# Jacobi vs tidal radius for star cluster

The tidal radius is defined in King (1962) as:

the value of r (the radius) at which f (the density profile) reaches zero...

This also referred by King as the "limiting radius" written as (Eq 3):

$$r_t=R_{GC}\left(\frac{M}{2M_g}\right)^{1/3}$$

where $$R_{GC}$$ is the galactocentric distance, $$M$$ is the cluster's mass, and $$M_{g}$$ is the galaxy's mass.

This can also be written as Pinfield et al. (1998) Eq (12):

$$r_t = \left( \frac{GM}{2(A-B)^2} \right)^{1/3}$$

where $$A$$ and $$B$$ are the Oort constants and $$M$$ is the total mass.

Now, I've seen the "tidal radius" defined as:

$$r_t=R_{GC}\left(\frac{M}{3M_g}\right)^{1/3}$$

in Chernoff & Weinberg (1990) Eq 2. Notice the 3 in the denominator instead of the 2 as above. According to the authors:

The tidal radius depends weakly upon the form of the mass distribution of the halo. The Eq. above is correct for the tidal stress of a Keplerian force field. On the other hand, for constant $$v_g$$, the factor $$3M_g$$ becomes $$2M_g$$.

This same quantity (using $$3M_g$$) is referred to as the "Jacobi radius" in Piatti (2015) Eq 4.

I've also seen the Jacobi radius written as:

$$r_J=\left(\frac{GM}{4\Omega^2-k^2}\right)^{1/3}$$

(where $$G, M, \Omega$$, and $$k$$ are the gravitational constant, the cluster mass, the circular, and the epicyclic frequencies of a near circular orbit, respectively) in Ernst et al. (2011) Eq (8).

According to Wikipedia, both the tidal and the Jacobi radius are the same thing.

My question: are these two radius the same quantity?

Here, $$r_J$$ is defined as:
$$r_J= R_0\left(\frac{m}{3M}\right)^{1/3}$$ The Jacobi radius $$r_J$$ (also, Roche or Hill radius) of an orbiting stellar system is expected to correspond to the observational tidal radius, the maximum extent of the satellite system. However this correspondence is only approximate.