Stellar dynamics
Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that the number of body formula_1 Typical galaxies have upwards of millions of macroscopic gravitating bodies and countless number of neutrinos and perhaps other dark microscopic bodies. Also each star contributes more or less equally to the total gravitational field, whereas in celestial mechanics the pull of a massive body dominates any satellite orbits. Connection with fluid dynamics! Stellar dynamics also has connections to the field of plasma  physics. The two fields underwent significant development during a similar time period in the early 20th century, and both borrow mathematical formalism originally developed in the field of fluid mechanics. In accretion disks and stellar surfaces, the dense plasma or gas particles collide very frequently, and collisions result in equipartition and perhaps viscosity under magnetic field. We see various sizes for accretion disks and stellar atmosphere, both made of enormous number of microscopic particle mass, formula_2 The system crossing time scale is long in stellar dynamics, where it is handy to note that formula_6 The long timescale means that, unlike  gas particles in accretion disks, stars in galaxy disks very rarely see a collision in their stellar lifetime. However, galaxies collide occasionally in galaxy clusters, and stars have close encounters occasionally in star clusters. As a rule of thumb, the typical scales concerned (see the Upper Portion of P.C.Budassi's Logarithmic Map of the Universe) are formula_2 Connection with Kepler problem and 3-body problem. At a superficial level, all of stellar dynamics might be formulated as an N-body problem by Newton's second law, where the equation of motion (EOM) for internal interactions of an isolated stellar system  of N members can be written down as, formula_11 Here in the N-body system, any individual member, formula_12 is influenced by the gravitational potentials of the remaining formula_13 members. In practice, except for in the highest performance computer simulations, it is not feasible to calculate rigorously the future of a large N system this way. Also this EOM gives very little intuition. Historically, the methods utilised in stellar dynamics originated from the fields of both classical mechanics and statistical mechanics. In essence, the fundamental problem of stellar dynamics is the N-body problem, where the N members refer to  the members of a given stellar system. Given the large number of objects in a stellar system, stellar dynamics can address both the global, statistical properties of many orbits as well as the specific data on the positions and velocities of individual orbits. 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. The gravitational potential, formula_14, of a system is related to the acceleration and the gravitational field, formula_15 by: formula_16 whereas the potential is related to a (smoothened) mass density, formula_17, via the Poisson's equation in the integral form formula_18 or the more common differential form formula_19 An example of the Poisson Equation and escape speed in a uniform sphere! Consider  an analytically smooth spherical potential formula_20 where formula_21 takes the meaning of the speed to "escape to the edge" formula_22, and formula_23 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 formula_24 by computing the corresponding density using the spherical Poisson Equation formula_25 where the enclosed mass formula_26 Hence the potential model corresponds to a uniform sphere of radius formula_27, total mass formula_28 with formula_29 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, tidal forces, and dynamical friction that influence the trajectories of the system's members. Relativistic Approximations. There are three related approximations made in the Newtonian EOM and Poisson Equation above. SR and GR. Firstly above equations neglect  relativistic corrections, which are of order of formula_30 as typical stellar 3-dimensional speed, formula_31 km/s, is much below the speed of light. Eddington Limit. Secondly non-gravitational force is typically negligible in stellar systems. For example, in the vicinity of a typical star the ratio of radiation-to-gravity force on a hydrogen atom or ion, formula_32 hence radiation force is negligible in general, except perhaps around a luminous O-type star of mass formula_33, or around a black hole accreting gas at the Eddington limit so that its luminosity-to-mass ratio formula_34 is defined by formula_35 . Loss cone. Thirdly a star can be swallowed if coming within a few Schwarzschild radii of the black hole. This radius of Loss is given by formula_36 The loss cone can be visualised by considering infalling particles aiming to the black hole within a small solid angle (a cone in velocity). These particle with small formula_37 have small angular momentum per unit mass formula_38 Their small angular momentum (due to ) does not make a high enough barrier near formula_39 to force the particle to turn around. The effective potential formula_40 is always positive infinity in Newtonian gravity. However, in GR , it nosedives to minus infinity near formula_41 if formula_42 Sparing a rigorous GR treatment, one can verify this formula_43 by computing the last stable circular orbit, where the effective potential is at an inflection point formula_44 using an approximate classical potential of a Schwarzschild black hole formula_45 Tidal disruption radius! A star can be tidally torn by a heavier black hole when coming within the so-called Hill's radius of the black hole, inside which a star's surface gravity yields to the tidal force from the black hole, i.e., formula_46 For typical black holes of formula_47 the destruction  radius formula_48 where 0.001pc is the stellar spacing in the densest stellar systems (e.g., the nuclear star cluster in the Milky Way centre). Hence (main sequence) stars are generally too compact internally and too far apart spaced to be disrupted by even the strongest black hole tides in galaxy or cluster environment. Radius of sphere of influence! A particle of mass formula_49 with a relative speed V will be deflected when entering the (much larger) cross section formula_50 of a black hole. This so-called sphere of influence is loosely defined by, up to a Q-like fudge factor formula_51 , formula_52 hence for a Sun-like star we have, formula_53 i.e., stars will neither be tidally disrupted nor physically hit/swallowed in a typical encounter with the black hole thanks to the high surface escape speed formula_54 from any solar mass star, comparable to the internal speed between galaxies in the Bullet Cluster of galaxies, and greater than the typical internal speed formula_55 inside all star clusters and in galaxies. Connections between star loss cone and gravitational gas accretion physics. First consider a heavy black hole of mass formula_56 is moving through a dissipational gas of (rescaled) thermal sound speed  formula_57 and density formula_58, then every gas particle of mass m will likely transfer its relative momentum formula_59 to the BH when coming within a cross-section of radius formula_60 In a time scale formula_61 that the black hole loses half of its streaming velocity, its mass may double by Bondi accretion, a process of capturing most of gas particles that enter its sphere of influence formula_62, dissipate kinetic energy by gas collisions and fall in the black hole. The gas capture rate is formula_63 where the polytropic index formula_64 is the sound speed in units of velocity dispersion squared , and the rescaled sound speed formula_65, allows us to match the Bondi spherical accretion rate, formula_66 for the adiabatic gas formula_67, compared to formula_68 of the isothermal case formula_69. Coming back to star tidal disruption and star capture by a (moving) black hole, setting formula_70, we could summarise the BH's growth rate from gas and stars, formula_71 with, formula_72 because the black hole consumes a fractional/most part of star/gas particles passing its sphere of influence. Gravitational dynamical friction. Consider the case that a heavy black hole of mass formula_56 moves relative to a background of stars in  random motion in a cluster of total mass formula_74 with a mean number density formula_75 within a typical size formula_76. Intuition says that gravity causes the light bodies to accelerate and gain momentum and kinetic energy (see slingshot effect). By conservation of energy and momentum, we may conclude that the heavier body will be slowed by an amount to compensate. Since there is a loss of momentum and kinetic energy for the body under consideration, the effect is called dynamical friction. After certain time of relaxations the heavy black hole's kinetic energy should be in equal partition with the  less-massive background objects. The slow-down of the black hole can be described as formula_77 where formula_78 is called a dynamical friction time. Dynamical friction time vs Crossing time in a virialised system! Consider a Mach-1 BH, which travels initially at the sound speed formula_79, hence its Bondi radius formula_62 satisfies formula_81 where the sound speed is formula_82 with the prefactor formula_83 fixed by the fact that for a uniform spherical cluster of the mass density formula_84, half of a circular period is the time for "sound" to make a oneway crossing in its longest dimension, i.e., formula_85 It  is customary to call the "half-diameter" crossing time formula_86 the dynamical time scale. Assume the BH stops after traveling a length of formula_87 with its momentum formula_88 deposited to formula_89 stars in its path, then the probability of a star being influenced by the BH's Bondi cross section is formula_90 More generally, the Equation of Motion of the BH at a general velocity formula_91 in the potential formula_92 of a sea of stars can be written as formula_93 where formula_94 is the number of deflection per "diameter" crossing time, formula_95 and the Coulomb logarithm modifying factor formula_96 discounts  friction on a supersonic moving BH with mass formula_97. As a rule of thumb, it takes about a sound crossing formula_98 time to "sink" subsonic BHs, from the edge to the centre without overshooting, if they weigh more than 1/8th of the total cluster mass. Lighter and faster holes can stay afloat much longer. More rigorous formulation of dynamical friction. The full Chandrasekhar dynamical friction formula for the change in velocity of the object involves integrating over the phase space density of the field of matter and is far from transparent. It reads as formula_99 where formula_100 is the number  of particles in an infinitesimal cylindrical volume of length formula_101 and a cross-section formula_102 within the black hole's sphere of influence. Like the "Couloumb logarithm" formula_103 factors in the contribution of distant background particles, here the factor formula_104 also factors in the probability of finding a background slower-than-BH particle to contribute to the drag. The more particles are overtaken by the BH, the more particles drag the BH, and the greater is formula_104. Also the bigger the system, the greater is formula_103. A background of elementary (gas or dark) particles can also induce dynamical friction, which  scales with the mass density of the surrounding medium, formula_107; the lower particle mass m is compensated by the higher number density n. The more massive the object, the more matter will be pulled into the wake. Summing up the gravitational drag of both collisional gas and collisionless stars, we have formula_108 Here the "beaten" fraction for gas and for stars are given by formula_109 where the factor formula_110 is the number of handshakes between a pair of stars without double-counting, the mean pair separation formula_111 is only about 40\% of the radius of the uniform sphere. Note also  the similarity of the formula_112 Mean free path! The mean free path of strong encounters in a typically formula_113 stellar system is then formula_114 i.e., it takes about formula_115 radius crossings for a typical star to come within a cross-section formula_116 to be deflected from its path completely. Hence the mean free time of a strong encounter is much longer than the crossing time formula_117. Weak encounters. Weak encounters have a more profound effect on the evolution of a stellar system over the course of many passages. The effects of gravitational encounters can be studied with the concept of  relaxation time. A simple example illustrating relaxation is two-body relaxation, where a star's orbit is altered due to the gravitational interaction with another star. Initially, the subject star travels along an orbit with initial velocity, formula_118, that is perpendicular to the impact parameter, the distance of closest approach, to the field star whose gravitational field will affect the original orbit. Using Newton's laws, the change in the subject star's velocity, formula_119, is approximately equal to the acceleration at the impact parameter, multiplied by the time duration of the acceleration. The relaxation time can be thought as  the time it takes for formula_119 to equal formula_118, or the time it takes for the small deviations in velocity to equal the star's initial velocity. The number of "half-diameter" crossings for an average star to relax in a stellar system of formula_122 objects is approximately formula_123 from a more rigorous calculation than the above mean free time estimates for strong deflection. The answer makes sense because there is no relaxation for a single body or 2-body system. A better approximation of the ratio of timescales is formula_124, hence the relaxation time for 3-body, 4-body , 5-body, 7-body, 10-body, ..., 42-body, 72-body, 140-body, 210-body, 550-body are about 16, 8, 6, 4, 3, ..., 3, 4, 6, 8, 16 crossings. There is no relaxation for an isolated binary, and the relaxation is the fastest for a 16-body system; it takes about 2.5 crossings for orbits to scatter each other. A system with formula_125 have much smoother potential, typically takes formula_126 weak encounters to build a strong deflection to change orbital energy significantly. Relation between friction and relaxation. Clearly that the dynamical friction of a black hole is much faster than  the relaxation time by roughly a factor formula_127, but these two are very similar for a cluster of black holes, formula_128 For a star cluster or galaxy cluster with, say, formula_129, we have formula_130. Hence encounters of members in these stellar or galaxy clusters are significant during the typical 10 Gyr lifetime. On the other hand, typical galaxy with, say, formula_131 stars, would have a crossing time formula_132 and their relaxation time is much longer than the age of the Universe. This justifies modelling galaxy potentials with mathematically smooth functions, neglecting two-body encounters throughout the lifetime of typical galaxies . And inside such a typical galaxy the dynamical friction and accretion on stellar black holes over a 10-Gyr Hubble time change the black hole's velocity and mass by only an insignificant fraction formula_133 if the black hole makes up less than 0.1% of the total galaxy mass formula_134. Especially when formula_135, we see that a typical star never experiences an encounter, hence stays on its orbit in a smooth galaxy potential. The dynamical friction or relaxation time identifies collisionless vs. collisional particle systems. Dynamics on timescales much less than the relaxation time is effectively collisionless because typical star  will deviate from its initial orbit size by a tiny fraction formula_136. They are also identified as systems where subject stars interact with a smooth gravitational potential as opposed to the sum of point-mass potentials. The accumulated effects of two-body relaxation in a galaxy can lead to what is known as mass segregation, where more massive stars gather near the center of clusters, while the less massive ones are pushed towards the outer parts of the cluster. A Spherical-Cow Summary of Continuity Eq. in Collisional and Collisionless Processes! Having gone through the details of the rather complex  interactions of particles in a gravitational system, it is always helpful to zoom out and extract some generic theme, at an affordable price of rigour, so carry on with a lighter load. First important concept is "gravity balancing motion" near the perturber and for the background as a whole formula_137 by consistently "omitting" all factors of unity formula_138, formula_139, formula_140 etc for clarity, "approximating" the combined mass formula_141 and being "ambiguous" whether the "geometry" of the system is a thin/thick gas/stellar disk or a (non)-uniform stellar/dark sphere with or without a boundary, and about the "subtle distinctions  among the kinetic energies from the local Circular rotation speed formula_142, radial infall speed formula_143, globally isotropic or anisotropic random motion formula_144 in one or three directions, or the (non)-uniform isotropic Sound speed formula_145 to emphasize of the logic" behind the order of magnitude of the friction time scale. Second we can with high confidence recap or "very loosely summarise" the various processes so far of collisional and collisionless gas/star or dark matter by Spherical cow style "Continuity Equation on any generic quantity Q" of the system: formula_146 where the formula_147 sign is generally negative except for the (accreting ) mass M, and the Mean free path formula_148 or the friction time formula_61 can be due to direct molecular viscosity from a physical collision Cross section, or due to gravitational scattering (bending/focusing/Sling shot) of particles; generally the influenced area is the greatest of the competing processes of Bondi accretion, Tidal disruption, and Loss cone capture, formula_150 E.g., in case Q is the perturber's mass formula_151, then we can estimate the Dynamical friction time via the (gas/star) Accretion rate formula_152 where we have applied the relations motion-balancing-gravity. In the limit the perturber is just 1  of the N background particle, formula_153, this friction time is identified with the (gravitational) Relaxation time. Again all Coulomb logarithm etc are suppressed without changing the estimations from these qualitative equations. For the rest of Stellar dynamics, we will consistently work on "precise" calculations through primarily "Worked Examples", by neglecting gravitational friction and relaxation of the perturber, working in the limit formula_154 as approximated true in most galaxies on the 14Gyrs Hubble time scale, even though this is sometimes violated for some clusters of stars or clusters of galaxies.of the cluster. A concise 1-page summary of some main equations  in Stellar dynamics and Accretion disc physics are shown here, where one attempts to be more rigorous on the qualitative equations above. Connections to statistical mechanics and plasma physics. The statistical nature of stellar dynamics originates from the application of the kinetic theory of gases to stellar systems by physicists such as James Jeans in the early 20th century. The Jeans equations, which describe the time evolution of a system of stars in a gravitational field, are analogous to Euler's equations for an ideal fluid, and were derived from the collisionless Boltzmann equation. This was originally developed by Ludwig  Boltzmann to describe the non-equilibrium behavior of a thermodynamic system. Similarly to statistical mechanics, stellar dynamics make use of distribution functions that encapsulate the information of a stellar system in a probabilistic manner. The single particle phase-space distribution function, formula_155, is defined in a way such that formula_156 where formula_157 represents the probability of finding a given star with position formula_158 around a differential volume formula_159 and velocity formula_160 around a differential velocity space volume formula_161. The distribution function is normalized (sometimes) such that integrating it over all positions and velocities will equal N, the total number of  bodies of the system. For collisional systems, Liouville's theorem is applied to study the microstate of a stellar system, and is also commonly used to study the different statistical ensembles of statistical mechanics. Convention and notation in case of a thermal distribution! In most of stellar dynamics literature, it is convenient to adopt the convention that the particle mass is unity in solar mass unit formula_162, hence a particle's momentum and velocity are identical, i.e., formula_163 formula_164 For example, the thermal velocity distribution of air molecules (of typically 15 times the proton mass per molecule) in a room  of constant temperature formula_165 would have a Maxwell distribution formula_166 formula_167 where the energy per unit mass formula_168 where formula_169 and formula_170 is the width of the velocity Maxwell distribution, identical in each direction and everywhere in the room, and the normalisation constant formula_171 (assume the chemical potential formula_172 such that the Fermi-Dirac distribution reduces to a Maxwell velocity distribution) is fixed by the constant gas number density formula_173 at the floor level, where formula_174 formula_175 The CBE! In plasma physics, the collisionless Boltzmann equation is referred to as the Vlasov equation, which is used to study the time  evolution of a plasma's distribution function. The Boltzmann equation is often written more generally with the Liouville operator formula_176 as formula_177 formula_178 where formula_179 is the gravitational force and formula_180 is the Maxwell (equipartition) distribution (to fit the same density, same mean and rms velocity as formula_181). The equation means the non-Gaussianity will decay on a (relaxation) time scale of formula_182, and the system will ultimately relaxes to a Maxwell (equipartition) distribution. Whereas Jeans applied the collisionless Boltzmann equation, along with Poisson's equation, to a system of stars interacting via the long range force of gravity, Anatoly  Vlasov applied Boltzmann's equation with Maxwell's equations to a system of particles interacting via the Coulomb Force. Both approaches separate themselves from the kinetic theory of gases by introducing long-range forces to study the long term evolution of a many particle system. In addition to the Vlasov equation, the concept of Landau damping in plasmas was applied to gravitational systems by Donald Lynden-Bell to describe the effects of damping in spherical stellar systems. A nice property of f(t,x,v) is that many other dynamical quantities can be formed by its moments, e.g., the total mass , local density, pressure, and mean velocity. Applying the collisionless Boltzmann equation, these moments are then related by various forms of continuity equations, of which most notable are the Jeans equations and Virial theorem. Probability-weighted moments and hydrostatic equilibrium! Jeans computed the weighted velocity of the Boltzmann Equation after integrating over velocity space formula_183 and obtain the Momentum (Jeans) Eqs. of a formula_184opulation (e.g., gas, stars, dark matter): formula_185 The general version of Jeans equation, involving (3 x 3) velocity moments is cumbersome. It only becomes useful or solvable if we could drop some of these moments, epecially drop the  off-diagonal cross terms for systems of high symmetry, and also drop net rotation or net inflow speed everywhere. The isotropic version is also called Hydrostatic equilibrium equation where balancing pressure gradient with gravity; the isotropic version works for axisymmetric disks as well, after replacing the derivative dr with vertical coordinate dz. It means that we could measure the gravity (of dark matter) by observing the gradients of the velocity dispersion and the number density of stars. Applications and examples. Stellar dynamics is primarily used to study the mass distributions within stellar systems and galaxies. Early examples of applying stellar  dynamics to clusters include Albert Einstein's 1921 paper applying the virial theorem to spherical star clusters and Fritz Zwicky's 1933 paper applying the virial theorem specifically to the Coma Cluster, which was one of the original harbingers of the idea of dark matter in the universe. The Jeans equations have been used to understand different observational data of stellar motions in the Milky Way galaxy. For example, Jan Oort utilized the Jeans equations to determine the average matter density in the vicinity of the solar neighborhood, whereas the concept of asymmetric drift came from studying the Jeans equations  in cylindrical coordinates. Stellar dynamics also provides insight into the structure of galaxy formation and evolution. Dynamical models and observations are used to study the triaxial structure of elliptical galaxies and suggest that prominent spiral galaxies are created from galaxy mergers. Stellar dynamical models are also used to study the evolution of active galactic nuclei and their black holes, as well as to estimate the mass distribution of dark matter in galaxies. A unified thick disk potential. Consider an oblate potential in cylindrical coordinates formula_186 where formula_187 are (positive) vertical and radial length scales. Despite its complexity, we can easily  see some limiting properties of the model. First we can see the total mass of the system is formula_188 because formula_189 when we take the large radii limit formula_190, so that formula_191 which implies formula_192, and formula_193 between zero and formula_24. To verify the above formula_195 being constants of motion in our spherical potential, we note formula_196 formula_197 for any "steady state" potential. formula_198 which reduces to formula_199 around the z-axis of any axisymmetric potential, where formula_200. Likewise the x and y components of the angular momentum are also conserved for a spherical potential. Hence formula_201. So for any  time-independent spherical potential (including our uniform sphere model), the orbital energy E and angular momentum J and its z-component Jz along every stellar orbit satisfy formula_202 Hence using the chain rule, we have formula_203 i.e., formula_204, so that CBE is satisfied, i.e., our formula_205 is a solution to the Collisionless Boltzmann Equation for our static spherical potential. A worked example on moments of distribution functions in a uniform spherical cluster! We can find out various moments of the above distribution function, reformatted as with the help of three Heaviside functions, formula_206 as required by the Virial Theorem. For  this self-gravitating sphere, we can also verify that the virial per unit mass equals the averages of half of the potential formula_207 Hence we have verified the validity of Virial Theorem for a uniform sphere under self-gravity, i.e., the gravity due to the mass density of the stars is also the gravity that stars move in self-consistently; no additional dark matter halo contributes to its potential, for example. A worked example of Jeans Equation in a uniform sphere! Jeans Equation is a relation on how the pressure gradient of a system should be balancing the potential gradient  for an equilibrium galaxy. In our uniform sphere, the potential gradient or gravity is formula_208 The radial pressure gradient formula_209 The reason for the discrepancy is partly due to centrifugal force formula_210 and partly due to anisotropic pressure formula_211 so formula_212 at the very centre, but the two balance at radius formula_213, and then reverse to formula_214 at the very edge. Now we can verify that formula_215 Here the 1st line above is essentially the Jeans equation in the r-direction, which reduces to the 2nd line, the Jeans equation in an anisotropic (aka formula_216) rotational (aka formula_217) axisymmetric (formula_218  ) sphere (aka formula_219) after much coordinate manipulations of the dispersion tensor; similar equation of motion can be obtained for the two tangential direction, e.g., formula_220, which are useful in modelling ocean currents on the rotating earth surface or angular momentum transfer in accretion disks, where the frictional term formula_221 is important. The fact that the l.h.s. formula_222 means that the force is balanced on the r.h.s. for this uniform (aka formula_223) spherical model of a galaxy (cluster) to stay in a steady state (aka time-independent equilibrium formula_224 everywhere) statically (aka with zero flow formula_225 everywhere). Note systems like accretion  disk can have a steady net radial inflow formula_226 everywhere at all time. A worked example of Jeans equation in a thick disk. Consider again the thick disk potential in the above example. If the density is that of a gas fluid, then the pressure would be zero at the boundary formula_227. To find the peak of the pressure, we note that formula_228 So the fluid temperature per unit mass, i.e., the 1-dimensional velocity dispersion squared would be formula_229 formula_230 Along the rotational z-axis, formula_231 formula_232 which is clearly the highest at the centre and zero at the  boundaries formula_227. Both the pressure and the dispersion peak at the midplane formula_234. In fact the hottest and densest point is the centre, where formula_235 A recap on worked examples on Jeans Eq., Virial and Phase space density! Having looking at the a few applications of Poisson Eq. and Phase space density and especially the Jeans equation, we can extract a general theme, again using the Spherical cow approach. Jeans equation links gravity with pressure gradient, it is a generalisation of the Eq. of Motion for single particles. While Jeans equation can be solved in disk systems, the most user -friendly version of the Jeans eq. is the spherical anisotropic version for a static formula_236 frictionless system formula_237, hence the local velocity spead formula_238 everywhere for each of the three directions formula_239. One can project the phase space into these moments, which is easily if in a highly spherical system, which admits conservations of energy formula_240 and angular momentum J. The boundary of the system sets the integration range of the velocity bound in the system. In summary, in the spherical Jeans eq., formula_241 which matches the expectation from the Virial theorem formula_242, or in other words, the formula_243 kinetic  energy of an equilibrium equals the average kinetic energy on circular orbits with purely transverse motion.