209.1 Introduction

The Pseudo-Newtonian Potentials (PNPs), which are constructed/proposed to replicate few general relativistics features approximately in Newtonian framework, are often used to study inner relativistic dynamics of the accretion flow around spacetime geometries describing black holes. For a general class of static spherically symmetric space time metrics \(ds^2 =-f(r)^\beta \, c^2 \, dt^2 + \frac{1}{f(r)^\beta } \, dr^2 + f(r)^{1-\beta } r^2 d\Omega ^2 \, ,\) where f(r) is the generic metric function, \(\beta \) is an arbitrary constant parameter, the PNP can be written as [1]

$$\begin{aligned} V_\mathrm{GN} = \frac{c^{2}(f^{\beta }-1)}{2}-\left( \frac{1-f^{2\beta -1}}{2 \, f^{2\beta -1}}\right) \left[ \frac{f^{2\beta }-1}{f \, \left( f^{2\beta -1}-1\right) }\dot{\, r}^{2}+r^{2}\,\dot{\Omega }^{2}\right] \, .\qquad \end{aligned}$$
(209.1)

The particle trajectories can be obtained by solving the Lagrangian equations for the potential given in (209.1). In the below, we studied particle trajectories for a well known naked sigularity spacetime - Janis-Newman-Winicour (JNW) metric for which \(\beta =\gamma \) and \(f(r)= 1- \frac{2r_s}{\gamma r}\) where, \(0 < \gamma \le 1\). The geodesic equations are

$$\begin{aligned} \ddot{r} = -c^2 \left( 1-\frac{2r_s}{\gamma r} \right) ^{3\gamma -1} \frac{r_s}{r^2} + \frac{2 \dot{r}^{2}}{\left( 1-\frac{2r_s}{\gamma r} \right) } \frac{r_s}{r^2} + \left[ r -\frac{r_s}{\gamma } (1 + 2 \gamma ) \right] \left( \dot{\theta }^{2} + \sin ^2 \theta \dot{\phi }^{2} \right) \, ,\nonumber \\ \end{aligned}$$
(209.2)

and

$$\begin{aligned} \ddot{\phi }= -\frac{2\dot{r} \, \dot{\phi }}{r} \left[ \frac{\gamma r-r_s (1 + 2 \gamma )}{\gamma r-2 r_s } \right] - 2 \cot \theta \, \dot{\phi }\, \dot{\theta }\, , \ddot{\theta }= -\frac{2\dot{r} \, \dot{\theta }}{r} \left[ \frac{\gamma r-r_s (1 + 2 \gamma )}{\gamma r-2 r_s } \right] + \sin \theta \, \cos \theta \, \dot{\phi }^2 \end{aligned}$$
(209.3)

For particle dynamics along circular orbit, \(\dot{r}=0\) and \(\ddot{r}=0\). The particle trajectories obtained by solving (209.2)–(209.3) are shown in Fig. 209.1.

Fig. 209.1
figure 1

Comparison of elliptic like trajectories of particle orbit in equatorial plane in JNW spacetime with those in Schwarzschild and Newtonian cases projected in the x-y plane. Solid and short-dashed lines corresponding to Newtonian and Schwarzschild cases, respectively. Long dotted-dashed curve in Fig. 209.1a, b, c, d, e, f are for \(\gamma = 0.2, 0.3, 0.4, 0.5, 0.7, 0.95\), respectively. The particle starts from apogee with \(r_a = 40 r_s\) with \(v_x = 0.0\) and \(v_y \equiv v_\mathrm{in} = 0.092\) (in units of c). For \(\gamma < 0.2\), no proper well defined elliptic like orbits exist with the orbital parameters chosen here

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209.2 Conclusions

The test particle dynamics along circular orbit in JNW space-time departs to those in Schwarzschild geometry. The stated deviation is larger for smaller \(\gamma \).