Zanstra method

Lore & Background

The Zanstra method was developed by Herman Zanstra in 1927. It is based on the assumption that the nebula is optically thick in the Lyman continuum, which implies that all ionizing photons from the central star are absorbed inside the nebula. For a pure hydrogen nebula, the ionization equilibrium requires that the number per unit time of ionizing photons from the central star is balanced by the rate of recombinations of protons and electrons to neutral hydrogen inside the Strömgren sphere. Ionizations can only be caused by photons with at least the frequency ν₀, corresponding to the ionization potential of hydrogen (13.6 eV). The luminosity of the central star is denoted by L_ν, and α_B is the recombination coefficient to the excited levels of hydrogen. The ratio between the number of photons emitted by the nebula in the Hβ line and the number of ionizing photons from the central star can be estimated using the effective recombination coefficient for Hβ, α_Hβ^eff.

Reader's Guide

The Zanstra method provides a way to determine the effective temperature of central stars of planetary nebulae by comparing observed fluxes. The Zanstra ratio is defined as Z = L_ν_s / (∫_{ν₀}^{∞} (L_ν / hν) dν) = hν_Hβ (α_Hβ^eff / α_B) (F_ν_s / F_Hβ), where F_ν_s and F_Hβ are the observed fluxes in the stellar reference frequency and in Hβ, respectively. Using this formula, the Zanstra ratio can be determined from observations. On the other hand, applying model stellar atmospheres, theoretical Zanstra ratios may be computed as a function of the central star's effective temperature, which can then be fixed by comparison with the observed value. The method is significant because it allows astronomers to infer the temperature of the central star without direct spectroscopic analysis, relying instead on nebular emission lines and the assumption of optical thickness in the Lyman continuum.

Did You Know?

• It assumes the nebula is optically thick in the Lyman continuum, so all ionizing photons are absorbed.
• The method uses the intensity ratio of a stellar reference frequency to the Hβ nebular line.
• The ionization potential of hydrogen relevant to the method is 13.6 eV.