Excited Dirac stars with higher azimuthal harmonic index

In this paper, we investigate the properties of the first excited state Dirac stars (DSs) with higher azimuthal harmonic index (specifically, the azimuthal harmonic indexes m D = 3/2, 5/2, 7/2), as well as the relationship between the ADM mass and angular momentum of Dirac stars with respect to frequency. Moreover, we find that the ergospheres of DSs appear at lower spinor field frequencies, and both the ergospheres and the distribution of the spinor field functions are asymmetric about the equatorial plane. Furthermore, we introduce the ground state scalar field and examine its impact on this system, which is known as the multi -state Dirac -boson stars (DBSs) model. We show various types of solution families for DBSs under both synchronized frequency omega and nonsynchronized frequencies and find that similar to DSs, the spinor field and the ergospheres of DBSs are also asymmetric about the equatorial plane, but the ergospheres appear at higher spinor field frequencies.