Uniform Rotation and Mass Shedding

Uniform rotation turns a spherical stellar sequence into a two-parameter surface of equilibria. Its rapid-rotation boundary is usually mass shedding: at the equator, a fluid element needs no pressure force to remain on its circular orbit and corotates with a circular geodesic. It can still be gravitationally bound; mass shedding is not the marginally bound orbit of test-particle dynamics. Finding that boundary is a numerical free-surface problem, not an algebraic correction to a nonrotating mass. This page states the physical condition and a reproducible calculation contract; it does not claim that a rotating-star code was executed here.

Required background. Killing Fields and Isometry Groups supplies the stationary and axial symmetries used to define energy, angular momentum, and angular velocity.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies the matter sources; The Einstein Equation supplies the field equations; Relativistic Matter and Compact Objects supplies the EOS closure; Formulations of General Relativity supplies gauge and constraint language; Computational Relativity supplies discretization and convergence practice.

Families of uniformly rotating equilibria

For a barotropic perfect fluid in circular motion, $u^\mu=u^t(t^\mu+\Omega\phi^\mu)$ with constant $\Omega$. A model is fixed by an EOS, a central enthalpy, and one rotation parameter; useful reparameterized families hold baryon mass $M_b$, angular momentum $J$, or spin frequency fixed. These are not interchangeable. A constant-$M_b$ track approximates secular spin-down without mass loss, whereas a fixed-frequency slice is useful for comparing observed pulsars.

The global quantities include gravitational mass $M$, baryon mass $M_b$, angular momentum $J$, equatorial and polar circumferences, kinetic energy $T$, and binding energy. “Maximum mass” must say which family and which stability condition are meant. A supramassive sequence has no nonrotating member but remains uniformly rotating and stationary until it reaches a stability boundary. The taxonomy and equilibrium surface are summarized in Paschalidis and Stergioulas 2017, §2.9.1.

The equatorial Kepler condition

For a sub-Keplerian star, pressure decreases outward and its force is outward: it supplies the support that centrifugal acceleration does not. In the Newtonian equatorial limit this is explicit,

$$\frac{1}{\rho}\frac{dp}{dr} =-\frac{GM}{r^2}+\Omega^2r.$$

At mass shedding the outward derivative of enthalpy vanishes at the equator, the free surface develops a cusp, and the outermost fluid follows a circular geodesic. Enthalpy is the useful diagnostic: for an EOS whose density vanishes at the surface, $dp/dr$ can tend to zero even below mass shedding. In stationary axisymmetric coordinates adapted to the Killing fields, with equatorial reflection symmetry and a regular radial coordinate, the radial circular-geodesic condition is

$$\partial_r g_{tt}+2\Omega_K\partial_r g_{t\phi} +\Omega_K^2\partial_r g_{\phi\phi}=0$$

at the equatorial surface. The prograde root is

$$\Omega_K= \frac{-\partial_r g_{t\phi} +\sqrt{(\partial_r g_{t\phi})^2 -\partial_r g_{tt}\,\partial_r g_{\phi\phi}}} {\partial_r g_{\phi\phi}}.$$

All metric derivatives and the surface location must come from the same converged equilibrium. Frame dragging and oblateness are already contained in this expression. Substituting a spherical radius into a Newtonian Kepler formula discards precisely the effects the rapid-rotation solve was meant to capture. The relativistic shedding criterion and other limiting sequences are given in Paschalidis and Stergioulas 2017, §2.9.3, Eqs. (74)–(75).

Inputs and documented environment

A rerun record begins with the EOS file, checksum, units, thermodynamic variables, valid density and temperature domain, crust join, and interpolation rule. It then records the target family, central enthalpy bracket, rotation parameter, coordinate gauge, surface representation, outer boundary, radial compactification, angular basis, symmetry assumptions, and stopping tolerances. The software name, commit or release version, compiler, numerical precision, library versions, and input files belong with the result.

For each point, store $M$, $M_b$, $J$, $\Omega$, equatorial and polar circumferences, central density, virial residuals, elliptic residual norms, and the mass-shedding indicator. If a sequence is located by continuation, record the predictor step and corrector tolerance. If a target such as $M_b$ is enforced by an outer solve, report that residual separately from the field iteration residual. A result without this environment and input boundary is a plot, not a reproducible numerical claim.

No rotating-star executable, EOS table, or frozen input set is registered on this page, and no computation has been rerun. A future implementation should provide a clean command that consumes the versioned input and emits both the equilibrium table and machine-readable convergence diagnostics.

Benchmarks, convergence, and uncertainty

The first benchmark is the zero-rotation limit against an independently integrated TOV sequence using the same EOS interpolation. The second is the slow-rotation overlap for $J$, moment of inertia, and the quadrupolar deformation. The third is a published rapid-rotation comparison. Nozawa and collaborators compared independent codes, coordinates, virial identities, and global quantities in Nozawa et al. 1998, §§3.1–3.3 and 5.3–5.5, Tables 8–14; matching a documented benchmark model is stronger than reproducing only one’s own sequence.

Use at least three radial–angular resolutions with the same physical inputs. Track fractional changes in $M$, $J$, circumferences, and $\Omega_K$, together with the GRV2 and GRV3 virial identities. Repeat after refining the EOS table or changing a thermodynamically consistent interpolation, because a small field residual cannot reveal EOS interpolation bias. Quote the observed convergence behavior rather than assuming formal spectral or finite-difference order. Near a cusp, smooth spectral convergence can degrade; a localized surface error may dominate the nominal bulk residual.

A useful uncertainty statement separates discretization, finite outer boundary or compactification, root-location tolerance, EOS interpolation, and input-EOS uncertainty. The last is physical-model uncertainty and should not be combined with numerical error as though both were sampling noise.

Failure modes and physical endpoints

Iteration failure can arise from an unresolved equatorial cusp, branch jumping, an inconsistent pair of target constraints, a central density outside the EOS domain, loss of thermodynamic monotonicity, or an inadequate exterior domain. Each is a numerical failure mode that must be diagnosed before assigning a physical endpoint. It does not by itself establish mass shedding. Conversely, satisfying the geodesic formula at one noisy grid point is insufficient unless the surface enthalpy gradient and resolution sequence behave consistently.

Mass shedding is not the only endpoint. A constant-$J$ or constant-$M_b$ family may encounter an axisymmetric stability turning point first. Viscous or gravitational-radiation-driven nonaxisymmetric instabilities can limit an astrophysical spin before either equilibrium boundary. The computed Kepler curve is therefore an existence boundary for stationary uniform rotation, not a population prediction.

Check your understanding

Ignore frame dragging and take a spherical exterior at the equator with $g_{tt}=-(1-2GM/r)$ and $g_{\phi\phi}=r^2$. What does the circular-geodesic condition give for $\Omega_K$?

Solution

Here $\partial_r g_{t\phi}=0$, $\partial_r g_{tt}=-2GM/r^2$, and $\partial_r g_{\phi\phi}=2r$. Therefore

$$-\frac{2GM}{r^2}+2r\Omega_K^2=0, \qquad \Omega_K^2=\frac{GM}{r^3}.$$

The familiar Kepler result is recovered as a limit. A rapidly rotating star is oblate and has $g_{t\phi}\ne0$, so this limiting substitution is a benchmark, not the production shedding formula.

References

• Nozawa, Takeshi, Nikolaos Stergioulas, Eric Gourgoulhon, and Yoshiharu Eriguchi. “Construction of Highly Accurate Models of Rotating Neutron Stars—Comparison of Three Different Numerical Schemes.” Astronomy and Astrophysics Supplement Series 132 (1998): 431–454. DOI.
• Paschalidis, Vasileios, and Nikolaos Stergioulas. “Rotating Stars in Relativity.” Living Reviews in Relativity 20 (2017): 7. DOI.