Stability of compact stars in a uniform density background cloud

Abstract We are discussing a scenario where a compact star (neutron star, NS) is embedded in a thin, uniform density background cloud (a remnant cloud after a supernova or a cloud generated from the late stages of a star e.g., a planetary nebula or asymptotic red giant phases) and its effect on the stability of the compact star. Due to the thin background cloud, the spacetime geometry is minimally deformed allowing us to employ the technique of minimal geometric decoupling (MGD). Assuming a uniform background cloud density simplifies the problem, and through the MGD method, one can take $$\Theta ^t_t = \Theta &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>Θ</mml:mi> <mml:mi>t</mml:mi> <mml:mi>t</mml:mi> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mi>Θ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , where $$\Theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> is the density of the cloud. The background cloud interacts with the compact star through a coupling strength $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . By varying $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , one can tune the cloud density to analyze the stability of the embedded compact star. We found that for $$\alpha &lt; 3 \times 10^{-5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , all the thermodynamic quantities are well-behaved, indicating a stable configuration. Once the coupling parameter exceeds $$3 \times 10^{-5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , the adiabatic index drops below $$\Gamma _{\text {max}}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Γ</mml:mi> <mml:mrow> <mml:mtext>max</mml:mtext> </mml:mrow> <mml:mo>′</mml:mo> </mml:msubsup> </mml:math> , triggering a gravitational collapse. Beyond this limit of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , the pressure and speed of sound also become non-physical. At the end, we have used the $$M-R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>-</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> curve generated from the solution to determine the radii of a few compact stars, namely PSR J1614-2230, PSR J0952-0607, GW190814, and GW200210. Furthermore, we have discussed the possibility of the secondary component of GW200210 i.e. the less massive compact object with an upper mass of $$3.3M_\odot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3.3</mml:mn> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:mrow> </mml:math> , which may be a stellar black hole with a Schwarzschild radius $$R_{\text {BH}} = 9.746$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mtext>BH</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>9.746</mml:mn> </mml:mrow> </mml:math> km. However, if the mass is $$2.83M_\odot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2.83</mml:mn> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:mrow> </mml:math> as observed, then its predicted minimum radius is 10.74 km, corresponding to $$\alpha = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . This radius is far beyond $$R_{\text {BH}} = 8.357$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mtext>BH</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>8.357</mml:mn> </mml:mrow> </mml:math> km and therefore is most probably a massive NS in the mass gap.