= Weathering feedback with fast thermal adjustment
{title2=$\epsilon\dot\theta=-\theta+\Gamma(P-1),\quad\dot P=v-P^\mu e^\theta$}
In this idealized greenhouse-<weathering> model, <temperature> adjusts rapidly when $\epsilon\ll1$, giving $\theta\simeq\Gamma(P-1)$. The remaining carbon balance has <equilibrium> $\mu\log P_*+\Gamma(P_*-1)=\log v$. For positive $\mu,\Gamma$, the left side has <derivative> $\mu/P+\Gamma>0$ and ranges from minus infinity to infinity, so there is exactly one positive <equilibrium> for every sustained positive source $v$.
<Linearization> of the reduced carbon equation gives <dimensionless> decay rate $v(\Gamma+\mu/P_*)$, hence dimensional relaxation time $t_C/[v(\Gamma+\mu/P_*)]$. Its <negative feedback> is increased <weathering> when either <temperature> or carbon <pressure> rises. The two-variable <equilibrium> is also linearly stable: its Jacobian has negative trace and determinant $v(\mu/P_*+\Gamma)/\epsilon>0$. A fast radiative response therefore does not imply a fast carbon-cycle response. Linearized radiative coefficients must be calibrated at a consistent reference state before numerical use.
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