in the external, vacuum R-N spacetime. We shall see in the following that due to the bulk space tidal effects on the matching of the internal and external spacetimes (see [@Ger-Maar:2001:] for details) we obtain a non-standard variety of relations of the effective potentials in the ECS interior and exterior.

Using the dimensionless radial coordinate expressed in terms of the gravitational radius ($r/G\mathcal{M}\rightarrow r$), and dimensionless tidal charge $b$, we obtain the relations

$$V{}_{\mathrm{eff}}^{\mathrm{int}} = \frac{r^2 R^2 \left[b Y - 2R (2b - 3R) Z\right]^2}{9 \left[b + (R-2) R\right] \left[(R - b) + R (2b -3R) Z\right]^2}, \label{EQellsb}$$

$$V{}_{\mathrm{eff}}^{\mathrm{ext}} = \frac{r^4}{b + (r - 2) r},$$

where $$\begin{aligned}
      Y &\equiv & R^2 \Delta(r) = \sqrt{r^2(b-2R)+R^4},\\
      Z &\equiv & R \Delta(R) = \sqrt{b+(R-2) R}.
    \end{aligned}$$

![The effective potential $V_{\mathrm{eff}}$ of both internal (dashed lines) and external spacetimes (full lines) given for the black-hole type ($b < 1$) and naked-singularity ($b > 1$) type of the external spacetime. All characteristic cases of its behaviour are presented for both ECS and non-extreme compact stars.[]{data-label="FIGVdiff"}](Fig2a.pdf "fig:"){width="\hsize"} ![The effective potential $V_{\mathrm{eff}}$ of both internal (dashed lines) and external spacetimes (full lines) given for the black-hole type ($b < 1$) and naked-singularity ($b > 1$) type 