 Cloud albedo susceptibility

For $\tau \propto N_d^{1/3}$ at fixed liquid-water content and thickness, and $A=\tau /(\tau +7)$, the derivative is $dA/d{N}_d=A(1-A)/(3N_d)$. For a fixed fractional number increase, the initial response is greatest at $A=1/2$. The exact finite-change expression is $A^{\prime }=sA/(1-A+sA)$ with $s=(N_d^{\prime }/N_d{)}^{1/3}$. Uncertain activation of additional particles and changes in water content limit direct application to real clouds.