\magnification=\magstep0

\input ../macros.tex

\hoffset -0.8cm
\voffset -1.0cm
\noblackbox
\nopagenumbers


$$
\eqalign {
\Phi({\bf r})\ &= \ -G \int_V { \rho({\bf r}^{\prime}) \over  
| {\bf r}^{\prime} - {\bf r}|  }\   d^3 {\bf r}^{\prime} \cr
{\bf F}({\bf r})\ &=\ -{\bf \nabla} \Phi ({\bf r})\ =\ G \int_V { {\bf r}^{\prime} - {\bf r} \over \ 
| {\bf r}^{\prime} - {\bf r}| ^3 }\  \rho ({\bf r}^{\prime})\ d^3 {\bf r}^{\prime} \cr }
$$

$$
\eqalign {
{\bf \nabla\, \cdot \, F}({\bf r})\ &=\ -4\pi G\rho({\bf r}) \cr
\nabla^2 \Phi({\bf r})\ &=\ 4\pi G\rho({\bf r}) \cr
\nabla^2 \Phi({\bf r})\ &=\ 0 \cr }
$$

$$
\eqalign {
4 \pi G M\ &=\ 4 \pi G \int_V \rho ({\bf r})\, d^3{\bf r}\ \cr
 &=\ \int_V - {\bf \nabla}\, \cdot\, {\bf F}({\bf r})\,
d^3{\bf r}\ =\ 
\int_A -{\bf F}({\bf r})\, \cdot\, d^2{\bf S}  \cr }
$$
$$
W\ =\ {1\over 2} \int_V \rho({\bf r})\, \Phi ({\bf r})\, d^3 {\bf r}\ =\
-{1\over 8\pi G} \int_V \, | \nabla \Phi |^2\, d^3 {\bf r}
$$

$$
{V_0^2\over r}\ =\ F_r\ =\ -\, {d\Phi\over dr} \ \ \ ; \ \ \ \Phi(r)\ =\ 
V_0^2 \ln r + const
$$

$$
\eqalign {
\rho_H (r)\ &=\ {Ma\over 2\pi r(r+a)^3}
\ \ ;\ \ \  \Phi_H (r)\ =\ -\, {GM\over (r+a)} \cr
\rho_J (r)\ &=\ {Ma\over 4\pi r^2(r+a)^2}
\ ;\ \ \  \Phi_J (r)\ =\  {GM\over a}\ln\left({r\over r+a}\right) \cr }
$$

$$
\rho_P (r)\ =\ \left({3M\over 4\pi b^3}\right) {\left( 1 + {r^2\over b^2}
\right)\!\!}^{-5/2}
\ \ ;\ \ \  \Phi_P (r)\ =\ -\, {GM\over \sqrt{r^2 + b^2}}
$$

$$ 
\eqalign {
-{\partial\,\Phi \over \partial z}\ =\ g_z\ &=\ 4\pi G \rho_0\ z \ \ \ \ {\rm (inside)}
\cr
 &=\ 2\pi G \Sigma \ \ \ \ \ \ \ {\rm (above)}
\cr}
$$

$$
V_c^2(R)\ =\ 4\pi G\,\Sigma_0\,R_d\, y^2\left[\, I_0(y)K_0(y) - I_1(y)K_1(y)\,\right]
$$

$$
\Sigma_K(R)\ =\ {aM\over 2\pi (R^2+a^2)^{3/2}} \ \ ; \ \ \
\Phi_K(R,z)\ =\ -\, {GM\over \sqrt{R^2+(a+|z|)^2} }
$$

$$
\Sigma_{T_n} (R)\ =\ {\left({d\over da^2}\right)\!\!}^{n-1}\!\!\!\Sigma_{K} (R)\ \ ;\ \ 
\Phi_{T_n} (R)\ =\ {\left({d\over da^2}\right)\!\!}^{n-1}\!\!\!\Phi_K
$$

$$
\eqalign {
\rho_{M} (R,z)\ &=\ \left({Mb^2\over 4\pi}\right)
{aR^2+(a+3B)(a+B)^2 \over [R^2+(a+B)^2]^{5/2}B^3 } \cr
\Phi_{M} (R,z)\ &=\ -\, {GM\over \sqrt{R^2+(a+B)^2} }\ \ \ ; 
\ \ B^2=z^2+b^2 \cr 
\rho_{S_n}(R,z)\ &=\ {\left({d\over db^2}\right)\!}^{n}\rho_{M} \
\ ;\ \ \Phi_{S_n}(R,z)\ =\ {\left({d\over db^2}\right)\!}^{n}\Phi_{M}
\cr }
$$

$$
{d\over dt}(m^{\alpha}v_i^{\alpha})\ =\ F_i^{\alpha}\ =\ 
-\, G m^{\alpha} \sum_{\beta\ne\alpha} m^{\beta}{r_i^{\alpha} - r_i^{\beta} \over
|{\bf r}^{\alpha} - {\bf r}^{\beta}|^3}
$$

$$
{1\over 2}\, {d^2\over dt^2}I_{i,j}\  =\ 
2\, K_{i,j} \ +\ W_{i,j}\ =\ 2\, T_{i,j}\ +\ \Pi_{i,j}\ +\ W_{i,j}
$$

$$
\eqalign {
I_{i,j}\ &=\ \int\!\rho\, r_i\, r_j\, d^3r\ \ \ \ \ \ \ \ \ \ =\ {\rm moment\ of\ inertia} \cr
K_{i,j}\ &=\ \int\!{\scriptstyle{1\over 2}}\, \rho\, \langle v_i v_j \rangle \, d^3r
\ \ \ \ \ \ =\ {\rm total\ KE} \cr 
T_{i,j}\ &=\ \int\!{\scriptstyle{1\over 2}}\, \rho\, \langle v_i\rangle\, 
\langle v_j \rangle \, d^3r\ \ \ =\ {\rm ordered\ KE} \cr
\Pi_{i,j}\ &=\ \int\!\!\rho\,\sigma_{i,j}^2\, d^3r \ \ \ \ \ \ \ \ \ \ \ \ =\ {\rm dispersion\ KE} \cr
W_{i,j}\ &=\ - {1\over 2} G \int\!\int \rho({\bf r})\, 
\rho({\bf r}^{\prime})\, {(r_i - r_i^{\prime})\, (r_j - r_j^{\prime}) \over
|{\bf r}^{\prime} - {\bf r}|^3} d^3 rd^3 r^{\prime}\ =\ PE \cr }
$$

$$
2K_{ij}\ +\ W_{ij}\ =\ 0
$$

$$
2K\ +\ W\ =\ 0
$$

$$
E\ =\ K\ +\ W\ =\ -K\ =\ {1\over 2} W
$$

$$
M_{tot}\ \simeq\ {<\!v^2\!>R_m \over 0.4\, G}
$$

$$
{V_o\over\sigma_o}\ =\ \sqrt{2\left[ (1-\delta )(1-\epsilon )^{-0.9}\ -\ 1 \right]}
$$
$$
{V_o\over\sigma_o}\ \simeq\ \sqrt{{\epsilon \over (1 - \epsilon )}}
$$

$$
v_x dt\, dv_x \left[ f(x,v_x,t) - f(x+dx,v_x,t)\right]\ =\ -\,v_x\, dt\,  dv_x {\partial f\over \partial x}\, dx 
$$

$$
dx\, {dv_x\over dt} dt\, \left[ f(x,v_x,t) - f(x,v_x+dv_x,t)\right] \ =\ -\, 
dx\,  dt\, {dv_x \over dt} {\partial f\over \partial v_x} dv_x
$$

$$
dx\, dv_x\, {\partial f\over \partial t} dt\ =\ -\, dt\, dx\, v_x\, 
{\partial f \over \partial x}\, dv_x \ -\ dx\, dt\, {dv_x\over dt} \,
{\partial f\over \partial v_x}\, dv_x
$$

$$
{\partial f \over \partial t}\, +\, v_x\, {\partial f\over\partial x}\, +\, 
{dv_x\over dt}\, {\partial f\over \partial v_x}\, =\, 0
$$

$$
{dv_x\over dt}\ =\ a_x \ =\ -\, {\partial \Phi\over \partial x}
$$

$$
{\partial f\over\partial t}\ +\ v_x\, {\partial f\over\partial x}\ -\ 
{\partial \Phi\over\partial x}\, {\partial f\over\partial v_x}\ =\ 0
$$

$$
{\partial f\over\partial t}\ +\ {\bf v}\, \cdot\, {\bf \nabla}f\ -\ 
{\bf \nabla}\phi\, \cdot\, {\partial f\over\partial {\bf v}}\ =\ 0
$$

$$
{d\, f\over d\, t}\ =\ {\partial f\over \partial t}\, {dt\over dt}\ +\
{\partial f\over \partial x}\, {dx\over dt}\ +\ 
{\partial f\over \partial v_x}\, {dv_x\over dt}\ =\ 0
$$

$$
{\partial n \over \partial t}\, +\, {\partial \big(n \langle v_x\rangle \big)
\over\partial x}\,  =\, 0
$$

$$
{\partial \langle v_x\rangle\over \partial t}\, +\, \langle v_x\rangle\,
{\partial \langle v_x\rangle\over \partial x}\ =\ -\,{\partial \Phi\over \partial x}\, -\, {1\over n}
{\partial (n\sigma_x^2)\over \partial x}\
$$

$$
{\partial\langle v_j\rangle\over\partial t}\ +\ \langle v_i\rangle 
{\partial\langle v_j\rangle\over\partial x_i}\ =\ 
-\ {\partial\Phi\over\partial x_j}\ -\ {1\over n}\, {\partial (n\sigma_{i,j}^2)\over\partial x_i}
$$
$$
{\partial {\bf v} \over \partial t}\ +\ \left({\bf v 
\cdot \nabla}\right)
{\bf v}\ =\ -\ {\bf \nabla}\Phi\ -\ {1\over\rho}\, {\bf \nabla}p
$$

$$
{1\over n}\, {d (n\sigma_r^2)\over dr}\, +\, {1\over r}\,
\Big[\, 2 \sigma_r^2 \, -\, (\,\sigma_{\theta}^2\, +\,
\sigma_{\phi}^2)\,\Big]\, -\, {\langle v_{\phi}\rangle^2\over r}\ =\ -\, {d \Phi\over dr}
$$

$$
{1\over n}\, {d (n\sigma_r^2)\over dr}\, +\, 2\beta\, {\sigma_r^2\over r}\,
 -\, {V_{\rm rot}^2\over r}\ =\ -\, {d \Phi\over dr}
$$

$$
V_{\rm rot}^2\ \ -\ \ \sigma_r^2\left({d\ln n\over d \ln r}\, +\, {d\ln (\sigma_r^2)\over d \ln r}\, +\, 2\beta \right)\ =\ 
{GM(<r)\over r}\ =\ V_c^{\, 2}
$$

$$
\eqalign {
{dI\over dt}\ &= \ \sum_{i=1}^3 {\partial I\over \partial x_i}\, {dx_i\over dt}
\ +\ \sum_{i=1}^3 {\partial I\over \partial v_i}\, {dv_i\over dt}\ =\ 0 \cr
&= \ {\bf \nabla}I \cdot {d{\bf x}\over dt}\ +\ {\partial I\over 
\partial {\bf v} } \cdot {d {\bf v} \over dt}\ =\ 0 \cr
&= \ {\bf v} \cdot \nabla I \ -\ \nabla\Phi \cdot {\partial I\over \partial
{\bf v} }\ =\ 0 \cr } 
$$

$$
\eqalign {
4\pi G\int f({\bf r,v})\, d^3{\bf v}\ &=\ 4\pi G\rho ({\bf r})\ \cr
&=\ \nabla^2\Phi ({\bf r})\ \cr }
$$

$$
{1\over r^2}\, {d\over dr}\left( r^2 {d\Phi\over dr}\right)\ =\ 
4\pi G \int f(\,{\scriptstyle {1\over 2}} v^2+\Phi\, ,|\, {\bf r}\times{\bf v}|\, )\, d^3{\bf v}
$$

$$
\eqalign {
{1\over r^2}\, {d\over dr}\left( r^2 {d\Psi\over dr}\right)\ &=\ 
-\,16\pi^2 G \int_0^{\sqrt{2\Psi}}\! f(\Psi - {\scriptstyle {1\over 2}} 
v^2)\, v^2\, dv \cr
&=\ -\,16\pi^2 G \int_0^{\Psi} f(E_r)\, \sqrt{2(\Psi - E_r)}\  dE_r \cr } 
$$

$$
f(E_r)\ =\  {1\over\pi^2\sqrt{8}}\left[{\int_0^{E_r} {d^2\rho\over d \, \Psi^2}\, 
{d\, \Psi\over\sqrt{E_r - \Psi}}\ +\ {1\over\sqrt{E_r}}{\left( {d\rho\over d\Psi}\right)\!}
_{\Psi = 0} }\right]
$$

$$
N(E_r)\, dE_r\ =\  16\pi^2\, f(E_r)\, \int_0^{r_{m}(E_r)}\!\! r^2 
\sqrt{2(\Psi(r) - E_r)}\, dr
$$

$$
\rho \ =\ 4\pi \int_0^{\infty}\!\! f(E_r)\, v^2\, dv \ =\ 4\pi F 
\int_0^{\sqrt{2\Psi}}\!\! (\Psi - {\scriptstyle {1\over 2}} v^2)^{n-{3\over 2}}
v^2\, dv
$$

$$
{1\over r^2}\, {d\over dr}\left( r^2 {d\Psi\over dr}\right)\ +\ 4\pi G c_n \Psi^n\ =\ 0
$$

$$
f(E_r)\ =\ {\rho_1\over (2\pi\sigma^2)^{3\over 2} } \ e^{E_r/\sigma^2}
$$

$$
{d\over dr}\left( r^2 {d\ln\rho\over dr}\right)\ =\ 
-\, {4\pi G\over\sigma^2}\, r^2 \rho
$$


$$
f(E_r)\ =\ {\rho_1\over (2\pi\sigma_0^2)^{3\over 2} } \ ( e^{E_r/\sigma_0^2} - 1 )
$$

$$
{d\over dr}\left( r^2 {d\Psi\over dr}\right)\ =\ 
-\,4\pi G \rho_1r^2\, 
{\left[{e^{\Psi/\sigma_0^2}{\rm erf}\left({\sqrt{\Psi}\over\sigma_0}\right) -
\sqrt{{4\Psi\over\pi\sigma_0^2}}\left(1\,+\, 
{2\Psi\over 3\sigma_0^2}\right)}\right]}
$$

$$
\eqalign {
|\Delta V_{\rm tot} |^2\ =\  \sum |\Delta V |^2 \ &=\ 
\int_{b_{min}}^{b_{max}}{\left({2Gm\over bV}\right)\!}^2t\ n V\, 2\pi b\ db \cr \cr
&=\ {8\pi G^2 m^2 n t\over V} \ \ln \Lambda \cr }
$$

$$
\eqalign {
t_{\rm relax}\ &\simeq\ 0.34\, {\sigma^3\over G^2 m \rho \ln\Lambda} \cr\cr
&\simeq\ {1.8 \times 10^{10}\ {\rm yr}\over\ln\Lambda } \ \sigma_{10}^3\ 
m_{\odot}^{-1}\ \rho_3^{-1} \cr }
$$

$$
t_{\rm relax}\ \simeq\ t_{\rm cross}\ {N\over 6 \ln N}
$$

$$
{d\, f\over d\, t}\ =\ \Gamma(f)\ =\ 
\int S({\bf w}-\Delta{\bf w}\, ,\, \Delta{\bf w})\,
f({\bf w}-\Delta{\bf w})\ -\ S({\bf w},\Delta{\bf w})\, 
f({\bf w})\ \ d^3{\bf w} 
$$
$$
{d\, f\over d\, t}\ =\ \Gamma(f)\ =\ 
- \sum_{i=1}^6 {\partial\over\partial w_i} [f({\bf w})\, D(\Delta  w_i)]\ +\ 
{\scriptstyle{1\over 2}}\!\!  \sum_{i,j=1}^6 {\partial^2\over\partial w_i \partial w_j} [f({\bf w})\, D(\Delta  w_i\Delta w_j)]
$$

$$
\eqalign {
D(\Delta v_{\parallel})\ &=\ -{16\pi^2 G^2 m_a(m+m_a)\ln\Lambda \over
v^2} \int_0^v v_a^2f_a(v_a)dv_a\ \cr
D(\Delta v_{\parallel}^2)\ &=\ {32\pi^2 G^2 m_a^2\ln\Lambda \over
3v} \left[ \int_0^v {v_a^4\over v^2}f_a(v_a)dv_a\ +\ v\!\int_v^{\infty}
v_a f_a(v_a)dv_a \right] \cr
D(\Delta v_{\perp}^2)\ &=\ {32\pi^2 G^2 m_a^2\ln\Lambda \over
3v} \left[ \int_0^v \! \left( 3v_a^2 - {v_a^4\over v^2}\right) f_a(v_a)dv_a\ +\ 2v\!\int_v^{\infty}\!\!\!\!
v_a f_a(v_a)dv_a \right] \cr }
$$

$$
D(\Delta E)\ =\ 16\pi^2 G^2 m m_a \ln\Lambda \left[ m_a\int_v^{\infty} v_af_a(v_a)dv_a\ -\ m \int_0^v {v_a^2\over v} f_a(v_a)dv_a \right]
$$

$$
{1\over n}\, {d \big(n\langle v_r^2\rangle\big)\over dr}\, +\, 2\beta\,
{\langle v_r^2\rangle \, \over r}\ =\ -\, {d \Phi\over dr}
$$

$$
-\, \langle v_r^2\rangle\, \Big({d\,\ln n\over d\,\ln r}\ +\ 
{d\,\ln \langle v_r^2\rangle \over d\,\ln r}\ +\ 2\beta \Big)\ =\ 
{G M(<\! r) \over r}\ =\ V_c^2
$$


\bye

