the orbit with the same $J$, $r_{min}$, and $r_{max}$ around the Schwarzschild black hole of the same mass as the SFNS, it can be shown from (\[A-f\]) and (\[cond1\]) that $T_{\scriptscriptstyle osc}^{\scriptscriptstyle SFNS}< T_{\scriptscriptstyle osc}^{\scriptscriptstyle Sch}$ and $v_{o}^{\scriptscriptstyle SFNS}> v_{o}^{\scriptscriptstyle Sch}$.

The orbital velocity in pericentre is also determined only by the metric function $A$, while the period of an oscillation and the radial velocity can be determined only if another metric function in (\[A-f\]), $f(r)$, is known; however, the latter cannot in general be exactly reconstructed in this way. One has $\mathrm{e}^{\Phi}\leq1$ for all $r>0$, in accordance with the expression (\[F-xi\]); the latter is equivalent to the condition $f(r)\geq{}A(r)$, which holds for any physically reasonable configuration except for wormholes. All that has been said allows us to conclude that scalar fields, even if they do not exist in nature, are useful in modelling spherically symmetric, self-gravitating, compact hairy objects, and thus are useful in interpreting the astronomical observations of bound orbits.

 An analytic example {#Sec4}
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In a purely analytical treatment of the quadratures (\[A-f\])–(\[V\]), it is more convenient to start with some specially chosen function $\xi(r)$ (or $\mathrm{e}^{\Phi(r)}$) instead of the field function $\phi(r)$, because each of the 