Secular phase integral for proportional mass growth ($\Phi (t)={\int }_0^{t}g(t\prime )dt\prime$)

If both precession and forcing in $\dot{z}=ig(t)(z-z_f)$ share a common mass factor while $z_f$ stays fixed, then $z=z_f+(z(0)-z_f)\exp (i{\int }_0^{t}g(t\prime )dt\prime )$. The Argand diagram circle is unchanged; only its traversal speed varies. With $g(t)=g_{f}t/t_p$, the phase is $g_f{t}^2/(2t_p)$.