A star having highly eccentric and compact orbit around a black hole experiences tidal distortion. The tidal radii of the black hole–star system are expressed as (Shiokawa et al. 2015; Coughlin et al. 2017)
(7)$$\begin{eqnarray*}
r_t = (\frac{M_{\rm BH}}{M})^{1/3} R,
\end{eqnarray*}$$where MBH is the mass of the black hole, M is the mass of the star, and R is the radius of the star. Stars orbiting at a distance smaller than the tidal radius are disrupted by tidal force of the black hole. The additional contribution to perihelion shift arising from tidal effect in terms of pericentre distance is given by (Will 2008)
(8)$$\begin{eqnarray*}
\delta \phi _{\rm prec}^{\rm tidal} = \frac{30 \pi }{(1+e)^5} k_2 (1+ \frac{3e^2}{2} + \frac{e^4}{8})(\frac{M_{\rm BH}}{M}) (\frac{{R}^5}{ r_{\rm p}^5}),
\end{eqnarray*}$$k2 in equations (8) is the Love number of a star that measures its deformability and susceptibility of its shape to change in response to a tidal potential (Poisson & Will 2014). The tidal Love number (kl) of degree l for polytropic stars has been studied by Yip & Leung (2017) through perturbative expansion of kl around the polytropic indices n = 0 and n  = 1. These are obtained for Newtonian stars whose stiffness is governed by polytropic index (n). In the general framework of Yip & Leung (2017), the Lane Emden equation is solved perturbatively by applying Scaled Delta Expansion method to develop perturbation series of the multipolar tidal Love number about two exactly solvable polytropic indices (n = 0, n  = 1). These perturbative expansions are then used to make a two-point Pade approximation of the Love number for n ϵ [0, 3] which is given as
(9)$$\begin{eqnarray*}
k_l({\it n}) = (5 - {\it n})^3 \frac{a_1 + {\it n}a_2 + {\it n}^2 a_3 + {\it n}^3 a_4}{1000 + {\it n}a_5 + {\it n}^2 a_6}.
\end{eqnarray*}$$The factor (5 − n) is empirically inserted to ensure that the Love number reduces to zero for n  = 5 that corresponds to infinite configuration. Here, ais are analytically determined coefficients of the perturbation series and these are taken from table 1 of Yip & Leung (2017). We consider polytropic models, n = 1 and n  = 3, for evaluating the Love number and obtain k2 (n = 1)  = 0.26 k2 (n = 3)  = 0.014. These values are then used to evaluate tidal contribution to the periastron shift. For the tidal effect, we also have to take into account the mass and radii of the stars. Structure of the nuclear star cluster in the vicinity of Sgr A* was reported by Do et al. (2010). A general consensus is that the cluster has preponderance of young main sequence stars (∼ 4 – 8Myr) over late-type stars (∼1 – 10Gyr). Presence of OB main-sequence stars was also reported by Allen, David & Hyland (2010), Ghez et al. (2003a), and Do et al. (2013). Here, we consider only the main-sequence stars for the mass–radius relation that is required for calculation of tidal contribution to periastron shift (see equation 8). Study of empirical mass–radius relation (MRR) of stars began to appear in literature after late twentieth century (Gimenez & Zamorano 1985; Harmanec 1988; Demircan & Kahraman 1991; Malkov 2007). The stellar radii are related to luminosity through Stefan–Boltzmann law which in turn is dependent on stellar masses. Unfortunately, there was no final consensus on what functions properly represent mass–luminosity relation (MLR) and MRR. It had been challenging to have a single MRR for main-sequence stars till the publication of Eker et al. (2015) in which the MLR and MRR for Galactic nearby main-sequence stars were reported. The MLR was revised with the help of power law L ∝ Mα for a sample of galactic main-sequence stars with mass and radii accurate up to ≤3 per cent and luminosities accurate up to ≤30  per cent. But there was no derivation of MRR to estimate radii for a given stellar mass. It was found that for LMSs (M  1M⊙) stellar radii have narrow distribution with mass but for massive stars (M > 1M⊙) the distribution was quite wide. Thus, there was no unique function to represent MRR. Eker et al. (2018) updated the data of 509 main-sequence stars in the solar neighbourhood and obtained interrelated MLR, MRR, and mass-effective temperature relation (MTR) in the mass range 0.179 ≤ (M/M⊙) ≤ 31. A single MRR was found for the LMSs with 0.179 ≤ (M/M⊙) ≤ 1.5. An MTR function was found for massive stars with 1.5 ≤ (M/M⊙) ≤ 31. The MLR was established for six mass ranges encompassing ultra-low-mass (0.179 ≤ (M/M⊙) ≤ 0.45) and very high-mass (7 ≤ (M/M⊙) ≤ 31) stars. Thus, by using Stefan–Boltzmann law (L  = 4$\pi R^2 \sigma T_{\rm eff}^4$), the MRR can be obtained for the massive stars with 1.5 ≤ (M/M⊙) ≤ 31. It is true that there are effects of metallicities of the stars on the MLR and hence finally on MRR (Eker et al. 2018). But due to absence of detailed model of stellar evolution and stellar atmosphere of the GC stars with different metallicities, we have assumed that the young GC stars have nearly the solar metallicities (see Lu et al. 2013 and references therein) and hence obey the interrelated MLR, MTR, and MRR functions of the main-sequence stars in the solar neighbourhood. The MRR for the LMSs [0.179 ≤ (M/M⊙) ≤ 1.5] is (Eker et al. 2018)
(10)$$\begin{eqnarray*}
\frac{R}{{\rm R}_{\odot }} \cong [0.438(\frac{M}{{\rm M}_{\odot }})^2 + 0.479 (\frac{M}{{\rm M}_{\odot }}) + 0.075].
\end{eqnarray*}$$For massive stars (1.5 ≤ (M/M⊙) ≤ 31), we use the interconnection between MLR, MTR, and MRR presented by Eker et al. (2018) as follows. The Stefan–Boltzmann law
(11)$$\begin{eqnarray*}
{\rm log}(\frac{L}{{\rm L}_{\odot }}) = 2 {\rm log}(\frac{R}{{\rm R}_{\odot })} + 4{\rm log}(\frac{T_{\rm eff}}{T_{{\rm eff}(\odot)}})
\end{eqnarray*}$$is used in conjunction with the M–Teff relation (MTR) for 1.5 ≤ (M/M⊙) ≤ 31
(12)$$\begin{eqnarray*}
{\rm log}(\frac{T_{\rm eff}}{1K})\simeq - 0.170({\rm log}(\frac{M}{{\rm M}_{\odot }}))^2 + 0.888 {\rm log}(\frac{M}{M_{\odot }}) + 3.671
\end{eqnarray*}$$and the MLR for the mass range 7 ≤ (M/M⊙) ≤ 31,
(13)$$\begin{eqnarray*}
{{\rm log}(\frac{L}{{\rm L}_{\odot }})\simeq 2.865 {\rm log}(\frac{M}{{\rm M}_{\odot }}) + 1.105}.
\end{eqnarray*}$$The result is the following MRR for massive stars,
(14)$$\begin{eqnarray*}
{\rm log}\Big (\frac{R}{{\rm R}_{\odot }}\Big) \cong [-0.3435 {\rm log}\Big (\frac{M}{{\rm M}_{\odot }}\Big) + 0.34 {{\rm log}\Big (\frac{M}{{\rm M}_{\odot }}\Big) }^2 + 0.7365]. \nonumber \\
\end{eqnarray*}$$We study the tidal contribution to periastron shift for both the low-mass (0.179 ≤ (M/M⊙) ≤ 1.5) and high-mass (7 ≤ (M/M⊙) ≤ 31) categories of young main-sequence stars near Sgr A*. For low-mass category, the contribution is studied for M/M⊙ = 0.18, 1.0, and 1.5, whereas for high-mass category it is estimated for M/M⊙ = 10, 20, and 30. Equation (8) can be written as
(15)$$\begin{eqnarray*}
\delta \phi _{\rm prec}^{\rm tidal} = \frac{30 \pi }{(1+e)^5} k_2 (1+ \frac{3e^2}{2} + \frac{e^4}{8})(\frac{M_{\rm BH}}{M}) (\frac{{{\rm R}_{\odot }}}{ r_p})^5 (\frac{R}{{\rm R}_{\odot }})^5.
\end{eqnarray*}$$Equations (10) and (14) have been used in equation (15) to estimate the tidal contributions to periastron shift arising from LMS and HMS. The actual astrometric size of the perihelion shift at periapsis is given by θ  = ($\delta \phi ^{\rm total}_{\rm prec}$) ($\frac{ r_{p} \sin i}{D}$), where D is the distance to the GC and i is the inclination of the orbital plane of the star with respect to the observer. Therefore, the actual rate of periastron shift is expressed as
(16)$$\begin{eqnarray*}
\dot{\theta }_{\rm prec} = \frac{(\delta \phi)^{\rm total}_{\rm prec} r_{p} \sin i}{\rm PD}.
\end{eqnarray*}$$Here, P is the orbital period for the stars, P  = ${ 2\pi a^{\frac{3}{2}}}/{\sqrt{GM_{\rm BH}}}$  = ${2\pi r_p^{\frac{3}{2}}}/{\sqrt{GM_{\rm BH}}(1-e)^{\frac{3}{2}}}$.
Citation: Demircan & Kahraman 1991
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