This paper (Birnboim & Dekel 2004; http://arxiv.org/abs/astro-ph/0302161) performs a global 1-D linear stability analysis of the virial shock in a self-gravitating dark matter halo. The authors relate the pressure and density of the post-shock gas (using $d\ln P/d\ln \rho = \gamma_{\rm eff}$) and perform a stability analysis on the perturbations of the pressure and gravitational forces, assuming hydrostatic equilibrium. $\gamma_{\rm eff}$ depends on the cooling rate; for an adiabatic ideal gas blob $\gamma_{\rm eff}=5/3$. They derive a critical $\gamma_{\rm eff}$ for the shock stability in presence of cooling; only for $\gamma_{\rm eff}>\gamma_{\rm crit}=1.43$ is the post-shock plasma stable. In presence of cooling, the effective adiabatic index is given by
$$\gamma_{\rm eff}=\gamma + \frac{r_s}{2 u_1} \frac{q}{e},$$
where $r_s$ is the shock radius, $u_1$ is the post-shock velocity ($<0$) and $q$ is the cooling rate and $e$ is the internal energy. This instability criterion can also be expressed in terms of the upstream variables and is roughly equivalent to $\frac{t_{\rm ff}}{t_{\rm cool}} \lesssim {\cal O}(1)$. A similar criterion was proposed by Rees & Ostriker 1977 (http://adsabs.harvard.edu/abs/1977MNRAS.179..541R) for efficient cooling of gas to form galaxies. They argued that if $t_{\rm cool}<t_{\rm ff} \sim t_{\rm snd}$, the gas cools to sub-virial temperatures ($10^4$ K), and the cool gas falls on a free-fall time, unimpeded by gas pressure.
Both these papers find that stable/quasi-steady, pressure supported IGM is formed only for a halo mass $\gtrsim 10^{12} M_\odot$. While RO77 and other subsequent papers envisioned a hot, shocked IGM, BD04 showed that a stable virial shock is formed only for $\gtrsim 10^{12} M_\odot$ halos, and that the gas accretes in the code mode (at $\sim 10^4$ K) for lower mass halos, as observed in recent numerical simulations. BD04 also performed 1-D Lagrangian simulations of shock formation with DM + gas, starting from cosmologically consistent initial conditions. Their numerical simulations agree with their linear stability analysis. The critical halo mass for forming stable shock ($\sim 10^{11.5} M_\odot$) is fairly insensitive to metallicity, angular momentum, baryon fraction, power spectrum, etc. An important effect that BD04 may have missed is feedback heating; in presence of feedback heating the virial shock can become stable as feedback heating provides the post-shock pressure to support the shock.
Recently Dekel and collaborators have argued that cold streams are responsible for star-forming galaxies (SFGs) with SFR~100$M_\odot {\rm yr}^{-1}$; however major mergers are likely responsible for the rarer sub-mm galaxies at z~2 with SFR~1000$M_\odot {\rm yr}^{-1}$. Major mergers are responsible for $< 1/3$ in total star formation.
$$\gamma_{\rm eff}=\gamma + \frac{r_s}{2 u_1} \frac{q}{e},$$
where $r_s$ is the shock radius, $u_1$ is the post-shock velocity ($<0$) and $q$ is the cooling rate and $e$ is the internal energy. This instability criterion can also be expressed in terms of the upstream variables and is roughly equivalent to $\frac{t_{\rm ff}}{t_{\rm cool}} \lesssim {\cal O}(1)$. A similar criterion was proposed by Rees & Ostriker 1977 (http://adsabs.harvard.edu/abs/1977MNRAS.179..541R) for efficient cooling of gas to form galaxies. They argued that if $t_{\rm cool}<t_{\rm ff} \sim t_{\rm snd}$, the gas cools to sub-virial temperatures ($10^4$ K), and the cool gas falls on a free-fall time, unimpeded by gas pressure.
Both these papers find that stable/quasi-steady, pressure supported IGM is formed only for a halo mass $\gtrsim 10^{12} M_\odot$. While RO77 and other subsequent papers envisioned a hot, shocked IGM, BD04 showed that a stable virial shock is formed only for $\gtrsim 10^{12} M_\odot$ halos, and that the gas accretes in the code mode (at $\sim 10^4$ K) for lower mass halos, as observed in recent numerical simulations. BD04 also performed 1-D Lagrangian simulations of shock formation with DM + gas, starting from cosmologically consistent initial conditions. Their numerical simulations agree with their linear stability analysis. The critical halo mass for forming stable shock ($\sim 10^{11.5} M_\odot$) is fairly insensitive to metallicity, angular momentum, baryon fraction, power spectrum, etc. An important effect that BD04 may have missed is feedback heating; in presence of feedback heating the virial shock can become stable as feedback heating provides the post-shock pressure to support the shock.
Recently Dekel and collaborators have argued that cold streams are responsible for star-forming galaxies (SFGs) with SFR~100$M_\odot {\rm yr}^{-1}$; however major mergers are likely responsible for the rarer sub-mm galaxies at z~2 with SFR~1000$M_\odot {\rm yr}^{-1}$. Major mergers are responsible for $< 1/3$ in total star formation.
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