is smaller than the Bondi accretion radius $R_{\rm
acc}=Gm_{12}/c_s^2$ ($c_s$ being the sound speed of the gas medium), the gas density close to the binary will be much higher than the background density due to accretion. This is normally the case for those BBHs embedded in common envelopes and AGN accretion disks. If we use $\tilde{n}\sim
n\,(R_{\rm acc}/a)^{3/2}$ to correct the gas density around the binary [@bondi52; @antoni19], we find that the timescale due to hydrodynamical drag becomes $$\begin{aligned}
    {T}_{\rm gas}\simeq&8.5\times10^{4}q^{-1}(1+q)^{-3}
\left(\frac{n}{10^{11}\,{\rm cm^{-3}}}\right)^{-1}\nonumber\\
&\times
\left(\frac{m_1}{10\,M_\odot}\right)^{-1}
    \left(\frac{c_s}{10^2\,{\rm km\,s^{-1}}}\right)^{3}\,{\rm years}.\label{eq:Tgas}\end{aligned}$$ Note that the new timescale does not depend on $a$ or $f$. A similar result can be found in [@bartos17 see their Fig. 2]. Moreover, in the last equation we have rescaled $n$ with $10^{11}\,{\rm cm^{-3}}$. Given this timescale, we derive that $$\begin{aligned}
    {\Gamma}&\simeq1.1\,\left(\frac{1+q}{2}\right)^{10/3}
\left(\frac{n}{10^{11}\,{\rm cm^{-3}}}\right)\nonumber\\
&\times\left(\frac{m_1}{10\,M_\odot}\right)^{-2/3}
\left(\frac{c_s}{10^2\,{\rm km\,s^{-1}}}\right)^{-3}
    \left(\frac{f}{3\,{\rm mHz}}\right)^{-8/3}.\label{eq:Gamm2}\end{aligned}$$ For illustrative purposes, we show in Figure \[fig:Gamm\] the dependence of the ${\Gamma}$ computed in the last equation on $m_1$ and $n$. The black dashed curve marks the location where ${\Gamma}=1$. Above it, gas dominates the dynamical evolution of a BBH, and hence the chirp signal is determined by gas dynamics, not GW radiation.

Equations (\[eq:Gamm\]) and (\[eq:Gamm2\]) suggest that for 