  The Galactic Center has been under intense scrutiny in the recent years
thanks to the unprecedented missions aiming at measuring the gas and star
dynamics near the supermassive black hole (SMBH) and at finding gravitational
wave (GW) signatures of inspiralling stellar black holes. In the crowded
environment of galactic nuclei, the two-body interactions alter the
distribution of stars on long timescales, making them drift in energy and
angular momentum. We present a simplified analytical treatment of the
scattering processes in galactic stellar nuclei, assuming all stars have the
same mass. We have discussed how the interplay between two-body relaxation and
gravitational wave emission modifies the slope of the inner stellar cusp within
the SMBH sphere of influence, and calculated the rates of tidal disruption
events (TDEs) and main-sequence extreme-mass ratio inspirals (MS-EMRIs) of
stars that are tidally disrupted by the SMBH. We find that typically the ratio
of the TDE and MS-EMRI rates is the square of the ratio of the tidal and
Schwarzschild radii. For our Galaxy, this implies that the rate of MS-EMRIs is
just about a percent of the TDE rate. We then consider the role of stars
injected on highly eccentric orbits in the vicinity of the SMBH due to Hills
binary disruption mechanism, and show that the MS-EMRI rate can almost approach
the TDE rate if the binary fraction at the SMBH influence radius is close to
unity. Finally, we discuss that physical stellar collisions affect a large area
of phase space.
