Zanstra method

Zanstra Method: A Tool for Measuring Stellar Temperatures in Planetary Nebulae

In the vast expanse of space, we encounter phenomena that challenge our understanding and require innovative methods to decipher their mysteries. One such method is the Zanstra method, developed by Herman Zanstra in 1927, which allows us to determine the temperature of central stars found within planetary nebulae.

How it Works

The Zanstra method assumes that the nebula is optically thick in the Lyman continuum. This means that all ionizing photons emitted by the central star are absorbed within the nebula itself. By utilizing the intensity ratio of a stellar reference frequency to a nebular line such as Hβ, we can estimate the central star’s effective temperature.

For a pure hydrogen nebula, the number of ionizing photons from the central star must equal the rate at which protons and electrons recombine to form neutral hydrogen within the Strömgren sphere of the nebula. The luminosity of the central star, denoted as Lν, and the recombination coefficient for hydrogen, αB, play crucial roles in this equation.

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 Zanstra method. This ratio helps us estimate important quantities such as effective recombination coefficients for Hβ (αHβeff) and the Zanstra ratio.

The Zanstra Ratio and Its Significance

The Zanstra ratio is defined by the following equation:

Z = (Lνs ∫ ν0 ∞ Lν hν dν) / (hνHβ αHβeff αB Fνs FHβ)

Here, Lνs is the luminosity of a chosen stellar reference frequency, Fνs and FHβ are the fluxes in that same frequency and Hβ respectively, and αB and αHβeff are recombination coefficients for hydrogen. The Zanstra ratio can be determined by observations, providing us with valuable insights into the central star’s characteristics.

Applying the Zanstra Method

By comparing observed Zanstra ratios to theoretical ones computed using model stellar atmospheres, we can fix the central star’s effective temperature. This comparison offers invaluable knowledge about these celestial bodies and deepens our comprehension of the universe.