Ly$\alpha$ Blobs from Cold Streams Undergoing Kelvin-Helmholtz Instabilities

We present an analytic toy model for the radiation produced by the interaction between the cold streams thought to feed massive halos at high redshift and their hot CGM. We begin by deriving cosmologically motivated parameters for the streams as they enter the halo virial radius, $R_{\rm v}$, as a function of halo mass and redshift. For $10^{12}M_{\odot}$ halos at $z=2$, we find the Hydrogen number density in streams to be $n_{\rm H,s}\sim (0.1-5)\times 10^{-2}{\rm cm}^{-3}$, a factor of $\delta \sim (30-300)$ times denser than the hot CGM density, while the stream radii are in the range $R_{\rm s}\sim (0.03-0.50)R_{\rm v}$. As the streams accelerate towards the halo centre, they become denser and narrower. The stream-CGM interaction induces Kelvin-Helmholtz Instability (KHI), which leads to entrainment of CGM mass by the stream and therefore to stream deceleration by momentum conservation. Assuming that the entrainment rates derived by Mandelker et al. 2019 in the absence of gravity can be applied locally at each halocentric radius, we derive equations of motion for the stream in the halo. Using these, we derive the net acceleration, mass growth, and energy dissipation induced by the stream-CGM interaction, as a function of halo mass and redshift, for different CGM density profiles. For the range of model parameters considered, we find that the interaction can induce dissipation luminosities $L_{\rm diss}>10^{42}~{\rm erg~s^{-1}}$ within $\le 0.6 R_{\rm v}$ of halos with $M_{\rm v}>10^{12}M_{\odot}$ at $z=2$, with the emission scaling with halo mass and redshift approximately as $\propto M_{\rm v}\,(1+z)^2$. The magnitude and spatial extent of the emission produced in massive halos at high redshift is consistent with observed Ly$\alpha$ blobs, though better treatment of the UV background and self-shielding is needed to solidify this conclusion.