In a deep-space astrophysical model, a high-energy cosmic ray proton enters a cold interstellar molecular cloud at a temperature of 50 K50\text{ K}50 K. The cloud is composed of hydrogen molecules which behave as an ideal gas.
As the proton travels through the cloud, it loses energy through a series of nnn random collisions with the cold gas molecules. In each collision, the proton's kinetic energy is reduced by a constant fraction α=0.22\alpha = 0.22α=0.22.
The kinetic energy EkE_kEk of the proton after nnn collisions is modeled by:
Ek=E0(1−α)n E_k = E_0 (1 - \alpha)^n Ek=E0(1−α)nwhere E0E_0E0 is its initial kinetic energy.
The proton is considered to have thermalised when its kinetic energy is reduced to the average kinetic energy of a gas particle in the interstellar cloud.
The proton enters the cloud with an initial kinetic energy of 8.0 keV8.0\text{ keV}8.0 keV.
Calculate the minimum number of collisions nnn required for this proton to become thermalised.
(Boltzmann constant kB=1.38×10−23 J K−1k_B = 1.38 \times 10^{-23}\text{ J K}^{-1}kB=1.38×10−23 J K−1, electron charge e=1.60×10−19 Ce = 1.60 \times 10^{-19}\text{ C}e=1.60×10−19 C)
