$$
  v_{1D}^2 = \sigma_r^2 
$$

$$
  v_{3D}^2 = 3 v_{1D}^2 = 3 \sigma_r^2 
$$

$$
  {\rm KE_{avg}} = {{1}\over{2}} m v^2 =  {{1}\over{2}} m ( 3 \sigma_r^2 )
$$

$$
  {\rm GPE } = - {{G M m}\over{r}}
$$

$$
  {\rm GPE_{avg} } \sim  - {{G M m}\over{({{1}\over{2}}R})}
$$

$$
  {{1}\over{2}} m ( 3 \sigma_r^2 ) \sim  {{1}\over{2}} {{G M m}\over{({{1}\over{2}}R})}
$$

$$
  m ( 3 \sigma_r^2 ) \sim  {{G M m}\over{({{1}\over{2}}R})}
$$







\bye
