Study of neutron stars instability using the Chandrasekhar’s criterion and realistic equations of

At the moment, the most robust constraints on the neutron star equation of state are based on measurements of the lower limit of the maximum neutron star mass. The importance of this region is evident from its direct connection with part of the high-density neutron star equation of state.

Figure 1.1: Internal structure of a neutron star

Relativistic Stars

The Tolman-Oppenheimer-Volkoff equations

The edge of the star is defined by zero pressure because at this pressure it cannot support materials against the gravitational pull from the interior. The mechanical equilibrium of the star matter is determined by these equations together with the equation of state, E=E(r), of.

Stellar Models

The stability of a compact object depends primarily on the interplay between the equation of state of the fluid interior and the relativistic field strength. The adiabatic index is a function of the baryon density and thus exhibits a radial dependence on the instability criterion (3.1).

Trial Function: ξ(r) = re ν/2
Trial Function: ξ(r) = r (
Trial Function: ξ(r) = r

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic mass instability and the trial function in Eq. 3.7), we obtain the trajectories of the critical/average adiabatic compactness index, as shown in Fig. 6.4. Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic mass instability and the trial function in Eq. 3.8), we obtain the trajectories of the critical/average adiabatic compactness index, as shown in Fig. 6.5.

Figure 6.1: Mass-Radius trajectory for M DI − 1 EoS

Figure 6.7: ⟨ γ ⟩ , γ cr − β trajectories for M DI − 2 EoS (using trial function 1) From Figure 6.7 we extract the data from the point where the critical adiabatic index is equal to the averaged one, which correspond to onset of the instability

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.5), we obtain the critical/average trajectories of the adiabatic index of compactness, as shown in Figure 6.17. Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.6), we obtain the critical/average trajectories of the adiabatic index of compactness, as shown in Figure 6.18.

Equation of State: N LD

Equation of State: HHJ − 1

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic mass instability and the trial function in Eq. 3.5), we obtain the trajectories of the critical/average adiabatic compactness index, as shown in Fig. 6.27.

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.7), we obtain the critical/average trajectories of the adiabatic index of compactness, as shown in Figure 6.29.

Trial Function: ξ(r) = r

Equation of State: HHJ − 2

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the trial function in Eq. 3.5), we obtain the critical/average adiabatic index compactness curves, as shown in Figure 6.32. Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the trial function in Eq. 3.6), we obtain the critical/average adiabatic index compactness curves, as shown in Figure 6.33.

Equation of State: Ska

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.5), we obtain the critical/average adiabatic index compactness trajectories, as shown in Figure 6.37. Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.6), we obtain the critical/average adiabatic index compactness trajectories, as shown in Figure 6.38.

Equation of State: SkI4

Equation of State: HLP S − 1

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the trial function in Eq. 3.5), we obtain the critical/average adiabatic index compactness curves, as shown in Figure 6.47. Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the trial function in Eq. 3.6), we obtain the critical/average adiabatic index compactness curves, as shown in Figure 6.48.

Equation of State: HLP S − 2

Equation of State: HLP S − 3

Equation of State: SCV BB

Equation of State: BS

Equation of State: P S

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.5), we obtain the critical/average trajectories of the adiabatic index of compactness, as shown in Figure 6.82.

Using the Chandrasekhar criterion in Eq. 3.1) for the dynamic instability of a mass and the test function in Eq. 3.7), we obtain the critical/average adiabatic index compactness trajectories, as shown in Figure 6.84.

Trial Function: ξ(r) = r

Equation of State: W

Equation of State: BGP

Equation of State: BL − 1

Equation of State: BL − 2

Equation of State: DH

Figure 6.86: Mass-Radius trajectory for W EoS

Trial Function: ξ(r) = r

Compactness parameter as a function of Pc/Ec ratio, 7.16b Instability onset (γins) as a function of Pc/Ec ratio, and 7.16c Instability onset(γins) as a function of compactness parameter. However, for Figure 7.16c, it is clear that the linear approximation describes very well the dependence of γins on the compactness parameter, β, exactly as we mentioned in Section 7.1.

Figure 7.16: 7.16a The compactness parameter as a function of the ratio P c / E c , 7.16b the onset of instability (γ ins ) as a function of the ratio P c / E c and 7.16c the onset of instability (γ ins ) as a function of the compactness parameter

Data from all trial functions

Maximum Mass, Maximum Radius and the onset of instability

The above finding is clearly expected for low values of the compactness parameter (since all equations of state converge for low density values). These correlations can help to impose constraints on the maximum value of the onset of instability and, of course, on the Pc/Ec ratio.

Internal structure of a neutron star

An equation of State (left) and the integration with TOV equations (right)

Mass-Radius trajectory for N LD EoS

Mass-Radius trajectory for HHJ − 1 EoS

Mass-Radius trajectory for HHJ − 2 EoS

Mass-Radius trajectory for Ska EoS

Mass-Radius trajectory for SkI4 EoS

Mass-Radius trajectory for HLP S − 1 EoS

Mass-Radius trajectory for SCV BB EoS

Mass-Radius trajectory for BS EoS

Mass-Radius trajectory for P S EoS

Mass-Radius trajectory for W EoS

Mass-Radius trajectory for BGP EoS

Mass-Radius trajectory for BL − 1 EoS

Mass-Radius trajectories for the selected EoS

Linear dependence using trial function 1

Non-linear dependence using trial function 1

Linear dependence from all trial functions

Non-linear dependence from all trial functions

Linear dependence using all trial functions

Non-linear dependence using all trial functions

Maximum mass and radius corresponding to the onset of instability