1. Li-Chin Yeh(1) and Ing-Guey Jiang(2) (1)Department of Applied Mathematics, National Hsinchu University of Education, Taiwan; (2)Department of Physics & Institute of Astronomy, National Tsing-Hua University, Hsin-Chu City, Taiwan

2. Abstract It is known that galactic centers could host single or binary black holes, and observations show that black- hole masses are correlated with the properties of galaxies. Moreover, in terms of center's density profiles, early-type galaxies can be classified into power-law or core galaxies. Therefore, in this project, we study the dynamics near galactic centers with given galactic dark-matter density profiles and binary black holes. Our results could constrain the properties of binary black holes or dark-matter density profiles in their host galaxies.

3. Density Profiles

4. Equations of Motion

5. Model A: n=1 equilibrium points on the x-axis

6. The First Property

7. Model B: equilibrium points on the x-axis

8. The 2nd Property

9. Conclusions Depending on density profiles, number of equilibrium points could be different, and combine with the general properties proved in Jiang & Yeh (2006), we have: For Model A, if below condition is satisfied, there are 6 equilibrium points (4 collinear and 2 triangular); if not, there are 5 equilibrium points (3 collinear and 2 triangular) For Model B, there are always 6 equilibrium points