Atmospheres, Vacuum Interfaces, and Tests

Conservative relativistic fluid schemes usually cannot represent exact vacuum because primitive recovery and characteristic speeds require positive density and pressure. An artificial atmosphere replaces vacuum with a low-density state and resets cells that fall below thresholds. This is a numerical regularization that injects or removes mass, energy, momentum, composition, and magnetic content. Its influence must be measured through floor sweeps and integrated budgets, especially for ejecta, winds, magnetospheres, and accretion rates whose physical densities approach the floor.

Helpful background. Formulations of General Relativity supplies spacetime coupling, Ordinary Differential Equations supplies evolution and tolerance concepts, Relativistic Matter and Compact Objects supplies material regimes, Computational Relativity supplies evidence standards, and Fluid and Kinetic Theory supplies conservation and MHD waves.

An atmosphere is a source term

Specify a density floor $\rho_{\rm atm}$, pressure or temperature prescription, velocity and magnetic treatment, reset threshold $f_{\rm thr}\rho_{\rm atm}$, and whether floors scale with position or time. Every reset changes the conserved state by

$$\Delta\mathbf U_{\rm atm}=\mathbf U_{\rm after}-\mathbf U_{\rm before}.$$

Accumulate this correction by component and region. A finite-volume flux update may be exactly conservative while atmosphere operations dominate the global error. Do not label floor material as physical ejecta or accretion.

For undensitized variables the rest-mass injection in cell $j$ is $\Delta M_{b,j}=V_j(D_{\rm after}-D_{\rm before})$, where $V_j$ is its proper spatial volume at the reset time. Sum signed changes for the net budget and absolute changes for the amount of material altered; large injection and removal can cancel in the net. For example, adding $10^{-8}$ of the stellar baryon mass per step over $10^5$ steps adds $10^{-3}$ in total. If the physical ejecta are also $10^{-3}$ of the star, that apparently tiny per-step reset is as large as the claimed signal. A budget normalized only by the bulk mass hides the problem.

Primitive-recovery failure and a valid low-density state are different events. A recovery fallback may invoke the atmosphere, but the log should preserve the reason, pre-reset state, repair magnitude, and location. Here $b^\mu=u_\nu{}^*F^{\nu\mu}$ is the comoving magnetic four-vector with $\sqrt{4\pi}$ absorbed, and $b^2=b^\mu b_\mu$; hence $b^2/\rho$ is dimensionless when $c=1$. This magnetization grows as density falls, so a floor also caps the physical regime. The reported domain of validity should state that cap and the magnetic convention.

Sweep floors and thresholds independently

Run a sequence such as $\rho_{\rm atm}/\rho_{\max}=10^{-8},10^{-9},10^{-10}$ where numerically resolvable, and vary the reset threshold separately. Refining the grid at one floor does not remove atmosphere bias. A physical observable $Q$ is supported only if it approaches a stable limit under both spatial refinement and decreasing floor, before primitive recovery or precision breaks down.

Measure cumulative injected baryon mass, energy and momentum, reset-cell volume, maximum magnetization, and contribution to the observable. For ejecta mass, change the density and velocity selection criteria and exclude tagged atmosphere material. For accretion, compare flux through several surfaces and subtract no background by an undocumented convention.

Benchmark ladder

Use a fixed-spacetime TOV star to isolate the surface treatment. For this record only set $G=c=M_\odot=1$, use the cold polytrope $p=K\rho^\Gamma$ with $K=100$, $\Gamma=2$, and central density $\rho_c=1.28\times10^{-3}$. Solve the TOV ODE independently in 80-digit arithmetic with relative and absolute tolerances $10^{-13}$, and use that stationary profile and metric as the reference. The radial computational domain is $0\leq r\leq120$, with regular parity at the center and non-inflow outflow data fixed to the atmosphere state at the outer boundary. One length unit is $GM_\odot/c^2=1.4766\,\mathrm{km}$ and one time unit is $GM_\odot/c^3=4.9255\,\mu\mathrm{s}$.

Here $\rho$ is rest-mass density, so use total energy density $e=\rho+p/(\Gamma-1)$ in the TOV equation. The radial coordinate is areal radius, and a shell’s proper volume is $V_j=4\pi\int_{r_{j-1/2}}^{r_{j+1/2}}r^2[1-2m(r)/r]^{-1/2}\,dr$. Using Euclidean shell volumes would corrupt the baryon budget. The independent reference uses the regular center expansion and surface matching described in Relativistic Stellar Equilibria. The atmosphere is not itself a hydrostatic continuation of the vacuum TOV exterior, so the test measures its effect on the star as well as discretization error.

Evolve the fluid in the Cowling approximation with fifth-order WENO reconstruction, an HLL flux, SSPRK3, and CFL $0.25$ to $t=100$ using $\Delta r=0.25,0.125,0.0625$. Cross every resolution with $\rho_{\rm atm}/\rho_c=10^{-8},10^{-9},10^{-10}$, reset cells below $10\rho_{\rm atm}$ to the cold EOS at $\rho_{\rm atm}$ and zero Eulerian velocity, and repeat primitive-recovery tolerances $10^{-10}$ and $10^{-12}$. Record all reset corrections rather than counting the initial exterior atmosphere as injected matter.

Compare the numerical density with exact shell averages using

$$E_1=\frac{ \sum_{\bar\rho_{{\rm ex},j}>100\rho_{\rm atm}} V_j|\rho_j-\bar\rho_{{\rm ex},j}|} {\sum_{\bar\rho_{{\rm ex},j}>100\rho_{\rm atm}} V_j\bar\rho_{{\rm ex},j}},$$

and also measure maximum fractional central-density drift and cumulative reset-induced baryon mass divided by the initial stellar baryon mass. Accept the finest, lowest-floor, tolerance-$10^{-12}$ run only if $E_1\leq10^{-2}$, central-density drift is at most $10^{-3}$, and injected baryon mass is at most $10^{-5}$ of the star’s initial baryon mass. Tightening the recovery tolerance may change each measure by no more than $10^{-4}$, and the two finest resolutions at the two lowest floors must agree in central density and stellar baryon mass within $1\%$.

Observed discrepancy: not available—specification only.

Begin the broader ladder with relativistic shock tubes and smooth Alfvén waves whose exact or high-accuracy solutions remain above the floor. The atmosphere should be inactive; any reset signals a solver failure. Then extend the TOV test to a dynamical spacetime and monitor baryon mass, central density, radial-mode spectrum, and surface drift across resolution and floor sweeps.

Use Bondi accretion to test an exterior flow with known steady mass flux, and a magnetized torus to test low-density funnels. Poudel et al. 2020, §II.C and §§III–V, Figures 1–11 compares conservative schemes with vacuum-relevant tests and motivates alternatives to a fixed atmosphere. Font 2008, §§4.1–4.3 summarizes standard relativistic hydrodynamic benchmarks.

For merger ejecta, require the unbound mass, kinetic energy, composition, and velocity distribution to converge under floor changes or replace the atmosphere with a validated vacuum-capable method. A large bulk remnant can converge while a small ejecta tail remains floor dominated; claims must be observable specific.

Alternatives and limitations

Positivity-preserving conservative corrections, cold-vacuum states, force-free matching, or schemes that evolve conserved variables without immediate primitive recovery can reduce atmosphere dependence. Each introduces its own admissibility and interface tests. There is no universal exact-vacuum solution for every GRMHD algorithm.

The physical regime must keep resolved fluid densities sufficiently above numerical regularization, or explicitly include the regularization uncertainty. Lowering the floor indefinitely is impossible because internal-energy cancellation, EOS range, magnetic dominance, and finite precision eventually cause breakdown. Report that lower limit rather than choosing only the lowest run that completes.

Environment and failure criteria

Pin EOS, floor formulas and units, reset threshold, velocity and magnetic reset, recovery and fallback, grid, AMR transfers, boundaries, integrator, precision, tracer policy, unbound criterion, and code revision. A clean interface could be vacuum-study --config tov-atmosphere.yaml --output run/, emitting reset logs, integrated corrections, fluid fields, observable sweeps, convergence data, and environment metadata. It specifies a rerun record without claiming supplied software.

Reject an observable if it scales with atmosphere mass, lacks a joint floor–resolution limit, or relies on unreported resets. Failure modes include counting reset material as ejecta, capping magnetization without recording energy change, an atmosphere moving in the wrong frame, floor activation in an exact shock benchmark, and mass convergence created by cancellation between boundary loss and floor injection.

Diagnostic

Ejecta mass appears converged with resolution but halves whenever the atmosphere density is reduced by a factor of ten. Bulk remnant mass is stable. Which claim is supported?

Solution

The bulk remnant may be supported within its separate error budget, but the ejecta mass is atmosphere dominated and is not converged. Lower the floor with adequate primitive recovery, track injected mass and ejecta tags, vary the unbound selection, or use a validated vacuum treatment. Spatial convergence at fixed regularization does not establish the physical vacuum limit.

References

• Font, J. A. (2008). “Numerical hydrodynamics and magnetohydrodynamics in general relativity.” Living Reviews in Relativity 11, 7. DOI.
• Poudel, A., Tichy, W., Brügmann, B., and Dietrich, T. (2020). “Increasing the accuracy of binary neutron star simulations with an improved vacuum treatment.” Physical Review D 102, 104014. DOI.