Stagnation pressure of imploding shells and ignition energy scaling of inertial confinement fusion targets

The stagnation pressure p(s) of imploding cylindrical (n = 2) and spherical (n = 3) shells is found to scale as p(s)/p(0) (proportional to) M-0(2(n+1)/(gamma +1)), where M-0 is the Mach number of the imploding shell and p(0) its maximum pressure. The result holds approximately for Mach numbers in the range 2 < M-0 < 25 relevant for inertial confinement fusion capsules and is of key importance for their ignition energy scaling. It is derived analytically on the basis of similarity solutions for an ideal gas with adiabatic exponent gamma.