Nuclear stellar cluster (NSCs) are known to exist around massive black holes(MBHs) in galactic nuclei. Two formation scenarios were suggested for theirorigin: (1) Build-up of NSCs from consecutive infall of stellar cluster and (2)Continuous in-situ star-formation. Though the cluster-infall scenario has beenextensively studied in recent years, the in-situ formation scenario have beenhardly explored. Here we use Fokker-Planck (FP) calculations to study theeffects of star formation on the build-up of NSCs and its implications fortheir long term evolution and their resulting structure. We use the FP equationto describe the evolution of several stellar populations, and add appropriatesource terms to account for the effects of newly formed stars. We show thatcontinuous star-formation even 1-2 pc away from the MBH can lead to thebuild-up of an NSC with properties similar to those of the Milky-way NSC. Wealso find that the general structure of the old stellar population in the NSCwith in-situ star-formation could be very similar to the steady-stateBahcall-Wolf cuspy structure. However, its younger stellar population do notyet achieve a steady state. In particular, formed/evolved NSCs with in-situstar-formation contain differential age-segregated stellar populations whichare not yet fully mixed. Younger stellar populations formed in the outerregions of the NSC have a cuspy structure towards the NSC outskirts, whileshowing a core-like distribution inwards; with younger populations havinglarger core sizes. In principal, such a structure can give rise to an apparentcore-like radial distribution of younger (up to 2-3 Gyrs) stars, as observed inthe Galactic center. Such an NSC still preserves an underlying stellar cusp ofolder stars, that can be potentially be missed by current observations ofred-giants.
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