(*EoS->p=kρ²,k=100km²,ρ_c=2.0*10^(18)kg/m³)
In[]:=
λ'[r]=1/(4*r(1+2*α*R[r]+α*r*R'[r]))*(2*(1+2*α*R[r])*(1-Exp[2*λ[r]])+((r²*Exp[2*λ[r]])/3)*(16*Pi*(2*ϵ[r]+3*p[r])+2*R[r]+3*α*R[r]^2)+(2*α*r*R'[r])/(1+2*α*R[r])*(2*(1+2*α*R[r])*(1-Exp[2*λ[r]])+(r²*Exp[2*λ[r]])/3*(16*Pi*ϵ[r]+R[r]+3*α*R[r]^2)+4*α*r*R'[r]))//Simplify
Out[]=
-2(-1+
2λ[r]

)(1+2αR[r])+
1
3
2λ[r]

r²(48πp[r]+2R[r]+3α
2
R[r]
+32πϵ[r])+
2rα
′
R
[r]-2-1+
2λ[r]

(1+2αR[r])+
1
3
2λ[r]

r²R[r]+3α
2
R[r]
+16πϵ[r]+4rα
′
R
[r]
1+2αR[r]
4r(1+2αR[r]+rα
′
R
[r])
In[]:=
ψ'[r]=1/(4*r(1+2*α*R[r]+α*r*R'[r]))*(r²*Exp[2*λ[r]](16*Pi*p[r]-α*R[r]^2)+2*(1+2*α*R[r])*(Exp[2*λ[r]]-1)-8*α*r*R'[r])//Simplify
Out[]=
2(-1+
2λ[r]

)(1+2αR[r])+
2λ[r]

r²(16πp[r]-α
2
R[r]
)-8rα
′
R
[r]
4r(1+2αR[r]+rα
′
R
[r])
In[]:=
R''[r]=(Exp[2*λ[r]]/(6*α))*(8*Pi*(3*p[r]-ϵ[r])+R[r])+(λ'[r]-ψ'[r]-2/r)*R'[r]//Simplify
Out[]=
1
6rα(1+2αR[r])
24
2λ[r]

πrp[r](1+2αR[r])+6α
′
R
[r](-1-
2λ[r]

+2rα
′
R
[r])-8
2λ[r]

πϵ[r](r-2r²α
′
R
[r])+
2λ[r]

α
2
R[r]
(2r+3r²α
′
R
[r])+R[r](
2λ[r]

r-16
2λ[r]

πrαϵ[r]+(
2λ[r]

(r²-12α)-12α)α
′
R
[r])
In[]:=
p'[r]=-(ϵ[r]+p[r])ψ'[r]
Out[]=
(-p[r]-ϵ[r])2(-1+
2λ[r]

)(1+2αR[r])+
2λ[r]

r²(16πp[r]-α
2
R[r]
)-8rα
′
R
[r]
4r(1+2αR[r]+rα
′
R
[r])
(*INITIALCONDITIONS​​ψ[0]=0,λ[0]=0,R[0]=R_c,R'[0]=0​​p=p[ϵ]->EoS​​R_c->r->inf->0​​​​MATCHINGCONDITIONS​​ψin[Rstar]=ψout[Rstar]​​λin[Rstar]=λout[Rstar]​​Rin[Rstar]=Rout[Rstar]​​R'in[Rstar]=R'out[Rstar]​​*)