Radial energy equation

As regards the last but one equation, the interaction terms are:

$$\frac{\partial{p_{\rm r}}}{\partial {t}} + \frac{1}{r^2} \fr... ...{\partial r} (r^2 F_{\rm r})- \frac{2F_{\rm t}}{r}= \nonumber$$

$$\bigg( \frac{\delta p_{\rm r}}{\delta t}\bigg)_{\rm drag}+\bigg( \frac{\delta p_{\rm r}} {\delta t}\bigg)_{\rm coll},$$ (31)

where

$$\bigg( \frac{\delta p_{\rm r}}{\delta t}\bigg)_{\rm drag}=-2... ...X_{\rm coll} \rho_{\star} \tilde{\sigma_{\rm r}}^2 \epsilon.$$ (32)

In order to determine $\epsilon$ we introduce the ratio $k$ of kinetic energy of the remaining mass after the encounter over its initial value (before the encounter); $k$ is a measure of the inelasticity of the collision: for $k=1$ we have the minimal inelasticity, just the kinetic energy of the liberated mass fraction is dissipated, whereas if $k<1$ a surplus amount of stellar kinetic energy is dissipated during the collision (tidal interactions and excitation of stellar oscillations). If we calculate the energy loss in the stellar system per unit volume as a function of $k$ we obtain

$$\epsilon=f_{\rm c}^{-1}[1-k(1-f_{\rm c})].$$ (33)

We divide by $p_{r }$ so that we get the logarithmic variable version of the equation. We make also the following substitution:

$$F_{\rm r}= 3p_{\rm r}v_{\rm r}$$

$$F_{\rm t}= 2p_{\rm t}v_{\rm t}$$ (34)

The resulting equation is

$$\frac{\partial \ln p_{\rm r}}{\partial t}+(u_{\star}+3v_{\rm r}) ... ...2}{r}\bigg( u_{\star}+3v_{\rm r}-2v_{\rm t} \frac{p_{\rm t}}{p_{\rm r}} \bigg)+$$

$$\frac{4}{5}\frac{ 2-\frac{p_{\rm t}} {p_{\rm r}}}{t_{\rm relax}}=... ...+\frac{1}{p_{\rm r}} \bigg( \frac{\delta p_{\rm r}} {\delta t}\bigg)_{\rm coll}$$ (35)