这是一份uwa西澳大学PHYS3003的成功案例

with $\sigma(M)$ and $\rho(M)$ given. Thus, subclumps of mass $f M$ having mean density $f^{1-3 \alpha} \rho(M)$ typically collide at velocity $\sigma(M)$ (i.e. the internal velocity dispersion of their parent clump) and the resulting shock-compressed layer typically fragments into subsubclunps of mass $f^{2} M$ having internal velocity dispersion $f^{2 \beta} \sigma(M)$. We can therefore substitute $m_{\text {FRAG }} \rightarrow f^{2} M$, $\sigma_{\mathrm{s}} \rightarrow f^{2 \beta} \sigma(M), \rho_{\mathrm{O}} \rightarrow f^{1-3 \alpha} \rho(M)$ and $v_{\mathrm{O}} \rightarrow \sigma(M)$ in Eqn. 1, to obtain
$$
f^{14 \beta+3 \alpha-5}\left(\frac{M}{M_{\bullet}}\right)^{6 \beta+3 \alpha-3}=\frac{G^{3} M_{*}^{2} \rho_{\bullet}}{\sigma_{\bullet}}
$$
Since this must be true for any $f$ and any $M$, both the exponents on the lefthand side of Eqn. 5 must be zero, which in combination with gives
FOR TURBULENT HIGH-MASS CLUMPS:
$\alpha \simeq \frac{1}{2}$
$$
\beta \simeq \frac{1}{4}, \quad \gamma \simeq \frac{7}{4}
$$

PHYS3003 COURSE NOTES ：

If we now require that $t_{\text {COUPLE }} \lesssim t_{\mathrm{FF}}=\left(3 \pi / 32 G \rho_{\mathrm{s}}\right)^{1 / 2}$, we obtain
$$
P_{\mathrm{S}} \equiv \rho_{\mathrm{s}} a_{\mathrm{s}}^{2} \gtrsim P_{\mathrm{COUPLE}} \simeq \frac{64 G}{3}\left[\frac{r_{\mathrm{D}} \rho_{\mathrm{D}}}{Z_{\mathrm{D}}}\right]^{2} \simeq 10^{5} \mathrm{~cm}^{-3} \mathrm{~K} k_{\mathrm{B}}
$$
where $k_{\mathrm{B}}$ is the Boltzmann constant. We emphasize two important aspects of this result. (i) Since in our model of dynamically triggered star formation $P_{\mathrm{S}}=\rho_{\mathrm{s}} a_{\mathrm{s}}^{2}=P_{\text {RMM }}=\rho_{\mathrm{o}} v_{\mathrm{O}}^{2}$, this is in effect a threshold for the ram pressure which will trigger rapid star formation in a turbulent cloud. (ii) $10^{5} \mathrm{~cm}^{-3} \mathrm{~K} k_{\mathrm{B}}$ is typical of the turbulent pressures found in star-forming molecular clouds; for a self-gravitating cloud, it corresponds to a critical surface-density
$$
\Sigma_{\text {COUPLE }} \simeq\left(\frac{P_{\text {COUPLE }}}{G}\right)^{1 / 2} \simeq 1.4 \times 10^{-2} \mathrm{~g} \mathrm{~cm}^{-2},
$$
or equivalently to a critical column-density $N_{\mathrm{H}{2}} \simeq 4 \times 10^{21} \mathrm{H}{2} \mathrm{~cm}^{-2}$.