How Large Can a Star Be?

In a recent article Schoen and Yau develop an inequality which shows that there exists a relationship between the minimum energy density ${\rho }_0$ in a star and the size $R$ of the star, of the form ${\rho }_0{R}^2\le \frac{\pi }{6}$. This article shows that this inequality is valid for two different measures of the size of the star and conjectures that the inequality can be improved to ${\rho }_0{R}^2\le \frac{3\pi }{32}$.