 Solution

For logarithmic forcing $F=a\ln (C_a/C_{a0})$, where $C_a$ denotes atmospheric carbon-dioxide concentration rather than heat capacity, $\dot{F}=(a/C_a){\dot{C}}_a$. Approximately constant $\alpha$ in $\dot{F}=\alpha E$ therefore requires a roughly constant conversion from emissions to persistent atmospheric burden and a concentration range small enough to hold $a/C_a$ fixed. Carbon removal must be negligible on the modeled horizon, or be replaced by a fixed retained fraction; a realistic multiscale carbon-cycle response generally violates the permanent-step approximation.

A one-tonne pulse at time $t_e$ is $E(t)=\delta (t-t_e)$ in tonne-normalized units. It gives a persistent forcing step $F=\alpha \Theta (t-t_e)$. Convolving that step with the thermal response yieldsThere is no instantaneous temperature jump for finite $C$, although forcing jumps. Both persistence and constant coefficients are deliberate simplifications, rather than an assertion that all emitted carbon dioxide remains airborne forever.