Tidal truncation validation - progenax — differentiable initial conditions for N-body simulations

progenax.tidal computes the Jacobi (tidal) radius and applies a sharp, differentiable tidal truncation. Test files: tests/validation/test_tidal_physics.py (9 tests, validation tier) and tests/unit/test_tidal.py (15 unit tests); figures: scripts/validate_tidal.py. The validation tier checks the physics against independent oracles — the inner Lagrange point of the full restricted three-body problem, the tidal force balance, and the analytic Plummer profile — not against the formula’s own algebra.

Key result: the King/Hill radius is the restricted-3-body L1 point

$r_J = R\,(m/3M_{\rm g})^{1/3}$ reproduces the inner Lagrange point L1 (solved numerically from the full rotating-frame force balance) across seven decades in mass ratio; the residual is the leading Hill correction, scaling as $(m/M_{\rm g})^{1/3}$ (measured slope 0.34) and reaching just $1.1\times10^{-3}$ for a Galactic globular ( $m/M_{\rm g}\sim10^{-7}$ ). The defining balance — self-gravity $Gm/r^2$ equal to tidal+centrifugal $3\Omega^2 r$ — is met at $r_J$ , the flat-rotation-curve (isothermal) radius is exactly $(3/2)^{1/3}$ larger than the Keplerian one, and the truncated bound mass tracks the analytic Plummer $M(<r)$ to $0.6\%$ — all differentiable in the truncation radius.

What is verified¶

Rows map to test_tidal_physics.py; Measured values are regenerated by scripts/validate_tidal.py.

Property	Tolerance (as tested)	Measured	Anchor
$r_J$ vs L1 Lagrange point ( $m/M_{\rm g}=10^{-7}$ )	rel $< 10^{-2}$	$1.1\times10^{-3}$	restricted 3-body L1
Hill correction scaling	log-log slope $\approx 1/3$	0.34	$(m/M_{\rm g})^{1/3}$
Tidal force balance $Gm/r_J^2 = 3\Omega^2 r_J$	rel $< 10^{-10}$	exact	point-mass host
Effective-potential L1 saddle	within $5\%$ of $r_J$	119.0 vs 119.5 pc	rotating-frame $\Phi_{\rm eff}$ max
Isothermal / Keplerian ratio	$(3/2)^{1/3}$ to 10^-6	1.1447	tidal tensor $2\Omega^2$ vs $3\Omega^2$
Truncated mass vs analytic Plummer $M(<r)$	max dev $< 0.02$	0.006	$M(<r)=Nr^3/(r^2+a^2)^{3/2}$
$\partial r_J/\partial M_{\rm c}$ (AD vs analytic)	rel $< 10^{-6}$	$4\times10^{-16}$	$\tfrac{R}{3}(3M_{\rm g})^{-1/3}M_{\rm c}^{-2/3}$
$\partial M_{\rm bound}/\partial r_t$ (surrogate)	median rel $< 0.2$ vs shell	0.02	analytic Plummer shell $dM/dr$
Truncation keeps static shape $N$	no dynamic indexing	True	JIT/vmap/grad-safe

Figures¶

Generated by scripts/validate_tidal.py (PASS/FAIL per panel; PNG + PDF vector).

The Jacobi radius is the L1 Lagrange point. (a) r_J = R(m/3M_{\rm g})^{1/3}
(line) overlays the inner Lagrange point of the full restricted three-body problem
(circles) across seven decades in mass ratio. (b) The relative error is the leading
Hill correction \propto(m/M_{\rm g})^{1/3} (measured slope 0.34), reaching only
1.1\times10^{-3} at a Galactic globular’s m/M_{\rm g}\sim10^{-7}. — Figure 1:**The Jacobi radius is the L1 Lagrange point.** (a) $r_J = R(m/3M_{\rm g})^{1/3}$ (line) overlays the inner Lagrange point of the full restricted three-body problem (circles) across seven decades in mass ratio. (b) The relative error is the leading Hill correction $\propto(m/M_{\rm g})^{1/3}$ (measured slope 0.34), reaching only $1.1\times10^{-3}$ at a Galactic globular’s $m/M_{\rm g}\sim10^{-7}$ .

The tidal force balance. (a) The cluster’s self-gravity Gm/r^2 (inward) equals
the tidal+centrifugal field 3\Omega^2 r (outward) exactly at r_J. (b) In the
rotating frame the effective potential along the cluster–galaxy line peaks at the L1
point (\approx r_J) — the escape barrier a star must climb to be stripped. — Figure 2:**The tidal force balance.** (a) The cluster’s self-gravity $Gm/r^2$ (inward) equals the tidal+centrifugal field $3\Omega^2 r$ (outward) exactly at $r_J$ . (b) In the rotating frame the effective potential along the cluster–galaxy line peaks at the L1 point ( $\approx r_J$ ) — the escape barrier a star must climb to be stripped.

Keplerian vs isothermal host. The flat-rotation-curve (isothermal) tidal radius
is (3/2)^{1/3}=1.145\times the point-mass (Keplerian) one at matched orbital
frequency — the tidal tensor carries 2\Omega^2 for a flat rotation curve versus
3\Omega^2 for a point mass. At a Galactic-globular orbit (8 kpc): 30.4 vs
26.6 pc. — Figure 3:**Keplerian vs isothermal host.** The flat-rotation-curve (isothermal) tidal radius is $(3/2)^{1/3}=1.145\times$ the point-mass (Keplerian) one at matched orbital frequency — the tidal tensor carries $2\Omega^2$ for a flat rotation curve versus $3\Omega^2$ for a point mass. At a Galactic-globular orbit (8 kpc): 30.4 vs 26.6 pc.

Truncating a Plummer cluster. (a) The truncated bound-mass fraction (points) lies
on the analytic Plummer enclosed mass M(<r)/M = r^3/(r^2+a^2)^{3/2} (line) to
0.6\%. (b) Applying the cut at r_t=r_J for a typical Galactic globular (fill
factor r_h/r_J\approx0.19): bound stars inside the Jacobi circle, tidally stripped
stars outside. — Figure 4:**Truncating a Plummer cluster.** (a) The truncated bound-mass fraction (points) lies on the analytic Plummer enclosed mass $M(<r)/M = r^3/(r^2+a^2)^{3/2}$ (line) to $0.6\%$ . (b) Applying the cut at $r_t=r_J$ for a typical Galactic globular (fill factor $r_h/r_J\approx0.19$ ): bound stars inside the Jacobi circle, tidally stripped stars outside.

Differentiability. (a) \partial r_J/\partial M_{\rm cluster} from autodiff
matches the closed-form derivative to machine precision. (b) The hard truncation
is still differentiable in r_t: the straight-through surrogate gradient of the bound
mass tracks the analytic Plummer shell mass dM/dr_t (median rel 0.02) — so r_t
(and any upstream parameter feeding it, e.g. via jacobi_radius) can be inferred by
gradient descent. — Figure 5:**Differentiability.** (a) $\partial r_J/\partial M_{\rm cluster}$ from autodiff matches the closed-form derivative to machine precision. (b) The **hard** truncation is still differentiable in $r_t$ : the straight-through surrogate gradient of the bound mass tracks the analytic Plummer shell mass $dM/dr_t$ (median rel 0.02) — so $r_t$ (and any upstream parameter feeding it, e.g. via `jacobi_radius`) can be inferred by gradient descent.

How to run¶

# physics tests (9 validation + 15 unit, ~2 s on CPU)
pytest tests/validation/test_tidal_physics.py tests/unit/test_tidal.py -q

# regenerate the five figures with PASS/FAIL tables
python scripts/validate_tidal.py

Honest scope¶

jacobi_radius is the point-mass / Hill form; the $(m/M_{\rm g})^{1/3}$ residual vs the exact L1 point (Figure 1) is the known leading correction, not an error.
apply_tidal_truncation is a sharp cut (mass $\to0$ beyond $r_t$ ); the gradient in $r_t$ is a straight-through logistic surrogate (it smooths only the backward pass, never the exact forward truncation). For a physically smooth King-model truncation, use the King profile directly.
Long-term tidal evolution (mass loss, tidal tails) is an N-body problem for gravax; this page validates the IC-level Jacobi radius and the truncation operator only.

References¶

King (1962) (tidal radius); Binney & Tremaine (2008), §8.3.1; Spitzer (1987), Dynamical Evolution of Globular Clusters; Baumgardt & Makino (2003) (fill factor). The utilities are in progenax/tidal.py.

References¶

King, I. R. (1962). The structure of star clusters. I. An empirical density law. The Astronomical Journal, 67, 471. 10.1086/108756
Baumgardt, H., & Makino, J. (2003). Dynamical evolution of star clusters in tidal fields. Monthly Notices of the Royal Astronomical Society, 340, 227–246. 10.1046/j.1365-8711.2003.06286.x