7.6 Maximum Mass, Maximum Radius and the onset
TF-3
2.0 2.2 2.4 2.6 2.8 3.0
2.5 3.0 3.5 4.0 4.5
Mmax(M☉) γins
(a)γins−Mmax
TF-3
10 11 12 13
2.5 3.0 3.5 4.0 4.5
Rmax(km) γins
(b)γins−Rmax
Figure 7.24: 7.24a The onset of instability(γins)as a function of the maximum mass and 7.24b the onset of instability(γins)as a function of the maximum radius
TF-4
2.0 2.2 2.4 2.6 2.8 3.0
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
Mmax(M☉) γins
(a)γins−Mmax
TF-4
10 11 12 13
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
Rmax(km) γins
(b)γins−Rmax
Figure 7.25: 7.25a The onset of instability(γins)as a function of the maximum mass and 7.25b the onset of instability(γins)as a function of the maximum radius
CHAPTER 8
Concluding Remarks and Discussion
In the present work, we have employed a large number of published realistic equations of state (24 in number) for neutron star matter based on various theoretical nuclear models. We have calculated both the effective averaged and critical adiabatic indices for each configuration and mainly focus on the adiabatic indices corresponding to the maximum mass configuration. The calculation recipe is as follow
• We solve the TOV equations, using the method as we mentioned in Section 2.3, in order to determine the mass-radius dependence, as well as the corresponding energy density and pressure configurations
• For each configuration we determine the critical and the averaged indices
Mainly, we are interested for the maximum mass, the corresponding radius, the ratioPc/Ec
and the corresponding compactness parameter, β, for each case. The onset of instability is found from the equality⟨γ⟩=γcr and the corresponding compactness parameter is denoted as βmax.
This method is the one of the two criteria which defines the stability limit. According to the second one, the stability limit is found by the equalitydM/dEc = 0, providing an additional value of the compactness parameter for the maximum mass configuration. As we already mention, the effective averaged and critical indices are functionals of the trial function. With this fact, 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 function the one which produces values of the compactness parameter, as close as possible, close to the second method. In particular, we found that the trial function in Eq. (3.7) is the optimal one, leading to an error, in the most of the cases, less than1%.
The majority of the equations of state, that we present and use here, reproduce the recent observation of two-solar massive neutron stars. This statement can be seen in Figure 7.1, where the black line represents the recent observation. It is obvious that the various predictions cover a wide range of the maximum neutron star masses and the corresponding radii.
We have displayed the dependence ofγins as a function of the compactness parameter for all the employed equations of state and using the four trial functions. We observed that for every trial function, the dependence ofγins as a function of the compactness parameter, is a linear one for high values of the compactness parameter or a non-linear for all the region. In each case, the results are slightly different. That is because we use different trial functions. As we mentioned before, the trial function in Eq. (3.7) is the optimal one, because it is leading to an error less than the other three trial functions, in the most of the cases.
The most distinctive feature, in Figures 7.18-7.19, is the remarkable unanimity of all equations of state and consequently the occurrence of a model-independent relation between γins and βmax, at least for any stable configuration. The above finding, clearly expected for low values of the compactness parameter (since all equations of state converge for low values of density).
However, at high densities of the equations of state, where there is a considerable uncertainty, these results were not obvious. In any case, as a consequence of the convergence, both for
131
low and high values of the compactness the majority of the points indicate the onset of the instability, located in the mentioned trajectory. In particular, we found that the expression
γins(β) =h+keβ·l (8.1)
whereh,kandlare the unknown parameters, reproduces very well the numerical results due to the use of realistic equations of state. Equation (8.1) is the relativistic expression for the onset of instability and can be considered as the relativistic generalization of the post-Newtonian approximation. The parametrization is provided in Table 7.3. The stable configurations, in- dependently of the equation of state, correspond to a universal relation between γins and compactness parameter. One can safely conclude thatγinsis an intrinsic property of the neu- tron stars (likewise the parameterβ) which reflects the relativistic effects on their structure.
In particular, γins depends linear with compactness parameter in the Newtonian and post- Newtonian regime but exhibits more complicated behaviour in the relativistic regime (high values of compactness parameter).
The above finding may help to impose constrain to the equation of state of neutron star matter.
For example, the accurate and simultaneously observation of possible maximum neutron star mass and the corresponding radius will constrain the maximum values of the compactness and consequently the maximum value of the critical adiabatic index. In any case, useful insights may be gained by the use of the expression (8.1).
In order to clarify further the effects of the trial functions,ξ(r), on the results we present the Figure 7.18c. In particular, we display the dependence of the onset of instability,γinsas a func- tion of the compactness parameter βmax using the selected trial functions in Eq. (3.5)-(3.8).
The most distinctive feature in this case, is the occurrence of an almost linear dependence (in the region under study) between the adiabatic index and the compactness parameter βmax. Obviously, the use of the trial function, ξ(r), affects mainly the values of γins (for the same βmax) but not the linear dependence.
We have also displayed in Figure 7.19a and 7.19b the dependence of βmax and γins on the ratioPc/Ec(which corresponds to the maximum mass configuration). In general, in the case of realistic equation of state,βmaxand γinsis an increasing function of the ratioPc/Ec without obeying in a specific formulae. However, we found that the expressions
β=a+b (Pc
Ec
) +c
(Pc
Ec
)2
(8.2) γins=d+f
(Pc
Ec
) +g
(Pc
Ec
)2
(8.3) reproduce very well the numerical results of the realistic equations of state. The parametriza- tion is provided in Table 7.2. These correlations may help to impose constraint to the maximum value of the onset of instability and of course the ratioPc/Ec. By imposing constraint to the ratioPc/Ecwe may consequently constraint the maximum density in the universe by the help of accurate measure of the maximum value of the compactness parameter.
In our efforts to find a better approximation for the relation between the onset of instability and the corresponding compactness parameter, we have displayed in Figures 7.20-7.21 a uni- versal trend line for all the trial functions. The results that we take have not a big difference from the every case independently. However, the trial function in Eq. (3.7) is still the optimal trial function producing results very close to the second method that we tested here.
As a follow-up to the previous argument, we have displayed in Figures 7.22-7.25 the depen- dence ofγinsonMmaxand the correspondingRmax. Obviously, in these cases, the dependence is almost random and consequently and is unlikely to impose constraints from these kind of correlations.
The previous findings have been also tested with linear dependences, as we presented in the previous chapter. The linear dependence concerns only the high values of the compactness parameter and not the whole area. For the sake of completeness and of course for the study in the high density region of neutron stars, where we can find very useful the relations that have been presented in the previous chapter for linear dependence, we have also presented the
linear relations for every trial function and for all trial functions together.
To be more specific, from recent observations of the GW170917 binary system merger, Bauswein et al [53] propose a method to constrain some neutron stars properties. In particular, they found that the maximum radius,Rmax, corresponding to the maximum mass, of the non-rotating max- imum mass configuration must be larger than9.6+0.15−0.04km. Almost simultaneously, Margalit and Metzger [54] combining electromagnetic and gravitational-wave information on the binary neutron star merger GW170817, constrain the upper limit of the maximum mass,Mmax, ac- cording toMmax ≤2.17M⊙. The combination of the two suggestions leads to an absolute maximum value of compactness which is equal toβmax = 0.333+0.001−0.005. The use of this value with the help of the Figures 7.10 (corresponding to the optimal trial function) and 7.13b will impose constraints both on the maximum values of the indexγinsand the ratioPc/Ec. Accord- ing to expression (8.1) with the parametrization in Table 7.3 for trial function 3, constraint on theγins can be imposed, which isγins,max= 3.725+0.028−0.134, correspondingly toβmax. Also, according to expression (8.3) with the parametrization in Table 7.2 for trial function 3, con- straint on thePc/Ec can be imposed, which is(Pc/Ec)max= 0.9474+0.014−0.066, correspondingly to βmaxandγins,max. Even more, a large number of realistic equations of state must be excluded.
In any case, further theoretical and observational studies, as well as refined combinations of them, are necessary before accurate, reliable and robust constraints to be inferred.
As a conclusion, we suggested a new method to constraint the neutron star equation of state by means of the stability condition introduced by Chandrasekhar [9]. We found that the pre- dicted critical adiabatic index, as function of the compactness parameter, for the most of the equations of state considered here (although they differ considerably at their maximum masses and in how their masses are related to radii) satisfies a universal relation. In particular, the exploitation of these results leads to a model-independent expression for the onset of instability as a function of the compactness parameter. The expression (8.1) reproduces very well this relation. The above finding may be added to the rest approximately EoS-independent rela- tions [55,56,57,58,59,60,61,62,63]. These universal relations break degeneracies among astrophysical observable and leading to a variety of applications. We state that additional theoretical and observational measurements of the bulk neutron star properties close to the maximum mass conjuration will help to impose robust constraints on the neutron star equa- tion of state or, at least, to minimize the numbers of the proposed equations of state.
CHAPTER 9
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7.1 The parametrization of the formulas (7.1)-(7.3) using realistic equations of state 126 7.2 The parametrization of the formulas (7.7)-(7.8) using realistic equations of state 127 7.3 The parametrization of the formula (7.9) using realistic equations of state . . . 127 7.4 The parametrization of the formulas (7.1)-(7.3) using realistic equations of state
and all trial functions . . . 127 7.5 The parametrization of the formulas (7.7)-(7.9) using realistic equations of state
and all trial functions . . . 127
143
APPENDIX A
Code: Mathematica
The code presented here is the universal code that used in this master thesis. Obviously, there are different results for every case and every trial function. However, we present here only the M DI−3 equation of state using the trial function in Eq. (3.7), which is the optimal one. Respectively with this case, we solve all the other equations of state using the four trial functions.
145
Study of neutron stars instability using the Chandrasekhar’s criterion
and realistic EoS
Case: MDI-3 EoS
Author: Koliogiannis Koutmiridis Polychronis Aristotle University of Thessaloniki
Department of Theoretical Physics Clear["Global`*"]
Equation of State
En[r_]:=Piecewise[{{15.55*P[r]^0.666+76.71*P[r]^0.247,P[r]≥0.155}, {0.00873+103.17338*(1-Exp[-P[r]/0.38527])+7.34979*(1-Exp[-P[r]/0.01211]), 9.34375*10^(-6)≤P[r]<0.155},
{0.00015+0.00203*(1-Exp[-344827.5*P[r]])+0.10851*(1-Exp[-7692.3076*P[r]]), 4.1725*10^(-8)≤P[r]<9.34375*10^(-6)},
{0.0000051*(1-Exp[-0.2373*10^10*P[r]])+0.00014*(1-Exp[-0.4021*10^8*P[r]]), 1.44875*10^(-11)≤P[r]<4.1725*10^(-8)},{10^(31.93753+10.82611*Log[10,P[r]]+
1.29312*Log[10,P[r]]^2+0.08014*Log[10,P[r]]^3+0.00242*Log[10,P[r]]^4+
0.000028*Log[10,P[r]]^5),6.3125*10^(-25)≤P[r]<1.44875*10^(-11)}}]
rmin=0.001;iv=2000;
Coefficients
c1=4*Pi/(QuantityMagnitude[UnitConvert[Quantity["SolarMass"],"Kilograms"]]*
QuantityMagnitude[UnitConvert[Quantity["SpeedOfLight"]^2,"(Kilometers/Second)^2"]])*
QuantityMagnitude[Quantity[1.6*10^41,"Joules/Kilometers^3"]];
c2=QuantityMagnitude[UnitConvert[Quantity["GravitationalConstant"],
"Kilometers^3/(Kilograms*Seconds^2)"]]*
QuantityMagnitude[UnitConvert[Quantity["SolarMass"],"Kilograms"]]/
QuantityMagnitude[UnitConvert[Quantity["SpeedOfLight"]^2,"(Kilometers/Second)^2"]]; Print["c1 = ",c1," and c2 = ",c2]
c1 = 11.2506 and c2 = 1.477
TOV Equations
deqm=M'[r]c1*10^(-6)*r^2*En[r];
deqp=P'[r]-c2*En[r]*M[r]/r^2*(1+P[r]/En[r])*(1+c1*10^(-6)*r^3*P[r]/M[r])*
(1-2*c2*M[r]/r)^(-1);
Initial Values listIv={};
For[i=1,i≤iv,i++,AppendTo[listIv,{deqm,deqp,M[rmin]0.0001,P[rmin]i, WhenEvent[P[r]≤4.1725*10^(-8),{rmax=r,"StopIntegration"}]}]];
Solution
listpmr={};listpr={};listmr={};listmrn={};listpmrn={}; For[i=1,i≤Length[listIv],i++,
s=NDSolve[listIv[[i]],{M,P},{r,rmin,Infinity},
Method{"DiscontinuityProcessing"False},MaxSteps100000];
ms=M[r]/.s[[1,1]]; ps=P[r]/.s[[1,2]]; msm=ms/.(rrmax)//Chop;
psm=ps/.(rrmax)//Chop;
If[psm≥0,
AppendTo[listpmr,{rmax,msm,i}]; AppendTo[listpr,{rmax,psm}]; AppendTo[listmr,{rmax,msm}];]]
listmrn=Select[listmr,0≤#[[2]]≤3&&5≤#[[1]]≤20&]; listpmrn=Select[listpmr,0≤#[[2]]≤3&&5≤#[[1]]≤20&];
Graphical Solution
bsp=BSplineFunction[listmrn,SplineDegree3];
frame[legend_]:=Framed[legend,FrameStyleBlue,RoundingRadius5,FrameMargins0] lp1=ListPlot[listmrn,FrameTrue,FrameLabel{"r (km)","M (M☉)"},
FrameStyleDirective[Blue,12],FrameTicks{{All,None},{All,None}}, LabelStyleDirective[Blue,Italic],PlotRange{{8,16},{0,3}},AxesTrue, PlotStyleRed,ImageSizeLarge,AspectRatio1,
PlotLegendsPlaced[LineLegend[{"MDI-4"},LegendFunctionframe,LegendMargins5], {Right,Top}]];
llp1=ParametricPlot[bsp[x],{x,0,1},FrameTrue,FrameLabel{"r (km)","M (M☉)"}, FrameStyleDirective[Blue,12],FrameTicks{{All,None},{All,None}},
LabelStyleDirective[Blue,Italic],PlotRange{{8,16},{0,3}},AxesTrue,PlotStyleRed, ImageSizeLarge,AspectRatio1,
PlotLegendsPlaced[LineLegend[{"MDI-4"},LegendFunctionframe,LegendMargins5], {Right,Top}]];
GraphicsGrid[{{lp1,llp1}},ImageSize->Full]
MDI-4
8 10 12 14 16
0.0 0.5 1.0 1.5 2.0 2.5 3.0
r(km) M(M☉)
MDI-4
8 10 12 14 16
0.0 0.5 1.0 1.5 2.0 2.5 3.0
r(km) M(M☉)
Maximum Mass
maxm=Sort[listpmrn,#1[[2]]>#2[[2]]&];
Print"Maximum Mass = ",maxm[[1,2]]," M☉ at Radius = ",maxm[[1,1]]," km with Pc = ", maxm[[1,3]]," Mev·fm-3"
Maximum Mass = 2.00832 M☉ at Radius = 10.919 km with Pc = 968 Mev·fm-3
Solution using the initial values of the TOV Equations
listPM={};
For[i=1,i≤Length[listpmrn],i++,
s1=NDSolve[listIv[[listpmrn[[i,3]]]],{M,P},{r,rmin,Infinity}, Method{"DiscontinuityProcessing"False},MaxSteps100000]; ms1=M[r]/.s1[[1,1]];ps1=P[r]/.s1[[1,2]];
AppendTo[listPM,{ms1,ps1,rmax}]];