Narrow Results

For the first time, we present Einstein’s cluster model in embedding class one spacetime. This paper shows that for any neutral configurations there is only one Einstein cluster solution in embedding class one. In fact, one can find two solutions where the first solution i.e. $g_{rr}=e^{\lambda }=1$ and $g_{tt}=e^{\nu }=C$ is an unphysical one as it has zero density profile as well as violates the Pandey–Sharma condition (i.e. not a class one solution). However, the second solution can describe matter distribution representing Einstein’s cluster which is in static and equilibrium as it satisfies the static stability criterion and TOV-equation. The second solution not only satisfies the above conditions, but also satisfies the energy conditions. The equation of state parameter ${\omega }_t$ is less than unity signifying that it can represent physical matters. Further, we have also shown that the Einstein’s clusters may also exhibit the properties of compact stars.

In this paper, we are willing to develop a model of an anisotropic star by choosing a new $g_{rr}$ metric potential. All the physical parameters like the matter density, radial and transverse pressure are regular inside the anisotropic star, with the speed of sound less than the speed of light. So the new solution obtained by us gives satisfactory description of realistic astrophysical compact stars. The model of this paper is compatible with observational data of compact objects like RX J1856-37, Her X-1, Vela X-12 and Cen X-3. A particular model of Her X-1 (Mass 0.98 $M_{\odot }$ and radius=6.7 km.) is studied in detail and found that it satisfies all the condition needed for physically acceptable model. Our model is described analytically as well as with the help of graphical representation.