Mathematica Black hole Joule-Thomson expansion in Mathematica

[djymndl07]

I am trying to recreate the $T-P$ plane of figure 3 in the article black hole Joule-Thomson in Mathematica. I am following the procedure but cannot obtain the graph. I don't know exactly where am i doing wrong. Can any one tell me how to fix it? My notebook is attached here.

 

[renormalize]

I am trying to recreate the $T-P$ plane of figure 3 in the article black hole Joule-Thomson in Mathematica. I am following the procedure but cannot obtain the graph. I don't know exactly where am i doing wrong. Can any one tell me how to fix it? My notebook is attached here.

Finding the zeros of your function:$$f\left(r\right)=1-\frac{2Mr^{2}}{\left(r^{2}+q^{2}\right)^{3/2}}+\frac{8\pi Pr^{2}}{3}\tag{1}$$requires solving a quintic (5th-order) polynomial in $r^2$. This is basically impossible in Mathematica using Solve. One alternative is to proceed numerically by using FindRoot to solve $f(r_H)=0$.
But for the $T$ vs. $P$ graph you're interested in, why bother trying to find $r_H$ at all? Just directly solve $f(r_H)=0$ for the pressure $P$ instead:$$P\left(r_{H},q,M\right)=-\frac{3}{8\pi r_{H}^{2}}+\frac{3M}{4\pi\left(r_{H}^{2}+q^{2}\right)^{3/2}}\tag{2}$$From your notebook, the temperature $T$ is then:$$T\equiv\frac{1}{4\pi}\frac{\partial f\left(r\right)}{\partial r}\left|_{r=r_{H}}\right.=\frac{4Pr_{H}}{3}-\frac{Mr_{H}\left(2q^{2}-r_{H}^{2}\right)}{2\pi\left(r_{H}^{2}+q^{2}\right)^{5/2}}\tag{3}$$or:$$T\left(r_{H},q,M\right)=-\frac{1}{2\pi r_{H}}+\frac{3Mr_{H}^{3}}{2\pi\left(r_{H}^{2}+q^{2}\right)^{5/2}}\tag{4}$$after substituting eq.(2) into eq.(3). With the specific forms of $P(r_H),T(r_H)$ given by eqs.(2,4) you can directly plot $T$ vs. $P$ by varying $r_H$ with ParametricPlot:

This more or less looks like the first curve in Fig. 3a of your reference paper, except that the peak amplitude is significantly smaller here and the peak occurs at a smaller value of $P$. Since I just used the equations from your notebook, you'll have to crosscheck your notebook with the paper to see why there's a discrepancy.