King (1962)

Abstract (paraphrased)¶

Re-examines the density distribution in globular clusters from new Mount Wilson / Palomar star counts. Jeans’ $r^{-4}$ outer law is shown to rest on unjustified assumptions and to be poorly supported by the data; instead the surface density drops to zero at a finite tidal radius, as expected from galactic tidal forces. King proposes a three-parameter empirical density law that fits the counts from centre to edge in clusters of all concentrations, and argues that the same law describes both globular clusters and Sculptor-type dwarf galaxies. This is the first of the King-model papers; the self-consistent lowered-isothermal dynamical model appears in King (1966) (Paper III).

The empirical density law (Eq. 2, verified)¶

For the outer parts, King fits

a surface density that reaches zero at the tidal radius $r_t$ (the intercept of $\sqrt{f}$ vs $1/r$). This finite cutoff — absent from a pure power law — is the empirical signature of galactic tidal truncation that the later King (1966) dynamical models reproduce self-consistently.

The tidal / limiting radius from galactic tides (§III, verified)¶

King estimates the tidal cutoff $r_t$ from galactic tidal forces. Following von Hoerner (1957), the instantaneous limiting radius of a cluster on a straight-line (radial) orbit is

where $R$ is the galactocentric distance and $M$, $M_g$ are the cluster and galaxy masses (Eq. 3). King explicitly notes a typographical error in von Hoerner’s Eq. 33 — a 2 printed in the numerator instead of the denominator.

For the more realistic case of a cluster on an elliptical orbit, the cluster is truncated most severely at perigalacticon $R_p = a(1-e)$, and the limiting radius there is (Eq. 11)

For the elongated orbits typical of globular clusters, King adopts the compromise formula (Eq. 12)

The classical Hill/Roche factor-3 form $r_J = R\,(M_{\rm cl}/3 M_{\rm gal})^{1/3}$ used in progenax is the circular-orbit ($e = 0$) limit of Eq. 11 for a point-mass host. Because of the $1/3$ power, none of the masses or the orbital eccentricity needs to be known accurately — King stresses that this weak dependence is “especially fortunate” given the crudeness of the inverse-square galactic-force approximation.

Use in progenax¶

• Tidal physics — the Jacobi / tidal radius. The point-mass factor-3 form $r_J = R\,(M_{\rm cl}/3 M_{\rm gal})^{1/3}$ is the $e=0$ case of King’s Eq. 11; the flat-rotation-curve factor-2 form follows from including the host’s mass-profile slope.

• progenax.tidal.jacobi_radius — the factor-3 (point-mass) tidal radius.

• Tidal truncation validation — validation of the tidal-truncation behaviour.

Notes¶

This is the empirical density-law paper. The self-consistent lowered-isothermal King model — the distribution function progenax actually integrates for King profiles and velocity DFs — is King (1966), Paper III; see King (1966). progenax cites King (1962) specifically for the tidal-radius estimate from galactic tides, not for the dynamical model.