Analytic Modeling of Hohlraum Drive for High Performing HDC capsules.*

We present here a fully analytic derivation for the temperature, T(t), drive history of a typical hohlraum (with DU walls) containing a high performing High-Density-Carbon (HDC) capsule at the National Ignition Facility (NIF).  A method to model the laser's complex temporal pulse shape is presented, which uses power laws, in time,  for T(t) that differ in two temporal regimes, the foot of the pulse and the main power pulse.  For each power law we use our published theory [Hammer & Rosen PoP 10, 1829 (2003)]. We match both the temperature T, as well as the absorbed diffusive ("Marshak wave") flux into the walls at the transition point between those two temporal regimes. Other complications, that are dealt with, are the dynamic motion of both the capsule and of the Laser Entrance Hole (LEH), and the time dependence of the conversion efficiency of turning laser power into x-ray drive.  Good agreement with complex simulations [e.g. Callahan et al PoP 27, 072704 (2020)] and with data ensues.