Joint PDF Model

Note

See Kim et al. (2020b) for details.

Cool outflow model

The cool outflow (\(T<2\times10^4{\rm K}\)) in the TIGRESS suite is well described by a model combining log-normal and generalized gamma distribution:

\[\tilde{f}_{M}^{\rm cool}(u,w) = A_c \left(\frac{v_{\rm out}}{v_{\rm out,0}}\right)^2 \exp\left[-\left(\frac{v_{\rm out}}{v_{\rm out,0}}\right)\right] \exp\left[-\frac{1}{2}\left(\frac{\ln(c_{\rm s}/c_{\rm s,0})}{\sigma}\right)^2\right]\]

where \(A_c=(\ln 10)^2/(2\pi\sigma^2)^{1/2}=2.12/\sigma\).

\[\frac{v_{\rm out,0}}{{\rm km/s}} = v0 \left(\frac{\Sigma_{\rm SFR}}{M_{\odot}{\rm kpc^{-2}yr^{-1}}}\right)^{0.23}+3\]

At \(|z|=H\), we adopt \(v0=25\), \(c_{s,0}=6.7{\rm km/s}\), and \(\sigma=0.1\). We found the same function form with \((v0,c_{s,0})=(45,7.5)\), (45,8.5), and (60,10) works reasonably well at \(|z|=2H\), 500pc, and 1kpc.

Hot outflow model

The hot outflow (\(T>5\times10^5{\rm K}\)) in the TIGRESS suite is well described by a model combining two generalized gamma distributions:

\[\tilde{f}_{M}^{\rm hot}(u,w) = A_h \left(\frac{v_{\mathcal{B},z}}{v_{\mathcal{B},0}}\right)^2 \exp\left[-\left(\frac{v_{\mathcal{B},z}}{v_{\mathcal{B},0}}\right)^4\right] \left(\frac{\mathcal{M}}{\mathcal{M}_0}\right)^3 \exp\left[-\left(\frac{\mathcal{M}}{\mathcal{M}_0}\right)\right]\]

where \(A_h\equiv 2(\ln 10)^2/\Gamma(1/2)=5.98\), \(v_{\mathcal{B},z}\equiv(v_{\rm out}^2+c_s^2)^{1/2}\), and \(\mathcal{M}=v_{\rm out}/c_s\).

\[\frac{v_{\mathcal{B},0}}{10^3{\rm km/s}} = 2.4 \left(\frac{\Sigma_{\rm SFR,0}^{1/2}}{2+\Sigma_{\rm SFR,0}^{1/2}}\right)+0.8\]

where \(\Sigma_{\rm SFR,0}\equiv \Sigma_{\rm SFR}/(M_{\odot}{\rm kpc^{-2}yr^{-1}})\)

We adopt \(\mathcal{M}_0=0.5\) irrespective of \(z\).