The above relation has been used in the synchrotron cooling expression in the paper "Behaviour of dissipative accretion flows around black holes" 2007MNRAS.376.1659D.
We know that the electron thermal velocity is v_e = (k T_e/m_e)^0.5 and ion thermal velocity is v_i = (k T_i/m_i)^0.5. If we say that ions and electrons have same kinetic energy then...m_e (v_e^2) = m_i (v_i^2)...and we arrive at the ralation...
T_e = (m_e/m_i)T_i
But under what assumptions can we arrive at the relation T_e = (m_e/m_i)^0.5 T_i ??
no matter what is the exact relation between T_e and T_i immediately behind the front (which is exactly as you have written T_e=(m_e/m_i)T_i), it gradually grows then due to heat conduction in electrons (at some point reaches (m_e/m_i)T_i as assumed in 2007MNRAS.376.1659D, and finallycomes into T_e=T_i -- see, for details Zeldovich & Raizer, Physics of Shock waves and High Temperature phenomena, Chapter VII, Section 12.
Thanks Yuri for your reply. Actually in the paper 2007MNRAS.376.1659D, the relation assumed is T_e = (m_e/m_i)^0.5 T_i.
It is established fact that in two temperature solution of accretion flow around black hole, far away from the black hole the relation T_e=T_i holds. As we approach the black hole, due to inefficient coulomb coupling, the electron temperature starts falling in comparison to the proton temperature and the relations that roughly hold between T_e and T_i would be T_e = (m_e/m_i)T_i or T_e = (m_e/m_i)^0.5 T_i.
Is my reasoning alright??
have to think -- I am not quite familiar with what's going on with T_e/T_i when you do approach black hole.. depends on heating sources of the electron, I guess.
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