Figure 1: Properties of a periodically intermittent source (see Section 2.2
of main text for source parameters). Panel (a) shows the velocity of the
shock in the ambient medium (top) and the radio luminosity of the cocoon
(bottom). Panel (b) shows the evolution of the cocoon radius (bottom
curve) and shock radius (top curve).
We now apply the above model to the case of a periodically bursting source.
For concreteness, we shall consider an initially small source which
undergoes 10,000yr long bursts that recur every 100,000yr. We take the
jet power to be 
during the bursts and negligibly
small at other times. The cocoon is assumed to be dominated by
relativistic material (i.e., 
) and we take a non-relativistic
equation of state for the shocked ISM/ICM shell (i.e., 
).
Finally, the following physically reasonable parameters are taken to
characterize the density profile of the ambient medium: 
, 
and 
. The qualitative behaviour
described below is insensitive to reasonable departures from these
canonical parameter values.
Figure 1 shows the results of a numerical integration of eqns. (1)-(2).
During the initial burst of activity (t<10,000yr) the source expands
self-similarly, i.e., the cocoon radius is a constant multiple of the shock
radius. This is the evolutionary phase that has been previously examined
by Falle (1991), B96, and Kaiser & Alexander (1997). As found in previous
works, 
. By assumption, the power source
switches off at t=10,000yr and the cocoon/shocked-shell system enters a
`coasting' phase akin to Taylor-Sedov expansion. It must be stressed that
the expansion during this coasting phase is still pressure-driven, as
opposed to being driven by the momentum of the shocked shell. Despite the
increased deceleration of the shocked shell, it still remains highly
supersonic for the entire period of this coasting phase (i.e., until
t=100,000yr). However, due to the drop in source pressure, the radio
luminosity of the cocoon will fall rapidly once the power source has turned
off. Thus, even though the basic source structure remains intact, a
coasting source will be significantly fainter than an active source. It is
also interesting to note that the source evolution is no longer
self-similar in the coasting phase.
The onset of the second burst of activity induces a rapid increase in
pressure leading to a subsequent increase in both the source expansion rate
and cocoon radio luminosity. Note that during the (brief) period in which
the contact discontinuity is accelerating (i.e., 
), this
interface will be subject to Rayleigh-Taylor instabilities that will tend
to mix material from the shocked shell with the relativistic cocoon
material. This mixing will produce time-dependence in the effective value
of 
, thereby affecting the detailed evolution of the cocoon. The
implications of this mixing will be addressed in future work.
After many bursts (or, more precisely, when the recurrence timescale is
short compared with the expansion timescale), the intermittent nature of
the source will be unimportant in determining its evolution. It will
behave as a constantly fed source with jet power 
, where f is the
fraction of time that the source is on with power 
. The
re-establishment of an approximately self-similar expansion in our
numerical experiment reflects this fact.