# The last stable orbits for spinning particles in Schwarzschild spacetime

## Abstract

In the pole-dipole approximation, the motion of a spinning compact object in curved spacetime is described by the Mathisson-Papapetrou-Dixon (MPD) equations. These equations do not yield a determinate system which have to be closed using a spin supplementary condition (SSC). We study generic bound orbits (circular and eccentric) in the spacetime of a Schwarzschild black hole and investigate how the spin changes the location of the last stable orbit (LSO). Taking into account generic values of the particle spin σ and bound orbits parametrized by the semi-latus rectum *p* and eccentricity *e*, our analysis revealed the existence of LSOs nearer (farther) to the black hole if σ > 0 (σ < 0), which is sustained by the spin-induced force.

In the pole-dipole approximation, the motion of a spinning compact object in curved spacetime is described by the Mathisson-Papapetrou-Dixon (MPD) equations. These equations do not yield a determinate system which have to be closed using a spin supplementary condition (SSC). We study generic bound orbits (circular and eccentric) in Schwarzschild spacetime and investigate how the spin changes the location of the last stable orbit (LSO). Taking into account generic values of the particle spin $\sigma$ and bound orbits parametrized by the semi-latus rectum $p$ and eccentricity $e$, our analysis revealed the existence of LSOs nearer (farther) to the black hole if $\sigma>0$ ($\sigma<0$) which is sustained by the spin-induced force.In the pole-dipole approximation, the motion of a spinning compact object in curved spacetime is described by the Mathisson-Papapetrou-Dixon (MPD) equations. These equations do not yield a determinate system which have to be closed using a spin supplementary condition (SSC). We study generic bound orbits (circular and eccentric) in the spacetime of a Schwarzschild black hole and investigate how the spin changes the location of the last stable orbit (LSO). Taking into account generic values of the particle spin *σ* and bound orbits parametrized by the semi-latus rectum *p* and eccentricity *e*, our analysis revealed the existence of LSOs nearer (farther) to the black hole if *σ *> 0 (*σ *< 0) which is sustained by the spin-induced force.

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*Proceedings of the Samahang Pisika ng Pilipinas*

**42**, SPP-2024-PB-10 (2024). URL: https://proceedings.spp-online.org/article/view/SPP-2024-PB-10.