Statistical mechanics of gravitating gas like galaxy
aa r X i v : . [ a s t r o - ph . GA ] J un MNRAS , 1–7 (2017) Preprint 14 September 2018 Compiled using MNRAS L A TEX style file v3.0
Statistical mechanics of gravitating gas like galaxy
Alexander B. Kashuba, ⋆ Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The most probable state of an infinite self-gravitating gas in the dynamical equi-librium is defined by ‘gravitational haziness’, a parameter representing many-bodyeffects and formally like the temperature in the case of thermal equilibrium. A kineticequation is constructed using a concept of statistical equipartition of the virial amongsubsystems of the self-gravitating gas. A closed equation for the gravitational potentialis conjectured as a special property of the kinetic equation. An equilibrium particledistribution function in the phase space, an analog of the Maxwell-Boltzmann weight,and a galaxy equation of state are found for all ‘gravitational haziness’. The first lawof a ‘hazydynamics’ (thermodynamics) states that the total mass of an astronomicalstellar collection is the sum of the Archimedes displaced mass and an excess ‘gobbled’mass determined by the ‘gravitational haziness’ and history.
Key words: galaxies: kinematics and dynamics, elliptical and lenticular, cD
Stellar dynamics, governed by the Newtonian gravity, hasdeveloped over the last century into a matured subjectSaslaw (1985). Noticeably, Jeans equation guides us throughcomplicated anisotropic kinematic properties of star clustersas discovered by Oort Saslaw (1985). Zwicky has used thevirial equilibrium to evaluate the dark matter in galaxiesZwicky (1933). Recently, computer simulations Barnes & Hut(1986) reveal novel and important details of the stellar dy-namics like the binary star formation, the core oscillation,the mass segregation, the star evaporation and the core col-lapse Spitzer (1987). Yet, a microscopic statistical mechan-ics capable to predict macroscopic properties of the globularclusters, elliptical galaxies and clusters of galaxies is lack-ing. Instead, astronomers are still relying on the empiricalde Vaucouleurs law for the galaxy surface brightness whereaswealth of data on globular clusters Harris (1996) is coded interms of the King’s model King (1966).This paper seeks to connect a more rigorous microscopicstatistical mechanics theory of the self-gravitating gas andthe astronomy. The random matrix theory Wigner (1957);Dyson (1962) and the spin glasses Binder & Young (1985) aremajor examples of strongly correlated systems with all roundinteractions irrespective of the distance. In the stellar dy-namics a similar collective phenomenon, the violent relax-ation Lynden-Bell (1967), has been observed in numericalsimulations. The state of a system in the dynamical equi-librium is distinct from the state in the thermal equilibrium ⋆ E-mail: alexander − [email protected] Staniscia et al (2010). Building on that, this paper finds akinetic equation with the kernel encoding both the strongcorrelations between and the equipartition of the Clausiusvirial among subsystems of the self-gravitating gas. An equi-librium particle distribution function in the phase space, thesolution of the kinetic equation and analog of the Maxwell-Boltzmann weight, is found to depend on a gravitationalhaziness, a variable formally like the temperature. Unlikecommon view as being finite, galaxies in this paper are in-finite pressurized intrinsically and disposed for an expan-sion. The implications for the astronomy are far-reaching.For elliptical galaxies and globular clusters, a new equa-tion of state, an analog of the Clapeyron gas equation, isfound. The surface brightness and the anisotropic velocityprofiles, new relations for rotation vs size, the core mass vsthe overall mass and the surface brightness vs pressure aswell as the super-massive black hole in the centre of a galaxyGenzel et al (1994); McConnell et al (2011) are all discussedin this paper.
The gravitational potential near a moving particle, a star,of the self-gravitating gas can be divided into a smooth part Φ (® r ) and a time dependent part φ ( ) (® r ( t )) rapidly fluctuatingin the co-moving frame. Many particles contribute to bothpotentials. For simplicity, let the gas of the size R consists of N particles with the same unit mass. Among particles, thenearest neighbor, constantly changing its identity, applies © A. Kashuba the random force with a typical dispersion: h (cid:16) ®∇ φ ( ) ( t ) (cid:17) i ∼ G / a ∼ N G ρ (® r ) Φ (® r ) Ra , (1)where a is the mean distance between the nearest particles.The average gravitational potential and the average density ρ (® r ) are related by the Poisson’s equation: ®∇ Φ (® r ) + π G ρ (® r ) = , (2)where G is the Newton constant. In statistical mechanics Ifind convenient to invert the sign of the gravitational poten-tial. All particles in the gas can be divided into N spheres ofinfluence with respect to a given particle. k -th sphere of in-fluence comprises approximately k particles at the distance k / a away. The amplitude of their random force depends ontheir mutual correlations while being isotropic on average.Inspired by unique cooperative phenomena when all partsof the system interact equally as described by the randommatrix theory Wigner (1957); Dyson (1962) and the spinglass theory Binder & Young (1985), I assume that the ran-dom force incoming from each sphere of influence is approx-imately the same and, therefore, simply multiply Eq.(1) by N . Also, the time it takes for a particle to cross the whole gasin a random wobbly trajectory is approximately the same asthe period of a classical mechanics orbit in the smooth po-tential Φ (® r ) Schwarzschild (1979). Thus, the time-dependentfluctuating force has a mean square: h∇ α φ ( ) ( t )∇ β φ ( ) ( t ′ )i = p Θ G ρ (® r ) Φ (® r ) δ αβ δ ( t − t ′ ) (3)where Θ is a dimensionless, velocity diffusion constant con-trolling the meandering of the real trajectory away from theorbit. Throughout this paper it is called the gravitationalhaziness. In terms of the Langevin stochastic dissipative dy-namics Kadanoff (2000) of a chosen particle: d ® rdt = ® v , d ® v dt = ®∇ Φ (® r ) + ®∇ φ ( ) ( t ) + ® f drag (® v , ® r ) , (4)this fluctuation force is generated by the stochastic actionMartin et al (1978); Dominicis & Peliti (1978): S [ φ ( ) ] = √ Θ ∫ dt p G ρ (® r ( t )) Φ (® r ( t )) ®∇ φ ( ) · ®∇ φ ( ) . (5)In neutral plasmas the inter-particle correlations are lim-ited spatially to within the Debye radius and the BalescuLenard kinetic equation determines the drag force. Simi-larly, for the self-gravitating gas the gravitational drag wasfound Chandrasekhar (1965). Unlike the non-linear BalescuLenard equation, the effective kinetic equation in the self-gravitating gas strongly correlated by all round interactionsis intrinsically one-particle and linear. Remember that inspin glasses the effective Thouless Anderson Palmer equa-tion describes one spin in the effective field and it solvesthe problem exactly Binder & Young (1985). Also, the equa-tion for the density of states in the random matrix theory islinear too and is exact for the large sizes of matrices.In thermal equilibrium the gravitational drag and therandom force fluctuations, Eq.(5), are related by the fluc-tuation dissipation theorem Callen & Welton (1951), thus,completing the definition of the Langevin dynamics Eq.(4).However, in the dynamical equilibrium it is rather thevirial equipartition that relates them. For the one-particle Langevin stochastic dissipative dynamics there are two ef-fective fields: the momentum diffusion Eq.(5) and the effec-tive gravitational drag different from the Chandrasekhar’sone. With all these in mind, the kinetic equation for theparticle distribution function ψ (® r , ® v ) in the phase space, theFokker-Plank equation Kadanoff (2000), reads: = ∂∂ t ψ + ® v · ∂∂ ® r ψ + d Φ d ® r · ∂∂ ® v ψ − √ Θ p G ρ (® r ( t )) ∂∂ v α (cid:18) L αβ ( v , Φ ) ∂∂ v β ψ + v α ψ (cid:19) , (6)where the upper line is the Liouville’s part and L αβ ( v , Φ ) isthe mean-field velocity diffusion kernel. Both L and ψ areunknown in Eq.(6) and this ambiguity will be dealt withby imposing physical conditions later on. The gravitationalhaziness Θ enters the kinetic equation in the same way asthe temperature does for a gas composed of large heavy par-ticles moving in the viscous gas of light particles and wherethe Stokes friction law applies. Perhaps similarly, stars aremoving through a haze of the gravitational field fluctuationscoupled to the density fluctuations. This form of the kineticequation, Eq.(6), does allow for the direct exchange of theheat between the gas of particles and the viscous gas and theentropy of the gas of particles can change both way unlikethe case of the Boltzmann H-theorem.The divergence form of the last term in the kinetic equa-tion, Eq.(6), ensures the conservation of the total number ofparticles, and hence the mass, of the gas: ∂ ρ∂ t + ∂∂ ® r · ρ ® u = . (7)The particle distribution function is normalized such as togive the following average mass density and the average localvelocity of the gas in motion: ρ (® r ) = ∫ ψ (® r , ® v ) d ® v , ρ (® r ) ® u (® r ) = ∫ ® v ψ (® r , ® v ) d ® v . (8)The most probable state of the self-gravitating gas in thedynamical equilibrium is the state in rest: ® u (® r ) = .The kernel L αβ ( v , Φ ) , already representing effects ofstrong correlations between remote subsystems, can be de-fined using a concept of local virialization of gas subsys-tems. Unlike the Maxwell-Boltzmann gas, a single particlehere is too small to form a subsystem. In the self-gravitatinggas, a subsystem possesses a property of local density asthe far away subsystems interact via the gravitational force.The Gibbs’s statistical physics is founded on the concept ofequipartition of the conserved properties, like energy, mass,charge, element, momentum or angular momentum, amongall subsystems Kadanoff (2000). Similarly, the total virial ofthe self-gravitating gas can be subdivided into the sum ofvirials of the local gas subsystems: T + U = ∫ d ® r ρ (® r ) (cid:18) h® v i(® r ) − Φ (® r ) (cid:19) , (9)where the double counting of the interaction is excluded.The mechanical global virial theorem states that T + U = .The equipartition of the virial is a stronger local statisticalphysics condition: h® v i(® r ) = Φ (® r ) . (10)For thick gravitational haziness, Θ → ∞ , the last line in MNRAS , 1–7 (2017) tatistical mechanics of galaxy Eq.(6) should vanish separately. A simple kernel: L αβ = δ αβ Φ (® r )/ , satisfies the virial theorem and the equiparti-tion of the virial, Eq.(10). In this leading approximation,the particle distribution function is the Maxwell’s ones: ψ (® v ) ∼ ρ (® r ) Φ − / exp (− ® v / Φ ) . Erroneously, I searched fora solution of the kinetic equation with this kernel usingMaxwell’s PDF multiplied by polynomials in the velocity.Whole Eq.(6) does allow for such a solution order by order,up to the fourth order terms, / Θ , satisfying the gas in restcondition. However, the kernel L αβ ( v , Φ ) has to be modifiedin the third order and such solution violates the time re-versal symmetry and is nonphysical. In this exercise makingwrong choices quickly leads to a contradiction whereas byimposing the following equation on the potential: ®∇ Φ + Φ + / Θ + Θ (cid:16) ®∇ Φ (cid:17) Φ = , (11)the longest series can be constructed. It is simply the Pois-son isotropic equation Eq.(2) for the gravitational potentialwhere the density of the gas, ρ (® r ) , is the sum of the last twoterms divided by π G . By writing the density in this form theequation of state of the self-gravitating gas is stated implic-itly. Despite Eq.(11) depends on one parameter Θ , the gravi-tational haziness, let the coefficient in front of the third termbe independent. If it is zero then Eq.(11) is the Lane-Emdenequation with the exponent k = + / Θ . For < k < , theLane-Emden equation describes the thermodynamic equilib-rium of a star of finite mass Chandrasekhar (1965). For therelativistic equation of state at k = , only the equilibriumat the Chandrasekhar mass limit is possible. Unlike that,the Lane-Emden equation for < k < ∞ describes an infi-nite collection of mass with the potential being the sum ofa regular and an oscillating parts. The regular part falls offat large distance as r − Θ /( + Θ ) . The third term, if switchedon, suppresses the oscillating part until it disappears alto-gether for Eq.(11). At even stronger third term the solutiondescribes a finite mass. In the parameter space, Eq.(11) rep-resents a special critical line. Its physical infinite isotropicsolution is: Φ (® r ) = (cid:18) + + Θ Θ ® r (cid:19) − Θ + Θ , (12)for the average gravitational potential of the gas and πρ (® r ) = (cid:18) + ® r Θ (cid:19) (cid:18) + + Θ Θ ® r (cid:19) − + Θ + Θ , (13)for the average density of the gas. Both are functions ofthe re-scaled radius vector ® r and depend on the three pa-rameters: the gravitational haziness Θ , the core radius andthe central density both set to one. For thick gravitationalhaziness, the density profile Eq.(13), shows also an interme-diate crossover radius r mid = √ Θ , much larger than thecore radius. Within r mid the density profile follows that ofthe Plummer model whereas the gravitational potential isthe Newtonian one. The case of thin gravitational haziness, Θ → , though, is more close to the thermodynamic equi-librium. The dynamical equilibrium is intrinsically unstableand it evolves towards the thermal equilibrium eventuallyStaniscia et al (2010).After this preliminary, a solution of the kinetic equation Eq.(6) can be written as the generating series: ψ (® r , ® v ) = ∫ d ® q ( π ) e i ® q ·® v (cid:18) ρ − q a q b T ab ( ) + i q a q b q c T abc ( ) + ... (cid:19) , (14)where the coefficients depends on the coordinates inside thegas and represent the moments of the velocity distribution: T ab ... c ( n ) (® r ) = ∫ v a v b ... v c ψ (® r , ® v ) d ® v . (15)The first coefficient, T a ( ) (® r ) = ρ u a , is zero for the gas in rest.Eq.(6) with the kernel: L ab ( v , Φ ) = Φ (® r ) δ ab / ++ ∂∂ v c (cid:18) v a v b v c v d − ® v A abcd (® v ) + ® v B abcd (cid:19) ∂∂ v d , (16)annihilates the next odd coefficient: T abc ( ) (® r ) = . Here, thenotations: B abcd = δ ab δ cd + δ ac δ bd + δ ad δ bc and A abcd ( ® x ) = x a x b δ cd + x a x c δ bd + x a x d δ bc ++ δ ab x c x d + δ ac x b x d + δ ad x b x c , (17)where ® x is either ® r or ® v , are used. Thus, the kinetic equationorder by order of the velocity moments expansion is reducedto a sequence of simpler equations. The first of them, thehydrostatic equilibrium equation, reads: ∂ a T ab ( ) = ρ (® r ) ∂ b Φ , (18)where the gradient of the Jeans pressure on the left handside balances the gravitational pull on the right hand side.The solution of Eq.(18) satisfying the virial equipartitioncondition: T aa ( ) (® r ) = ρ (® r ) Φ (® r )/ , reads: T ab ( ) (® r ) = ρ (® r ) Φ (® r ) δ ab + r a r b / Θ + ® r / Θ , (19)where the gravitational potential and the density are givenby Eqs.(12,13). Divided by the density, the Jeans pressuretensor, Eq.(19), gives the velocity dispersion as the functionof coordinates. It reveals a kinematic anisotropy between theradial and the transverse velocity dispersions. While beingequal inside r mid they drastically diverge for r > r mid . Onthe outskirts of the gas particles follow the high apogee or-bits. Therefore, r mid may be called the kinematic anisotropyradius, r anis = r mid . For thin gravitational haziness, Θ → ,the average particle motions degenerate into the straight ra-dial ones almost everywhere. Projected velocity dispersionprofiles for elliptical galaxies have been measured using theplanetary nebula spectrograph Romanowsky et al (2003) inqualitative agreement with Eq.(19).The second reduced kinetic equation reads: ∂ d T abcd ( ) = T ab ( ) ∂ c Φ + T ca ( ) ∂ b Φ + T bc ( ) ∂ a Φ . (20)Its solution is not unique and we need some physical condi-tion. For large distances away from the centre, r ≫ , thestellar motions degenerates into the straight radial lines witha negligible velocity diffusion. Thus, the stationary and col-lisionless kinetic equation, the upper line of Eq.(6), appliesat r ≫ . Its solution depends largely on the particle energynear the edge singularity of a semi-circular law: ψ (® r , ® v ) = Θ ( + Θ ) π ® r q Φ (® r ) − v r δ ( ) (® v ⊥ ) , (21)where the particle energy is negative, v r < Φ (® r ) , for closed MNRAS , 1–7 (2017)
A. Kashuba orbits. Eq.(21) can be conveniently rewritten in the samerepresentation as that of the series Eq.(14): ψ (® r , ® v ) = ρ (® r ) ∫ d ® q ( π ) e i ® q ·® v J (cid:18)q Φ (® r ) (cid:0) ® q · ® r (cid:1) /® r (cid:19)q Φ (® r ) (cid:0) ® q · ® r (cid:1) / ® r , (22)where r ≫ and J ( x ) is the Bessel function. For arbitrarydistances, the expansion of Eq.(22) in powers of ® q providesthe physical condition to select a proper solution of Eq.(20): T abcd ( ) (® r ) = (cid:18) + + Θ Θ ® r (cid:19) − − Θ + Θ (cid:18) A abcd (® r ) Θ + + Θ ( Θ ) r a r b r c r d + + Θ + Θ r ( + Θ ) B abcd (cid:19) . (23)The particle distribution function in the dynamical equi-librium satisfies separately the Liouville’s and the collisionparts of the kinetic equation, like it does in the thermalequilibrium. Leaving the collision kernel L , that annihilatesthe odd velocity terms, unresolved, the particle distribu-tion function in the dynamical equilibrium, an analog of theMaxwell-Boltzmann weight, can be found in all orders, usingthe Liouville’s part alone, to consists of two terms: ψ (® r , ® v ) = π ( + Θ ) q Φ (® r ) − ® v − + Θ Θ [® r × ® v ] ++ π √ π + Θ + Θ Γ (cid:16) + Θ (cid:17) Γ (cid:16) + Θ (cid:17) (cid:18) Φ (® r ) − ® v (cid:19) + Θ , (24)where the gravitational potential is given by Eq.(12). Bothterms depend only on the integrals of particle motion, theenergy and the angular momentum, in accordance with theJeans theorem. The first term vanishes for thick gravita-tional haziness whereas the last term vanishes for thin grav-itational haziness. In the momentum space the particle dis-tribution function develops an integrable divergence near asurface of an elongated ellipsoid marking the escape velocity.Physically, this critical surface in the phase space seedsstrong inter-particles correlations in the self-gravitating gasas was anticipated in the beginning of this section. Indeed, ifthere were no two-particle correlations then the usual Boltz-mann kinetic collision integral would apply:St ψ = AL G m ρ √ Φ ∂ ∂ ® v ψ = AL G m ρ √ Φ (√ Φ − |® v |) ψ, (25)where m is the typical mass of particles, ρ is the central massdensity, A ∼ is the constant, L is the so-called Coulomblogarithm and ® r is in the centre for simplicity. We see thatthe rate of relaxation is quadratically divergent upon ap-proaching the critical surface. This divergence overcomesany dilution of the gas and signals a growth of inter-particlecorrelations at least near the critical surface.The hydrodynamic equation for the self-gravitating gasin motion, an analog of the Navier-Stokes equation: ∂ u α ∂ t + ® u · ®∇ u α + ρ ∇ β t αβ ( ) = ∇ α Φ − √ Θ p G ρ u α , (26)is dominated by the many-body gravitational drag, the lastterm, unlike the local one Chandrasekhar (1965). Both thegravitational potential and the density depend on the motion in a non-linear way, see the Poisson’s and the continuityequations Eqs.(2,7). The Jeans pressure: T αβ ( ) = t αβ ( ) + ρ u α u β ,though, is determined by a secondary equation: dt αβ ( ) dt + ∂ γ t αβγ ( ) + t αβ ( ) ®∇ · ® u + t αγ ( ) ∂ γ u β + t βγ ( ) ∂ γ u α = √ Θ p G ρ (cid:18) ρ (cid:16) u α u β − ® u δ αβ (cid:17) − t γγ ( ) + ρ Φ (cid:19) . (27)And so on. The gas motion induces the odd moments T ( n + ) .For the self-gravitating gas the Navier-Stokes equation isactually a series of coupled equations encoding the equationof state in the dynamical equilibrium. In the remaining, the statistical mechanics of the self-gravitating gas will be projected into the night sky. Remem-ber that Eqs.(12,13) describe an infinite collection of massin the dynamical equilibrium and it should merge into a uni-form outer space density at large radius R . Galaxies are nodifferent from industrial cylinders containing gas, they areeither sealed or at ambient pressure. The stellar pressure ismediated by stars constantly joining or leaving a collection.For Θ → ∞ , the Plummer model has a well defined mass con-fined in the core: M core = √ . It is a fraction of the overallmass encircled by a large radius R : M core / M ( R ) = R − /( + Θ ) . (28)In astronomy the situation when M core ≈ M ( R ) is called thecore collapse. In a typical galaxy, like large elliptical galaxy,the mass resides on the outskirts. For thin gravitational hazi-ness, Θ → , the galaxy core becomes tiny. Alternatively, thetotal mass can be related to the outer space density ρ ( R ) as: M ( R ) = π R ρ ( R ) ( + Θ ) , (29)i.e. the Archimedes displaced mass times the gravitationalhaziness enhancement factor which, for Θ → ∞ , may becapped at the logarithm of the ratio of R to the core size.Mechanically, the total mass encircled by a large radius R : GM ( R ) = Θ + Θ R Φ ( R ) , (30)is proportional to the gravitation potential on the outskirts Φ ( R ) . The gravitational potential is a good measure of theline of sight velocity dispersion h v los i( r ) = Φ ( r )/ , within thekinematic anisotropy radius r anis , see Eq.(19).Let a stellar collection be confined by a spherical wall ofradius R , much larger than the kinematic anisotropy radius r anis . Let it exerts a pressure p on the wall. In this case, thegalaxy equation of state can be found from Eqs.(19,30): M ( R ) = π G Θ pR . (31)It looks like the Clapeyron thermodynamic gas equation, pV = NT , although, the pressure here builds up for Θ → .The gravitational haziness is determined by the ratio of thetotal gravitational energy, which is not extensive in the ther-modynamic sense, to the work needed to inflate the galaxy.Interestingly, at constant mass and gravitational haziness MNRAS , 1–7 (2017) tatistical mechanics of galaxy the galaxy equation of state follows the relativistic adiabaticlaw of the cosmic microwave background radiation.Alternatively, a sealed stellar collection in vacuum canbe sustained by a slow rotation of the gas as a whole ratherthan by ambient pressure. Inspecting the kinetic equationEq.(6) there are two modifications in the rotating referenceframe with a frequency Ω . First, the gravitational force ®∇ Φ is augmented by the centrifugal and the Coriolis forces. Sec-ond, the virial is no longer Φ / but rather Φ / − Ω ® r andthe force is no longer the gradient of the virial. Due to theprominence of the virial in the structure of the kinetic equa-tion, the potential Φ (® r ) is better to be redefined into thevirial. Then, the Poisson’s equation reads: ®∇ Φ + π G ρ { Φ , Ω } + Ω = , (32)where the gas density is now a functional of the virial Φ and the rotation frequency Ω . The last term comes from thedifference between the virial and the gravitational potentialand it preserves the isotropy of Eqs.(2,11). The sign of thisterm is significant. If the series of the density functional inpowers of Ω is weaker than this last term then Eq.(32) de-scribes a finite mass with a sharp boundary at some largeouter radius R ( Ω ) . For slow rotation, Ω → , it can be eval-uated using the Eq.(11) instead of the correct ρ { Φ , Ω } : R ( Ω ) ∼ Ω −( + Θ )/( + Θ ) . (33)For thick gravitational haziness, Θ → ∞ , the boundary out-skirts of the self-gravitating gas rotate according to the Ke-pler’s law whereas for thin gravitational haziness, Θ → , theboundary rotation velocity is constant. The elliptical galax-ies do slowly rotate to keep stars with smaller and lighterones rotating faster than the larger and heavier ones. Thethicker is the gravitational haziness the slower large ellip-tical galaxies rotate. Unlike that and owing to living in agalaxy halo, globular clusters are sustained by an ambienthalo density, and are typically spherical systems.Let a compact self-gravitating gas be confined to withinits core. For thick gravitational haziness this is possible, seeEq.(28). Such an object is stable and needs neither ambi-ent media no rotation. It has no outer radius and is com-pletely defined by just two parameters: the central veloc-ity dispersion, h® v i( ) , and the central energy density, ǫ ( ) .To an outside observer it will look like a central mass, M ,spreading out the Newton gravitation field Φ ( r ) = GM / r .From Eqs.(12,13,19) we find: M = r π h® v i G √− G ǫ , r core = h® v i√− π G ǫ . (34)Inspecting astronomical data on vastly disparate objects,from small cold molecular clouds near the Sun, through glob-ular clusters orbiting in the Galaxy halo to huge ellipticalgalaxies, Eqs.(34) seem to be universal and the energy den-sity ǫ is close to that of the cosmic microwave backgroundradiation. It looks like a mechanical equilibrium. Almost asif the ambient cosmic microwave background radiation, apump, pressurizes the stellar motions in a galaxy, a ball.Despite the Universe being apparently transparent the grav-itational lensing phenomenon suggests that the cosmic mi-crowave background radiation might apply a pressure andenergize the Universe mechanically. Eq.(34), known as theFaber-Jackson law Faber & Jackson (1976), relates the total luminosity of an elliptic galaxy to its line of sight velocitydispersion. For typical extended galaxy it is derived fromEqs.(29,30) with the energy density ǫ and the velocity dis-persion, h® v i , being defined on the outskirts: M = r π ( Θ ) / ( + Θ ) h® v i G √− G ǫ . (35)While for thick gravitational haziness the ambient pressurepenetrates all the way towards the centre, a thin gravita-tional haziness system does resist mechanically. It balancesthe pressure on the boundary and may build up an internalpressure on its own.The cores in nearby elliptical galaxies have been spa-tially resolved in observations and the surface brightnessprofile, proportional to the projected surface mass density,see Appendix, has been determined. Using Eqs.(31,A3) themaximum central surface mass density: µ max = r Θ p π G (cid:18) + Θ Θ R (cid:19) − / ( + Θ ) , (36)is found to be proportional to the square root of the ambi-ent pressure p . For thick gravitational haziness the centralsurface density as well as the central pressure, see Eq.(A4),increases more sharply with the galaxy size R over the coresize than that for the thin gravitational haziness.A schematic and mechanistic picture of the galactic evo-lution follows from the statistical mechanics. It is possiblethat all galaxies were born non-rotating elliptical and sus-tained by the ambient pressure back then. As the Universeexpands and the pressure drops the galaxies are expandingaccording to the equation of state, Eq.(31), while some closeby galaxies get mutual rotations. If remaining non-rotating,and hence blown up, galaxies are dispersing stars and feedingup the more compact rotating ones or, alternatively, settledown into the pressurized centre of a cluster of galaxies thenthe Darwin selection, survival of the rotating species, definesthe galaxy evolution. If the pace of the Universe expansionis faster than a rotating galaxy can accommodate then itmay puncture too. Dispersed stars, leaving the parent ellip-tical galaxy on the equator of the rotation, are burdened bythe gravitation and form bending spiral arms that representordered exiting orbits in mechanical equilibrium as distinctfrom the remaining elliptical galaxy in the chaotic dynamicalequilibrium.This last example turns the statistical mechanics of self-gravitating gas into a ‘hazydynamics’ confronted with ques-tions like: if two close by galaxies have widely different grav-itational haziness will they exchange something? does thegravitational haziness thickens towards the core? The lawsof thermodynamics Kadanoff (2000) helps us in usual statis-tical physics. The first law of ‘hazydynamics’ should relatemasses by distinguishing them, unlike the Newton universalgravitation, in two categories (by rewriting Eq.(29)): M total = M Archimedes + M excess . (37)The first category is either the Archimedes displaced mass orthe mass sustained by the rotation or the outside pressure.Unlike that, the second category is an excess featureless gob-bled mass, like the mass inside binaries. It is proportionalto the gravitational haziness and keeps record of the historyof the star collection. Also, dividing the mass into pieceswill decrease the gravitational haziness whereas assembling MNRAS , 1–7 (2017)
A. Kashuba the mass from dust lumps into stars in particular and thegravitational collapse in general will increase it.At high- z distance the pressure of the cosmic microwavebackground radiation decreases as the fourth power of theUniverse scale. An elliptical galaxy in the dynamical equi-librium at constant mass and gravitational haziness ex-pands coherently with the Universe, see Eq.(31), and its sur-face brightness depends only on the gravitational haziness: µ ∼ √ Θ , not on z . For a disk galaxy, phase transitions andhydrodynamic instabilities may slow down this expansion.Recently, dramatic phenomena in many galaxies havebeen observed and interpreted as a super-massive blackhole residing in the centre McConnell et al (2011). Nomi-nal black hole is associated with the gravitational collapsein the end of the star evolution Chandrasekhar (1965). Thesuper-massive black hole is a different object and, as thispaper may suggest, not necessary. It might be rather a sta-ble self-gravitating gas of nominal black holes in dynamicalequilibrium growing by the process of the mass segregationwhereby heavy black holes sink into the centre. The reverseprocess of evaporation Spitzer (1987) vanishes for Θ → ∞ and the cross section for a direct collision of two black holesis small. Such compact gas of black holes at infinite gravi-tational haziness is readily described by Eqs.(34). However,the energy density ǫ is by twenty or so orders of magnitudelarger than that of the cosmic microwave background radi-ation. The super-massive black hole is found to be heavierin galaxies defined by thin gravitational haziness Seth et al(2014). The Kepler orbit of a probe star around the cen-tral object shows a perigee precession that depends on both Θ , ∞ and r core , and is opposite to that of the Einsteingeneral relativity effect. The non-relativistic statistical mechanics theory of the self-gravitating gas in the dynamical equilibrium is developed.The particle distribution function in the phase space, ananalog of the Maxwell-Boltzmann weight, and the associatedequation for the gravitational potential are found using theproper kinetic equation. The first two hydrodynamic equa-tions, an analog of the Navier-Stokes equation, are writtendown. All macroscopic static and kinematic properties ofthe self-gravitating gas depend on the gravitational hazinessvariable. The equation of state for a globular star collectionin the dynamical equilibrium, like elliptical galaxy, is found.The physics of the core collapse phenomenon is clarified. Forelliptical galaxies the relations between the rotation and thesize, between the core and the overall masses and betweenthe surface brightness and the pressure are all found. In theAppendix, the galaxy surface brightness profile, replacingthe de Vaucouleurs law, is provided. A survey of the maxi-mum surface brightness, Eq.(36), and the gravitational hazi-ness of non-rotating elliptical galaxies can possibly providea map of the ambient pressure field in the local universe.
APPENDIX A: SURFACE BRIGHTNESS
Close galaxies are clearly resolved with the internal struc-ture being salient. In this case Eq.(13) is directly applicable. Distant galaxies, though, reveal light from the outskirt andcan end in either sharp or a smooth manner. Lacking thequantitative theory of the rotating self-gravitating gas andthe corresponding elliptical deformation I provide here twomodels for the surface brightness of a galaxy. First, the sur-face density, the mass projected along the line of sight, of aglobular stellar collection given by the bulk density Eq.(13)with a sharp cutoff radius R reads: µ ( r ) = dMdA ( r ) = ∫ √ R − r −√ R − r ρ (cid:16)p r + z (cid:17) dz = √ R − r (cid:16) + + Θ Θ r (cid:17) + Θ + Θ (cid:18) Θ + ( + Θ ) r Θ + ( + Θ ) R (cid:19) + Θ + Θ + (cid:18) + r Θ (cid:19) F (cid:20) , + Θ + Θ , , − ( + Θ )( R − r ) Θ + ( + Θ ) r (cid:21)(cid:21) , (A1)where r is the radius on the astronomical plate away fromthe centre and perpendicular to the line of site. F [ a , b , c , x ] isthe usual hypergeometric function. The second model adoptsa soft density cutoff at large outer radius R : µ ( r ) = dMdA ( r ) = ∫ ∞−∞ ρ (cid:16)p r + z (cid:17) + (cid:0) r + z (cid:1) / R dz = r π Θ + Θ Γ (cid:16) − + Θ (cid:17) Γ (cid:16) + Θ + Θ (cid:17) R Θ (cid:16) + + Θ Θ r (cid:17) − + Θ × (cid:18) + Θ − R R + r F (cid:20) , , − + Θ + Θ , Θ + ( + Θ ) r ( + Θ )( R + r ) (cid:21)(cid:19) + π R √ R + r − R Θ sin π + Θ (cid:16) − + + Θ Θ R (cid:17) + Θ + Θ , (A2)where R ≫ r anis . For low- z distances the surface brightnessis the product of light to mass ratio, homogeneous in thedynamical equilibrium up to weak kinetic effects like themass segregation, and the surface density. In both models,the surface density profile in the inner region, emerging inthe limit R → ∞ , is the same: µ ( r ) = µ max + r / Θ (cid:0) + ( + Θ ) r / Θ (cid:1) − / ( + Θ ) , (A3)where the maximum central surface density: µ max = s Θ p max ( + Θ ) G + Θ + Θ Γ (cid:16) − + Θ (cid:17) Γ (cid:16) + Θ + Θ (cid:17) , (A4)is determined by the central stellar Jeans pressure p max . Re-markably, for the bulk density Eq.(13) this projected surfacedensity is given by more or less the same formula. Moreover,there exists a potential in the two dimensions: Φ ( ) (® r ) = π Θ µ max (cid:16) + ( + Θ ) ® r / Θ (cid:17) / ( + Θ ) , (A5)such that this potential and the surface density are relatedby the two dimensional Poisson’s equation: ∆ ( ) Φ ( ) − πµ (® r ) = . (A6) MNRAS , 1–7 (2017) tatistical mechanics of galaxy REFERENCES
Barnes J. E. and Hut P., 1986, Nature 324, 466Binder K. and Young A. P., 1986, Rev. Mod. Phys. 58, 801Callen H. B. and Welton, T. A., 1951, Phys. Rev., 83, 34Chandrasekhar S., 1965,
Principles of Stellar Dynamics , (Dover,New York)C. De Dominicis and Peliti L., 1978, Phys. Rev. B, 18, 353Dyson F. J., 1962, J. Math. Phys., 3, 1199Faber S. M. and Jackson R. E., 1976, Astrophys. J. 204, 668Genzel R., Hollenbach D. and Townes C. H., Rep. Prog. Phys.,1994, 57, 417Harris W. E., 1996, Astron. J., 112, 1487Kadanoff L. P., 2000,
Statistical Physics (World Scientific)King I. R., 1966, Astron. J., 71, 64Lynden-Bell D., 1967, Mon. Not. R. Astron. Soc., 136, 101Martin P. C., Siggia E. D. and Rose H. A., 1978, Phys. Rev. A,8, 423McConnell N. J., Chung-Pei Ma, Gebhardt K., Wright S. A., Mur-phy J. D., T. R. Lauer, J. R. Graham and D. O. Richstone,2011, Nature, 480, 215Romanowsky A. J., Douglas N. G., Arnaboldi M., Kuijken K.,M. R. Merrifield, N. R. Napolitano, M. Capaccioli and K. C.Freeman, 2003, Science, 301, 1696Saslaw W. C., 1985, ‘Gravitational Physics of Stellar and GalacticSystems’, Cambridge, Cambridge University PressSchwarzschild M., 1979, Astrophys. J., 232, 236Seth A. C., Bosch R., S. Mieske, H. Baumgardt, M. den Brok, J.Strader, N. Neumayer, I. Chilingarian, M. Hilker, R. McDer-mid, L. Spitler, J. Brodie, M. J. Frank and J. L. Walsh, 2014,Nature, 513, 398Spitzer L. Jr, 1987,
Dynamical Evolution of Globular Clusters (Princeton, Princeton University Press)Staniscia F., Turchi A., Fanelli D., Chavanis P. H., De Ninno G.,2010, Phys. Rev. Lett. 105, 010601Wigner E. P., 1957, Ann. Math., 65, 203Zwicky F., 1933, Helv. Phys. Acta, 6, 110This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000