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| - Accelerated post-AGB evolution, initial-final mass relations, and the star-formation history of the Galactic bulge
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| - Aims. We study the star-formation history of the Galactic bulge, as derived from the age distribution of the central stars of planetary nebulae that belong to this stellar population. Methods. The high resolution imaging and spectroscopic observations of 31 compact planetary nebulae are used to derive their central star masses. We use the Blöcker post asymptotic giant branch (post-AGB) evolutionary models, which are accelerated by a factor of three in this case to better fit the white dwarf mass distribution and asteroseismological masses. Initial-final mass relations (IFMR) are derived using white dwarfs in clusters. These are applied to determine original stellar masses and ages. The age distribution is corrected for observational bias as a function of stellar mass. We predict that there are about 2000 planetary nebulae in the bulge. Results. The planetary nebula population points at a young bulge population with an extended star-formation history. The Blöcker tracks with the cluster IFMR result in ages, which are unexpectedly young. We find that the Blöcker post-AGB tracks need to be accelerated by a factor of three to fit the local white dwarf masses. This acceleration extends the age distribution. We adjust the IFMR as a free parameter to map the central star ages on the full age range of bulge stellar populations. This fit requires a steeper IFMR than the cluster relation. We find a star-formation rate in the Galactic bulge, which is approximately constant between 3 and 10 Gyr ago. The result indicates that planetary nebulae are mainly associated with the younger and more metal-rich bulge populations. Conclusions. The constant rate of star-formation between 3 and 10 Gyr agrees with suggestions that the metal-rich component of the bulge is formed during an extended process, such as a bar interaction.
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