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 _ { 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 _ { H,s } \sim ( 0.1 - 5 ) \times 10 ^ { -2 } { 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 _ { s } \sim ( 0.03 - 0.50 ) R _ { 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 . ( 45 ) 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 _ { diss } > 10 ^ { 42 } ~ { } { erg~ { } s ^ { -1 } } within \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } 0.6 R _ { v } of halos with M _ { v } > 10 ^ { 12 } { M } _ { \odot } at z = 2 , with the emission scaling with halo mass and redshift approximately as \propto M _ { 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 .