We derive a tidal potential for a self-gravitating fluid star orbiting Kerr black hole along a timelike geodesic extending previous works by Fishbone and Marck. In this paper, the tidal potential is calculated up to the third and fourth-order terms in $R/r$, where $R$ is the stellar radius and $r$ the orbital separation, in the Fermi-normal coordinate system following the framework developed by Manasse and Misner. The new formulation is applied for determining the tidal disruption limit (Roche limit) of corotating Newtonian stars in circular orbits moving on the equatorial plane of Kerr black holes. It is demonstrated that the third and fourth-order terms quantitatively play an important role in the Roche limit for close orbits with $R/r \agt 0.1$. It is also indicated that the Roche limit of neutron stars orbiting a stellar-mass black hole near the innermost stable circular orbit may depend sensitively on the equation of state of the neutron star.Comment: Correct typo

The tidal disruption limit of neutron stars moving around a black hole is numerically studied in the framework of Newtonian gravity. The black hole is described as a point particle, and the neutron stars are modeled using polytropic equations of state. In this paper, we focus both on the Jeans problem, in which the system is axisymmetric and the stars located along a symmetric axis are momentarily static, and on the Roche problem, in which stars are in a corotating circular orbit on the equatorial plane. We find that these two approaches provide qualitatively the same numerical results, which are the following. (i) For given values of the central density and radius, neutron stars with softer equations of state are more fragile with respect to tidal disruption. However, for given values of the mass and radius, the strength with respect to tidal disruption depends very weakly on the equations of state. (ii) The tidal disruption limit determined in the so-called tidal approximation for the black hole tidal field yields an error of magnitude of order R/L, where R and L are the neutron star radius and the separation of two stars. (iii) The third-order term of the tidal field plays an important role in accurately determining the tidal disruption limit for R/L ∼ O(0.1). The dependence of the tidal disruption limit for black hole-neutron star binary systems on the equations of state and the mass ratio can be obtained from the study for the momentarily static case. This indicates that the momentarily static approaches should be helpful as a first step in obtaining a full understanding of the tidal disruption limit in general relativity.

Background Poor oral health conditions are known to affect frailty in the elderly. Diabetes is a risk factor for both poor oral health and frailty, and, therefore, oral health status may affect frailty in diabetic patients more than in the general population. The purpose of this study was to evaluate the influence of oral health on frailty in elderly patients with type 2 diabetes. Methods Patients with type 2 diabetes aged 75 years or older were included in this retrospective study. Eligible patients were surveyed by questionnaire for frailty, oral health status, and cognitive and living functions. Factors influencing pre-frailty, frailty, and individual frailty score categories were evaluated. Results Of the 111 patients analyzed, 66 cases (59.5%) were categorized as robust, 33 cases (29.7%) as pre-frailty, and 12 cases (4.5 %) as frailty. The oral frailty index, the cognitive and living functions score, and BMI were found to be factors influencing pre-frailty or frailty. In the evaluation of individual frailty score categories, BMI only had an influence on those with a frailty score ≤ 2. The cognitive and living functions score was a factor influencing those with frailty scores ≤ 3. The oral frailty index was found to have a significant influence on all frailty score categories. Conclusion Poor oral health has an influence on frailty in patients with type 2 diabetes aged ≥ 75. In this patient population, as frailty progresses, the impact of oral health on frailty may increase. Trial registration: This study was retrospectively registered in UMIN-CTR (UMIN000044227).

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