The Microscopic Basis of Hydrostatic Equilibrium

8/13/2019 The Microscopic Basis of Hydrostatic Equilibrium

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The Microscopic Basis of Hydrostatic

Equilibrium

R.E. Salvino

4408 Cherry Valley Drive

Rockville MD 20853, USA

R.D. Puff

Department of Physics, Box 351560

University Of Washington

Seattle WA 98195, USA

Revised: 6 Apr 2014

Abstract

We provide the first microscopically-based derivations of hydro-static equalibrium by analyzing a large but finite, single componentsystem of interacting particles. Initially, we provide a derivation basedon an effective mean-field approximation for the classical gravitationalinteraction to establish the notation. We then repeat the derivationfor the exact treatment of the two-body classical gravitational interac-tion which provides the exact hydrostatic condition. We show that the

familiar Newtonian equation of hydrostatic equilibrium follows from acompletely uncorrelated approximation for the pair correlation func-tion of the finite system of particles. Our results establish a consistencycondition on particle correlations between those that enter the equa-tion of state and those that enter the gravitational contribution to theequilibrium condition. In particular, use of the Newtonian equationimposes severe constraints on allowable forms of the equation of state.In addition, our results have two important consequences for stellarstructure:

(A) the Chandrasekhar mass limit for white dwarfs no longer followsdirectly from the equation of hydrostatic equilibrium and a newlyestablished limit must be based upon the exact equation

(B) both the general relativistic equation of hydrostatic equilibriumand the stellar collapse scenario must be considered to be due toa mean-field approximation for interparticle interactions

Current address: 9 Thomson Lane, 15-06 Sky@Eleven, Singapore 297726

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Keywords: hydrostatic equilibrium, many-body theory, non-relativistic

field theory, stellar structure, white dwarf

1 Introduction

The Newtonian equation of hydrostatic equilibrium is

dPhdr

= Gm(r)M(r)

r2 (1.1)

where Ph is the hydrostatic pressure, m(r) is the mass density, and M(r)is the total mass within the sphere of radius r . This equation has a centralplace in classical physics. For instance, pedagogically, it is used on vari-ous levels of physics education due to its sophisticated yet relatively simplestructure [1]; it is one of the fundamental equations in the theory of stellarstructure [2]; and it is a connecting link between relativistic and classicalgravitational theory since it provides the weak gravitational field limit ofthe general relativistic Tolman-Oppenheimer-Volkoff (TOV) equation of hy-drostatic equilibrium [3]. While the machinery for a microscopically-basedderivation of this equation has been in place for over 50 years, the deriva-tion of Equation (1.1), to our knowledge, is based solely on macroscopic

physics [5].Such a macroscopic approach leads inevitably to an approximation in

which two-body interactions are given a mean-field treatment. Consider asphere of radius R that contains a collection of particles, and within thatsphere imagine a concentric sphere of radiusr (see Figure 1). The interac-tions between the particles that are located within the radius r and thosein the spherical shell between r and R are replaced by the mean-field in-teraction that places all the particles within r at the origin. For the 1/rpotential, this is not an approximation at all, but is actually an exact state-ment but only for those particles outside the radius r. The interactionsof particles within the radius r with other particles within the radius r

cannot be exactly described by the mean-field prescription; the interactionsof particles within the spherical shell between r and R with other parti-cles in the spherical shell also cannot be exactly described by the mean-fieldprescription. Nevertheless, classical potential theory does treat all these re-gions in the same manner and, consequently, produces a mean-field result.

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These non-mean-field aspects of the 1/r Coulomb potential provide the de-

scription of the elecron gas that goes beyond the mean-field treatment of aCoulomb gas which ultimately depends upon particle correlations that aremissing in the mean-field treatment. In the gravitational case, the mag-nitude of the interaction is tiny by comparison to the electrostatic case, soneglecting particle correlations due to the gravitational interaction is a goodapproximation unless the particle density becomes very large. So for verydense systems, going beyond the mean-field treatment for the gravitationalinteraction is a necessity. Of greater importance, and relevant to low andhigh particle densities alike, particle correlations due to other interactionsthat may be present or due to particle statistics can not be ignored.

Figure 1: A collection ofNparticles constrained to a spherical volume definedby the radius R. A variable inner concentric sphere is defined by the radius r.Particles inside region A (0 r r) interact with particles in the shell region B(r< r R) as if all particles in region A were located at the origin. For the 1/rpotential, this is an exact statement. However, this is not an exact treatment forparticles in region A interacting with other particles in region A or for particles inthe spherical shell Binteracting with other particles in the spherical shell B.

The macroscopic derivation of Eq. (1.1) makes no reference to any par-ticular form for, or conditions on, the hydrostatic pressure Ph. It is simplyassumed that Ph represents the exact hydrostatic pressure and providesthe mechanism by which the gravitational compression is balanced. Noth-ing further is needed to derive the equation of equilibrium. In particular,

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all discussions based on issues regarding equations of state are irrelevant

to establishing Eq. (1.1). Similarly, our microscopically-based derivationsrequire that Ph be identified as the exact hydrostatic pressure and no dis-cussions regarding issues surrounding equations of state need be made. Wedo, however, provide the proper microscopically-based definitions of the hy-drostatic pressure which aid in the proper identification of the gravitationalcontribution to the equation of equilibrium.

Within the non-relativistic quantum field theoretical description of alarge but finite many-particle system, we provide a microscopically-basedderivation of (1.1) from a mean-field approximation for the gravitationalinteraction. This derivation also serves to establish the notation, physicalmeaning, and identification of all quantities. By using the Newtonian two-

body gravitational potential as a proper two-body interaction within thissame theoretical framework, we then derive the exact non-relativistic equa-tion of hydrostatic equilibrium and show explicitly that (1.1) is the result ofa completely uncorrelated approximation for the many-particle correlationfunction, in fact, the Hartree approximation. In addition, identification ofthe microscopic form for the hydrostatic pressure provides the explicit lim-iting form that is required when the mean-field equation is used. Finally, weoutline three specific additional impacts of these two basic results on stel-lar structure: the impact on white dwarf structure and the Chandrasekharmass limit, including a possible connection to recently discovered super-Chandrasekhar-mass-limit white dwarf stars; identifying general relativisticstellar structure as a necessarily mean-field theory; and noting that the con-

sequent stellar collapse scenario is also necessarily based upon a microscopicmean-field approximation that neglects particle correlations.

2 Derivation of the Newtonian Equation of Hydrostatic

Equilibrium

From the many-body theory viewpoint, the microscopic basis of the stan-dard macroscopic approach treats the Newtonian gravitational interactionas an effective mean-field on the system in question 1,

H= HNG+ HG,eff (2.1)

1The coupling of an external potential to the particle density is the simplest and,consequently, the most widely used approach in areas such as linear response theory,perturbation theory, and transport theory. See, for example, ref. [6].

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HNG =

d3r(r, t)

22

2m(r, t)

+1

2

d3rd3rVNG(r, r

)(r, t)n(r, t)(r, t) (2.2)

HG,eff=

d3rn(r, t)Ueff(r

) (2.3)

where n(r, t) = (r, t)(r, t) is the many-particle number density operatorand (r, t) and (r, t) are the second quantized field operators, and the

subscript NG (non-gravitational) refers to all parts of the total hamilto-nian that do not explicitly contain the gravitational interaction: the kineticenergy and a non-gravitational two-body interaction. This system of self-gravitating particles is necessarily a finite system and we denote the regionof space containing the N-particle system by the symbol . While the ac-tual form of the fields is not needed in this analysis, we simply note thatthe fields vanish in the region of space exterior to the region containingthe N-particle system and satisfy appropriate boundary conditions on thesurface bounding the region . Consequently, volume integrals containingthe many-particle fields may be interpreted as extending over all space orover the restricted space . The non-gravitational interaction, VNG(r, r

),that is included in HNG is treated as a true 2-body interaction and ulti-mately appears in the equation of state for the pressure. It may be a singleinteraction or a sum of two-body interactions, but it does not contain thegravitational two-body interaction. The effective one-body potential is justthe Newtonian gravitational potential

Ueff(r) = Gm

d3r

m(r)

|r r| (2.4)

Ueff(r) = G

M(r)

r + 4

r

drrm(r)

(2.5)

wherem(r) =mn(r, t) is the mass density andM(r) = 4r0

drr2m(r)

is the total mass within the radius r. Eq. (2.5) follows from (2.4) by ex-panding the inverse distance in spherical harmonics.

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The equations of motion, iA/t= [A, H] where A is a second quan-

tized operator, yield for the momentum current [7]

mj(r, t)

t = TNG(r, t) mn(r, t)Ueff(r) (2.6)

Substituting the explicit form for the effective potential produces

mj(r, t)

t = TNG(r, t)

Gmn(r)M(r)

r2 r (2.7)

The momentum density current operator, TNG(r, t), contains no explicitgravitational contributions, all explicit gravitational contributions are con-tained solely in the external potential. In equilibrium, this reduces to

TNG(r, t) = Gm(r)M(r)

r2 r (2.8)

Equilibrium averages for the finite system are defined in the same manneras the bulk equilibrium averages, except that the statistical density opera-tor for the finite system is formulated by means of finite system operators.Although the system is finite, the standard bulk thermodynamics will apply

as long as the ratio of the surface thermodynamic terms of the system tothe corresponding volume thermodynamic terms of the system is small; fora spherically symmetric system of radius R with surface tension and pres-sure P, this requires that /(3P R) is small. The equilibrium averages inthis section are specifically mean-field averages since they utilize the Hamil-tonian (2.1). It also follows that the equilibrium average of an operator willintroduce a dependence on the gravitational coupling constantG even if theoperator itself is independent ofG.

While a finite system does not possess translational symmetry, it maystill be rotationally invariant; for a spherically symmetric system, we knowthat TNG(r, t) = F(r)I where F(r) is a function of r only and I is the

unit tensor. As we noted above, we also identify the mass density for thesingle-component system as m(r) =mn(r, t) so that eq. (2.8) becomes

TNG(r, t) =dF

drr=

Gm(r)M(r)

r2 r (2.9)

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Comparing (2.9) with (1.1) identifies the function F(r) with Ph(r), that

is, the non-gravitational contribution to the equilibrium stress tensor is thehydrostatic pressure TNG(r, t) =Ph(r)I

Ph(r) =2

3ke(r)G,eff

1

12

d3r r

dVNG(r)

dr

1

1

d

r+

1

2(1 + )r

n

r

1

2(1 )r

r+

1

2(1 + )r

G,eff

(2.10)

where ke is the kinetic energy operator and G,eff indicates that thethermal averages utilize Eq. (2.1), H= HNG+ HG,eff. For translationally

invariant systems, n n2g(r r) where g(r r) is the infinitesystem pair correlation function and the -integral gives a simple factor of2.

Since the gravitational interaction is not a true external field but is anapproximate treatment for the two-body interaction, Ph(r) is not the fullthermodynamic pressure, it is that part of the stress tensor that has noexplicit dependence on the gravitational interaction. The thermodynamicpressure does not make an explicit appearance in these equations at all. Weknow that the thermodynamic pressure and stress tensor are related by

T(r, t) = TNG(r, t) + TG(r, t) =Pth(r)I (2.11)

where TG(r, t) is the part of the stress tensor that depends explicitly onthe gravitational interaction. The equations of motion then show

T(r, t) = Pth(r) = 0 (2.12)

TNG

(r, t) = TG

(r, t) = Gm(r)M(r)

r2 r (2.13)

Using the full stress tensor as the microscopic basis as in (2.12), the equationof hydrostatic equilibrium is the very simple equation Pth(r) = 0, whichis just the thermodynamic statement of mechanical equilibrium. In fact, the

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thermodynamic condition of mechanical equilibrium is the underlying basis

for the condition of hydrostatic equilibrium, since (2.9) follows from (2.12).The term involving TG(r, t) has been separated from the non-gravitationalpart of the stress tensor as in (2.13), it has been given a special mean-fieldtreatment as a fictitious external gravitational field and it enters the equa-tions of motion as a fictitious external force. As a result, the simplicityof the thermodynamic statement is lost. Now the explicit dependence onthe gravitational interaction no longer appears in the remaining part of thestress tensor, the non-gravitational part, which is the microscopic basis ofthe hydrostatic pressure Ph(r). Consequently, Ph(r) has no explicit depen-dence on the gravitational interaction. It does, however, contain an implicitdependence on the gravitational interaction through the thermal average

which typically manifests itself in Ph(r) through a dependence on the localdensity by means of an equation of state.This completes our demonstration that the Newtonian equation of hy-

drostatic equilibrium follows from a mean-field treatment and identifies thepressure in that equation as a hydrostatic pressure and not the full ther-modynamic pressure. It is this derivation from a one-body external potentialthat justifies calling Eq. (1.1) a mean-field result. While this does providea microscopic derivation of (1.1), we would like to treat the two-body grav-itational interaction on an equal footing with other 2-body interactions anddrop the idea of an effective external interaction. This permits an exacttreatment of the two-body gravitational interaction as described in relationto Figure 1.

3 Derivation Of The Exact Equation Of Hydrostatic

Equilibrium

We begin by considering a single component collection of interactingparticles, each of mass m, that are described by the Hamiltonian

H= HNG+ HG (3.1)

HNG =

d3r(r, t)

22

2m(r, t)

+1

2

d3rd3rVNG(r, r

)(r, t)n(r, t)(r, t) + . . .(3.2)

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HG = 1

2

d3rd3rVG(r, r

)(r, t)n(r, t)(r, t)

= 1

2

d3rd3r

Gm2

|r r|(r, t)n(r, t)(r, t) (3.3)

whereVG = Gm2/|r r| is the Newtonian two-body gravitational poten-tial. As we mentioned in Section 2, the system is necessarily a finite systemwith the particles self-confined to a finite region of space . In the aboveequations, (r, t) and (r, t) are the second-quantized field operators andn(r, t) = (r, t)(r, t) is the particle number density operator appropri-

ate for the finite system. The spatial integrations may be interpreted asrestricted to the region containing theN-particle system or over all spacesince the fields are defined to be zero outside the region . HNGcontains thekinetic energy term as well as all non-gravitational interactions that describethe many-particle system. Since the Newtonian gravitational interaction isa two-body interaction, we explicitly include a two-body non-gravitationalinteraction, however, the ellipsis in Equation (3.2) indicates that higher or-dern-body interactions may also be included. Equation (3.3) represents thepair-wise Newtonian gravitational interaction of the system of particles.

Such a system of particles obeys the many-particle conservation lawswhich are derived from iA/t = [A, H] and the standard commutation

rules [8]. Consequently, the conservation of particle number is

n(r, t)

t = j(r, t) (3.4)

the conservation of momentum is

mj(r, t)

t = T(r, t) (3.5)

and the conservation of energy is

(r, t)

t = j

(r, t) (3.6)

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In the above equations, j(r, t) is the number density current, mj(r, t) is

the momentum density, T(r, t) is the momentum density current tensor orstress tensor, (r, t) is the Hamiltonian density, and j

(r, t) is the energy

density current.We focus our attention on Equation (3.5), the momentum conservation

equation. The characteristic many-body treatment would group the grav-itational two-body potential with other two-body potentials and extract asingle finite-system divergenceless stress tensor. This, however, would ob-scure the connection with the differential equation form of the hydrostaticequilibrium condition which treats the gravitational interaction separatelyfrom other two-body interactions. Consequently, we concentrate on theform of T(r, t) and not on the extraction of the stress tensor, T(r, t).

As for the Hamiltonian, the momentum density current tensor has explicitnon-gravitational and gravitational parts

T(r, t) =TNG(r, t) +TG(r, t) (3.7)

so that Equation (3.5) may be written as

mj(r, t)

t = TNG(r, t) TG(r, t) (3.8)

TG(r, t) =

d3rVG(r, r

)(r, t)n(r, t)(r, t)

= Gm2

d3r r r

|r r|3(r, t)n(r, t)(r, t) (3.9)

By simply applying the commutation relations, the two-body operator onthe right-hand side of (3.9) may be written as

(r, t)n(r, t)(r, t) = n(r, t)n(r, t) n(r, t)(r r) (3.10)

where(x) is the usual delta function. In equilibrium,< mj(r, t)/t >=0so the equilibrium momentum conservation law (3.8) becomes

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< TNG(r, t)>= (3.11)

Gm2

d3r r r

|r r|3(< n(r, t)n(r, t)> < n(r, t)> (r r))

As we stated in Section 2, the equilibrium average for the finite systemtakes the same form as the bulk system equilibrium average except that alloperators are finite system operators. However, the finite system Hamil-tonian operator will now be given by (3.1) rather than (2.1). As alsostated in Section 2, while this system retains its rotational invariance, itis not translationally invariant. However, if we were dealing with a trans-

lationally invariant system (nuclear matter, for example), the quantity(< n(r, t)n(r, t) > < n(r, t) > (r r )) would be independent ofthe center of mass and

(< n(r, t)n(r, t)> < n(r, t)> (r r)) =n2g(|r r|) (3.12)

wheren is the bulk particle density N/V and g(|r r|) is the well-knownradial distribution function. Consequently, in strict analogy with (3.12), wewill define the finite system 2-point distribution function using the same

symbol with a subscript F S(finite system) as follows 2:

(< n(r, t)n(r, t)> < n(r, t)> (r r)) =

< n(r, t)> gFS(r,r) (3.13)

where we have explicitly indicated that the finite system 2-point distributionfunction gFS(r, r) is not, in general, a function of |r r| only. Since the

2Equations (3.12) and (3.13) describe pair correlations in the bulk and finite systems,

respectively. See ref. [9]. For an alternative, non-standard definition of the pair correlationfunction, see ref. [10]. For a brief overview of the pair correlation function and its relationto computer simulations of molecular order, see ref. [11]. For extensive discussions of therole of the pair correlation function in the structure of fluids and its direct relation toexperimental measurements, see ref. [12] or ref. [13].

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mass density of the system is simply the particle mass times the particle

number density,

m(r) =m < n(r, t)> (3.14)

substituting (3.13) into (3.12) and using (3.14) gives

< TNG(r, t)>= Gm(r)

d3r

r r

|r r|3m(r

)gFS(r, r) (3.15)

Now, to connect with the Newtonian equation of hydrostatic equilibrium, wewill assume the system is spherically symmetric and the non-gravitationalpart of the momentum density current has the form

< TNG(r, t)>=F(r)I= Ph(r)I (3.16)

where we have made use of the result in Section 2 that F(r) is the hydrostaticpressure,F(r) =Ph(r), Iis the unit tensor, and the exact Ph(r) is given by

Ph(r) =23

ke(r) 112

d3r r dV

NG(r)dr

1

1

d n(x)n(y)gFS(x,y)

(3.17)

n(x)n(y)gFS(x,y) (x)n(y)(x) (3.18)

x r+1

2(1 + )r (3.19)

y r 1

2(1 )r (3.20)

where ke is the kinetic energy operator and the thermal averages utilize

the exact Hamiltonian Eq. (3.1), H =

HNG +

HG. Consequently, (3.15)becomes

Ph(r) = Gm(r)

d3r

r r

|r r|3m(r

)gFS(r, r) (3.21)

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This may be more conveniently written as

Ph(r) =Gm(r)

d3r

1

|r r|

m(r

)gFS(r,r) (3.22)

Equation (3.22) is the exact equation of hydrostatic equilibrium for spher-ically symmetric systems (systems without spherical symmetry require eq.(3.15)). As shown in Eq. (3.17),Ph(r) is the thermal average of that partof the stress tensor that has no explicit dependence on the gravitational in-teraction. As in the mean-field derivation of Section 2, Ph(r) is not the full

thermodynamic pressure since the explicit dependence on the gravitationalinteraction has been separated from the stress tensor and treated in a sep-arate manner. The relationship of the stress tensor to the thermodynamicpressure is the same as eq. (2.11)

T(r, t) = TNG(r, t) + TG(r, t) =Pth(r)I (3.23)

except that (3.23) contains the exact treatment of these quantities and not amean-field approximation. The equations of motion then provide the exactanalogues to the mean-field equations (2.12) and (2.13)

T(r, t) = Pth(r) = 0 (3.24)

TNG(r, t) = TG(r, t)

= Gm(r)

d3r

r r

|r r|3m(r

)gFS(r,r) (3.25)

As we found in the mean-field case of Section 2, eq. (3.24) shows that

the thermodynamic statement of mechanical equilibrium holds in the ex-act treatment. For both the mean-field and exact treatments, the equationof hydrostatic equilibrium is just a re-phrasing of the thermodynamic con-dition of mechanical equilibrium: the equation of hydrostatic equilibriumfollows from the thermodynamic condition of mechanical equilbrium. Eq.

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(3.25) shows the separation of the explicit dependence on the gravitational

interaction from the non-gravitational part of the stress tensor for the exacttreatment. The exact treatment of the non-gravitational part of the stresstensor produces the functionPh(r) and may be indentified as the exact formfor the hydrostatic pressure, Ph(r).

The presence of the pair correlation function gFS(r,r) in (3.22) requiressome comment. First, it is important to recognize thatgFS(r, r) is relatedto pair correlations for the fully interacting many-particle system and notthose due simply to the gravitational interaction. One may successfullyargue that the gravitational interaction is negligible compared to the short-range interactions for the fully interacting system and will contribute signif-icantly to pair correlations only over large distances where gFS(r, r

) 1.

That, however, does not mean that gFS(r

,r

) may be set equal to unityin (3.22). We are not claiming that the gravitational interaction alters thepair correlations of the system, although under certain high density circum-stances it may; but we are claiming that the presence of pair correlationsalters the manner in which the gravitational interaction enters the hydro-static equilibrium condition. Even quantum mechanical free particles havepair correlation functions which differ from unity. Second, in a related issue,any improvements to the equation of state that require something beyonda mean-field approximation for pair correlations must also be included inthe gravitational term of Eq. (3.22). There is a consistency requirement inEq. (3.22): the pair correlation function that corresponds to the equationof state used on the left-hand side of (3.22) must be used on the right-hand

side of that equation. Third, the short-range correlations make a significantimpact on the amount of mass a given equation of state can uphold. As anextreme example, contrast the mean-field approximationgFS(r, r

) 1 witha simplified hardcore approximation, gFS(r, r

) (|r r| hc), where(x) = 1 forx >0, 0 otherwise andhc is the hardcore diameter. The pres-ence of this hardcore exclusion permits the pressure to uphold an arbitrarilylarge amount of mass; no matter how much mass is involved, it will notbe enough to force the hardcores to interpenetrate. While real systems donot include such an idealized hardcore, the realistic cores will substantiallyenhance the pressures ability to uphold mass. In addition, the short-rangerepulsion ingFS(r,r

) also reduces the gravitational compression by cutting

off the short distance divergence of the gravitational two-body interaction.Fourth, we should point out that gFS(r,r

) is not necessarily a functionof |r r| only unless the system under consideration is a bulk or infinitesystem; for finite systems, we might expectgFS(r, r) to be well representedby the infinite system function away from system surfaces and boundaries

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M(r) = 4 r0

drr2

m(r) (4.3)

As might be expected, comparing (4.2) with the results of Section 2 pro-vides an alternative approach for identifying the one-body effective externalpotential, Ueff(r). By taking the gradients on the left hand side of (4.2),we finally arrive at Equation (1.1)

dPhdr

r= Gm(r)M(r)

r2 r (4.4)

where r is the unit radial vector.While Eq. (4.4) is formally equivalent to the result obtained in Section 2,

it is important to recognize that (4.4) places an additional self-consistencyconstraint on the pressure Ph(r). The result obtained in Section 2 sug-gests that the pressure Ph(r) may contain particle correlations describedbygFS(x,y). But the derivation based upon the exact equation of hydro-static equilibrium, Eq. (3.22), requires gFS(r,r

) = 1 for all values of rand r. Consequently, we are required to use gFS(x,y) = 1 in Ph(r) forself-consistency and so the pressure Ph(r) must be given by

Ph(r) =2

3ke(r)

1

12

d3r r

dVNG(r)

dr

1

1

d

n

r+

1

2(1 + )r

n

r

1

2(1 )r

(4.5)

where, as before, ke(r) is the kinetic energy density. This means thatthe non-gravitational interactions can contribute, at most, terms that areexplicitly of second order in the density to Newtons equation of hydrostaticequilibrium. If the equation of state forPh(r) contains terms of higher orderin the density, then they must come from the kinetic energy density term

or the approximation gFS(r, r

) = 1 cannot be used. Classically, the kineticenergy density is given by

ke(r) =3

2n(r)kBT (classical result) (4.6)

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This is consistent with gFS(r,r) = 1. However, the quantum mechanical

form for the kinetic energy density may not be consistent withgFS(r,r

) =1. For example, we know that the bulk pair correlation function for freefermions is not unity but is given by

g(r, r) =g(|r r|) = 1

3j1(kF|r r

|)

kF|r r|

2

(4.7)

where kF = (32n0)

1/3 and n0 is the average particle density, n0 = N/V.Although this is an infinite system result, we expect that the finite sys-tem result may not be too different away from boundaries and regions of

large density gradients. In any case, we should expect that gFS(r, r

) = 1for all r and r in general for quantum mechanical systems. Consequently,we must additionally conclude that the Newtonian equation of hydrostaticequilibrium is rigorously applicable only to systems describable by classi-cal statistical mechanics. Furthermore, if the pressurePh(r) is required tocontain a radiative T4 term,

Ph(r) = nkBT+1

4aT4 (4.8)

then we must have

1

4aT4 =

1

12

d3r r

dVNG(r)

dr

1

1

d

n

r+

1

2(1 + )r

n

r

1

2(1 )r

(4.9)

This means that the non-gravitational interactions described by VNG(r)

may be required to depend on thermophysical parameters in model depen-dent statements. In the case of the radiative T4 term, the r dependencein the particle density n must also be interpreted as an r dependence inthe temperature T. This may be considered an artifact of the conventionalwisdom that states that a temperature gradient is established by means ofa density gradient [14,15].

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Although the mean-field approximation gFS(r, r) = 1 appears to place

severe constraints on the equation of state that may be used for Ph(r),we may explicitly incorporate this mean-field result in Equation (3.22) bysimply adding and subtracting the gFS(r, r

) = 1 term,

Ph(r) = Gm(r)M(r)

r2 r

+ Gm(r)

d3r

1

|r r|

m(r

)(gFS(r, r) 1) (4.10)

As mentioned in relation to Figure 1 in the introduction, gravitational po-tential theory can not produce Eq. (4.10) or, equivalently, Eq. (3.22) sinceit introduces a mean-field treatment of the gravitational two-body interac-tion. Potential theory provides exact results for the region exterior to theN-particle system, but it necessarily introduces the Hartree mean-field ap-proximation for particle interactions in the interior region of the N-particlesystem and will reproduce Eq. (4.4) not Eq. (4.10).

5 Conclusion

We have approached the derivation of the equation of hydrostatic equi-librium from a microscopic many-particle point of view. This perspectivehas explicitly demonstrated that the equation of hydrostatic equilibrium issimply a re-statement of the thermodynamic condition of mechanical equi-librium. It also allows us to treat the classical gravitational interaction inthe many-particle system as a true two-body interaction and not as a mean-field one-body external potential. Our result, either (3.22) or (4.10), clearlyshows the non-Newtonian nature of the exact equation. Our discussion inSection 4 demonstrates that the mean-field approximation gFS(r,r

) = 1has important repercussions on the allowable equations of state for Ph(r).In many cases, an assumption of an ideal gas equation of state for the pres-sure can utilize the Newtonian hydrostatic equation (1.1) and Eqs. (4.5) and(4.9) show how terms of second order in the density or the radiative T4 termmay be included. However, a more accurate equation of state will requirea better approximation for the 2-point distribution function gFS(r, r

) in

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(4.10), if only for self-consistency. For example, identifying gFS(r, r) = 1

as corresponding to the gas pressure Ph0(r), then self-consistency demands

Ph(r) Ph0(r)

=

Gm(r)

d3r

1

|r r|

m(r

)(gFS(r, r) 1) (5.1)

where dPh0(r)/dr = Gm(r)M(r)/r2. If the pressure assumes the form

of a simple extension beyond the ideal gas law, a modified van der Waalspressure,Ph(r) = PHS(r) an(r)

2 where PHS(r) is a hard sphere equa-

tion of state, thengFS(r, r) must take the form appropriate for that systemof hard spheres while the van der Waals attractive component may be ac-commodated by the second order term in Eq. (4.5). If the self-consistencycondition (5.1) can not be satisfied, then (4.10) must be approached abinitio.

The mean-field nature of the Newtonian equation of hydrostatic equi-librium has two important consequences, specifically for stellar evolution.First, we note that the form of the Newtonian equilibrium equation, partic-ularly the appearance ofM(r) in eq. (1.1), is fundamental to the derivationof the Chandrasekhar mass limit for white dwarf stars 4 [16]. The derivationdepends crucially on the ability to manipulate the Newtonian equilibrium

equation into a direct method of calculating the stellar mass,

Mmf(r) = r2m(r)

G

dPh(r)

dr (5.2)

where the subscript mf indicates that (5.2) is a mean-field result. For thecase of a polytropic equation of state, this approach provides the classicanalysis and yields the Lane-Emden equation 5. However the exact equa-tion of hydrostatic equilibrium, either (3.22) or (4.10), closes this mathemat-ical route because the physical content of the exact hydrostatic condition

4For a concise p edagogical treatment, see ref. [5]. Improvements to the basic line of

development, such as that in ref. [17], concentrate on the equation of state.5The basic line of development identifies the electron degeneracy as the only substantial

mechanism for producing pressure to counter-balance gravitational contraction. However,the mass of the star is due almost exclusively to the mass of the nuclei. To includethe nuclei in the analysis, the density of the electrons is eliminated in favor of the totalmass density of the star by means of a simple proportionality, ne(r) =N0m(r)/e, the

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is different from that of the Newtonian condition. The presence of parti-

cle correlations and the consequent 2-point distribution function gFS(r, r

)in the exact hydrostatic condition does not permit such a straightforwardconnection with the stellar mass

M(r) = r2

Gm(r)

dPh(r)

dr

+r2

d3r

d

dr

1

|r r|

m(r

)(gFS(r, r) 1) (5.3)

In addition, the presence of the 2-point distribution function gFS(r, r) in

the hydrostatic equation indicates that incorporating the nuclei into theanalysis solely by means of the mass density is no longer adequate.6 It mustbe remembered that our derivation is valid for a single-component many-particle system; the white dwarf analysis must be based upon a completetwo-component model (electrons and nuclei). This takes the form of a par-tial pressure decomposition, Ph(r) = Ph,ee(r) +Ph,nn(r) +Ph,en(r) wherethe subscriptn denotes nuclei and the subscripte denotes electron; the low-est order approximation identifies the stellar mass and mass density with

proportionality constant being the ratio of Avogadros number to the mean molecularweight per free electron in the star. This maneuver is required to use the one-component

Newtonian equation (1.1) for the two-component system: it is this substitution that allowsthe electron degeneracy pressure to be considered as a function of the total mass densityof the star [5,18]

6In the simplest model for utilizing the one-component result, either (3.22) or (4.10), fora two-component system as in the standard treatment of white dwarf stars (see ref. [5]), thepressure may still be identified as that due to electron degeneracy while the gravitationalterm is that due primarily to the nuclei. This gravitational term, however, now explicitlycontains the 2-point distribution function for the nuclei. To revert to the standard whitedwarf star treatment requires the additional assumption of completely uncorrelated nuclei.This would seem to be a severe constraint. More importantly, the exact two-componentequation of hydrostatic equilibrium will contain contributions corresponding to (3.22) or(4.10) from both electrons and nuclei as well as an additional electron-nuclei interactionterm. This suggests the standard white dwarf treatment has three distinct items to ad-dress: (1) the importance of the 2-point distribution function for nuclei, (2) the importanceof the electron-nuclei interaction contribution to the equation of hydrostatic equilibrium,and (3) the importance of the purely electron contribution apart from the pressure equa-tion of state. These issues are distinct from others that may have already been identifiedand addressed, such as corrections to the pressure equation of state and corrections for anon-static, rotating star.

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the mass and mass density contribution from the nuclei and identifies the

pressure with the partial pressure contribution from the electrons,

M(r) Mn(r) m(r) m,n(r) Ph(r) Ph,ee(r) (5.4)

so that the hydrostatic equilibrium condition is

Ph,ee(r) Gm,n(r)Mn(r)

r2 r

+ Gm,n

(r) d3r 1|r r|

m,n

(r)(gFS,nn

(r, r) 1) (5.5)

Eq. (5.5) shows that the correlation function in (5.3) for the white dwarfstar should be interpreted as the nuclei-nuclei correlation function. Theapproximation defined by (5.4), in conjunction with the Hartree approxima-tion for the nuclei pair correlation functiongFS,nn(r, r

) 1, then producesthe basis for the standard white dwarf structure analysis. While there maybe other considerations that limit the mass of a white dwarf, such as ro-tational inertia for rotating dwarf stars, the standard analysis for a staticwhite dwarf does not follow without the use of the mean-field Newtonianequation 7. This does not mean that a mass limit does not exist, but it does

mean that such a limit must be re-established based upon Eq. (5.5) ratherthan Eq. (5.2).

Finally, as noted in the introduction, the general relativistic hydrostaticcondition, the Tolman-Oppenheimer-Volkoff equation,

dPh(r)

dr =

G(m(r) + Ph(r)/c2)(M(r) + 4Ph(r)r

3/c2)

r(r 2GM(r)/c2) (5.6)

is expected to reduce to the classical equation in the weak-field limit. Theweak gravitational field limit has no bearing on the other interactions that

7

At this point, it may be relevant to note the appearance of super-Chandrasekhar-mass white dwarf stars in the literature related to supernovae. For instance, see ref. [19]and ref. [20]. Because a static, non-rotating star is the starting point for deriving theChandrasekhar mass limit, the theoretical mechanisms that explain the existence of super-Chandrasekhar-mass white dwarf stars appear to focus on rotation. Our results indicatethat this may not be necessary.

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[7] N.S. Gillis, Ph.D. Dissertation (University of Washington, Seattle,

1966).

[8] R.D. Puff and N.S. Gillis, Ann. Phys. 46, 364 (1968).

[9] S.W. Lovesey,Condensed Matter Physics: Dynamic Correlations(Ben-jamin Cummings, Menlo Park CA, 1986).

[10] A.L. Fetter and J.D. Walecka,Quantum Theory of Many-Particle Sys-tems(McGraw-Hill, New York, 1971).

[11] M. Bishop and C. Bruin, Am. J. Phys. 52, 1106 (1984).

[12] D.L. Goodstein,States of Matter(Dover, New York, 1985).

[13] N.H. March and M.P. Tosi,Atomic Dynamics in Liquids(Dover, NewYork, 1991).

[14] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon, Oxford,1987).

[15] R.E. Salvino, Phys. Rev. A, 41, 4236 (1990).

[16] S. Chandrasekhar, Astrophys. J. 74, 81 (1931).

[17] T. Hamada and E.E. Saltpeter, Astrophys. J. 134, 683 (1961).

[18] F. Mohling,Statistical Mechanics: Methods and Applications

(Publish-ers Creative Services, New York, 1982).

[19] D.A. Howell, M. Sullivan, P.E. Nugent, R.S. Ellis, A.J. Conley, D. LeBorgne, R.G. Carlberg, J. Guy, D. Balam, S. Basa, D. Fouchez, I.M.Hook, E.Y. Hsiao, J.D. Neill, R. Pain, K.M. Perrett, and C.J. Pritchet,Nature443, 308 (2006).

[20] R.A. Scalzo, G. Aldering, P. Antilogus, C. Aragon, S. Bailey, C. Bon-gard, C. Buton, M. Childress, N. Chotard, Y. Copin, H.K. Fakhouri, A.Gal-Yam, E. Gangler, S. Hoyer, M. Kasliwal, S. Loken, P.E. Nugent, R.Pain, E. Pecontal, R. Pereira, S. Perlmutter, D. Rabinowitz, A. Rau,

G. Rigaudier, K. Runge, G. Smadja, C. Tao, R.C. Thomas, B. Weaver,C. Wu, Astrophys J. 713, 1073 (2010).

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The Microscopic Basis of Hydrostatic Equilibrium