Can a particle’s motion distinguish scale-dependent Planck stars from renormalization group improved Schwarzschild black holes?

We investigate the differences between a test timelike particle's motions around scale-dependent Planck stars and its motions around renormalization group improved Schwarzschild black holes in detail. By introducing the positive or negative sign of the parameter s (with s=-1 and s=+1 denoting scale-dependent Planck stars and renormalization group improved Schwarzschild black holes, respectively), we present the circular orbital characteristics of both scale-dependent Planck stars and renormalization group improved Schwarzschild black holes. This shows the existence of stable and unstable circular orbits and the dependence of the innermost stable circular orbit on a dimensionless parameter || (, _), where || comes from the theory of nonperturbative renormalization groups, is the identification of the cutoff for the distance scale, and _ are two new dimensionless normalized parameters. In comparison to the black holes, the parameter has more influence on stable and unstable circular orbits for the scale-dependent Planck stars. Specifically, we calculate the particle's relativistic periastron advance for the two models and give a preliminary bound on scale-dependent Planck stars and the renormalization group improved Schwarzschild black holes by using the result of the S2 star's precession with observations from GRAVITY. The bound we obtain in the present work is s||= (-0. 801. 53) 10^92. By using the shadow result from EHT, the bound on s|| is improved by up to 3 or 4 orders of magnitude, which is -1. 3610^88||2. 3210^89. It suggests that a primordial black hole or even a mini--black hole can give a tighter bound on two spacetimes in the future. While we need to break the parameters' degeneracy for the two models with future observations, the negative value in our bound suggests that the existence of Planck stars may be possible. We also intensively compare the particle's periodic orbits around Planck stars with those around the black holes. It is found that small variations in _ make the particle's orbits alternate back and forth from periodic to quasiperiodic orbits, or even jump to nonbound orbits in two spacetimes. We discuss briefly the gravitational waveforms generated from the periodic motions for a small body which orbits a supermassive Planck star or a supermassive black hole, which exhibit some distinct phases and amplitudes for the two models. Our results show that it might be possible to aid in the identification between two spacetimes by using the particle's motion.