**Publication date:** Jul 2023

**Abstract:**

We present a model of a slowly rotating Tolman VII (T-VII) fluid sphere, at second order in the angular velocity. The structure of this configuration is obtained by integrating numerically the Hartle-Thorne equations for slowly rotating relativistic masses. We consider a sequence of models where we vary the parameter $R/R_{mathrm{S}}$ , where R is the radius of the configuration and $R_{mathrm{S}}$ is its Schwarzschild radius, representing an adiabatic and quasi-stationary contraction by progressively reducing the radius while keeping the angular momentum and gravitational mass constant. We determined the moment of inertia I, mass quadrupole moment Q, and the ellipticity ɛ, for various configurations. Similarly to previous results for Maclaurin and polytropic spheroids, in slow rotation, we found a change in the behavior of the ellipticity when $R/R_{mathrm{S}}$ reaches a certain critical value. Based on our analysis for the T-VII solution, we found variations of $mathcal{O}(10%)$ in the $I-mathcal{C}$ and $Q-mathcal{C}$ relations, and $mathcal{O}(1%)$ variation in the I – Q relation, with respect to the universal fittings proposed for realistic neutron stars (NSs). Our results suggest that the T-VII solution can be considered a rather good approximation for the description of the interior of NSs.

**Authors:**

Posada, Camilo; Stuchlík, Zdeněk;