only [@Kot-Stu-Tor:2008:CLASQG:]. The radii of the photon circular orbits are given by $$r_{\mathrm{ph}\pm} = \frac{3}{2}\left(1 \pm \sqrt{1-\frac{8}{9}b}\right). \label{EQph}$$ However, only the outer photon circular geodesic at $r_\mathrm{ph+}$ is physically relevant and we see immediately that $(b < 0)$ there is $r_{\mathrm{ph}}>3$.

For braneworld neutron stars with a positive tidal charge ($b > 0$) both black-hole and naked singularity R-N spacetimes are relevant [@Kot-Stu-Tor:2008:CLASQG:]. In the black-hole type spacetimes ($b \leq 1$), the photon circular orbits are given by the relation (\[EQph\]), but only the outer solution corresponding to an unstable orbit can be astrophysically relevant since the surface of the compact star has to be located above radius of the outer horizon. In the naked singularity spacetimes ($b > 1$) two photon circular orbits are can be astrophysically relevant when $b < 9/8$, the inner one being stable, the outer one — unstable. In the spacetimes with $b > 9/8$, no photon circular orbits can exist. Therefore, in the braneworld spacetimes, the existence of ECS is governed by the interplay of the behaviour of the internal and external effective potential of the motion and will be determined and classified in the following section.

Null geodesics of internal and external spacetimes 