  Can a dynamically robust bosonic star (BS) produce an (effective) shadow that
mimics that of a black hole (BH)? The BH shadow is linked to the existence of
light rings (LRs). For free bosonic fields, yielding mini-BSs, it is known that
these stars can become ultra-compact - i.e., possess LRs - but only for
perturbatively unstable solutions. We show this remains the case even when
different self-interactions are considered. However, an effective shadow can
arise in a different way: if BSs reproduce the existence of an innermost stable
circular orbit (ISCO) for timelike geodesics (located at $r_{\rm ISCO}=6M$ for
a Schwarzschild BH of mass M), the accretion flow morphology around BHs is
mimicked and an effective shadow arises in an astrophysical environment. Even
though spherical BSs may accommodate stable timelike circular orbits all the
way down to their centre, we show the angular velocity along such orbits may
have a maximum away from the origin, at $R_{\Omega}$; this scale was recently
observed to mimic the BH's ISCO in some scenarios of accretion flow. Then: (i)
for free scalar fields or with quartic self-interactions, $R_{\Omega}\neq 0$
only for perturbatively unstable BSs; (ii) for higher scalar self-interactions,
e.g. axionic, $R_{\Omega}\neq 0$ is possible for perturbatively stable BSs, but
no solution with $R_{\Omega}=6M$ was found in the parameter space explored;
(iii) but for free vector fields, yielding Proca stars (PSs), perturbatively
stable solutions with $R_{\Omega}\neq 0$ exist, and indeed $R_{\Omega}=6M$ for
a particular solution. Thus, dynamically robust spherical PSs can mimic the
shadow of a (near-)equilibrium Schwarzschild BH with the same M, in an
astrophysical environment, despite the absence of a LR, at least under some
observation conditions, as we confirm by comparing the lensing of such PSs and
Schwarzschild BHs.
