Tidal physics

A star cluster orbiting in a host galaxy’s potential is tidally limited: stars beyond a critical radius — the Jacobi radius $r_J$, also called the tidal radius — are stripped by the differential pull between the cluster’s gravity and the galaxy’s. progenax provides two closed-form estimators of $r_J$ — jacobi_radius (point-mass host) and jacobi_radius_isothermal (flat-rotation-curve host) — and a differentiable utility, apply_tidal_truncation, that “removes” stars with $|\mathbf{r}| > r_t$ from the IC by zeroing their mass.

This chapter derives the Jacobi-radius formula from the restricted three-body problem, gives the two host-potential forms progenax implements, and documents the truncation utility.

The Jacobi radius¶

For a cluster of mass $M_{\mathrm{cl}}$ on a circular orbit at galactocentric radius $R$ in a host galaxy with enclosed mass $M_{\mathrm{gal}}(<R)$, the Jacobi radius is

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The numerical factor in the denominator carries the host’s mass-profile dependence through $\mathrm{d}\ln M_{\mathrm{gal}}/\mathrm{d}\ln R$:

• Point-mass host ($M_{\mathrm{gal}}(<R) =$ const): the derivative is 0, giving the factor 3, $r_J = R\,(M_{\mathrm{cl}}/3 M_{\mathrm{gal}})^{1/3}$ — the classical Hill/Roche radius King, 1962.

• Singular isothermal halo / flat rotation curve ($M_{\mathrm{gal}}(<R) \propto R$, so $v_c^2 = G M_{\mathrm{gal}}/R$ is constant): the derivative is 1, giving the factor 2, $r_J = R\,(M_{\mathrm{cl}}/2 M_{\mathrm{gal}})^{1/3}$. The factor of 2 (rather than 3) is the signature of the logarithmic potential.

progenax implements these as two separate functions, each with a fixed factor — there is no callable-host-profile API:

• jacobi_radius(M_cluster, M_galaxy, R_galactic) — the point-mass factor-3 form $r_J = R\,(M_{\mathrm{cl}}/3 M_{\mathrm{gal}})^{1/3}$ (King (1962); Binney & Tremaine 2008 §8.3).

• jacobi_radius_isothermal(M_cluster, V_circ, R_galactic, G) — the flat-rotation-curve factor-2 form, parameterised by the host’s circular velocity $V_{\mathrm{circ}}$ rather than its enclosed mass. Algebraically $r_J = (G\,M_{\mathrm{cl}}/2\Omega^2)^{1/3}$ with $\Omega = V_{\mathrm{circ}}/R$, equivalent to the N-body-calibrated relation of Baumgardt & Makino (2003) Eq. 1.

Derivation sketch¶

The derivation comes from the restricted three-body problem in the rotating frame co-rotating with the cluster’s circular orbit. In that frame, the effective potential is

with $\Omega_R^2 = G M_{\mathrm{gal}}(<R)/R^3$ the angular velocity of the cluster’s circular orbit. The Lagrange points $L_1, L_2$ along the line connecting the cluster centre to the galactic centre define the distance $r_J$ at which a test particle’s effective potential is saddle-shaped — the boundary beyond which it can be lost to the host. Solving $\partial \Phi_{\mathrm{eff}}/\partial r = 0$ to lowest order in $r_J/R$ yields (1).

Higher-order corrections in $r_J/R$ are negligible for $r_J/R \lesssim 0.1$ (typical for Galactic clusters at $R \gtrsim 1$ kpc) and become significant only for very nearby clusters or very massive ones (globular clusters near the Galactic centre, ultra-compact dwarfs near their hosts).

Corrections for eccentric orbits¶

For a cluster on an eccentric orbit, the Jacobi radius varies with orbital phase: it is smallest at perigalacticon (where tidal stripping is most effective) and largest at apogalacticon. progenax’s default treatment uses the perigalacticon Jacobi radius — the most restrictive — to set the IC truncation. This is conservative: the cluster will not lose any additional mass during evolution from the initial condition.

For a more sophisticated treatment, evaluate jacobi_radius at the two orbital extremes (passing the enclosed galaxy mass at each radius) and phase-average:

from progenax.tidal import jacobi_radius

r_peri = jacobi_radius(M_cl, M_gal_peri, R_peri)  # M_gal_peri = M_gal(<R_peri)
r_apo  = jacobi_radius(M_cl, M_gal_apo,  R_apo)   # M_gal_apo  = M_gal(<R_apo)
r_J_avg = 0.5 * (r_peri + r_apo)                  # Phase-averaged

Whether to use peri, apo, or a phase average depends on the science target. For long-term evolution studies (Gyr timescales) the perigalacticon value sets the long-term mass-loss rate. For short-term (“snapshot”) studies the apogalacticon value is more representative of the cluster’s instantaneous extent.

Tidal truncation¶

apply_tidal_truncation(positions, velocities, masses, r_t) “removes” stars beyond a truncation radius $r_t$ — but rather than boolean-indexing them out (which would collapse the array shape), it sets the mass of every star with $r > r_t$ to zero. Those zero-mass “ghost” particles then contribute nothing to any mass-weighted quantity (energy, virial ratio, centre of mass) while keeping the array length fixed at $N$:

from progenax.tidal import apply_tidal_truncation

positions, velocities, masses_trunc, keep_mask = apply_tidal_truncation(
    positions, velocities, masses, r_t=r_J,
)
# All four outputs have length N (shape-preserving):
#   positions, velocities : unchanged (truncated stars left in place)
#   masses_trunc          : masses with r > r_t entries set to 0
#   keep_mask             : bool (N,), True where r <= r_t

Implementation detail. The forward pass is an exact hard Heaviside cut. The mass-zeroing is wrapped in a @jax.custom_jvp straight-through surrogate: the backward pass replaces the delta-function derivative of the step with a logistic bump (width grad_width * r_t, default 5%), so $r_t$ — and any upstream parameter feeding it (e.g. via jacobi_radius) — stays differentiable. Because the output shape is static, the function is fully jit / vmap / grad safe; use keep_mask to filter number-based downstream quantities that would otherwise still see the zero-mass ghosts.

Fill-factor: $r_h / r_J$¶

A useful dimensionless quantity is the fill-factor

↗ model card

which measures how “full” the cluster is relative to its tidal limit. Galactic globular clusters have observed $\mathcal{F}$ in the range 0.05–0.3 Küpper et al., 2011. (The fill-factor / “tidally filling” terminology is later community usage — it is a definitional ratio, not a result tabulated by any single source; King (1966)’s single-mass models supply the tidal radius $r_t$ but no fill-factor data.) progenax’s fill_factor_to_r_h(fill_factor, r_J) utility computes the half-mass radius corresponding to a target fill factor — useful when the science specifies the tidal-truncation regime rather than the absolute size.

Composition with King profile¶

The King (1966) profile already has a built-in tidal radius $r_t = \xi_t \cdot r_c$ (King profile). Passing a King profile through apply_tidal_truncation is therefore redundant if $r_t = r_J$, which is the equilibrium configuration King implicitly assumes. For non-equilibrium configurations — King-like clusters that are not in tidal equilibrium with their current galactocentric position — apply_tidal_truncation adds the secondary truncation explicitly.

For Plummer and EFF (which extend to infinity), tidal truncation must be applied externally; it is not built in.

Domain of validity¶

1. Point-mass cluster. (1) treats the cluster as a point mass for the host’s tidal field. Corrections for the cluster’s extended mass distribution are $\mathcal{O}((r_J/R)^2)$ and become important only for very large clusters.

2. Static host potential. Galactic disc/halo potential is assumed constant during the cluster’s orbit. For times longer than the galaxy’s secular evolution timescale ($\sim 1$–10 Gyr), this assumption breaks down.

3. Single component. (1) uses a single $M_{\mathrm{gal}}(<R)$. For galaxies with multi-component potentials (disc + halo + bulge), compute the total enclosed mass yourself and pass it to jacobi_radius (which takes a scalar M_galaxy, not a callable).

4. No cluster-cluster interactions. Two clusters in orbit around each other (e.g. mutually-bound binary clusters) require a different treatment beyond the scope of this utility.

Implementation, validation & references¶

• In code: src/progenax/tidal.py (jacobi_radius, jacobi_radius_isothermal, apply_tidal_truncation, fill_factor_to_r_h). See the tidal API.

• Validated in: tidal truncation — the regression suite for $r_J$ and the shape-preserving truncation.

• Primary sources: the Jacobi-radius derivation is standard textbook material (Murray & Dermott Solar System Dynamics §3; Binney & Tremaine 2008 §8.3); the point-mass factor-3 form traces to King (1962) and the N-body-calibrated isothermal form to Baumgardt & Makino (2003). The isothermal-halo approximation and fill-factor comparison sample follow King (1966) and Küpper et al. (2011). Full notes in the bibliography.