The Ionization Structure of Planetary Nebulae: III. The Ionization of Helium

In planetary nebulae the helium to hydrogen abundance ratio, by numbers of atoms, is Y = 0.1 to 0.2; the abundance of heavier elements, by numbers of atoms relative to hydrogen, is Z ∼ 10⁻³. The heavier elements may be neglected when considering the ionization of hydrogen and helium, but are of importance in considering the thermal balance. Three main regions are considered: In region (i), that is, the innermost region, the most abundant ions are He⁺² and H⁺. The He⁺²−He⁺ ionization equilibrium is calculated by the method of Paper I. Recombinations of He⁺² to He⁺ produce sufficient quanta to keep the hydrogen ionized. With Z = 0, the electron temperature in region (i) would be very high ($$T_e\geqslant7.5\times10^4\enspace ^\circ\text K$$ with no collisional excitation), but with Z ∼ 10⁻³ an efficient cooling process is shown to be excitation of allowed transitions in lithium-like ions such as C⁺³, giving T_e ≲ 2 × 10⁴ °K. In region (ii) the most abundant ions are He⁺ and H⁺. The He⁺−He⁰ and H⁺−H⁰ ionization equilibria are calculated on solving coupled equations. At lower star temperatures (T_δ < 5 × 10⁴ °K for Y = 0.2), the inner part of region (ii) contains He⁺ and H⁺ and the outer part contains He⁰ and H⁺. At higher star temperatures the transitions from He⁺ to He⁰ and from H⁺ to H⁰ occur at approximately the same point. In region (iii) none of the gas is ionized. This region exists only if the total amount of gas is sufficient to produce absorption of all ionizing stellar quanta.