The Dark Stars
Problem 1
Estimate how much will the period of rotation of the Earth around the Sun change in one year due to gravitational capture of dark matter particles.
Problem 2
Estimate the rate of energy outcome due to the WIMP-annihilation process using the parameter values $m_{WIMP}=100GeV$ and $\langle \sigma v \rangle_{ann}=3\cdot 10^{-26} cm^3/sec$ for the annihilation cross section.
WIMP-annihilation in unit volume produces produces amount of energy equal to \[Q_{ann}=\left<\sigma v\right>\rho_\chi^2/m_\chi\simeq 10^{-29} \frac{\mbox{erg}}{\mbox{cm}^3/\mbox{s}}\frac{\left<\sigma v\right>}{3\times 10^{-26}\mbox{cm}^3/\mbox{s}} \left(\frac{n}{\mbox{cm}^{-3}}\right)^{1,6} \left(\frac{100\mbox{GeV}}{m_\chi}\right),\] where $\rho_\chi$ is the dark matter density inside the star and $n$ is that of hydrogen.
Problem 3
It is theorized that the dark matter particles' annihilation processes could be a competitive energy source in the first stars. Why did those processes play an important role only in the early Universe and why are they not important nowadays?
Number of particles reproduced in the reactions of the dark matter particles annihilation is proportional to the latter's density squared. In early Universe the density was considerably higher than today.
