-1}]} .

Using Failed to parse (Missing texvc executable; please see math/README to configure.): n

as the number of nucleons (in this case, protons), we can introduce the degree of ionization Failed to parse (Missing texvc executable; please see math/README to configure.): 0\leq x \leq1 
so Failed to parse (Missing texvc executable; please see math/README to configure.): n_e=xn

, and the numerical density of neutral hydrogen is Failed to parse (Missing texvc executable; please see math/README to configure.): n_e=(1-x)n . With a cross section Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_0

(which has units of area) and the number of ionizing photons per area per second Failed to parse (Missing texvc executable; please see math/README to configure.): J
the ionization rate Failed to parse (Missing texvc executable; please see math/README to configure.): N_I
is:

Failed to parse (Missing texvc executable; please see math/README to configure.): N_I=\alpha_0 n_h J

For simplicity we will consider only the geometric effects on Failed to parse (Missing texvc executable; please see math/README to configure.): J

as we get further from the ionizing source flux Failed to parse (Missing texvc executable; please see math/README to configure.): S_*

, so we have an inverse square law:

Failed to parse (Missing texvc executable; please see math/README to configure.): J(r)=\frac{3 S_*}{4 \pi r^3}

We are now in position of calculating the Stromgren Radius Failed to parse (Missing texvc executable; please see math/README to configure.): R_S , from the balance between the recombination and ionization

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{4 \pi}{3} (nx)^2 \beta_2 R_S^3 = S_*

and finally remembering that the region is considered as fully ionized (x=1):

Failed to parse (Missing texvc executable; please see math/README to configure.): R_S=\left( \frac{3}{4 \pi} \frac{S_*}{n^2 \beta_2} \right)^{\frac{1}{3}}

This is the radius of a region ionized by a type O-B star.