Using the Chandrasekhar criterion in Eq. (3.1) for the dynamical instability of a mass and the trial function in Eq. (3.8), we get the critical/averaged adiabatic index-compactness trajecto- ries, as shown in Figure 6.120.
<γ>
γcr
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0 2 4 6 8
β
<γ>,γcr
Figure 6.120: ⟨γ⟩, γcr−β trajectories forAP R−2EoS (using trial function 4) From Figure 6.120 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. These data are
⟨γ⟩=γcr= 3.3377 and βmax= 0.2746 (6.168) Comparing the Eq. (6.168) and (6.164), we can see that we have an accuracy at86.73%.
CHAPTER 7
Results
We employ a large number of published realistic equations of state for neutron star matter based on various theoretical nuclear models. In Figure 7.1 we display the mass-radius relation of neutron stars using the realistic equations of state.
8 10 12 14 16 18 20
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
r(km) M(M☉)
MDI-1 MDI-2 MDI-3 MDI-4 NLD HHJ-1 HHJ-2 Ska
SkI4 HLPS-1 HLPS-2 HLPS-3 SCVBB BS WFF-1 WFF-2
PS W BGP BL-1 BL-2 DH APR-1 APR-2
Figure 7.1: Mass-Radius trajectories for the selected EoS
We have calculated both the effective averaged and the critical adiabatic indices (as shown in Chapter 6), for each configuration, and mainly focus on the adiabatic indices corresponding to the maximum mass configuration. As we mentioned before, the onset of instability is found by the equalityγcr =⟨γ⟩. Although, there is a second criterion, which defines the stability limit according to the equality dM/dEc = 0, providing an additional value of β for the maximum mass configuration. Now, in general, sinceγcrand⟨γ⟩are functionals of the trial functionξ(r), we expect that the calculated values of the compactness parameter, for the two methods, will not coincide. In these cases, we will consider as the most optimum trial functionξ(r)the one that produces values ofβ, as close as possible, close to the second method. In the next sections, we will present the analysis for the four trial functions that we used in order to determine the onset of instability and our choice for the optimal one.
109
7.1 Trial Function: ξ(r) = re
ν/2Grouping the data from trial function (3.5) for the instability points, we have the trajectories (showing only the lines from the critical adiabatic indices)
●●
●●●●
●●●
●
●
●
●
●
●
●
●
●●
●
● ●
●●
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0 1 2 3 4 5 6
β γcr
MDI-1 MDI-2 MDI-3 MDI-4 NLD HHJ-1 HHJ-2 Ska
SkI4 HLPS-1 HLPS-2 HLPS-3 SCVBB BS WFF-1 WFF-2
PS W BGP BL-1 BL-2 DH APR-1 APR-2
●<γ>=γcr
Figure 7.2: γins−β trajectories for all EoS (using trial function 1)
We also present here just a few of them (the first five) with both the lines of critical and averaged indices, because we have a large amount of EoS and it would be incomprehensible.
●●
●●●
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0 2 4 6 8 10 12
β γcr
MDI-1 MDI-2 MDI-3 MDI-4 NLD
●<γ>=γcr
Figure 7.3: γins−β trajectories for the first five EoS (using trial function 1)
Using the trial function 1, as shown in Eq. (3.5), we extract equations for the correlation between the compactness parameter and the ratioPc/Ec, the points of the onset of instability1 and the ratio Pc/Ec and the points of the onset of instability and compactness parameter2. These equations exhibit a linear dependence on the independent parameter, respectively in
1The onset of instability, where⟨γ⟩=γcr, will be symbolized asγins.
2With the term “compactness parameter” we refer to the compactness parameter corresponding to the maximum mass configuration.
each case, according to the equations
β=a (
b+Pc
Ec
)
(7.1) γins=c
( d+Pc
Ec
)
(7.2)
γins=f(β−g) (7.3)
wherea,b,c,d,fandgare the unknown parameters of the equations.
Using the data from the realistic equations of state, in the case of trial function (3.5), we have the following plots
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.24 0.26 0.28 0.30 0.32
Pc/Ec
β
(a)β−Pc/Ectrajectory
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2.5 3.0 3.5 4.0
Pc/Ec γins
(b)γins−Pc/Ectrajectory
0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36
0 1 2 3 4
β
γins
(c)γins−βtrajectory
Figure 7.4: 7.4a The compactness parameter as a function of the ratio Pc/Ec, 7.4b the onset of instability (γins) as a function of the ratio Pc/Ec and 7.4c the onset of instability(γins)as a function of the compactness parameter
The corresponding equations, according to the Eq. (7.1)-(7.3), are β= 0.095
( 2.3 +Pc
Ec
)
(7.4) γins= 2.144
(
0.7768 +Pc
Ec
)
(7.5) γins= 22.9197 (β−0.1468) (7.6) From the Figures 7.4a and 7.4b it is obvious that the linear dependence is not the best ap- proximation. This is happening because the dispersion of the points is large causing the linear
fitting to have important errors. On the other hand, the linear approximation in the Figure 7.4c describes very well the dependence of theγinsfrom the compactness parameter,β. In this case, the errors are not important.
Although the linear approximation is the easiest way, and in many cases the best way, to approach these relations, there is another way to describe these ones, and as it shows, a more efficient. We found that there are equations that exhibit a non-linear dependence on the independent parameter [51,52], respectively in each case, according to the equations
β=a+b (Pc
Ec
) +c
(Pc
Ec
)2
(7.7) γins=d+f
(Pc
Ec
) +g
(Pc
Ec
)2
(7.8)
γins=h+keβ·l (7.9)
wherea,b,c,d,f,g,h,k andlare the unknown parameters of the equations.
Using the data from the realistic equations of state, in the case of trial function (3.5), we have the following plots
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Pc/Ec
β
(a)β−Pc/Ectrajectory
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4
Pc/Ec γins
(b)γins−Pc/Ectrajectory
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0 1 2 3 4
β
γins
(c)γins−βtrajectory
Figure 7.5: 7.5a The compactness parameter as a function of the ratio Pc/Ec, 7.5b the onset of instability (γins) as a function of the ratio Pc/Ec and 7.5c the onset of instability(γins)as a function of the compactness parameter
The corresponding equations, according to the Eq. (7.7)-(7.9), are β= 0.0113 + 0.7185
(Pc
Ec
)
−0.4445 (Pc
Ec
)2
(7.10) γins= 1.439 + 2.7096
(Pc
Ec
)
−0.3217 (Pc
Ec
)2
(7.11) γins= 1.2842 + 0.048e12.7742β (7.12)