Embedded class solutions compatible for physical compact stars in general relativity

We have explored a family of new solutions satisfying Einstein’s field equations and Karmarkar condition. We have assumed an anisotropic stress-tensor with no net electric charge. Interestingly, the new solutions yield zero values of all the physical quantities for all even integer $n > 0$. However, for all $n >0$ ($n \neq $ even numbers) they yield physically possible solutions. We have tuned the solution for neutron star Vela X-1 so that the solutions matches the observed mass and radius. For the same star we have extensively discussed the behavior of the solutions. The solutions yield a stiffer equation of state for larger values of n since the adiabatic index increases and speed of sound approaches the speed of light. It is also found that the solution is physically possible for Vela X-1 if $1.8 \leq n < 7$ (with $n\neq 2,4,6$). All the solutions for $n \geq 7$ violates the causality condition and all the solutions with $0 < n < 1.8$ lead to complex values of transverse sound speed $ v_{t}$. The range of well-behaved n depends on the mass and radius of compact stars