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 ( $$M=1.51M_\odot , ~R=10.9$$ km). For this particular star, the solution is well behaved in all respects for $$8 \le n \le 20$$ . 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 ( $$2.01M_\odot , ~11$$ km), EXO 1785-248 (1.3 $$M_\odot $$ , 8.85 km), and Her X-1 (0.85 $$M_\odot $$ , 8.1 km) are specified by the index n with limits $$24 \le n \le 54$$ , $$1.5 \le n \le 4$$ , and $$0.8 \le n \le 2.7$$ , respectively.

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