IMRPhenomNSBH torus ratio and Kerr ISCO radius

Scope and status

Specification nsbh-torus-isco/v1 covers the Kerr ISCO radius and baryonic torus-mass ratio that follow the already gated compactness/disruption root. It does not qualify remnant/QNM fits, amplitude branches, phase, or the whole waveform.

Inputs, conventions, and options

The black hole is body 1, the nonspinning neutron star body 2, \(q=m_1/m_2\ge1\), \(\chi\) is the aligned black-hole spin, \(C\) is neutron-star compactness, and \(\xi_{\rm tide}\) is the largest positive root of Thompson Appendix A Eq. (10). The ISCO radius is in units of the black-hole mass. The ratio is dimensionless and is relative to the initial neutron-star baryonic mass, not gravitational mass.

The frozen Cartesian gate has \(q\in\{1,2,5,10,15\}\), \(\chi\in\{-0.5,0,0.5\}\), and \(\Lambda\in\{0,1,10,300,800,2600,5000\}\): 105 cases. The compactness and root are the published scalar expressions specified in nsbh-compactness-disruption-root/v1. Both native backends are checked on CPU FP64.

Mathematics and components

Thompson et al. (2020), Appendix A Eqs. (11)–(12) give, at the paper's printed decimal precision,

[ Z_1=1+(1-\chi^2)^{1/3}\left[(1+\chi)^{1/3}+(1-\chi)^{1/3}\right], \quad Z_2=\sqrt{3\chi^2+Z_1^2}, ] [ \bar r_{\rm ISCO}=3+Z_2-\operatorname{sign}(\chi) \sqrt{(3-Z_1)(3+Z_1+2Z_2)}, ] [ T_{\rm raw}=0.296\,\xi_{\rm tide}(1-2C) -0.171\,qC\bar r_{\rm ISCO}. ]

The paper prints \(T_{\rm raw}=M_{b,\rm torus}/M_{b,\rm NS}\), but this fit can be negative. The native carrier's physical-mass adapter is \(T_{\rm carrier}=\max(0,T_{\rm raw})\) and its torus flag is \(T_{\rm carrier}>0\). The gate reports the unclipped equation and checks the adapter separately; it does not attribute clipping to Eq. (11). For \(\chi=0\), Eq. (12) gives \(\bar r_{\rm ISCO}=6\) and its local slope is \(-4\sqrt{2/3}\); the native implementations use that linear expansion only for \(|\chi|<10^{-4}\) to keep autodiff regular.

Implementation mapping

The backend-free, high-precision scalar reference is src/diffgw_reference/nsbh_torus.py, sharing only the independent scalar compactness/root reference. Torch and JAX use their respective conversions.kerr_isco_radius and approximants.imrphenom_nsbh.nsbh_torus_mass_ratio helpers. The latter is the exact component used in native amplitude classification.

Evidence and limitations

Before native comparison, the gate fixes maximum absolute errors \(2\times10^{-13}\) for ISCO radius and \(2\times10^{-12}\) for raw and clipped torus ratios. It checks the sign/clip relation and both positive- and zero-torus cases. A \(70\)- versus \(90\)-digit scalar check must differ by at most \(10^{-30}\) for all values. These are equation/component checks, not a public-binary or physical-accuracy claim. A source error common to both implementations remains possible despite the independent scalar evaluation.

Reproduce the checks

Run JAX_ENABLE_X64=1 PYTHONPATH=src pytest -q tests/test_nsbh_torus_reference.py with the reference, torch, and jax extras on CPU. The measured result is in the 27 September 2026 validation receipt.