CQW is an educational resource for theoretical physics and astrophysics, field theory, relativity and cosmology. we post a new question every wednesday for students to get and for teachers to stay in shape.

Wednesday, April 9, 2014

braking by scattering

can you estimate the time-scale on which a cluster of galaxies slows down due to the kinetic Sunyaev-Zel'dovich effect? (PS: there's something remarkable about the solution if one assumes small velocities and if CQW didn't make a mistake)

if an electron inside the cluster moves with a velocity $\beta$ relative to the CMB, it feels a difference in CMB temperature by a factor $1\pm\beta$ between the forward and backward directions. this translates into an effective radiation pressure $P$ which slows down the electron. the electron momentum is given by $m\upsilon$ in the nonrelativistic limit, and the radiation pressure is equal to a third of the energy density of the blackbody, i.e. $p = \sigma_{SB}T^4/(3c)$. this pressure acts on the Thomson cross-section $\sigma_T$ of the electron. for small velocities, it feels a radiation pressure proportional to $(1+\beta)^4$ from the front and $(1-\beta)^4$ from the back, therefore the resulting effective pressure is $(1+\beta)^4-(1-\beta)^4\simeq8\beta$ to lowest order. putting everything together gives as a time scale \begin{equation} \Delta t = \frac{3}{8}\frac{mc^2}{\sigma_T\sigma_{SB}T^4} \end{equation} with a Newtonian equation of motion $P\sigma_T \simeq mc\beta/\Delta t$.

the numerical value would be $\simeq10^6/H_0$. CQW think's it's remarkable that the time scale is the same one as if the electron would try to radiate away its rest mass at the CMB-temperature through the Thomson-cross section area, as the velocity of the cluster drops out - after all, the Feynman diagrams for both processes are identical.

if an electron inside the cluster moves with a velocity $\beta$ relative to the CMB, it feels a difference in CMB temperature by a factor $1\pm\beta$ between the forward and backward directions. this translates into an effective radiation pressure $P$ which slows down the electron. the electron momentum is given by $m\upsilon$ in the nonrelativistic limit, and the radiation pressure is equal to a third of the energy density of the blackbody, i.e. $p = \sigma_{SB}T^4/(3c)$. this pressure acts on the Thomson cross-section $\sigma_T$ of the electron. for small velocities, it feels a radiation pressure proportional to $(1+\beta)^4$ from the front and $(1-\beta)^4$ from the back, therefore the resulting effective pressure is $(1+\beta)^4-(1-\beta)^4\simeq8\beta$ to lowest order. putting everything together gives as a time scale

ReplyDelete\begin{equation}

\Delta t = \frac{3}{8}\frac{mc^2}{\sigma_T\sigma_{SB}T^4}

\end{equation}

with a Newtonian equation of motion $P\sigma_T \simeq mc\beta/\Delta t$.

the numerical value would be $\simeq10^6/H_0$. CQW think's it's remarkable that the time scale is the same one as if the electron would try to radiate away its rest mass at the CMB-temperature through the Thomson-cross section area, as the velocity of the cluster drops out - after all, the Feynman diagrams for both processes are identical.

ReplyDelete