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Qualitative and quantitative features of orbits of massive particles and photons moving in Wyman geometry

The Wyman's solution depends on two parameters, the mass $M$ and the scalar charge $σ$. If one fixes $M$ to a positive value, say $M_0$, and let $σ^2$ take values along the real line it describes three different types of spacetimes. For $σ^2 >0$ the spacetimes are naked singularities, for $σ^2 = 0$ one has the Schwarzschild black hole of mass $M_0$ and finally for $-M_0^2 \leq σ^2 < 0$ one has wormhole spacetimes. In the present work, we shall study qualitative and quantitative features of orbits of massive particles and photons moving in the naked singularity and wormhole spacetimes of the Wyman solution. These orbits are the timelike geodesics for massive particles and null geodesics for photons. Combining the four geodesic equations with an additional equation derived from the line element, we obtain an effective potential for the massive particles and a different effective potential for the photons. We investigate all possible types of orbits, for massive particles and photons, by studying the appropriate effective potential. We notice that for certain values of $σ^2 >0$, there is an infinity potential wall that prevents both massive particles and photons ever to reach the naked singularity. We notice, also, that for certain values of $-M_0^2 \leq σ^2 < 0$, the potential is finite everywhere, which allows massive particles and photons moving from one wormhole asymptotically flat region to the other. We also compute the radial timelike and null geodesics for massive particles and photons, respectively, moving in the naked singularities and wormholes spacetimes.