Differential Rotation and Hypermassive Stars

Differential rotation can support equilibria whose baryon mass exceeds the largest uniformly rotating mass for the same EOS. Such a hypermassive star is not a new permanent branch: magnetic winding, turbulence, viscosity, neutrino cooling, and gravitational radiation redistribute angular momentum and remove thermal support. Its computed mass depends strongly on the imposed rotation law and topology. The useful calculation is therefore a family with an explicit law, EOS, entropy prescription, and stability test—not a universal percentage added to the nonrotating maximum.

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

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 finite-temperature and composition closure; Formulations of General Relativity supplies gauge and constraint language; Computational Relativity supplies equilibrium and evolution checks.

Rotation laws are physical inputs

For circular barotropic flow, the Euler equation is integrable if $F(\Omega)=u^t u_\phi$ depends only on $\Omega$. A widely used one-parameter choice is the relativistic $j$-constant law

$$F(\Omega)=A^2(\Omega_c-\Omega),$$

where $\Omega_c$ is the central angular velocity and $A$ sets a differential- rotation length scale. In the Newtonian limit, with cylindrical radius $\varpi$,

$$\Omega(\varpi)=\frac{\Omega_c A^2}{A^2+\varpi^2}.$$

Thus $A\to\infty$ approaches uniform rotation, while small $A/R_e$ produces a steep profile. This law is convenient and captures one qualitative remnant feature, but it is not derived from merger transport. Other laws can permit different shear layers, extrema, or nearly constant specific angular momentum. Comparing masses across laws without matching a physical angular-momentum profile is not a universal test. The integrability condition and common laws are reviewed in Paschalidis and Stergioulas 2017, §§2.4–2.5, Eqs. (38)–(45).

The name does not mean that specific angular momentum is constant everywhere at finite $A$. In this Newtonian limit,

$$j(\varpi)=\varpi^2\Omega =\frac{\Omega_c A^2\varpi^2}{A^2+\varpi^2}, \qquad \frac{dj}{d\varpi} =\frac{2\Omega_c A^4\varpi}{(A^2+\varpi^2)^2}.$$

For positive $\Omega_c$, $j$ increases outward and approaches a constant only far from the axis. It passes the Newtonian Rayleigh test for local axisymmetric centrifugal instability while still having $d\Omega/d\varpi\lt0$. The latter shear can supply energy to magnetic or nonaxisymmetric instabilities; one stability test does not replace the others.

Hypermassive support and its limits

Call a configuration hypermassive when its conserved baryon mass exceeds the maximum available to cold, uniformly rotating equilibria of the same EOS and composition convention. Differential rotation can place more centrifugal support where it is useful and can produce spheroidal or quasi-toroidal density distributions. Early relativistic equilibrium and evolution calculations demonstrated stable examples on dynamical timescales in Baumgarte, Shapiro, and Shibata 2000, pp. L29–L32 and Figs. 1–2. That existence result does not establish long-term survival or a law-independent maximum.

A merger remnant adds thermal pressure, trapped or evolving composition, nonaxisymmetry, magnetic fields, and a massive disk. The quantity compared with a cold uniformly rotating ceiling must therefore exclude ejecta and disk mass consistently and distinguish gravitational from baryon mass. As the star cools and angular momentum is transported outward, it may settle below the uniform-rotation limit, cross a collapse boundary, or shed matter. Which outcome occurs is an evolution question, not an equilibrium label.

Turning points remain useful diagnostics but have weaker force than in the uniform barotropic theorem. Dynamical evolutions of constant-$J$ sequences find collapse onset near, but not exactly at, the equilibrium turning point for the studied $j$-constant families; the setup and qualification are given in Weih, Most, and Rezzolla 2018, §§2–4 and Figs. 1–3. A result for that family should not be promoted to arbitrary entropy gradients or rotation laws.

Inputs and environment boundary

A documented equilibrium run must freeze the cold or finite-temperature EOS, table checksum and domain, composition and entropy prescription, crust join, interpolation, central enthalpy, rotation-law functional form, $A/R_e$ or its replacement parameters, target $J$ or baryon mass, and topology branch. It must record the coordinate gauge, surface representation, domain mapping, radial and angular resolution, outer boundary, iteration tolerances, and software commit, compiler, precision, and libraries.

For a remnant-inspired family, the input profile should come from a named simulation time and specify how it was azimuthally averaged and mapped into an integrable stationary law. An arbitrary fit to $\Omega(\varpi)$ can erase baroclinicity and meridional flow; the mismatch should be reported rather than hidden in a fitted parameter. Store $M$, $M_b$, $J$, kinetic-to-binding-energy ratio, equatorial and polar circumferences, maximum-density location, virial residuals, and a rotation-profile residual for every model.

The analytic rotation law above specifies a family; this page does not present a new equilibrium sequence or remnant simulation. Reproducing one requires versioned inputs, an executable environment, equilibrium outputs, constraint and virial diagnostics, and, for stability claims, the mapped evolution data and perturbation prescription.

Benchmarks, convergence, and uncertainty

First take $A/R_e$ successively larger and recover the uniform-rotation sequence. Then take zero rotation and recover the TOV solution with the same EOS interpolation. Reproduce at least one published polytropic equilibrium from Baumgarte, Shapiro, and Shibata, translating their units and definition of $A$ explicitly. A separate dynamical benchmark should evolve a stable control without secular drift before comparing a candidate near collapse.

Use at least three radial–angular resolutions and follow global quantities, surface position, maximum-density location, elliptic residuals, and GRV2/GRV3. Quasi-toroidal configurations need sufficient angular and radial coverage of the off-center maximum. Stability evolutions require a grid-spacing sequence, atmosphere study, constraint norms, conserved baryon mass and angular momentum, and a collapse diagnostic such as apparent-horizon formation. Repeat with smaller imposed perturbations to distinguish a physical unstable mode from a finite-amplitude trigger.

Numerical uncertainty includes discretization, EOS interpolation, domain truncation, profile fitting, root location, and the mapping between equilibrium and evolution grids. Physical spread across plausible rotation laws and thermal profiles is a separate model uncertainty and will generally exceed a well-converged field-solve tolerance.

Failure modes and interpretation

A major numerical failure mode is branch jumping between spheroidal and quasi-toroidal families as a continuation parameter changes. Other failures include an unresolved surface cusp, loss of monotonic EOS interpolation, extrapolation beyond the table, poor resolution at an off-center density maximum, ill-conditioned target constraints, and artificial angular-momentum loss in evolution. Nonconvergence is not a stability boundary.

Corotation points, where a mode pattern speed equals local $\Omega$, can enable shear instabilities even at modest $T/|W|$. Magnetic winding and the magnetorotational instability act on timescales absent from an unmagnetized stationary model. Hence equilibrium existence, dynamical stability over a few rotations, and astrophysical lifetime are three distinct conclusions.

Check your understanding

For the Newtonian limit of the $j$-constant law, at what cylindrical radius is $\Omega=\Omega_c/2$? What happens to that radius as the law approaches uniform rotation?

Solution

Set

$$\frac{A^2}{A^2+\varpi^2}=\frac12.$$

This gives $\varpi=A$. As $A/R_e\to\infty$, the half-frequency radius moves far outside the star, so $\Omega\simeq\Omega_c$ throughout the material. The answer also shows why quoting $A$ without normalizing it to a stellar length does not specify the degree of differential rotation.

References

• Baumgarte, Thomas W., Stuart L. Shapiro, and Masaru Shibata. “On the Maximum Mass of Differentially Rotating Neutron Stars.” The Astrophysical Journal Letters 528 (2000): L29–L32. DOI.
• Paschalidis, Vasileios, and Nikolaos Stergioulas. “Rotating Stars in Relativity.” Living Reviews in Relativity 20 (2017): 7. DOI.
• Weih, Lukas R., Elias R. Most, and Luciano Rezzolla. “On the Stability and Maximum Mass of Differentially Rotating Relativistic Stars.” Monthly Notices of the Royal Astronomical Society: Letters 473 (2018): L126–L130. DOI.