\magnification=\magstep1

$$
{\rm mean\ free\ path\ } L \quad = \quad 
         {{1}\over{\pi \thinspace {(2 r)}^2 \thinspace n}}
$$

$$
{\rm before\ approach\ } E \quad = \quad KE_{A} \ + \ KE_{B} \ + \  GPE 
       \quad = \quad 
           {{1}\over{2}} m v^2 
               \ + \ 
           {{1}\over{2}} m v^2 
               \ - \ 
           {{G m m}\over{\infty}}
$$

$$
E_{\rm initial} \quad = \quad m v^2 
$$


$$
{\rm during\ approach\ } E \quad = \quad KE_{A} \ + \ KE_{B} \ + \  GPE 
       \quad = \quad 
           {{1}\over{2}} m v'^2 
               \ + \ 
           {{1}\over{2}} m v'^2 
               \ - \ 
           {{G m m}\over{R_{\rm min}}}
$$

$$
{{G m^2}\over{R_{\rm min}}} \quad = \quad {{1}\over{2}} \thinspace E_{\rm initial}
$$

$$
R_{\rm min} \quad = \quad {{2 G m}\over{v^2}}
$$

$$
{\rm duration\ of\ force\ } \Delta t \quad = \quad {{b}\over{v}}
$$

$$
{\rm size\ of\ force \ } F \quad = \quad {{G m_1 m_2}\over{b^2}}
$$

$$
{\rm impulse \ } \quad  m_1 \thinspace \Delta v_y \quad = \quad
                    F \ \Delta t
                  \quad = \quad 
                    {{G m_1 m_2}\over{b^2}} \ \left( {{b}\over{v}} \right)
$$

$$
\Delta v_y \quad = \quad {{G m_2}\over{b \thinspace v}}
$$

$$
{\rm distance\ travelled\ } \Delta y \quad = \quad
                     ({\rm velocity}) \ ({\rm time}) 
                       \quad = \quad 
                     \left( {{G m_2}\over{b \thinspace v}} \right) \ 
                     \left( {{b}\over{v}} \right)
$$

$$
                           \Delta y \quad = \quad    
               {{G m_2}\over{v^2}}
$$

$$
        2 \thinspace  \Delta y \quad \ll \quad b  \qquad \Rightarrow \qquad 
              {{2 \thinspace G m_2}\over{v^2}} \quad \ll \quad b
$$

$$
{\rm volume\ } \ V \quad = \quad ({\rm length\ }) \ ({\rm area\ of\ ring})
    \quad = \quad 
            ( v \thinspace t ) \ (2 \pi b \thinspace db) 
$$

$$
N \quad = \quad V \thinspace n \quad = \quad 
          ( 2 \pi \thinspace v t \thinspace b \thinspace db)  \ (n)
$$


$$
{\rm contribution\ at\ impact\ parameter\ } b \quad = \quad 
    \left( {\rm number\ of\ encounters\ } \right) \   
    \left( {\Delta {v_y}^2 } \right) 
$$

$$
\phantom{{\rm contribution\ parameter\ } b} = \quad 
    \left( 2 \pi \thinspace v t \thinspace n \thinspace b \thinspace db \right) \   
    \left( {{G m}\over{v b}} \right)^2
$$


$$
{\rm total\ } | {(\Delta v_y)}^2 | \quad = \quad 
    \int_{b_{\rm min}}^{b_{\rm max}} {
      \left( 2 \pi \thinspace v t \thinspace n \thinspace b \thinspace db \right) \   
      \left( {{G m}\over{v b}} \right)^2
    }
$$


$$
{\rm total\ } | {(\Delta v_y)}^2 | \quad = \quad 
   \left( {{2 \pi G^2 m^2 n}\over{v}} \right) \  t \ 
      \int_{b_{\rm min}}^{b_{\rm max}} { {{db}\over{b}} }
$$

$$
{\rm total\ } | {(\Delta v_y)}^2 | \quad = \quad 
    \left( {{2 \pi G^2 m^2 n}\over{v}} \right) 
       \  
        \thinspace t
       \
       \left( \ln{{b_{\rm max}}\over{b_{\rm min}} } \right)
$$

$$
\Lambda \quad \equiv \quad \left( { {b_{\rm max}}\over{b_{\rm min}} } \right)
$$

$$
\ln{\Lambda} \quad \equiv \quad \ln{ \left( { {b_{\rm max}}\over{b_{\rm min}} } \right) }
$$


     
$$
 | {(\Delta v_y)}^2 | \quad = \quad | v^2 |
$$

$$
 v^2 \quad = \quad 
    \left( {{2 \pi G^2 m^2 n}\over{v}} \right) 
       \  
        \thinspace t
       \
       \ln {\Lambda}
$$

$$
t_{\rm relax} \quad = \quad 
      {{v^3}\over{2 \pi \thinspace G^2 m^2 \thinspace n \thinspace \ln{\Lambda}}}
$$

      




 


\bye
