  The tidal love number determines a star's deformability rate in the presence
of gravitational potential and depends on the star's internal structure. In
this work, we investigate two significant prospects on tidal love number : (i)
the influence of the polytropic index of stars on the tidal love number and ,
(ii) how tidal love number affects the pericenter shift of S-stars near Sgr A*
which is an important probe for strong-field tests of gravitational theories.
We consider S-stars orbiting Sgr A* at a pericenter distance of 45 au to 500
au, well below the S-2 orbit. The S-stars have polytropic indices of the range,
n = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, and 4.5, and eccentricity, e = 0.9
inclined at $i=90 ^{\circ}$. The tidal love number is estimated for multipole
moments $l=2$ and $l=3$. It has been found that the tidal love number decreases
as the polytropic index increases. Additionally, the tidal love number for the
multipole moment of $l=2$ is dominant over that of $l=3$. The tidal distortion
effect also causes a greater pericenter shift in compact orbit S-stars with
lower polytropic indices and tidal love number having multipole moment $l=2$.
The estimated results offer relevant insights for testing general relativity
and its alternative theories in the vicinity of Sgr A*
