Estimate of the total kinetic power and age of extragalactic jet by its cocoon dynamics: The case of Cygnus A

We examine the constraints imposed on the total kinetic power ($L_{\rm j}$) and the age ($t_{\rm age}$) of relativistic jets in FR II radio sources in a new way. We solve the dynamical expansion of its cocoon embedded in the intra-cluster medium (ICM) and obtain the analytic solution of its physical quantities in terms of $L_{\rm j}$ and $t_{\rm age}$. The estimate of $L_{\rm j}$ and $t_{\rm age}$ is done by the comparison of the model and the observed shape of the cocoon. The analysis is focused on Cygnus A and we find that (i) the source age is $3 {\rm Myr} <t_{\rm age}<30 {\rm Myr}$ and (ii) the total kinetic power of the jet is estimated as $0.2\times 10^{46}{\rm erg s^{-1}} <L_{\rm j}<1\times 10^{48} {\rm erg s^{-1}}$ which is larger than 1% of the Eddington luminosity of Cygnus A.