Zanstra method
Encyclopedia
The Zanstra method is a method to determine the temperature of central stars of planetary nebula.
It was developed by Herman Zanstra
in 1927.
It is assumed that the nebula is optically thick in the Lyman continuum which means that all ionizing photons from the central star are absorbed inside the nebula.
Based on this assumption, the intensity ratio of a stellar reference frequency to a nebula line such as Hβ can be used to determine the central star's effective temperature.
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of the nebula. Ionizations can only be caused by photons having at least the frequency
, corresponding to the ionization potential of hydrogen which is 13.6eV:
Here,
is the radius of the Strömgren sphere and 
are the number densities of protons and electrons, respectively. The luminosity of the central star is denoted by 
and 
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 then be estimated:
where
is the effective recombination coefficient for Hβ.
Given a stellar reference frequency
, the Zanstra ratio is defined by
with
and 
being the fluxes in the stellar reference frequency and in Hβ, respectively. Using the second formula, the Zanstra ratio can be determined by observations.
On the other hand, applying modell stellar atmospheres, theoretical Zanstra ratios may be computed in dependence of the central star's effective temperature which may be fixed by comparison with the observed value of the Zanstra ratio.
It was developed by Herman Zanstra
Herman Zanstra
Herman Zanstra was a Dutch astronomer.Zanstra was born near Heerenveen in Friesland. In 1917 he graduated with an Engineer's degree in chemical engineering from the Delft Institute of Technology...
in 1927.
It is assumed that the nebula is optically thick in the Lyman continuum which means that all ionizing photons from the central star are absorbed inside the nebula.
Based on this assumption, the intensity ratio of a stellar reference frequency to a nebula line such as Hβ can be used to determine the central star's effective temperature.
Zanstra method for a nebula of hydrogen
For a pure hydrogen nebula, the ionization equilibrium states that the number per unit time of ionizing photons from the central star has to be balanced by the rate of recombinations of protons and electrons to neutral hydrogen inside the Strömgren sphereStrömgren sphere
In theoretical astrophysics, a Strömgren sphere is the sphere of ionized hydrogen around a young star of the spectral classes O or B. Its counterpart in the real world are the H II-regions, a type of emission nebula, the most prominent of which is the Rosette Nebula...
of the nebula. Ionizations can only be caused by photons having at least the frequency
, corresponding to the ionization potential of hydrogen which is 13.6eV:Here,
is the radius of the Strömgren sphere and are the number densities of protons and electrons, respectively. The luminosity of the central star is denoted by and 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 then be estimated:
where
is the effective recombination coefficient for Hβ.Given a stellar reference frequency
, the Zanstra ratio is defined bywith
and being the fluxes in the stellar reference frequency and in Hβ, respectively. Using the second formula, the Zanstra ratio can be determined by observations.On the other hand, applying modell stellar atmospheres, theoretical Zanstra ratios may be computed in dependence of the central star's effective temperature which may be fixed by comparison with the observed value of the Zanstra ratio.