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

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