Galaxies Lecture 02

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Timescale for collisional (star-star) relaxation

Suppose Star A passes Star B with impact parameter . This will deflect the trajectory of Star A in a direction to the original trajectory. We’ll say that a trajectory is significantly altered when . Integrating over all possible , and integrating over all pairs of stars, we will get an estimate of the timescale for star-star interactions in a galaxy.

Using , we have:

then using , and , we have:

Substituting ,

This breaks down when , giving us a minimum interaction distance

For , and , is of order 1 AU.\

Note, by the way, that:

Anyway, we wanted to integrate over all stars. We expect that if we throw a star through a galaxy that its net deflection because stars are probably distributed symmetrically. In order to get a real measure for the interactions going on in a galaxy, we want to calculate :

And we define . Using the definition of , we have:

And since , we have

and represents the condition for a galaxy to have lost its “memory” of its initial conditions. Using and , we have

The time for a star to cross a galaxy is , and the number of crossings required to relax is . Thus, the relaxation time is

To evaluate whether various galaxies have relaxed, we’ll make a table:


v (km/s)

R (pc)

t cross (yrs)

t relax (yrs)


Old Open Cluster: Pleideas (50)





Globular Cluster ()



Milky Way ()


Dwarf Galaxy ()


Galaxy Cluster ()



The takeaway point here is that stars in large galaxies still remember their original trajectories.

An important number to remember is . If a galaxy is relaxed, we may expect a thermalized velocity distribution of stars, but depending on the geometry of a galaxy, this may only apply within velocities along a particular axis.

When galaxies are ripped apart, streams of stars can be torn off into moving groups which can be identified by their common velocities.