In addition to the normal tidal forces which act to stretch an orbitting body along the radial action of gravity, tidal shear results when an object (like a globular cluster) differentially orbits around a massive body (that is, the near edge is moving in orbit more quickly than the far edge). The degree of shear depends on the relation between the differential rotation and the self-gravity of the system. In practice, we are in a rotating frame (our sun) which is also rotating, so we either need to find an inertial frame of reference or account for the Coriolis forces.
In a rotating frame, the force per unit mass is:
where R is the distance from the galactic center.
where , where A and B are the Oort constants. We may then write the condition for stability as:
where is the distance across the orbitting body. Futhermore, the acceleration from self-gravity on the cluster is:
Thus, the condition for stability is:
If a cluster were hypothetically in the solar neighborhood, this would come out to be:
In fact, as has been stated many times, the density in the solar neighborhood is , so such objects in our neighborhood would be tidally stable.
For another case, suppose we are in an inertial frame (we are the galaxy). In this case, we have no Coriolis term:
where is the circular angular velocity of our galaxy around the center of mass. Using , we have the condition for stability is:
If we assume we have a flat rotation curve, then , so we have
For a dwarf galaxy at 50 kpc, this comes to:
We can estimate the density of a dwarf galaxy: it has stars and has a diameter of about , so we have , so this galaxy would be on the border of stable. Anything closer (like the Sagittarius Dwarf) will get totally ruined. By the way, the “235” in the above equations is in units of km, s, and pc’s.