$\kappa$-Dirac stars

In this paper, we construct a Dirac star model composed of $\kappa$ pairs of spinor fields. The azimuthal harmonic indeces $m$ of these spinor fields are half-integers, and they satisfiy $-(\kappa-\frac{1}{2})1$, among these $\kappa$ pairs of spinor fields, there exist spinor fields with azimuthal harmonic indices $m>\frac{1}{2}$, and all spinor fields still conform to the same Dirac field equation. Different families of solutions are distinguished by the value of $\kappa$, so we named these solutions $\kappa$-Dirac stars. We obtain solutions for $\kappa=1,2,3,4,5,6$ by using numerical methods. Additionally, we compute their ADM mass $M$, Noether charge $Q$, and binding energy $E$, and illustrate how these quantities change with the spinor field's frequency $\omega$ for different $\kappa$. We observe significant differences between solutions for $\kappa>1$ and the $\kappa=1$ case. Furthermore, we provide the energy density distribution of the Dirac stars, wherein for $\kappa>1$ scenarios, the Dirac stars exhibit a spherical shell-like structure. Moreover, we employ three-dimensional diagrams to intuitively depict how $\kappa$ influences the combination of spinor fields to form a spherically symmetric configuration.