A common way to study models of stars is to choose an equation of state $\varepsilon (p)$, and then plot a mass-radius curve. A point on the mass-radius curve is found by choosing a central pressure, that is, the pressure at the center of the star, and then solving the Tolman-Oppenheimer-Volkoff equation $$\frac{\mathop{dP}}{\mathop{dr}} = -\frac{G}{r^2}\left( \varepsilon(r) + P(r)\right) \left(m(r) + 4\pi r^3 P(r) \right) \left(1 - \frac{2 G m(r)}{r}\right)^{-1}$$ with $$\frac{\mathop{dm}}{\mathop{dr}} = 4\pi r^2 \varepsilon(r)$$ with your given equation of state. The solution of these pair of equations gives you a mass and a radius (the radius $R$ is defined such that $P(R) = 0$).

My question is, can the mass-radius curve self intersect? That is, can there be two stars made of material with the same equation of state with the same mass and same radius, but with different central pressures, and thus different distributions of mass?

I have seen and calculated a lot of stars and mass-radius curves with a lot of different equations of state EoS (hadronic, quark-matter, with and without strangeness, with and without constant pressure phase transitions, polytropic,...) and I have not encountered stable stars with different central pressures but the same compactness.

One very basic stability criterion of TOV-equation is: $dM_R/dP_c>0$ so the stars beyond the maximum mass in a M-$P_c$ are mostly unstable. (With some exceptions “third family” of compact stars... arxiv.org/abs/astro-ph/0001467). Typical M-$P_c$ curves would look similar to this

The plot above shows mass over central baryon density ($\rho_0=0.17\mathrm{ fm}^3$), but since $P_c$ and $\rho_c$ are related via the EoS a $M-P_c$ plot would look similar.

Since for a given EoS $P_c$ is the only input paramter which determines the solution of the ToV equation I do not think you would get to the same mass and radius with a different pressure. So to conclude for stable stars with resonable pressures mass and radius are not equal for different pressures. Maybe there are some crazy EoS or some extended GR-Theories where this might be the case but not in the usual GR-setting with physical EoS.

With maybe one exeption: at very very high pressures ($10^7 \mathrm{MeV}/\mathrm{fm}^3$) for polytropic EoS or other EoS which are given analytically and not by interpolation, where you can calculate with such unrealistic pressures there seems to be an interresting asymptotic behavior: for a given EoS and very high pressures there seems to be a compacness (M/R)/ a point in the M-R plane to which the solutions converge. I have not shown this from the ToV equations structure it is just a feature I noticed working with polytropic equations of state and also with an ideal Fermi gas EoS. This behavior takes place in a regime where the solutions are unstable and the pressures are way above anything reasonable. This solutions are unstable an most likely a mathematical feature of the ToV differential equation. BUT you could say just looking at the M-$P_c$/M-R curves that such solutions have different central pressures but assymptotically the same Mass/Radius. Below three plots of the situation for a polytrope and the pure fermigas EoS of neutrons.

I have done all this plots/calculations myself but I have seen plots with this behaviour for extreme pressures in at least one other paper in case of the neutron gas/pure fermigas EoS.

If someone has literature or insight in this behaviour of the ToV equation at ultra high pressures I would be very interessted.

EDIT: I searched a bit on the arXiv and found a paper (arxiv.org/abs/gr-qc/0304012) which looks at polytropes and the ToV equation. There you can see the same high pressure asymptotic. This spiral structure at the ultra high pressure regime is typical for polytropes and is not a numerical error. In the paper of J. Mark Heinzle et. al. they actually quantized this effekt in their last mass-Radius-curve theorem: Theorem 6.4. (Spiral structure of the (M,R)-diagram. There are actually more references in this paper dealing with that phenomenon and it seems to be a general feature in the ultra high pressure regime. It is not surprising that one can find it for the ideal neutron gas, since in the high pressure regime it can be described by an ultra relativistic polytrope.

• +1 I may borrow your first plot to illustrate my answer from yesterday. Commented Aug 9, 2016 at 6:29
• I may add that the EoS of that plot are all hadronic and a bit old; most of them are from the time where the 2 solar mass neutron stars where unknown. Modern EoS would be stiffer to get to 2 solar masses. Information on the used EoS can be found in my B.Sc. Thesis alongside with that plot. theorie.ikp.physik.tu-darmstadt.de/nhq/downloads/thesis/…
– N0va
Commented Aug 9, 2016 at 11:04
• @M.J.Steil, thanks for your answer! I agree that all equation of states that I have seen mass-radius curves for have not had self-intersections. Then I plotted a mass radius curve for a different e.o.s and found a self intersection. I think it's just numerical error, but I wanted to check to see if it was possible. I'm looking into it further. I am also curious about the high pressure behavior that you mention at the end of your comment, so I hope somebody else can offer a comment. Usually by the time you get to high enough central pressures, computation times go through the roof anyway. Commented Aug 9, 2016 at 20:12
• @ Physics_Plasma: I currently use the explicit embedded RK Prince-Dormand (8, 9) method of the GSL to solve the ToV eq. with an inital step size of $10^{^-60}$ km. This method solves the ToV equation in around 200 steps (taking way below one second to run) even at high pressures of $10^7$MeV/fm$^3$. The step size is adapted to keep the relative and absolute error small. I found that high pressure behavior using old implementations of mine and with the recent one in C++. But maybe one can not realy trust those methods at such P; anyway this asymptotic behaivior is not of physical relevance.
– N0va
Commented Aug 9, 2016 at 20:55
• I edited my answer and put in a reference for that high pressure asymptotic; " It's not a bug, it's a feature".
– N0va
Commented Aug 10, 2016 at 8:46