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We study bound orbits near the centres of static, spherically symmetric,asymptotically flat configurations of a self-gravitating scalar field minimallycoupled to gravity. In our approach, a nonlinear scalar field is considered asan idealized model of dark matter, and the main examples that we have in mindare the centres of galaxies. We consider both scalar field black holes (SFBHs)and scalar field naked singularities (SFNSs). It turns out that the shape andparameters of a bound orbit depend crucially on the type of configuration. Thelapse metric function of a SFNS and, consequently, the effective potential of amassive test particle with zero angular momentum have a global minimum. A SFNShas a static degenerated orbit on which a test particle, having zero angularmomentum and the minimum of its energy, remains at rest at all times. Thisimplies that there exists a spherical shell consisting of cold gas or dust,which for a distant observer can look like the shadow of a black hole. We alsostudy the shape of noncircular bound orbits close to the centres of SFNSs andshow that their angles of precession are negative.

Bound orbits near scalar field naked singularities