star or with the star displaced with respect to the evacuated cavity. Such cavities might be created by stellar winds and supernovae. The resulting images more closely resemble many actual H II-regions than the original model. Mathematical basis Let's suppose the region is exactly spherical, fully ionized (x=1), and composed only of hydrogen, so that the numerical density of protons equals the density of electrons (). Then the Strömgren radius will be the region where the recombination rate equals the ionization rate. We will consider the recombination rate of all energy levels, which is is the recombination rate of the n-th energy level. The reason we have excluded n=1 is that if an electron recombines directly to the ground level, the hydrogen atom will release another photon capable of ionizing up from the ground level. This is important, as the electric dipole mechanism always makes the ionization up from the ground level, so we exclude n=1 to add these ionizing field effects. Now, the recombination rate of a particular energy level is (with ): where is the recombination coefficient of the nth energy level in a unitary volume at a temperature , which is the temperature of the electrons in kelvins and is usually the same as the sphere. So after doing the sum, we arrive at where is the total recombination rate and has an approximate value of Using as the number of nucleons (in this case, protons), we can introduce the degree of ionization so , and the numerical density of neutral hydrogen is . With a cross section (which has units of area) and the number of ionizing photons per area per second , the ionization rate is For simplicity we will consider only the geometric effects on as we get further from the ionizing source (a source of flux ), so we have an inverse square law: We are now in position to calculate the Stromgren radius from the balance between the recombination and ionization and finally, remembering that the region is considered as fully ionized (x = 1): This is the radius of a region ionized by a type O-B star. See also Reionization Gunn–Peterson trough References Astrophysics 1939 in science