A family of well-behaved Karmarkar spacetimes describing interior of relativistic stars

We present a family of new exact solutions for relativistic anisotropic stellar objects by considering a four-dimensional spacetime embedded in a five-dimensional pseudo Euclidean space, known as Class I solutions. These solutions are well behaved in all respects, satisfy all energy conditions, and the resulting compactness parameter is also within Buchdahl limit. The well-behaved nature of the solutions for a particular star solely depends on the index n. We have discussed the solutions in detail for the neutron star XTE J1739-285 ( M=1.51M_ , ~R=10.9 km). For this particular star, the solution is well behaved in all respects for 8 ≤ n ≤ 20 . However, the solutions with $$n<8$$ possess an increasing trend of the sound speed and the solutions belonging to $$n>20$$ disobey the causality condition. Further, the well-behaved nature of the solutions for PSR J0348+0432 ( 2.01M_ , ~11 km), EXO 1785-248 (1.3 M_ , 8.85 km), and Her X-1 (0.85 M_ , 8.1 km) are specified by the index n with limits 24 ≤ n ≤ 54 , 1.5 ≤ n ≤ 4 , and 0.8 ≤ n ≤ 2.7 , respectively.