Tidal radius from the outskirts (B7)

A cluster’s tidal (Jacobi) radius $r_t$ marks where the Galaxy strips stars, and it is written in the outskirts — the sparse, count-limited edge of the density profile. This stretch demo reuses the Poisson number-density channel (B11) to recover $r_t$ from those few outer stars, and turns it into the cluster’s Galactocentric distance.

The physics¶

The Elson–Fall–Freeman (1987) Elson et al., 1987 profile $\rho(r) = (1 + r^2/a^2)^{-\gamma/2}$ is sharply truncated at $r_t$ . The scale radius $a$ is pinned by the inner profile (where almost all the stars are), so it is held fixed; $r_t$ is read from the outer counts. Bins are placed out past $r_t$ so the outermost are empty: a too-large $r_t$ predicts stars in those empty bins (penalized by the Poisson $-\mu$ term), a too-small $r_t$ cannot explain the observed outer stars. The Poisson channel is honest there; a Gaussian-on-log-count would be ill-defined. (A steep halo self-truncates — its density at $r_t$ is already $\sim 0$ , so $r_t$ sits in the noise; this uses a shallow, YMC-typical $\gamma=2.5$ so stars reach $r_t$ .)

Inputs and assumptions¶

The fit recovers one parameter, the truncation/tidal radius $r_t$ ; the scale radius, the profile family, and both masses in the Jacobi conversion are assumed known. The mass assumptions matter most — they, not the count statistics, would dominate a real $R_{\rm gal}$ error budget.

Table 1:Model inputs

Input	Meaning and role	Status (fiducial)
$r_t$	EFF truncation = Jacobi/tidal radius — the science target, read from the count-limited outer bins and converted to $R_{\rm gal}$ .	recovered (12.0 pc)
$a$	EFF scale radius; pinned by the inner profile and held fixed (jointly fitting $(a,r_t)$ from counts is degenerate).	known / fixed (1.0 pc)
$\gamma$	EFF outer slope; shallow enough that stars reach $r_t$ (a steep $\gamma$ self-truncates and hides $r_t$ ).	known / fixed (2.5)
$M_{\rm gal}$	Interior Galaxy (point-mass) for the Jacobi $\to R_{\rm gal}$ conversion.	known / fixed ( $5\times10^{10}\,M_\odot$ )
$M_{\rm cl}$	Cluster mass entering $R_{\rm gal}=r_t(3M_{\rm gal}/M_{\rm cl})^{1/3}$ — summed from the IMF draw, treated as exactly known.	known / fixed ( $\sim10^4\,M_\odot$ , derived)
IMF	Maschberger ( $\alpha=2.3$ , 0.08– $100\,M_\odot$ ) — fixes $M_{\rm cl}$ ; the mass spectrum does not enter the count fit (positions only).	known / fixed
$N$ , bins, $r_{\rm hi}$ factor, boxes, MLE	$2\times10^4$ stars; 22 geometric bins from $r_{\rm lo}=0.2$ pc to $1.3\,r_t$ (so outer bins are empty — the Poisson $-\mu$ term is what pins $r_t$ ); 3000-pt cumulative-EFF grid; $r_t\in(6,18)$ pc; 3 Adam starts.	numerical choices

Important

The scale radius $a$ is held fixed, and the constraint lives in the empty outer bins. Jointly fitting $(a,r_t)$ from number counts alone is degenerate — the normalized profile shape trades them off — so $a$ is asserted to be pinned by the star-rich inner profile and $r_t$ is read purely from the outskirts (bins extending to $1.3\,r_t$ , mostly empty, supply $\sim$ 93% of the Fisher information via the Poisson $-\mu$ penalty). The conversion to Galactocentric distance then assumes $M_{\rm gal}$ and $M_{\rm cl}$ are error-free, so $\sigma(R_{\rm gal})/R_{\rm gal}=\sigma(r_t)/r_t$ inherits only the counting uncertainty — in reality a cluster-mass error (M/L, unseen low-mass stars, remnants) would dominate.

Result — freshly run, ALL PASS¶

Measured 2026-06-12 ( $N=2\times10^4$ , $\gamma=2.5$ , true $r_t=12$ pc; wall $\approx 5$ s).

quantity	value
$r_t$	$12.01 \pm 0.04$ pc (truth 12.0, pull +0.24)
$r_t$ Fisher info from the outer bins ( $r>0.6\,r_t$ )	93% — the count-limited outskirts pin $r_t$
forecast	$\sigma(r_t)\propto N^{-1/2}$

The per-bin Fisher information spikes at the truncation edge (panel b): $r_t$ is constrained almost entirely by the handful of outermost bins, which is precisely why the Poisson treatment of those low-count bins matters.

From $r_t$ to the Galactocentric distance¶

The recovered $r_t$ is the Jacobi/tidal radius King, 1962,

r_t = R_{\rm gal}\left(\frac{M_{\rm cl}}{3 M_{\rm gal}}\right)^{1/3} \;\Longrightarrow\; R_{\rm gal} = r_t\left(\frac{3 M_{\rm gal}}{M_{\rm cl}}\right)^{1/3},

(1)

with $\sigma(R_{\rm gal})/R_{\rm gal} = \sigma(r_t)/r_t$ . For the measured $M_{\rm cl}\approx1.2\times10^4\,\Msun$ and a representative interior $M_{\rm gal}=5\times10^{10}\,\Msun$ , the recovered $r_t$ gives

R_{\rm gal} = 2.79 \pm 0.01\ \mathrm{kpc},

(2)

(round-trips through jacobi_radius). So a cluster’s orbit is encoded in its faint outer star counts.

Figure¶

$Tidal radius from the outskirts (scripts/demo_tidal_radius.py, ALL PASS). (a) The EFF number-density profile (counts, \sqrt N errors) with the MLE fit and the recovered truncation \hat r_t. (b) The per-bin Fisher information for r_t on a log-r axis: it spikes at the truncation edge — 93% of the constraint comes from the count-limited outskirts. (c) Forecast \sigma(r_t)\propto N^{-1/2}, annotated with the derived Galactocentric distance.$

Figure 1:Tidal radius from the outskirts (scripts/demo_tidal_radius.py, ALL PASS). (a) The EFF number-density profile (counts, $\sqrt N$ errors) with the MLE fit and the recovered truncation $\hat r_t$ . (b) The per-bin Fisher information for $r_t$ on a log- $r$ axis: it spikes at the truncation edge — 93% of the constraint comes from the count-limited outskirts. (c) Forecast $\sigma(r_t)\propto N^{-1/2}$ , annotated with the derived Galactocentric distance.

Caveats¶

Warning

$a$ is fixed, $r_t$ recovered. The scale radius is pinned by the inner profile and held; jointly fitting $(a, r_t)$ from counts alone is degenerate (the normalized shape trades them off). The headline is the truncation.
Sharp truncation, complete census. The EFF cut is hard and every star is counted; real outskirts have foreground/membership contamination and a soft, orbit-phase-dependent edge.
Point-mass Galaxy for the Jacobi step. $M_{\rm gal}$ is a fixed representative interior mass; a realistic mass profile and an eccentric orbit shift $R_{\rm gal}$ . The point is the method — outer counts → $r_t$ → orbit.
$M_{\rm cl}$ assumed exactly known. Only $\sigma(r_t)$ propagates into $\sigma(R_{\rm gal})$ ; a real cluster-mass uncertainty would dominate it.
$\gamma$ and the EFF family are fixed at truth. A wrong outer slope or a non-EFF profile would bias $r_t$ (only a shallow enough $\gamma$ lets stars reach $r_t$ ); profile-family misspecification is not explored here (contrast B6).
3-D radii, complete outskirts. True 3-D radii are binned (no projection to surface density), and the result leans heavily on a few near-empty outer bins — robustness to a handful of interlopers there is not quantified.

How to run¶

env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_tidal_radius.py

References¶

The EFF profile is Elson et al. (1987); the tidal/Jacobi radius is King (1962). The Jacobi-radius derivation and progenax’s apply_tidal_truncation are on the tidal & substructure page; the Poisson count channel is shared with B11.

References¶

Elson, R. A. W., Fall, S. M., & Freeman, K. C. (1987). The structure of young star clusters in the Large Magellanic Cloud. The Astrophysical Journal, 323, 54–78. 10.1086/165807
King, I. R. (1962). The structure of star clusters. I. An empirical density law. The Astronomical Journal, 67, 471. 10.1086/108756

The physics¶

Inputs and assumptions¶

Result — freshly run, ALL PASS¶

From rtr_trt​ to the Galactocentric distance¶

Figure¶

Caveats¶

How to run¶

References¶

From $r_t$ to the Galactocentric distance¶