detuning $\Delta_\text{s}$. The results of this fit are: FSR $= 8.7673410 \text{ GHz} \pm 5 \text{ kHz}$ (Note the statistical fit error was 460 Hz. The quoted error reflects the precision of a frequency measurement used to generate 9 GHz error signal.), $\kappa_{\text{ext}}/\kappa = 0.243 \pm 0.003$, $\omega_\text{m}/2\pi = 261150.3 \pm 0.9 $ Hz, and $\Delta_\text{s}/2\pi = -880 \pm 250$ Hz. The precise value of the FSR adds an overall offset to $\Delta_\text{p}$ (i.e. a horizontal shift in Fig. \[fig2\]b). The estimate of FSR from this fit is significantly more precise than our independent estimate of 8.84 GHz based on cavity length. The ratio $\kappa_{\text{ext}}/\kappa$ simultaneously scales the optical spring and damping strength. This can be independently estimated as $\kappa_{\text{ext}}/\kappa = 0.2$ from measuring the cavity ringdown and the fraction of the incident light lost in the cavity with the laser tuned on resonance (64 % in this case). This estimate is lower than the fit value by 20%, which we attribute to imperfect cavity mode matching and that the membrane position varies by $\sim 10$ nm during measurements, which can affect the cavity finesse [@Sankey2010Strong]. We allow the bare mechanical frequency to float because we find that it can 