We present a class of solutions for a spherically symmetric anisotropic matter distribution in Vaidya and Tikekar spheroidal geometry. Making use of the Vaidya and Tikekar (VT) metric ansatz (J Astrophys Astron 3:325, 1982) for one of the metric functions $$g_{rr}$$ , we obtain the unknown metric function $$g_{tt}$$ by utilizing the Karmakar’s embedding condition which makes the master equation tractable. The model parameters are fixed by utilizing the matching conditions of the interior spacetime and the exterior Schwarzschild solution at the boundary of the star where the radial pressure vanishes. The closed-form interior solution of the Einstein field equations thus obtained is shown to be capable of describing ultra-compact stellar objects where anisotropy might develop. The current estimated masses and radii of some other pulsars are utilized to show that the developed model meets all the requirements of a realistic star. The stability of the model is analyzed. The dependence of the curvature parameter K of the VT model, which characterizes a departure from homogeneous spherical distribution, is also investigated.

中文翻译：

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在 Vaidya 和 Tikekar 球体几何中满足 Karmarkar 条件的嵌入类 I 的各向异性致密恒星模型

我们提出了 Vaidya 和 Tikekar 球体几何中球对称各向异性物质分布的一类解。利用 Vaidya 和 Tikekar (VT) 度量 ansatz (J Astrophys Astron 3:325, 1982) 作为度量函数 $$g_{rr}$$ 之一，我们获得未知度量函数 $$g_{tt}$ $ 通过利用使主方程易于处理的 Karmakar 嵌入条件。利用内部时空与径向压力消失的恒星边界处的外部施瓦西解的匹配条件，确定模型参数。由此获得的爱因斯坦场方程的封闭形式的内部解被证明能够描述可能产生各向异性的超致密恒星物体。一些其他脉冲星的当前估计质量和半径被用来表明开发的模型满足现实恒星的所有要求。分析了模型的稳定性。VT 模型的曲率参数 K 的相关性也进行了研究，该参数表征了与均匀球面分布的偏离。