# Upper Mass Limit of Quark Stars

While there is no confirmation that quark stars exist, is there any theoretical limit analogous to (but different from) the Tolman–Oppenheimer–Volkoff limit for neutron stars?

In other words, what is the maximum pressure for quark matter?

• The limit would depend on the flavor mix and would be smaller than that for neutron stars. – dmckee May 17 '14 at 0:57
• Estimates for the limit for neutron stars vary by a factor of about two; since we know even less about the equation of state for quark matter, I'd expect any quark-star mass limit to be even more poorly constrained. – rob May 17 '14 at 1:24
• @dmckee Structure of Quark Stars 2012 in table 1 say u,d,s quark stars have an upper limit of 2 solar mass and the limit for neutron stars is "same". Hydrostatic Equilibrium of Hypothetical Quark Stars 1970 says the limit is "much less" for quark stars (also u,d,s) than neutron stars. So I'm confused. I would think the newer reference is more correct, but I don't see why the limit would be the same for both. And the newer reference is saying quark stars aren't bound by gravity in table 1. – DavePhD May 18 '14 at 12:29
• Dave, I'm no expert in this area and will read your link with interest. My comment was fairly naive, and basically assumed that neutron degenerate equilibrium was be similar to a $u$, $d$ equilibrium but perhaps a little lower (at low momentum all the heavier quark content is fairly small, so probably similar thinking to the 1970 paper); and that higher heavy quark content would be denser could only make things more compact. That is to say I neglected the electric field issues that the 2012 abstract talks about. – dmckee May 18 '14 at 17:41

The upper mass limit for a quark star depends on your assumptions and ranges between 1 and 2 solar masses (cf. this paper (arXiv link) from 2001). It seems to me that the reason for the similarity to neutron stars' mass range is that it both compact objects satisfy the TOV equation, $$\frac{dp}{dr}=-\frac{G}{r^2}\left[\rho+\frac{p}{c^2}\right]\left[M+4\pi r^2\frac{p}{c^2}\right]\left[1-\frac{2GM}{rc^2}\right]^{-1}$$ but with different equations of state.
For the quark star, according to the aforemention paper, the pressure is defined as $$p(\mu)=\frac{N_f\mu^4}{4\pi^2}\left[1-2\frac{\alpha_s}{\pi}-\left(G+N_f\ln\frac{\alpha_s}\pi+\left(11-\frac23N_f\right)\ln\frac{\bar{\Lambda}}{\mu}\right)\frac{\alpha_s^2}{\pi^2}\right]$$ where $G\simeq10.4-0.536N_f+N_f\ln N_f$, $\alpha_s$ the strong coupling, $N_f$ the number of flavors (often taken as 3), $\mu$ the chemical potential, and $\bar{\Lambda}$ the renormalization subtraction point (my understanding of this term is minimal, but it seems to change the size of the mass-radius relation, but not the shape).