The old nuclear star cluster in the Milky Way: dynamics, mass, statistical parallax, and black hole mass

Chatzopoulos S., Fritz T. K., Gerhard O., Gillessen S., Wegg C., Genzel R., Pfuhl O., 2015, MNRAS, 447, 948 (to the paper)
To model the NSC we worked with ~2500 radial velocities (velocities along the line of sight) and ~10000 proper motions (perpendicular to the line of sight ) plus stellar number counts from T. Fritz et al. (submitted). Before this work the best model in the literature for the NSC was spherical Jeans models (really simple) and therefore we decided for this project to apply 2-Integral distribution functions (DF) (in simple words, a flattened system) that possess considerable advantages over Jeans modeling but are also much more complicated to implement. A DF gives the number of stars in phase space as a function of the position and velocity. Therefore if one knows the DF of a system he knows the position and velocity of every star in this system. The best indicator for the quality of the modeling was the good prediction of the Velocity Profiles (VP) since those include all the moments of the system. The 2-I DF gave us overall very good results (Figure 2) and makes this model, the best model of the NSC in the literature. Next I describe in more detail the steps of the project.

We started by fitting a parametric oblate gamma-model (this is an axisymetric stellar system model) to the surface density (SD) as a function of the l proper motion (Galactic latitude) and b proper motion (Galactic longitude) coordinate. The shape of the SD indicates an inflection point at about 100'' from the center. After noticing this we switched to a 2-component gamma-model for the SD. The use of 2 flattened models is advantageous because it allowed for a non constant axial ratio. Having the density allows us to calculate the total potential as a sum of the 2 model potentials plus the black hole potential. The goal was to calculate the DF that corresponds to this potential. For this we used the Hunter & Qian (Hunter & Qian 1993) contour integral method. This method is a generalization of Eddington's formula (describes spherical systems) and it takes place in the complex potential plane. The method gives us the even part of the DF. To test the DF we compare its first moments with Jeans modeling. By applying axisymmetric Jeans modeling we constrained the distance R0 to the NSC, the mass of the NSC and the SMBH mass giving new estimations. The HQ method gives us only the even part of the DF which contribute only to the density. In order to add rotation an odd part is needed. Adding an odd part to the DF is equivalent with reversing the sense of rotation of some stars. This won't affect the mass of the system nor the projected dispersions. Figure 2 shows how our best model predicts the VHs of proper motions for some combinations of cells. The results are overall very good. The model is able to predict the shape of the VPs in b direction (Schodel et al. (2009) tried to fit those with a Gaussian but the result was not good) and the double peak shape in l direction.

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