# 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?

In the book Galactic Dynamics by Binney and Tremaine (second edition) there is a whole section explaining the difference between the Jacobi radius and the tidal radius (page 677-chapter 8).

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.

Five reasons are subsequently given to explain why the correspondence is only approximate.

• If they had one good reasons, they wouldn't need five ;-)
– uhoh
Jan 9 '19 at 13:13

In the paper "A million binaries from Gaia eDR3: sample selection and validation of Gaia parallax uncertainties" El-Badry et al (2021) the Jacobi radius, in the context of orbiting binary White Dwarf's, is defined as the separation between two orbiting binaries beyond which the Galactic tidal field dominates a binary’s internal acceleration. The relation given for the Jacobi radius in the solar neighbourhood is

$$rJ = 1.35 \textrm{ pc} \times \frac{M_{tot}}{M}$$

$$M_{tot}$$ is the mass of the binary system and $$M$$ is the mass of the sun. It's a bit strange that the Jacobi radius is defined in terms of the sun but I think it is just an approximation of the effective range in which two objects are gravitationally bound in an orbit. The paper references (Jiang & Tremaine 2010) for the relationship.

thanx Gabriel for adding the detailed definition to my reply.. an additional definition of the tidal radius hereafter in this definition it describes molecular clouds rather than stellar systems..again notice the difference (2 is in the numerator).. there is no reference on where he got this formalism from..this is from Tan 2000 eq (9) check also eq (8)(http://adsabs.harvard.edu/abs/2000ApJ...536..173T