= 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 yields
$$
\boxed{T(t)-T_0=\frac\alpha\lambda\left[1-e^{-k_T(t-t_e)}\right]\Theta(t-t_e).}
$$
There 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.
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