Global Star Formation

Figure 3.1: Taken from Figs.6 and 7 of Kennicutt (1998). Left: The $x$-axis means the total (HI+H$_2$) gas density and the $y$-axis does the global star formation rate. Right: The $x$-axis means the total (HI+H$_2$) gas density divided by the orbital time-scale. The $y$-axis is the same.

The star formation rate is estimated by the intensity of H$\alpha$ emission (Kennicutt, Tamblyn, & Congdon 1994) as

$${\rm SFR}(M_\odot {\rm yr}^{-1})=\frac{L({\rm H}\alpha)}{1.26\times 10^{41}{\rm erg\ s^{-1}}},$$ (3.2)

which is used for normal galaxies. While in the starburst galaxies which show much larger star formation rate than the normal galaxies, FIR luminosity seems a better indicator of star formation rate

$${\rm SFR}(M_\odot {\rm yr}^{-1})=\frac{L({\rm FIR})}{2.2\ti... ...rg\ s^{-1}}} =\frac{L({\rm FIR})}{5.8\times 10^{9}L_\odot}.$$ (3.3)

Kennicutt (1998) summarized the relation between SFRs and the surface gas densities [Fig.3.1 (left)] for 61 normal spiral and 36 infrared-selected starburst galaxies. As seen in Figure 3.1, the star formation rate averaged over a galaxy ( $\Sigma_{\rm SFR} (M_\odot\ {\rm yr}^{-1}\, {\rm kpc}^{-2})$), which is called the global star formation rate, is well correlated to the average gas surface density $\Sigma_{\rm gas}(M_\odot\, {\rm pc}^{-2})$. He gave the power of the global Schmidt law as $n = 1.4 \pm 0.15$. That is,

$$\Sigma_{\rm SFR}\simeq (1.5\pm 0.7)\times 10^{-4}\left(\fra... ...^{1.4\pm 0.15} M_\odot\ {\rm yr}^{-1}\, {\rm k\,pc}^{-2}.$$ (3.4)

The fact that the power is not far from 3/2 seems to be explained as follows: Star formation rate should be proportional to the gas density ( $\Sigma _{\rm gas}$) but it should also be inversely proportional to the time scale of forming stars in respective gas clouds, which is essentially the free-fall time scale. Remember the fact that the free-fall time given in equation (2.26) is proportional to $\tau_{\rm ff}\propto 1/(G\rho)^{1/2}$. Therefore

$$\rho_{\rm SFR}\propto \rho_{\rm gas}\times (G\rho_{\rm gas})^{1/2} \propto \rho_{\rm gas}^{3/2},$$ (3.5)

where $\rho_{\rm gas}$ and $\rho_{\rm SFR}$ are the volume densities of gas and star formation. He found another correlation between a quantity of gas surface density divided by the orbital period of galactic rotation and the star formation rate [Fig.3.1 (right)]. Although the actual slope is equal to 0.9 instead of 1, the correlation in Fig.3.1(right) is well expressed as

$$\Sigma_{\rm SFR} \simeq 0.017\Sigma_{\rm gas}\Omega_{\rm gas} = 0.21 \frac{\Sigma_{\rm gas}}{\tau_{\rm arm-to-arm}},$$ (3.6)

where $\Omega_{\rm gas}$ represents the angular speed of galactic rotation. This means that 21 % of the gas mass is processed to form stars per orbit. These two correlations [eqs. (3.4) and (3.6)] implicitly ask another relation of $\Omega_{\rm gas}\propto \Sigma_{\rm gas}^{1/2}$.