Energy-driven outflow

An outflow is energy driven when cooling of the shocked wind is slow. Its hot bubble stores mechanical energy and expands by pressure work on the swept-up shell.

A quantitative momentum boost needs an energy-transfer convention as well as a momentum input. For one-speed outflow diagnostics, ${\dot{p}}_s={\dot{M}}_s{v}_s$ and ${\dot{E}}_s={\dot{M}}_s{v}_s^2/2$. A driving wind similarly has ${\dot{E}}_w={\dot{p}}_w{v}_w/2$. If a fraction $f_E={\dot{E}}_s/{\dot{E}}_w$ appears in the shell's inferred kinetic energy flux, thenThus the estimate below assumes a wind momentum input of order $L/c$ and an order-one shell energy fraction, with $f_E\sim 1/2$ giving its stated factor. Large momentum transfer is compatible with energy conservation because the swept shell moves much more slowly than the driving wind. Cooling, thermal energy retained inside the bubble and the observational definition of the shell rates change the coefficient.

A momentum boost is a shell momentum flux exceeding the direct input momentum flux. An energy-conserving wind of speed $v_w$ driving a shell of speed $v_s\ll v_w$ can give ${\dot{p}}_s/(L/c)$ of order $v_w/(2v_s)$.