\magnification=\magstep1

$$
F \quad = \quad {{G M m}\over{r^2}} 
$$

$$
{{m v^2}\over{r}} \quad = \quad {{G M m}\over{r^2}} 
$$

$$
v^2 \quad = \quad {{G M}\over{r}}
$$


$$
v \quad = \quad \sqrt{ {{G M}\over{r}} } \quad = \quad \sqrt{GM} \ \ r^{-1/2}
$$

$$
v^2(R) \quad = \quad {{G M_{\rm enc}}\over{R}}
$$

$$
M_{\rm enc} \quad = \quad \int_0^R {  {4 \pi r^2 \thinspace dr} \ \rho(r) }
$$

$$
v \quad \propto \quad R
$$

$$
\rho(r) \quad = \quad \rho_0 \ r^{\alpha}
$$

$$
M_{\rm enc} \quad = \quad \int_0^R {  {4 \pi r^2 \thinspace dr} \ \rho_0 \ r^{\alpha} } 
   \quad = \quad 4 \pi \rho_0 \ \int_0^R \ r^{2+\alpha} \thinspace dr 
$$

$$
M_{\rm enc} \quad = \quad \left( {{4 \pi \rho_0}\over{3 + \alpha}} \right) \ R^{3 + \alpha} 
$$

$$
v^2 \quad = \quad {{G M_{\rm enc}}\over{R}} \quad 
            \propto \quad {{R^{3 + \alpha}}\over{R}}
            \quad \propto \quad R^{2 + \alpha}
$$


$$
\rho(r) \quad = \quad \rho_0 \ r^{-2.2}
$$

$$
\mu \quad = \quad -k \ r
$$

$$
\mu \quad \propto \quad \ln{(I)}
$$

$$
\ln{(I)} \quad \propto \quad -k \ r \qquad \quad \Rightarrow  \qquad 
     I(r) \quad \propto \quad e^{-kr}
$$





\bye

