Imprints of the Lorentz-symmetry breaking on the precessing jet nozzle of M87*

The approximately 11-year jet precession period observed in M87* strongly suggests that the supermassive rotating black hole with a tilted accretion disk, which could provide a powerful constraint for confining the parameters of black hole. In this paper, our aim is to utilize the observations of M87* to preliminarily constrain the parameters of the rotating black hole in Bumblebee gravity by modeling the motion of the tilted accretion disk particle with the spherical orbits. We compute spherical orbits and ISSOs, demonstrating that the conserved quantities energy E and angular momentum L depend on ( r, a , ℓ, ζ ), exhibiting distinct behaviors for prograde and retrograde orbits. For prograde orbits, the ISSO radius r ISSO decreases with spin parameter a and LSB parameter ℓ and increases with the tilt angular ζ , whereas the opposite trends occur for retrograde orbits. Angular analysis shows that θ oscillates within ( π / 2 − ζ , π / 2 + ζ ) , while ϕ increases approximately linearly, enabling the determination of the oscillation period T θ , azimuthal accumulation ϕ ( T θ )/ π , and precession angular velocity ω t . Using the observed jet precession period T = 11.24 ± 0.47 years with a fixed tilt ζ = 1 . 25 ∘ , the orbital radius r / M ranges from (5.73,25.15) for prograde and (6.16,26.46) for retrograde orbits, increasing with a or ℓ. Comparisons with Kerr limits ( r / M = 14.12 prograde, 16.1 retrograde) suggest that r / M > 16 may indicate a non-vacuum Bumblebee vector field. Incorporating the EHT shadow θ s h = 42 ± 3 μ as further constrains r / M to (5.82,22.61) and (6.17,24.74), with discrepancies of 0.05 ∼ 1.96. These results highlight the dominant role of a and establish a framework for constraining black hole properties within alternative gravity theories.