Editing Stellar dynamics (section)

== Concept of a gravitational potential field ==
Stellar dynamics involves determining the gravitational potential of a substantial number of stars. The stars can be modeled as point masses whose orbits are determined by the combined interactions with each other. Typically, these point masses represent stars in a variety of clusters or galaxies, such as a [[Galaxy cluster]], or a [[Globular cluster]]. Without getting a system's gravitational potential by adding all of the point-mass potentials in the system at every second, stellar dynamicists develop potential models that can accurately model the system while remaining computationally inexpensive.<ref name=":12">{{Cite book | title=Galactic Dynamics| last1=Binney| first1=James| last2=Tremaine| first2=Scott |publisher=Princeton University Press |year=2008 |isbn=978-0-691-13027-9 | location=Princeton|pages=35, 63, 65, 698}}</ref> The gravitational potential, <math>\Phi</math>, of a system is related to the acceleration and the gravitational field, <math>\mathbf{g}</math> by:
<math display="block">\frac {d^{2}\mathbf {r_{i}} }{dt^{2}}}=\mathbf {\vec {g}} =-\nabla _{\mathbf {r_{i}} }\Phi (\mathbf {r_{i}} ),~~\Phi (\mathbf {r} _{i})=-\sum _{k=1 \atop k\neq i}^{N}{\frac {Gm_{k}}{\left\|\mathbf {r} _{i}-\mathbf {r} _{k}\right\|}, </math>
whereas the potential is related to a (smoothened) mass density, <math>\rho </math>, via the [[Poisson's equation]] in the integral form 
<math display="block"> \Phi(\mathbf {r}) = - \int {G \rho(\mathbf{R}) d^3\mathbf{R} \over \left\|\mathbf {r}-\mathbf {R} \right\|} </math>
or the more common differential form 
<math display="block">\nabla^2\Phi = 4\pi G \rho. </math>

=== An example of the Poisson Equation and escape speed in a uniform sphere ===

Consider an analytically smooth spherical potential
<math display="block"> \begin{align}
\Phi(r) & \equiv \left(-V_0^2\right) + \left[{r^2 -r_0^2 \over 2r_0^2}, ~~ 1 -{r_0 \over r} \right]_{\max} \!\!\!\! V_0^2  \equiv \Phi(r_0)-{V_e^2(r) \over 2},  ~~\Phi(r_0) = - V_0^2 , \\
\mathbf{g} &= -\mathbf{\nabla} \Phi(r) = -\Omega^2 r H(r_0 - r) - { G M_0 \over r^2}H(r-r_0), ~~\Omega={V_0 \over r_0}, ~~M_0 = {V_0^2 r_0 \over G},\end{align}</math>
where <math> V_e(r) </math> takes the meaning of the speed to "escape to the edge" <math> r_0</math>, and <math>\sqrt{2}V_0 </math> is the speed to "escape from the edge to infinity".  The gravity is like the restoring force of harmonic oscillator inside the sphere, and Keplerian outside as described by the Heaviside functions.

We can fix the normalisation <math> V_0 </math> by computing the corresponding density using the spherical Poisson Equation
<math display="block"> G\rho = {d \over 4 \pi r^2 dr} {r^2 d\Phi \over dr} = { d (G M) \over 4 \pi r^2 dr} = {3 V_0^2 \over 4 \pi r_0^2}H(r_0-r), </math>
where the enclosed mass 
<math display="block"> M(r) = {r^2 d\Phi \over G dr} = \int_0^{r} dr \int_0^{\pi} (r d\theta) \int_0^{2 \pi} (r \sin\theta d\varphi) \rho_0 H(r_0-r) = \left. M_0 x^3\right|_{x={r \over r_0}}.</math>

Hence the potential model corresponds to a uniform sphere of radius <math> r_0 </math>, total mass <math> M_0 </math> with
<math display="block"> {V_0 \over r_0} \equiv \sqrt{4\pi G \rho_0 \over 3} = \sqrt{G M_0 \over r_0^3}. </math>

=== Key concepts ===
While both the equations of motion and Poisson Equation can also take on non-spherical forms, depending on the coordinate system and the symmetry of the physical system, the essence is the same: 
The motions of stars in a [[galaxy]] or in a [[globular cluster]] are principally determined by the average distribution of the other, distant stars. The infrequent stellar encounters involve processes such as relaxation, [[Mass segregation (astronomy)|mass segregation]], [[tidal force]]s, and [[dynamical friction]] that influence the trajectories of the system's members.<ref>{{Cite journal|last1=de Vita| first1=Ruggero|last2=Trenti|first2=Michele|last3=MacLeod|first3=Morgan|date=2019-06-01|title=Correlation between mass segregation and structural concentration in relaxed stellar clusters|journal=Monthly Notices of the Royal Astronomical Society|  volume=485|issue=4|pages=5752–5760 |doi=10.1093/mnras/stz815 | doi-access=free| arxiv=1903.07619 | issn=0035-8711}}</ref>