The Ionization Structure of Planetary Nebulae: V. Radii, Luminosities and Problems of Evolution

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Abstract
The forty-seven planetaries of Paper IV are divided into two Classes: (a) with He II lines and (b) without He II lines. From results obtained in IV, the ionized hydrogen mass M of Class (a) nebulae is expected to be equal, or nearly equal, to the total nebular hydrogen mass MH . For fourteen of the nebulae considered electron densities Ne can be deduced from relative intensities of forbidden lines, and nebular radii R ( Ne ) and values of M can then be obtained using measured H β surface brightnesses. Of the eight Class (a) nebulae for which M can be deduced in this way, six have M close to 0.38 M⊙ . Assuming this value of M , and using measured surface brightnesses, radii R ( M ) are obtained for thirty-seven Class (a) nebulae. Most Class (b) nebulae have values of M less than 0.38 M⊙ ; this is a consequence of their being optically thick. Radii R ( Ne ) are obtained for six Class (b) objects. Deduced distances, r = R / θ where θ is the angular radius, are compared with distance estimates obtained by other methods. Satisfactory agreement is obtained. Using the results of Paper IV, central star temperatures T and luminosities L are obtained for forty planetaries and are plotted on the [log T , log( L / L⊙ )] H−R diagram. An evolutionary track is obtained: starting at [4.5, 1.8], T and L increase to [4.8, 4.4], then T increases at approximately constant L to [5.0, 4.4] and finally L decreases to [5.0, 2.0]. This whole process takes place in a time of 5 × 10 4 years. The evolution of the nebulae is discussed. They are optically thick for radii R < 0.06 parsecs, optically thin for 0.06 < R < 0.6 and again optically thick for R > 0.6 parsecs. The final optically thick stage is a consequence of the final luminosity drop. These results provide a satisfactory explanation of the apparent size distribution obtained by O'Dell, assuming all nebulae to be optically thin. The evolutionary track shows that the central stars are evolving towards the white dwarfs, the final luminosity drop being a consequence of the onset of degeneracy. The average central star mass Ms is equated to the average white dwarf mass, giving MS = 0.6 M⊙ . The average nebular mass, including helium, is 0.6 M⊙ and the average mass of the stars from which planetaries are formed is therefore 1.2 M⊙ . These stars belong to the Disk Population, which indicates an age of ∼ 5 × 10 9 years. This is in satisfactory agreement with the age to be expected for a highly evolved star of mass 1.2 M⊙ . The central star radii Rs are calculated and are found first to increase and then to decrease. During the later stages of the evolution of the central stars the luminosities to be expected from the rate of change of the total stellar energies are found to be greater than the observed photon luminosities by factors of about 30. Processes of neutrino emission are probably important.

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