  Deriving the energetics of remnant and restarted active galactic nuclei
(AGNs) is much more challenging than for active sources due to the complexity
in accurately determining the time since the nucleus switched-off. I resolve
this problem using a new approach that combines spectral ageing and dynamical
models to tightly constrain the energetics and duty-cycles of dying sources.
Fitting the shape of the integrated radio spectrum yields the fraction of the
source age the nucleus is active; this, in addition to the flux density, source
size, axis ratio, and properties of the host environment, provides a constraint
on dynamical models describing the remnant radio source. This technique is used
to derive the intrinsic properties of the well-studied remnant radio source B2
0924+30. This object is found to spend $50^{+14}_{-12}$ Myr in the active phase
and a further $28^{+6}_{-5}$ Myr in the quiescent phase, have a jet kinetic
power of $3.6^{+3.0}_{-1.7}\times 10^{37}$ W, and a lobe magnetic field
strength below equipartition at the $8\sigma$ level. The integrated spectra of
restarted and intermittent radio sources is found to yield a 'steep-shallow'
shape when the previous outburst occurred within 100 Myr. The duty-cycle of B2
0924+30 is hence constrained to be $\delta < 0.15$ by fitting the shortest time
to the previous comparable outburst that does not appreciably modify the
remnant spectrum. The time-averaged feedback energy imparted by AGNs into their
host galaxy environments can in this manner be quantified.
