## Wednesday, May 14, 2014

### cluster cooling by immersion into the CMB

on what time scale would the intra-cluster medium of a cluster cool down by Compton-scattering of the cold cosmic microwave background (ignoring thermal replenishment by the negative specific heat)?

1. there are a number of ways of solving this question: perhaps the easiest is to link the thermal SZ-effect to the kinetic one by assigning thermal velocities to the electrons: $m_ec^2\beta^2/2=3k_BT_e/2$. in this way the electrons inside the cluster feel a larger radiation pressure from the CMB in their direction of motion and a smaller radiation pressure from the backward direction:

\Delta p =
\frac{\sigma_{SB}T^4}{3c}\left[(1+\beta)^4-(1-\beta)^4\right] \simeq
\frac{8}{3}\frac{\sigma_{SB}T^4}{c}\beta

to first order. substituting into a Newtonian equation of motion

F = \sigma_{Th} \Delta p = mc\Delta\beta/\Delta t

yields

\Delta t = \frac{3}{8}\frac{m_ec^2}{\sigma_{Th}\sigma_{SB}T^4}

for the time scale $\Delta t$ in which one observes an order unity change in $\beta$. again, the time scale is equal to the time needed for radiating the electron rest mass through the Thomson cross-section at the CMB temperature, just in the case of the kinetic SZ-effect.

2. a more "thermodynamic" derivation could look like this one: using the first law one would write $\mathrm{d}U=-p\mathrm{d}V$ with a spherical volume $V=4\pi r^3/3$. the force needed for a volume change is $F=\sigma_{Th}p=\mathrm{d}U/\mathrm{d}V$, and consequently

m_ec\frac{\Delta\beta}{\Delta t} = F = \sigma_{Th}p = \sigma_{Th}\frac{\mathrm{d}U}{\mathrm{d}r}\frac{\mathrm{d}r}{\mathrm{d}V}

using a Newtonian equation of motion for an electron on the surface of the sphere. this yields again

\Delta t = \frac{3m_ec^2}{\sigma_{Th}\sigma_{SB}T^4}