Analysis of the fractional relativistic polytropic gas sphere.

Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman-Oppenheimer-Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from the general relativity framework. In the modified Riemann Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions in solving FTOV equations yields a limited physical range to the convergent power series solution. Therefore, combining the two techniques of Euler-Abel transformation and Padé approximation has been applied to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters ([Formula: see text]), we calculated twenty fractional gas models for the polytropic indexes n = 0, 0.5, 1, 1.5, 2. Investigating the impacts of fractional and relativistic parameters on the models revealed fascinating phenomena; the two effects for n = 0.5 are that the sphere's volume and mass decrease with increasing [Formula: see text] and the fractional parameter ([Formula: see text]). For n = 1, the volume decreases when [Formula: see text] = 0.1 and then increases when [Formula: see text] = 0.2 and 0.3. The volume of the sphere reduces as both [Formula: see text] and [Formula: see text] increase for n = 1.5 and n = 2. We calculated the maximum mass and the corresponding minimum radius of the white dwarfs modeled with polytropic index n = 3 and several fractional and relativistic parameter values. We obtained a mass limit for the white dwarfs somewhat near the Chandrasekhar limit for the integer models with small relativistic parameters ([Formula: see text], [Formula: see text]). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit = 1.63348 M⊙ at [Formula: see text] and [Formula: see text].

Bartsocas-Papas syndrome with variable expressivity in an Egyptian family.

Bartsocas-Papas syndrome (BPS) is an autosomal recessively inherited sublethal popliteal pterygium condition characterized by intrauterine or neonatal death, severe popliteal webbing, oligosyndactyly, ankyloblepharon, orofacial clefts, intraoral filiform bands and genital anomalies. Internal organ involvement has seldom been identified. We report on a 3 years old female patient of healthy first cousin parents with BPS. She presented with orofacial clefting, severe popliteal webs, club feet, oligosyndactyly of the toes, hypogenitalism and normal hands and internal organs. Ankyloblepharon and filiform bands between the alveolar ridges were evident at birth. Pedigree analysis revealed a more severely affected female sib, who died a few minutes after birth with additional manifestations including near complete lip fusion without oral cleft, complete syndactyly in both hands and an omphalocele. Linkage was excluded to the IRF6 gene; a candidate gene implicated in the Van der Woude and popliteal pterygium syndromes, with overlapping features with BPS. To our knowledge, this is the 5th surviving patient with this syndrome in the literature. In this report, we also discuss the proposed pathogenetic mechanisms for BPS and compare our patients with similarly described cases as well as overlapping spectrum of other popliteal pterygium syndromes. Our findings provide further evidence of intrafamilial clinical heterogeneity in families with BPS.