Applied Mathematical Sciences, Vol. 13, 2019, no. 19, 911 - 955 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2019.98122 Tidal Interactions and Principle of Corresponding States: from Micro to Macro Cosmos. A Century after van der Waals’ Nobel Prize R. Caimmi Physics and Astronomy Department, Padua University 1 Vicolo Osservatorio 3/2, I-35122 Padova, Italy This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd. Abstract The current paper was aimed to honor the first centennial of Jo- hannes Diderik van der Waals (VDW) awarding Nobel Prize in Physics. VDW theory of ordinary fluids is reviewed in the first part of the pa- per, where special effort is devoted to the equation of state and the law of corresponding states. In addition, a few mathematical features involving properties of cubic equations are discussed, for appreciating the intrinsic beauty of VDW theory. A theory of astrophysical fluids is shortly reviewed in the second part of the paper, grounding on the tensor virial theorem for subsystems, and an equation of state is formu- lated with convenient choices of reduced variables. Additional effort is devoted to selected density profiles, namely a simple guidance case and two cases of astrophysical interest. Given the analogy between macrogas reduced isoenergetics and VDW reduced isothermals, a phase transition (gas-stars) is assumed to take place in astrophysical fluids, similarly to a phase transition (vapour-liquid) observed in ordinary fluids. In this framework, the location of gas clouds, stellar systems, galaxies, clus- ter of galaxies, on the plane scanned by reduced variables, is tenta- tively assigned. A brief discussion shows how VDW’ two great discov- eries, namely a gas equation of state where tidal interactions between molecules are taken into account and the law of corresponding states, 1 Affiliated up to September 30th 2014. Current status: Studioso Senior. Current posi- tion: in retirement due to age limits.
The current paper was aimed to honor the first centennial of Jo-hannes Diderik van der Waals (VDW) awarding Nobel Prize in Physics.VDW theory of ordinary fluids is reviewed in the first part of the pa-per, where special effort is devoted to the equation of state and thelaw of corresponding states. In addition, a few mathematical featuresinvolving properties of cubic equations are discussed, for appreciatingthe intrinsic beauty of VDW theory. A theory of astrophysical fluidsis shortly reviewed in the second part of the paper, grounding on thetensor virial theorem for subsystems, and an equation of state is formu-lated with convenient choices of reduced variables. Additional effort isdevoted to selected density profiles, namely a simple guidance case andtwo cases of astrophysical interest. Given the analogy between macrogasreduced isoenergetics and VDW reduced isothermals, a phase transition(gas-stars) is assumed to take place in astrophysical fluids, similarly toa phase transition (vapour-liquid) observed in ordinary fluids. In thisframework, the location of gas clouds, stellar systems, galaxies, clus-ter of galaxies, on the plane scanned by reduced variables, is tenta-tively assigned. A brief discussion shows how VDW’ two great discov-eries, namely a gas equation of state where tidal interactions betweenmolecules are taken into account and the law of corresponding states,
1Affiliated up to September 30th 2014. Current status: Studioso Senior. Current posi-tion: in retirement due to age limits.
912 R. Caimmi
related to microcosmos, find a counterpart with regard to macrocosmos.In conclusion, after more than a century since the awarding of the NobelPrize in Physics, VDW’ ideas are still valid and helpful today for a fullunderstanding of the universe.
Keywords: gas: ideal, real - gas: equation of state - galaxies: evolution -dark matter: haloes
1 Introduction
More than one century ago (1910), the Nobel Prize in Physics was awardedto Johannes Diderik van der Waals (hereafter quoted as VDW). In his doctoralthesis (1873) the ideal gas equation of state was generalized for embracingboth the gaseous and the liquid state, where these two states of aggregationnot only merge into each other in a continuous manner, but are in fact of thesame nature. With respect to ideal gases, the volume of the molecules and theintermolecular tidal forces were taken into account.
VDW equation of state was later reformulated in terms of reduced (di-mensionless) variables (1880), which allows the description of all substances interms of a single equation. In other words, the state of any substance, definedby the values of reduced volume, reduced pressure, and reduced temperature,is independent of the nature of the substance. This result is known as the lawof corresponding states.
VDW equation of state, in dimensional and reduced form, served as aguide during experiments which ultimately led to hydrogen (1898) and helium(1908) liquefaction. The Cryogenic Laboratory at Leiden had developed underthe influence of VDW’s theories. For further details on VDW’s biography aninterested reader is addressed to specific textbooks e.g., [21].
The current paper was intended to be written in honor of the first centennialof VDW awarding Nobel Prize in Physics2. Ideal and VDW equation of state,both in dimensional and reduced form, are reviewed, and a number of featuresare analysed in detail, in Section 2. Counterparts to ideal and VDW equationsof state for astrophysical fluids, or macrogases, are briefly summarized andcompared with the classical formulation in Section 3. The discussion and theconclusion are drawn in Section 4 and 5, respectively.
2 Equation of state of ordinary fluids
Let ordinary fluids be conceived as fluids which can be investigated inlaboratory. The simplest description is provided by the theory of ideal gas,
2It is a revised and improved version of a previous, unpublished paper available on thearxiv site (arxiv:1210.3688v1).
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 913
where the following restrictive assumptions are made: (i) particles are identicalspheres; (ii) the number of particles is extremely large; (iii) the motion ofparticles is random; (iv) collisions between particles or with the wall of thebox are perfectly elastic; (v) interactions between particles or with the wall ofthe box are negligible.
Ideal gas equation of state may be written under the form e.g., [15] Chap. IV,§42:
pV = NkT ; (1)
where p is pressure, V volume, T temperature, N particle number, and kBoltzmann constant.
In getting a better description of ordinary fluids, the above assumption (v)is relaxed and tidal interactions between particles, due to charge distribution,are taken into consideration. VDW’ generalization of the equation of state ofideal gases, Eq. (1), reads [25]:(
p+ AN2
V 2
)(V −NB) = NkT ; (2)
where A and B are constants which depend on particle nature.More specifically, the presence of an attractive interaction between particles
reduces both force and frequency of particle-wall collisions: the net effect is areduction of pressure, proportional to square numerical density, expressed asA(N/V )2. On the other hand, the whole volume of the box, V , is not accessibleto particles, in that they are represented as identical spheres: the free volumewithin the box is V − NB, where NB is the covolume. In particular, T = 0implies V = NB, hence the covolume per particle, B, may be conceived as thevolume filled by a single sphere in the limit of zero absolute temperature.
Let 2r0 be the interparticle distance when the interaction energy is null(positive values being related to lower distances, implying repulsion, and neg-ative values to larger distances, implying attraction). Accordingly, r0 may beconceived as the effective radius of a single sphere, which implies an effectivevolume, V0 = (4π/3)r3
0. It can be seen B exceeds V0 by a factor of 4. Forfurther details, an interested reader is addressed to specific textbooks e.g., [15]Chap. VII, §§72-74.
The features of both ideal and VDW isothermal i.e. constant-temperaturecurves, or isothermals, were described in an earlier investigation [5], to be con-ceived as the parent paper. A comparison between VDW and empirical isother-mals with equal temperature shows satisfactory agreement only in presence ofa sole phase (gas or liquid) i.e. for sufficiently low or large volumes and/orsufficiently high temperatures. If, on the other hand, two phases coexist, thepressure of saturated vapour maintains constant as volume changes, yielding ahorizontal real isothermal. Real isothermals are less and less extended for in-creasing temperature, until a single point is attained, Pc ≡ (Vc, pc, Tc), which
914 R. Caimmi
is defined as critical point. The special VDW isothermal where the criticalpoint lies, is defined as critical isothermal.
Parameters appearing in VDW equation of state, Eq. (2), may be expressedin terms of coordinates of critical point, as:
Vc = 3NB ; (3)
Tc =8
27
A
B
1
k; (4)
pc =1
27
A
B2; (5)
Zc =pcVc
NkTc
=3
8; (6)
where, in general, the compressibility factor:
Z =pV
NkT; (7)
defines the degree of departure from the behaviour of ideal gases, for which Z =1, according to Eq. (1). For further details, an interested reader is addressedto specific textbooks e.g., [22] Chap. XII, §20; [15] Chap. VIII, §85; and to theparent paper [5].
Ideal and VDW isothermals with equal temperature are plotted in Fig. 1for different values of T/Tc with respect to the parent paper [5]. Special regionsand loci on the Clapeyron plane, (OV p), are represented in Fig. 2 for VDWisothermals. The comparison between real and VDW isothermals with equaltemperature, T/Tc = 20/23, is shown in Fig. 3.
For simplifying both notation and calculations, the (dimensionless) reducedvariables are defined as e.g., [15] Chap. VIII, §85:
/V =V
Vc
; /p =p
pc
; /T =T
Tc
; (8)
where the coordinates of the critical point are expressed by Eqs. (3), (4), and(5). Accordingly, ideal gas equation of state, Eq. (1), and VDW equation ofstate, Eq. (2), reduce to:
/p /V =8
3/T ; (9)(
/p+3
/V 2
)(/V − 1
3
)=
8
3/T ; /V >
1
3; (10)
respectively. Additional features related to VDW isothermals are outlined inAppendix A. The intersections between real and VDW isothermals of equaltemperature are analysed and discussed in Appendix B. The limit of absolutezero temperature is considered in Appendix C.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 915
Figure 1: Ideal (left panel) and VDW (right panel) isothermals with equaltemperature (from bottom to top), T/Tc = 20/23, 20/22, 20/21, 20/20, 20/19,20/18. No extremum point exists above the critical isothermal, T/Tc = 1.
The locus of intersections between VDW and real isothermals of equaltemperature is represented in Fig. 2 as a dashed trifid curve, where the left,the right, and the middle branch correspond to /VA, /VE, and /VC, shown inFig. 3, respectively. The branching point coincides with the critical point. Thelocus of VDW isothermal extremum points is represented in Fig. 2 as a dottedbifid curve starting from the critical point, where the left and the right branchcorresponds to minimum and maximum points, respectively.
A fluid state can be represented in reduced variables as ( /V , /p, /T ), whereone variable may be expressed as a function of the remaining two, by use ofideal gas reduced equation of state, Eq. (9), or VDW reduced equation of state,Eq. (10). The formulation in terms of reduced variables, Eq. (8), makes relatedequation of state universal i.e. it holds for any fluid. Similarly, the Lane-Emden equation expressed in polytropic (dimensionless) variables describesthe whole class of polytropic gas spheres, with assigned polytropic index, inhydrostatic equilibrium e.g., [10] Chap. IV, §4).
The coordinates of intersection points between real and VDW isothermals
916 R. Caimmi
of equal temperature and extremum points of VDW isothermals, for selectedtemperatures below the critical value, are listed in Table 1 where special efforthas been devoted to the lower neighbourhood of the critical point. More specif-ically, the following dimensionless parameters have been evaluated vs temper-ature, /T : lower volume limit, /VA, where liquid and vapour phase coexist;extremum point (minimum) volume, /VB; intermediate volume, /VC, where liq-uid and vapour phase coexist, for which pressure equals its counterpart relatedto corresponding lower and upper volume limit; extremum point (maximum)volume, /VD; upper volume limit, /VE, where liquid and vapour phase coexist;extremum point (minimum) pressure, /pB; pressure, /pA = /pC = /pE, related tohorizontal real isothermal; extremum point (maximum) pressure, /pD.
Tidal interactions between particles appear in Eq. (2) via parameters Aand B. To gain more insight, the product on the left-hand side therein can bedeveloped grouping the terms in A, B, AB, and combining with Eqs. (3), (5),(7), and (8). The result is:
pV
(1 +
9 /V − /p /V 2 − 3
3 /p /V 3
)= NkT ; (11)
Z =pV
NkT=
(1 +
9 /V − /p /V 2 − 3
3 /p /V 3
)−1
; (12)
where the effect of tidal interactions relates to the second term within brackets.More specifically, two additional parameters i.e. critical volume and criticalpressure, appear in the above mentioned term. Accordingly, VDW equationof state exhibits three variables, V , p, T , and two parameters, Vc, pc, relatedto the effects of tidal interactions between particles.
In terms of reduced variables, Eq. (11) via Eq. (6) reads:
/p /V
(1 +
9 /V − /p /V 2 − 3
3 /p /V 3
)=
8
3/T ; (13)
where the term within brackets tends to zero as /V → 1/3 i.e. volume reducesto covolume, which implies null temperature, as expected.
3 Equation of state of astrophysical fluids
3.1 General considerations
Let macrogases be defined as astrophysical fluids confined by gravitation.Let two-component system be taken into consideration. For assigned densityprofiles, the virial theorem can be formulated for each subsystem, where thepotential energy is the sum of the self potential energy of the component under
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 917
Table 1: Values of dimensionless parameters, /T , /VA, /VB, /VC, /VD, /VE, /pB, /pC,/pD, within the range, 0.85 ≤ /T ≤ 0.99, using a step, ∆ /T = 0.01. Additionalvalues are computed near the critical point, to increase the resolution. Allvalues equal unity on the critical point. Index captions: A, C, E - intersectionsbetween VDW and real isothermals of equal temperature; B - extremum pointof minimum; D - extremum point of maximum. Extremum points are relatedto VDW isothermals, while their real counterparts are flat in presence of bothliquid and vapour phase. To save aesthetics, 01 on head columns stands forunity and 9.99̄ on bottom left stands for 9.999.
where i and j denote inner and outer subsystem3, respectively, Ekin is kineticenergy, Esel, Etid, and Evir are self, tidal, and virial potential energy, respec-tively. For further details, an interested reader is addressed to the parent paper[5].
Astrophysical fluids differ from ordinary fluids on two main respects, namely(i) the latter are collisional while the former could be collisionless, which im-plies rectilinear and curvilinear trajectories, respectively, and (ii) macrogasescannot be bounded by rigid walls, which implies evaporation. The assump-tion of closed systems needs nonzero pressure on related boundaries, whereevaporating macroparticles are forced to be reflexed as in rigid walls.
Observables in ordinary fluids (e.g., volume, pressure, temperature) withincylindric boxes may be changed acting on a movable circular wall or piston,by adding or subtracting mechanical energy or work. Conceptually, nothingchanges if the box and the piston are thought to be spherical and the surfaceof the piston to be variable4.
By analogy, astrophysical fluids must necessarily be conceived as two-component fluids, where one subsystem, G, is the macrogas under consid-eration, and the other one, P, acts as a piston for changing the observables.Energy can be added to or subtracted from P subsystem (and then to G sub-system via tidal interaction) by changing the mass, MP, and/or the volume,VP, and/or the density profile, ρP, and determining the virial equilibrium con-figuration via the virial theorem for subsystems. Let related observables bedefined as macrovolume, VU, macropressure, pU, and macrotemperature, TU,U = G, P.
Macrovolume is merely the volume filled by macrogas. Macropressure hasnecessarily dimensions of force per unit area or energy per unit volume, re-gardless of (dimensionless) shape factors and profile factors. If macroparticlesare conceived as mass points, macrotemperature may be defined as in ordinaryfluids:
TU =1
3
2(EU)kin
NUk=
1
3
mU(σU)2
k; U = G,P ; (15)
where N is total number of macroparticles, m = M/N mean macroparticlemass, σ rms velocity, and k Boltzmann constant. For typical stellar systems,
3For sake of simplicity, configurations with intersecting boundaries shall not be consideredin the current investigation.
4A simpler example relates to ordinary fluids inside conical boxes with a changing-surfacepiston moving along the axis of the cone.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 919
m = 1m�, σ = 144 km/s, macrotemperature is T = 1062 K. The large value ofmacrotemperature with respect to temperature in ordinary fluids arises fromthe large value of mean macroparticle mass with respect to mean particle massin ordinary fluids.
Strictly speaking, Eq. (15) holds for collisional fluids, where the stresstensor is isotropic. For collisionless fluids, the stress tensor is in generalanisotropic. In any case, it is diagonal in the reference frame where the co-ordinate axes coincide with the principal axes of inertia. Accordingly, themacrotemperature tensor may be defined as:
in the following, attention shall be restricted to macrotemperature i.e. thetrace of macrotemperature tensor.
For assigned subsystems, (U, V) = (G, P), (P, G), the virial theorem reads:
−(EU)sel − (EUV)tid = −(EUV)vir = 2(EU)kin ; (19)
according to Eq. (14). For further details, an interested reader is addressed toearlier investigations [16] [1] [7] [8].
The combination of Eqs. (15) and (19) yields:
−1
3
(EU)sel
VU
VU
[1 +
(EUV)tid
(EU)sel
]= NUkTU ; (20)
in terms of macrotemperature. The following definition of macropressure:
pU = −1
3
(EU)sel
VU
; (21)
translates the virial theorem for subsystems into an equation of state, as:
pUVU
[1 +
(EUV)tid
(EU)sel
]= NUkTU ; (22)
with regard to (U, V) macrogases, (U, V) = (G, P), (P, G). Accordingly, theratio:
ZU =pUVU
NUkTU
=
[1 +
(EUV)tid
(EU)sel
]−1
; (23)
may be conceived as macrogas compressibility factor. In this view, Eqs. (22)and (23) are macrogas counterparts of Eqs. (11) and (12), respectively, whichhold for VDW gas.
920 R. Caimmi
In the limit of an infinitely extended V subsystem, VV → +∞, (EUV)tid →0, Eq. (22) has the same formal expression as the ideal gas equation of state,Eq. (1), which, in turn, results from the VDW equation of state, Eq. (2), inabsence of tidal action between molecules.
Strictly speaking, covolume, NUBU, should be taken into consideration,which should imply VU > NUBU instead of VU > 0. In any case, NUBU � VU,for astrophysical fluids, assuming either BG = V�, where V� is the solar vol-ume, or BU = (4π/3)[(RU)g]
3/NU, where (RU)g = 2GMU/c2 (c velocity of light
in baryonic matter vacuum) is the gravitational radius e.g., [14] Chap. IX, §97,of macrogas under consideration. For this reason, covolume shall be neglectedin the following.
From this point on, attention shall be restricted to the special case ofhomeoidally striated density profiles with similar and similarly placed bound-aries, where coordinate axes coincide with principal axes of inertia.
3.2 Homeoidally striated density profiles with similarand similarly placed boundaries
For homeoidally striated density profiles with similar and similarly placedboundaries, where coordinate axes coincide with principal axes of inertia, selfpotential energy and tidal potential energy take the explicit expression e.g.,[5]:
(EU)sel = −G(MU)2
aU
ΞU(νU)sel
[(νU)mas]2Σ ; U = G,P ; (24)
(EUV)tid = −G(MU)2
aU
ΞU(νUV)tid
[(νU)mas]2Σ ; (U,V) = (G,P), (P,G) ; (25)
where G is constant of gravitation, M mass, a = a1 truncation radius alongthe major semiaxis, Ξ = Ξ1 dimensionless truncation radius along the majorsemiaxis, νmas and νsel are profile factors i.e. depending only on the densityprofile, νtid is a profile factor which, in addition, depends on the mass ratio,m = MP/MG, and on the homodirection axis ratio, y = aP/aG = (aP)r/(aG)r,r = 1, 2, 3, and Σ is a shape factor i.e. depending only on the boundary, whereΣ = 2 in the special case of spherical symmetry e.g., [8] [3] [4].
The substitution of Eqs. (24) and (25) into (20) after little algebra yields:
1
3
G(MU)2
aU
FUV = NUkTU ; (U,V) = (G,P), (P,G) ; (26a)
FUV =ΞU(νU)sel
[(νU)mas]2
[1 +
(νUV)tid
(νU)sel
]Σ ; (26b)
where tidal effects from V subsystem relate to the profile factor, (νUV)tid.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 921
In addition, Eqs. (21), (22), and (23), take the explicit form:
pU =1
4π
G(MU)2
(aU)4ε21ε31
ΞU(νU)sel
[(νU)mas]2Σ ; U = G,P ; (27)
pUVU
[1 +
(νUV)tid
(νU)sel
]= NUkTU ; (U,V) = (G,P), (P,G) ; (28)
ZU =pUVU
NUkTU
=
[1 +
(νUV)tid
(νU)sel
]−1
; (U,V) = (G,P), (P,G) ; (29)
where εpq = ap/aq are homosurface axis ratios. For further details, an in-terested reader is addressed to the parent paper [5], keeping in mind thatmacropressure therein has a different definition where neither profile factorsnor shape factors are involved.
It is worth remembering Eqs. (28) and (29) exhibit a similar formal expres-sion with respect to VDW gases, Eqs. (11) and (12), respectively, where theeffects of tidal interactions relate to the second term within brackets. Morespecifically, two additional parameters i.e. mass ratio and homodirection axisratio appear in the above mentioned term. Accordingly, macrogas equation ofstate exhibits three variables, VU, pU, TU, and two parameters, m, y, relatedto the effects of tidal interactions between subsystems.
For an arbitrary macroisothermal curve, TU = const, or macroisothermal,the macrogas equation of state, Eq. (28), may be expressed as:
pUVU
MU
[1 +
(νUV)tid
(νU)sel
]=kTU
mU
; (U,V) = (G,P), (P,G) ; (30)
where 0 < VU < +∞ and (νUV)tid/(νU)sel ≥ 0 in the case under considera-tion of homeoidally striated density profiles with similar and similarly placedboundaries, as shown below. For homogeneous spherical configurations, Ξ = 1,νmas = Ξ3 = 1, νsel = 3Ξ5/10 = 3/10, εpq = 1, Σ = 2, and Eq. (27) reduces to:
(pU)hsp =1
4π
3
5
G(MU)2
(aU)4; (31)
where the index, hsp, means homogeneous sphere. Related self potential en-ergy, via Eq. (21), is:
[(EU)sel]hsp = −3
5
G(MU)2
aU
; (32)
which may be considered as a limiting value for the following reasons.Gravitational potential energy of a matter distribution (in absolute value)
may be conceived as a binding energy which, ipso facto, is increasing for in-creasing concentration at fixed total mass and major semiaxis. Accordingly,
922 R. Caimmi
both inhomogeneous and aspherical configurations increase the binding energyprovided density profiles with positive slope are excluded as unphysical. In thisview, homogeneous spherical configurations are related to a minimum bindingenergy, −[(EU)sel]hsp ≤ −(EU)sel.
On the other hand, the tidal action exerted from the external homeoid ofthe outer subsystem on the inner one is null owing to Newton’s theorem, whichimplies (Eij)tid/(Ei)sel = (νij)tid/(νi)sel > 0. In addition, the tidal action ex-erted from the inner subsystem on the outer one implies (Eji)tid/(Ej)sel =(νji)tid/(νj)sel > 0. This is why the presence of a second subsystem en-hances binding energy with respect to a single subsystem. In conclusion,(EUV)tid/(EU)sel = (νUV)tid/(νU)sel > 0, (U,V) = (G,P), (P,G).
According to the above considerations, the left-hand side of Eq. (28) via(20)-(22) satisfies the following condition:
pUVU
[1 +
(νUV)tid
(νU)sel
]= −1
3[(EU)sel + (EUV)tid]
=1
3
G(MU)2
aU
ΞU(νU)sel
[(νU)mas]2
[1 +
(νUV)tid
(νU)sel
]Σ >
1
3
3
5
G(MU)2
aU
; (33)
or:ΞU(νU)sel
[(νU)mas]2
[1 +
(νUV)tid
(νU)sel
]Σ >
3
5; (34)
which depends on the density profiles (including shape) and, in addition, onthe homodirection axis ratio, y, and the mass ratio, m, via (νUV)tid.
Let (VU, pU) be an assigned state on a selected macroisothermal, TU =const. An increasing macrovolume, VU, or major semiaxis, aU, at constantmass, MU, implies a decreasing left-hand side of Eq. (28), via Eq. (27), whichmay be compensated by increasing concentration and/or asphericity via Eq. (34),with due account taken of a changing tidal interaction due to the factor,[1 + (νUV)tid/(νU)sel], appearing in Eq. (28), until an infinite macrovolume isattained. In particular, [Ξνsel/(νmas)
2]Σ = 3/(5 − n) for polytropic spheresof polytropic index equal to n (0 ≤ n ≤ 5), Σ = π for flattened oblatespheroids (ε21 = 1, ε31 → 0), and Σ → +∞ for elongated prolate spheroids(ε21 = ε31 → 0).
A decreasing macrovolume, VU, or major semiaxis, aU, at constant mass,MU, implies an increasing left-hand side of Eq. (28), via Eq. (27), which maybe compensated by decreasing concentration and/or asphericity via Eq. (34),with due account taken of a changing tidal interaction via the factor, [1 +(νUV)tid/(νU)sel], appearing in Eq. (28), until a minimum macrovolume is at-tained, which makes the ending point of the macroisothermal. In fact, lowermacrovolumes would imply density profiles with positive slope, unless the as-sumption of mass conservation is released. By increasing mass, a macroisother-
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 923
mal can be continued until null macrovolume (or volume equal to covolume)is asymptotically attained.
Accordingly, astrophysical fluids lying on macroisothermals must be con-ceived as open systems i.e. variable macroparticle number, contrary to ordi-nary fluids, which can be conceived as closed systems i.e. constant particlenumber. Then a more germane formulation of macrogas equation of state,Eq. (30), reads:
pU
ρU
[1 +
(νUV)tid
(νU)sel
]=kTU
mU
; (U,V) = (G,P), (P,G) ; (35)
where mean density, ρ = M/V , appears instead of total mass, M .Dividing both sides of Eq. (28) where (U, V) = (P, G), by their counterparts
where (U, V) = (G, P), yields:
YpYV1 + (νPG)tid/(νP)sel
1 + (νGP)tid/(νG)sel
=NP
NG
YT ; (36)
in terms of the dimensionless variables:
Yp =pP
pG
; YV =VP
VG
; YT =TP
TG
; (37)
or, using Eqs. (15) and (27):
Yp =m2
y4
ΞP
ΞG
(νP)sel
(νG)sel
[(νG)mas
(νP)mas
]2
; YV = y3 ; YT =NG
NP
φ ; (38)
m =MP
MG
; y =aP
aG
; φ =(EP)kin
(EG)kin
=(EPG)vir
(EGP)vir
; (39)
where the mass ratio, m, has not to be confused with the mean macroparticlemass, m. The combination of Eqs. (15) and (37)-(39) yields:
TP =mP
mG
φ
mTG ; (40)
where mP = mG without loss of generality, provided P macrogas is consideredonly for the tidal potential induced on G macrogas.
Owing to Eqs. (38) and (39), Eq. (36) by use of (26b) takes the explicitform:
m2
y
FPG
FGP
= φ ; (41)
in terms of the dimensionless variables [5]:
Xp = m2 ; XV =1
y; XT = φ ; (42)
924 R. Caimmi
where the dimensionless factors, F , depend on the density profiles and, inaddition, on the variables, m, y, via the profile factors, νtid.
In the case under consideration of homeoidally striated, similar and simi-larly placed ellispoids, the dependence on the shape implies the same factor, Σ,for both FPG and FGP, which disappears in the ratio, FPG/FGP. Accordingly,Eq. (41) may be conceived as shape independent.
The additional restriction of coinciding scaled density profiles, fU = ρU/ρ†U,
ρ†U = ρU(r†U), r†U selected scaling radius along a fixed direction, U = G,P;coinciding scaled truncation radii, ΞU = RU/r
†U, RU truncation radius along
the same direction, U = G,P; and coinciding homodirection axes, y = 1, makesEq. (41) reduce to:
m = φ ; y = 1 ; (43)
for a formal demonstration, an interested reader is addressed to the parentpaper [5].
Due to appearence of parameters from both G and P macrogases, Eq. (36)or equivalently Eq. (41) may be conceived as a macrogas reduced equation ofstate, depending only on dimensionless parameters. More specifically, macro-gas reduced equation of state, Eq. (41), depends on three dimensionless vari-ables, namely mass ratio, m, homodirection axis ratio, y, and kinetic energyratio, φ, which is the counterpart of VDW reduced equation, Eq. (13), forVDW gases.
For assigned macrogases, the locus, φ = const, on the dimensionless plane,(Oy−1m2), may be conceived as a fractional isoenergetic curve, or fractionalisoenergetic, in the sense that kinetic energy ratio, φ, instead of macrotem-perature ratio, YT, maintains constant. It can be seen the latter alternativeimplies a subdomain where configurations have no physical meaning. For fur-ther details, an interested reader is addressed to Appendix D.
A generic point, (XV, Xp), on a fractional isoenergetic, XT, is connected toan infinity of configurations where homodirection axis ratio, y, mass ratio, m,kinetic energy ratio, φ, remain unchanged. Accordingly, macrogas equationsof state, Eqs. (26), are “entangled”: for fixed aU, MU, (EU)kin = (3/2)NUkTU,related counterparts aV, MV, (EV)kin = (3/2)NVkTV, are inferred via y, m, φ,by use of Eq. (39). Keeping in mind P subsystem is conceived as a piston forchanging the state of the other one, attention shall mainly be restricted to Gsubsystem.
With regard to a generic curve on (OXVXp) plane, and a generic subdo-main, XV1 ≤ XV ≤ XV2 , related area below the curve reads:
S12 =∫ XV2
XV1
Xp dXV =∫ y−1
2
y−11
m2 d1
y=∫ y1
y2
m2
y2dy ; (44)
where, in particular, mass, MG, and radius, aG, can be left unchanged while
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 925
MP and aP vary via Eqs. (39) and (42). Accordingly, Eq. (44) translates into:
S12GM2
G
aG
=∫ aP1
aP2
GM2P
a2P
daP ; (45)
which may be conceived as the work performed on G macrogas via P macrogasto leave MG and aG unchanged passing from (MP1 , aP1) to (MP2 , aP2).
A similar result holds for a generic curve on the (OX−1V X−1
p ) plane, wherethe counterpart of Eq. (45) exhibiting indexeses, G and P, reversed, may beconceived as the work performed on P macrogas via G macrogas to leave MP
and aP unchanged passing from (MG1 , aG1) to (MG2 , aG2).Macrogas equation of state, Eq. (28) via Eqs. (15) and (26) exhibits three
variables, aG, MG, (EG)kin, and two parameters, m, y, which are fixed bythe point selected on the fractional isoenergetic, where a third parameter, φ,remains unchanged. If, in addition, attention is restricted to isoenergetics,(EG)kin = (3/2)NGkTG = const, where mass conservation, MG = const, NG =const, takes place, then TG = const, and the remaining variable, aG, can bedetermined via Eq. (26).
It is worth emphasizing fractional isoenergetics, XT = const, and fractionalmacroisothermals, YT = (NG/NP)φ = (mP/mG)(φ/m) = const via Eq. (40),are equivalent provided mass conservation takes place in both macrogases. Thesame holds for isoenergetics, (EU)kin = (3/2)NUkTU = const via Eq. (15), andmacroisothermals, TU = const.
For assigned shape, density profiles, and truncation radii, isoenergetics ormacroisothermals can be determined along the following steps.
(i) Start from a generic configuration among (m, y).
(ii) For assigned φ = XT, via Eq. (41) plot related fractional isoenergetic on(OXVXp) plane.
(iii) For assigned mass, MG, and kinetic energy, (EG)kin = NGkTG, via Eq. (26)determine major semiaxis, aG and via Eq. (39) determineMP, aP, (EP)kin =NPkTP.
(iv) Return to (iii) and change m, y, along the fractional isoenergetic leavingMG, NGkTG, unchanged.
(v) Using Eq. (27), determine pG, VG; pP, VP; and plot related isoenergeticson (OVGpG); (OVPpP); plane, respectively.
(vi) Return to (ii) and repeat the procedure until a family of fractional isoen-ergetics is plotted on (OXVXp) plane, and a family of isoenergetics isplotted on (OVGpG) and (OVPpP) plane for different NGkTG and NPkTP,respectively.
926 R. Caimmi
Accordingly, a selected isoenergetic on the plane, (OVGpG), Eq. (28), cor-responds to a fractional isoenergetic on the dimensionless plane, (OXVXp),Eq. (41), and vice versa, conformly to Eqs. (37)-(40).
The procedure outlined above defines a special way of making macroiso-thermal changes of state, (VG, pG, TG)→ (VG+∆VG, pG+∆pG, TG), and in thissense it is not restrictive. With regard to ordinary fluids, it is equivalent tovary mechanical and thermal energy by assigned amounts making isothermalchanges of state, (V, p, T )→ (V + ∆V, p+ ∆p, T ).
The definition of dimensionless variables, YV, Yp, YT, expressed by Eq. (37),aims to a closer analogy between ordinary and astrophysical fluids with respectto the parent paper [5] where attention was restricted to XV, Xp, XT, expressedby Eq. (42), and fractional isoenergetics were conceived as macroisothermals.
Fractional isoenergetics were derived in the parent paper [5] for a wideamount of cases, with regard to different kind of macrogases, namely UU, HH,and HN/NH. More specifically, UU is a simple guidance case where no criticalpoint occurs; conversely, HH and HN/NH are cases of astrophysical interestwhere critical points take place. Accordingly, isoenergetics can be inferredfrom the above mentioned fractional isoenergetics.
Critical fractional isoenergetics, hosting a critical point i.e. a horizontalinflexion point, take place in macrogases for sufficiently steep density profiles:fractional isoenergetics on one side exhibit two extremum points (maximumand minimum), while fractional isoenergetics on the other side exhibit no ex-tremum point, similarly to reduced VDW isothermals. On the contrary, criticalfractional isoenergetics are absent for sufficiently mild density profiles, wherefractional isoenergetics show a nonmonotonic trend. For further details, an in-terested reader is addressed to the parent paper [5] and an earlier investigation[9].
The existence of a phase transition moving along a selected fractional isoen-ergetic in presence of extremum points (model fractional isoenergetic), wherethe path is a horizontal line (transition fractional isoenergetic), must neces-sarily be assumed as a working hypothesis, due to the analogy between VDWisothermals and model fractional isoenergetics. Unlike VDW equation of state,Eq. (2), macrogas equation of state, Eq. (30), is not analytically integrable,which implies the procedure, used for determining a selected fractional isoen-ergetic, must be numerically performed.
The main steps are the following.
(i) Calculate intersections, VGA, VGC, VGE; VGA < VGC < VGE; between thegeneric horizontal line on the plane, (OVGpG), pG = const, and the curverelated to theoretical macrogas equation of state, within the range, pGB <pG < pGD, where B and D denote extremum points of minimum andmaximum, respectively.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 927
(ii) Calculate area of regions, ABC and CDE.
(iii) Find the special value, pG = pGC, which makes regions of equal area.
(iv) Trace the transition fractional isoenergetic as a horizontal line connectingpoints, (VGA, pGA), (VGC, pGC), (VGE, pGE), pGA = pGC = pGE.
For further details, an interested reader is addressed to the parent paper [5].In order to preserve the analogy with ideal and VDW gases, the tidal potentialenergy has to be negligible and comparable, respectively, with regard to selfpotential energy, in the formulation of the virial theorem for subsystems andrelated equation of state concerning macrogas of interest.
For assigned density profiles allowing critical macroisothermal, Eqs. (28),(U, V) = (G, P), (36), and (41), may be translated into reduced variables, as:
/pG /VG1 + (νGP)tid/(νG)sel
1 + [(νGP)tid]c/(νG)sel
=MG
(MG)c
/TG ; (46)
/pG =pG
(pG)c
; /VG =VG
(VG)c
; /TG =TG
(TG)c
; (47)
/Yp /YVFc =m
mc
/YT ; (48)
/Yp =Yp
Ypc
; /YV =YV
YVc
; /YT =YT
YTc
; (49)
Fc =1 + (νPG)tid/(νP)sel
1 + [(νPG)tid]c/(νP)sel
1 + [(νGP)tid]c/(νG)sel
1 + (νGP)tid/(νG)sel
; (50)
/Xp /XVFc = /XT ; (51)
/Xp =Xp
/Xpc
; /XV =XV
/XVc
; /XT =XT
/XTc
; (52)
where the index, c, denotes critical point, and dimensionless variables, X, aredefined by Eq. (42).
In the limit of an infinitely extended P subsystem, y → +∞, (νPG)tid → 0,(νGP)tid → 0, Eqs. (46) and (48) reduce to:
/pG /VG =
{1 +
[(νGP)tid]c(νG)sel
}MG
(MG)c
/TG ; (53)
/Yp /YV =1 + [(νPG)tid]c/(νP)sel
1 + [(νGP)tid]c/(νG)sel
m
mc
/YT ; (54)
which can be related to ordinary gases filling a fixed volume. For furtherdetails, an interested reader is addressed to the parent paper [5].
928 R. Caimmi
The above result, expressed by Eq. (46), may be written in parametric formusing Eqs. (27) and (28), as:
/pG =/M2
G
/a4G
; /VG = /a3G ; /TG =
/MG
/aG
χc ; (55)
/aG =aG
(aG)c
; /MG =MG
(MG)c
; χc =1 + (νGP)tid/(νG)sel
1 + [(νGP)tid]c/(νG)sel
; (56)
where /aG and /MG may be conceived as reduced radius and reduced mass,respectively.
Reduced macropressure and reduced macrotemperature, as functions ofreduced macrovolume, read:
/pG = /M2G /V
−4/3G ; /TG = /MG /V
−1/3G χc ; (57)
where the dependence on the path along a fractional isoenergetic, φ = const,occurs via χc. The dependence of reduced macropressure, /pG, and reducedmacrotemperature, /TG, on reduced macrovolume, /VG, for assigned reducedmass, /MG, and parameter, χc, is shown in Fig. 4 where macrogas state isdefined by the intersection of two selected curves together with related pointon fractional isoenergetic, (m, y, φ).
The dependence of reduced macrotemperature i.e. χc, on reduced coordi-nate, /XV, for fixed reduced volume and reduced mass, or reduced coordinate,/XT, is owing to density profiles and shall be determined for UU, HH, HN/NHmacrogases in the following sections.
3.3 UU macrogases
UU macrogases exhibit flat density profiles, which is equivalent to poly-tropes of index, n = 0 e.g., [10] Chap. IV, §4, [2] [6], but implies negativedistribution functions for stellar fluids [24]. For this reason, UU macrogasesare of little astrophysical interest and can be considered as a simple guidancecase.
Fractional isoenergetics (XT = const) may be explicitly expressed as [5]:
Xp =XT
XV
[1 + Ψ(XV, XT)] ; (58a)
Ψ(XV, XT) =
X5VΦ2(XV,XT)
2XT− X3
VΦ(XV,XT)
XT
√X4
VΦ2(XV,XT)
4+ XT
XV;
0 < XV ≤ 1 ;Φ2(XV,XT)
2X5VXT
− Φ(XV,XT)X2
VXT
√Φ2(XV,XT)
4X6V
+ XT
XV;
1 ≤ XV < +∞ ;
(58b)
Φ(XV, XT) =
52
1X2
V− 3
2−XT ; 0 < XV ≤ 1 ;
1−(
52X2
V − 32
)XT ; 1 ≤ XV < +∞ ;
(58c)
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 929
limXV→0+
Xp = limXV→0+
XT
XV
= +∞ ; limXV→+∞
Xp = limXV→+∞
XT
XV
= 0 ; (58d)
limXT→0+
Xp = 0 ; limXT→+∞
Xp = +∞ ; (58e)
where Xp, XV, XT, are dimensionless variables defined by Eq. (42).Plotting fractional isoenergetics, Xp(XV), for assigned XT, on (OXVXp)
plane, shows two extremum points (minimum and maximum, respectively)but no critical point within the domain, 0 < XT < +∞ [5]. On the otherhand, φcrit ≈ 10, mcrit ≈ 10, for HH and HN/NH macrogases [5]. Thenφnorm = 10 shall be arbitrarily assumed as normalization value, together withynorm = 1.186944 implying mnorm = 7.290778 via Eqs. (58a)-(58c). The resultis:
XVn =1
ynorm
= 0.8425 ; Xpn = (mnorm)2 = 53.15545 ;
XTn = φnorm = 10 ; (59)
and reduced variables:
/XV =XV
XVn
; /Xp =Xp
Xpn
; /XT =XT
XTn
; (60)
shall be used in place of their counterparts related to critical values.Fractional isoenergetics, /X−1
p ( /X−1V ), are plotted in Fig. 5 for /X−1
T = 20/23,20/22, 20/21, 20/20, 20/19, 20/18, from bottom to top. The contribution fromthe term, /X−1
T / /X−1V , is shown by dotted curves. More specifically, dividing
both sides of Eq. (58a) by their counterparts related to (XVn, Xpn, XTn) yields:
/Xp =/XT
/XV
1 + Ψ(XV, XT)
1 + Ψ(XVn, XTn); (61)
where the inverse of the whole product and the first factor on the right-handside are represented in Fig. 5 as full and dotted curves, respectively. The reasonfor plotting /X−1
p vs /X−1V instead of /Xp vs /XV shall be clarified in dealing with
HH and HN/NH macrogases.In the case under consideration of flat density profiles, without loss of
generality, scaled truncation radii can be assumed as ΞU = RU/r†U = 1, U =
G,P. Accordingly, profile factors reduce to:
(νU)mas = 1 ; (νU)sel =3
10; U = G,P ; (62)
(νGP)tid =
310
my3
; 1 ≤ y < +∞ ;310m(
52− 3
2y2)
; 0 < y ≤ 1 ;(63)
(νPG)tid =
310
1m
(52− 3
21y2
); 1 ≤ y < +∞ ; ;
310
y3
m; 0 < y ≤ 1 ;
(64)
930 R. Caimmi
for further details, an interested reader is addressed to the parent paper [5].The special case of coinciding volumes, y = 1, implies (νGP)tid = (3/10)m;(νPG)tid = (3/10)(1/m); as expected.
In addition, Eqs. (26b) and (56) reduce to:
FGP =
310
(1 + m
y3
)Σ ; 1 ≤ y < +∞ ;
310
[1 +m
(52− 3
2y2)]
Σ ; 0 < y ≤ 1 ;(65)
FPG =
310
[1 + 1
m
(52− 3
21y2
)]Σ ; 1 ≤ y < +∞ ;
310
(1 + y3
m
)Σ ; 0 < y ≤ 1 ;
(66)
χc =
1+m/y3
1+mnorm/y3norm; 1 ≤ y < +∞ ;
1+m[5/2−(3/2)y2]1+mnorm/y3norm
; 0 < y ≤ 1 ;(67)
where y = aP/aG.The dependence of reduced macrotemperature, /TG, on reduced variable,
/XV, is shown in Fig. 6 for /M−1G /V
1/3G = 1 and /X−1
T = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top. A similar trend is exhibited for
critical fractional isoenergetic, /X−1T = 1, and /M−1
G /V1/3
G = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top.
3.4 HH macrogases
HH macrogases exhibit central cusp and vanishing density at infinite dis-tance [13], and have been proved to be consistent with nonnegative distributionfunctions in an acceptable parameter range [11].
Fractional isoenergetics on (O /XV /Xp) plane show weak dependence on scaledtruncation radii, ΞG, ΞP, including the limit, ΞG → +∞, ΞP → +∞, whereformulation is far simpler. For this reason, attention shall be restricted to infi-nite scaled truncation radii, which imply infinitesimal scaling radii, r†U, and/orinfinite truncation radii, RU, U = G,P, along an arbitrary direction. Accord-ingly, profile factors reduce to [5]:
(z−1)4[−2(2z + 1) ln z + (z − 1)(z + 5)] ; z 6= 1 ;
−323
; z = 1 ;(71)
w(ext)(z) =
{− 64
(z−1)4[2z(z + 2) ln z − (z − 1)(5z + 1)] ; z 6= 1 ;
−323
; z = 1 ;(72)
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 931
Table 2: Values of scaling fractional mass, m† = M(r†P)/M(r†G), fractionalmass, m, scaling fractional radius, y†, fractional truncation radius, y, andfractional energy, φ, related to critical point i.e. the horizontal inflexion pointon the critical fractional isoenergetic, for selected fractional scaled truncationradii, ΞG/ΞP, with regard to HH density profiles.
Eqs. (26b) and (68)-(73) into (41), are plotted in Fig. 7 for /X−1T = 20/23, 20/22,
20/21, 20/20, 20/19, 20/18, from bottom to top, where cases, ΞG/ΞP = 0.25,0.50, 1.00, 2.00, 4.00, are superimposed. The dashed curve (including centralbranch) is the locus of intersections between HH fractional isoenergetics andhorizontal lines yielding regions of equal area. The dotted curve is the locusof HH fractional isoenergetic extremum points. Plotting /X−1
p vs /X−1V yields
fractional isoenergetics similar to VDW isothermals shown in Fig. 2, whereextremum points are lying below the horizontal inflexion point.
Critical points, (XVc, Xpc, XTc), can be inferred from Table 2 via Eq. (42).
More specifically, values of scaling fractional mass, m† = M(r†P)/M(r†G), frac-tional mass, m, fractional scaling radius, y†, fractional truncation radius, y,and fractional energy, φ, related to critical point i.e. the horizontal inflexionpoint on the critical fractional isoenergetic, are listed in Table 2 for selectedfractional scaled truncation radii, ΞG/ΞP, with regard to HH density profiles.
An inspection of Table 2 discloses weak dependence of critical parameterson the ratio, ΞG/ΞP, within the range considered, leaving aside y via Eq. (73).Accordingly, critical parameters might be conceived as independent of (infinite)scaled truncation radii to a first extent.
An inspection of Fig. 7 discloses weak dependence of fractional isoenergeticson the ratio, ΞG/ΞP, within the range considered. Accordingly, fractionalisoenergetics might be conceived as independent of (infinite) scaled truncationradii to a first extent.
932 R. Caimmi
The dependence of reduced macrotemperature, /TG, on reduced variable,
/XV, is shown in Fig. 8 for /M−1G /V
1/3G = 1 and /X−1
T = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top. A similar trend is exhibited for
critical fractional isoenergetic, /X−1T = 1, and /M−1
G /V1/3
G = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top.
An inspection of Fig. 8 discloses weak dependence of reduced macrotemper-ature on the ratio, ΞG/ΞP, within the range considered. Accordingly, reducedmacrotemperature might be conceived as independent of (infinite) scaled trun-cation radii to a first extent.
3.5 HN/NH macrogases
HN/NH macrogases exhibit central cusp and vanishing density at infinitedistance, where HN means inner H density profile, related to G macrogas,and outer N density profile, related to P macrogas, y = aP/aG ≥ 1, and NHmeans inner N and outer H, y = aP/aG ≤ 1. N density profiles decline moreslowly with respect to above considered H, yielding infinite mass within infiniteradius [18] [19] [20]. HN macrogases have been proved to be consistent withnonnegative distribution functions in an acceptable parameter range [17] byuse of a previously stated theorem [12].
Fractional isoenergetics on (O /XV /Xp) plane show weak dependence on scaledtruncation radii, ΞG, ΞP, including the limit, ΞG → +∞, ΞP → +∞, whereformulation is far simpler. For this reason, attention shall be restricted to infi-nite scaled truncation radii, which imply infinitesimal scaling radii, r†U, and/orinfinite truncation radii, RU, U = G,P, along an arbitrary direction. Accord-ingly, profile factors reduce to [5]:
(νU)mas → +∞ ; (νU)sel = 36 ; ΞU → +∞ ; (74)
U ={
G ; NH macrogasesP ; HN macrogases
(νGP)tid =
−98m†w
(ext)HN (z) ; 1 ≤ y < +∞ ;
−98m†
y†w
(int)NH (z) ; 0 < y ≤ 1 ;
(75)
(νPG)tid =
−98
y†
m†w
(int)HN (z) ; 1 ≤ y < +∞ ;
−98
1m†w
(ext)NH (z) ; 0 < y ≤ 1 ;
(76)
w(int)HN (z) =
{− 64
(z−1)3(−2z ln z + z2 − 1) ; z 6= 1 ;
−643
; z = 1 ;(77)
w(ext)HN (z) =
{− 64
(z−1)2
(z+1z−1
ln z − 2)
; z 6= 1 ;
−323
; z = 1 ;(78)
w(int)NH (z) =
{− 64z
(z−1)2
(z+1z−1
ln z − 2)
; z 6= 1 ;
−323
; z = 1 ;(79)
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 933
Table 3: Values of scaling fractional mass, m† = M(r†P)/M(r†G), fractionalmass, m, scaling fractional radius, y†, fractional truncation radius, y, andfractional energy, φ, related to critical point i.e. the horizontal inflexion pointon the critical fractional isoenergetic, for selected fractional scaled truncationradii, ΞG/ΞP, with regard to HN/NH density profiles.
Eqs. (26b) and (74)-(81) into (41), are plotted in Fig. 9 for /X−1T = 20/23,
20/22, 20/21, 20/20, 20/19, 20/18, from bottom to top, where cases, ΞG/ΞP =0.25, 0.50, 1.00, 2.00, 4.00, are superimposed. The dashed curve (includingcentral branch) is the locus of intersections between HN/NH fractional isoen-ergetics and horizontal lines yielding regions of equal area. The dotted curveis the locus of HN/NH fractional isoenergetic extremum points. Plotting /X−1
p
vs /X−1V yields fractional isoenergetics similar to VDW isothermals shown in
Fig. 2, where extremum points are lying below the horizontal inflexion point.
Critical points, (XVc, Xpc, XTc), can be inferred from Table 3 via Eq. (42).
More specifically, values of scaling fractional mass, m† = M(r†P)/M(r†G), frac-tional mass, m, fractional scaling radius, y†, fractional truncation radius, y,and fractional energy, φ, related to critical point i.e. the horizontal inflexionpoint on the critical fractional isoenergetic, are listed in Table 3 for selectedfractional scaled truncation radii, ΞG/ΞP, with regard to HN/NH density pro-files.
An inspection of Table 3 discloses weak dependence of critical parameterson the ratio, ΞG/ΞP, within the range considered, leaving aside y via Eq. (81).Accordingly, critical parameters might be conceived as independent of (infinite)scaled truncation radii to a first extent.
934 R. Caimmi
An inspection of Fig. 9 discloses weak dependence of fractional isoenergeticson the ratio, ΞG/ΞP, within the range considered. Accordingly, fractionalisoenergetics might be conceived as independent of (infinite) scaled truncationradii to a first extent.
The dependence of reduced macrotemperature, /TG, on reduced variable,
/XV, is shown in Fig. 10 for /M−1G /V
1/3G = 1 and /X−1
T = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top. A similar trend is exhibited for
critical fractional isoenergetic, /X−1T = 1, and /M−1
G /V1/3
G = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top.
An inspection of Fig. 8 discloses weak dependence of reduced macrotemper-ature on the ratio, ΞG/ΞP, within the range considered. Accordingly, reducedmacrotemperature might be conceived as independent of (infinite) scaled trun-cation radii to a first extent.
3.6 Critical curves
VDW critical isothermal and HH, HN/NH, critical fractional isoenergeticare compared in Fig. 11, where the dashed curve is the same as in Fig. 2. Ac-cordingly, vapour and liquid phase of ordinary fluids coexist within the bell-shaped region bounded by the dashed curve. Both HH and HN/NH criticalfractional isoenergetic are more extended along horizontal direction with re-spect to VDW critical isothermal, which implies a more flattened counterpartof the above mentioned bell-shaped region. The critical point, by definition,reads ( /XVc , /Xpc , /XTc) ≡ (1, 1, 1).
4 Discussion
Tidal interactions between neighbourhing bodies span across the whole ad-missible range of lengths in nature: from, say, atoms and molecules to galaxiesand clusters of galaxies i.e. from micro to macro cosmos. Ordinary fluids arecollisional, which makes the stress tensor be isotropic and the velocity dis-tribution obey Maxwell’s law. Tidal interactions (electromagnetic in nature)therein act between colliding particles e.g., [15] Chap. VII, §74). Astrophysi-cal fluids could be collisionless, which makes the stress tensor be anisotropicand the velocity distribution no longer obey Maxwell’s law. Tidal interactions(gravitational in nature) therein act between single particles and the systemas a whole e.g., [5].
In both cases, an equation of state can be formulated in reduced variables:the VDW equation for ordinary fluids and an equation which depends ondensity profiles for astrophysical fluids. For sufficiently mild density profiles,fractional isoenergetics are characterized by the occurrence of two extremum
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 935
points, similarly to isothermals where a transition from liquid to gaseous phasetakes place, or vice versa. For sufficiently steep density profiles, the criticalfractional isoenergetic exhibits a single horizontal inflexion point, which definesthe critical point. Fractional isoenergetics below and above the critical one,show two or no extremum point, respectively, in complete analogy with VDWisothermals. In any case, the existence of an equation of state in reducedvariables implies the validity of the law of corresponding states for macrogaseswith assigned density profiles.
For astrophysical fluids, the existence of a phase transition must necessarilybe assumed as a working hypothesis by analogy with ordinary fluids. The phasetransition has to be conceived between gas and stars, and the (O /X−1
V /X−1p )
plane may be divided into three parts, namely
(i) a region bounded by the critical fractional isoenergetic on the left of thecritical point, and the locus of onset of phase transition on the right ofthe critical point, where only gas exists;
(ii) a region bounded by the critical fractional isoenergetic on the left of thecritical point, the locus of onset of phase transition on the left of thecritical point, and the vertical axis, where only stars exist;
(iii) a region bounded by the locus of onset of phase transition, and the hor-izontal axis, where gas and stars coexist.
The locus of onset of phase transition, not shown in Fig. 11 for reasons ex-plained above, is similar to its counterpart related to ordinary fluids, rep-resented by the bell-shaped curve in Fig. 11, but more extended along thehorizontal direction.
In this view, elliptical and S0 galaxies lie on (ii) region unless hosting hotinterstellar gas, and the same holds for globular clusters; spiral, irregular, anddwarf spheroidal galaxies lie on (iii) region, and the same holds for cluster ofgalaxies; gas clouds where stars never formed lie on (i) region, and the sameholds for hypothetical galaxies where stars never formed.
5 Conclusion
Van der Waals’ two great discoveries, more specifically a gas equation ofstate where tidal interactions between molecules are taken into account andthe law of corresponding states, related to microcosmos, find a counterpartwith regard to macrocosmos. After more than a century since the awarding ofthe Nobel Prize in Physics, van der Waals’ ideas are still valid and helpful today for a full understanding of the universe.
936 R. Caimmi
References
[1] P. Brosche, R. Caimmi, L. Secco, The tensor virial theorem for subsystems,Astronomy and Astrophysics, 125 (1983), 338-341.
[2] R. Caimmi, A special case of two-component polytropes with rigid rotation,Astronomy and Astrophysics, 159 (1986), 147-156.
[3] R. Caimmi, The Potential Energy Tensors for Subsystems. II. Mass Distri-butions with Ellipsoidal, Similar, and Coaxial Strata, Astrophysical Journal,419 (1993), 615-621. https://doi.org/10.1086/173512
[4] R. Caimmi, The potential-energy tensors for subsystems. III. Mass distri-butions with Ellipsoidal and Confocal Strata, Astrophysical Journal, 441(1995), 533-548. https://doi.org/10.1086/175380
[5] R. Caimmi, A Principle of Corresponding States for Two-Component,Self-Gravitating Fluids, Serbian Astronomical Journal, 180 (2010), 19-55.https://doi.org/10.2298/saj1080019c
[6] R. Caimmi, Pontential energies and potential-energy tensors for subsys-tems: general properties, Applied Mathematical Sciences, 10 (2016), 2749-2787. https://doi.org/10.12988/ams.2016.67217
[7] R. Caimmi, L. Secco, P. Brosche, The tensor virial theorem for subsystems.II, Astronomy and Astrophysics, 139 (1984), 411-416.
[8] R. Caimmi, L. Secco, The potential energy tensors for subsystems, Astro-physical Journal, 395 (1992), 119-125. https://doi.org/10.1086/171635
[9] R. Caimmi, T. Valentinuzzi, The Fractional Virial Potential Energy inTwo-Component Systems, Serbian Astronomical Journal, 177 (2008), 15-38. https://doi.org/10.2298/saj0877015c
[10] S. Chandrasekhar, An Introduction to the Study of the Stellar Structure,University of Chicago Press, 1939.
[11] L. Ciotti, The Analytical Distribution Function of Anisotropic Two-Component Hernquist Models, Astrophysical Journal, 471 (1996), 68-81.https://doi.org/10.1086/177954
[12] L. Ciotti, S. Pellegrini, Self-consistent two-component models of ellipticalgalaxies, Monthly Notices of the Royal Astronomical Society, 255 (1992),561-571. https://doi.org/10.1093/mnras/255.4.561
[13] L. Hernquist, An analytical model for spherical galaxies and bulges, As-trophysical Journal, 356 (1990), 359-364. https://doi.org/10.1086/168845
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 937
[14] L. Landau, E. Lifchitz, Theorie du Champs, Mir, Moscow, 1966.
[15] L. Landau, E. Lifchitz, Physique Statistique, Mir, Moscow, 1967.
[16] D.N. Limber, Effects of intracluster gas and duct upon the virial theorem,Astrophysical Journal, 130 (1959), 414-428. https://doi.org/10.1086/146733
[17] M. Lowenstein, R.E. White III, Prevalence and Properties of Dark Matterin Elliptical Galaxies, Astrophysical Journal, 518 (1999), 50-63.https://doi.org/10.1086/307256
[18] J.F. Navarro, C.S. Frenk, S.D.M. White, Simulations of X-ray clusters,Monthly Notices of the Royal Astronomical Society, 275 (1995), 720-740.https://doi.org/10.1093/mnras/275.3.720
[19] J.F. Navarro, C.S. Frenk, S.D.M. White, The Structure of Cold DarkMatter Halos, Astrophysical Journal, 462 (1996), 563-575.https://doi.org/10.1086/177173
[20] J.F. Navarro, C.S. Frenk, S.D.M. White, A Universal Density Profile fromHierarchical Clustering, Astrophysical Journal, 490 (1997), 493-508.https://doi.org/10.1086/304888
[21] Nobel Lectures, Nobel Lectures, Physics 1901-1921, Elsevier PublishingCompany, Amsterdam, 1967.
[22] A. Rostagni, Meccanica e Termodinamica, Libreria Universitaria di G.Randi, Padova, 1957.
[23] M.R. Spiegel, Mathematical Handbook, Schaum’s Outline Series, McGraw-Hill, Inc., New York, 1968.
[24] P.O. Vandervoort, The equilibrium of a galactic bar, Astrophysical Jour-nal, 240 (1980), 478-487. https://doi.org/10.1086/158253
[25] J.D. van der Waals, Over de Continuited Van Den Gas-en Vloeistoftoes-tand, Doctoral Thesis, Leiden University, Leiden, 1873.
Appendix
A Additional features of VDW isothermals
on the Clapeyron reduced plane
The equation of a generic VDW isothermal on the Clapeyron reduced plane,
938 R. Caimmi
(O /V /p), is e.g., [15] Chap. VIII, §85; [5]:
/p =8 /T
3 /V − 1− 3
/V 2; (82)
and the first and the second derivative with respect to /V read:(∂ /p
∂ /V
)/T
= − 24 /T
(3 /V − 1)2+
6
/V 3; (83)
(∂2 /p
∂ /V 2
)/T
=144 /T
(3 /V − 1)3− 18
/V 4; (84)
where, for assigned /T , the domain of the function, /p( /V ), is /V > 1/3, /V = 1/3is a vertical asymptote, and /p = 0 is a horizontal asymptote. In the specialcase of the critical point, ( /V, /T, /p) ≡ (1, 1, 1), the above mentioned derivativesare null, as expected.
The extremum points, via Eq. (83), are defined by the relation:
f( /V ) =(3 /V − 1)2
4 /V 3= /T ; (85)
which is satisfied on the critical point, as expected. The function on theleft-hand side of Eq. (85) has two extremum points: a minimum at /V = 1/3(outside the physical domain) and a maximum at /V = 1, where /T = 1.Accordingly, Eq. (85) is never satisfied for /T > 1, which implies no extremumpoint for related isothermals, as expected. The contrary holds for /T < 1,where it can be seen that the third-degree equation associated to Eq. (83)has three real solutions, related to extremum points. One lies outside thephysical domain, which implies /V ≤ 1/3. The remaining two are obtainedas the intersections between the curve, y = f( /V ), expressed by Eq. (85), andthe straight line, y = /T , keeping in mind that f(1/3) = 0, f(1) = 1, andlim /V→+∞ f( /V ) = 0.
The third-degree equation associated to Eq. (83), may be ordered as:
/V 3 − 9a /V 2 + 6a /V − a = 0 ; (86a)
a =1
4 /T; (86b)
where, with regard to the standard formulation e.g., [23] Chap. 9:
x3 + a1x2 + a2x+ a3 = 0 ; (87)
the discriminants of Eq. (86a) are:
Q =3a2 − a2
1
9= a(2− 9a) ; (88)
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 939
R =9a1a2 − 27a3 − 2a3
1
54=a(1− 18a+ 54a2)
2; (89)
D = Q3 +R2 =a2(1− 4a)
4; (90)
where D = 0 in the special case of the critical isothermal ( /T = 1, a = 1/4),D < 0 for /T < 1, and D > 0 for /T > 1. Accordingly, three (at least twocoincident) real solutions exist if D = 0, three different real solutions if D < 0,one real and two complex coniugate if D > 0. A real solution, /V0, always liesoutside the physical domain.
The three real solutions (D ≤ 0) may be expressed as e.g., [23] Chap. 9:
/V1 = 2√−Q cos
(π +
θ
3+
0π
3
)− 1
3a1 ; (91a)
/V2 = 2√−Q cos
(π +
θ
3+
2π
3
)− 1
3a1 ; (91b)
/V3 = 2√−Q cos
(π +
θ
3+
4π
3
)− 1
3a1 ; (91c)
θ = arctan
(√−DR
); (91d)
where a1 = −9a and, in the special case of critical isothermal, a = 1/4, Q =−1/16, R = −1/64, D = 0, which implies /V0 = min( /V1, /V2, /V3), /VA = /VB =/VC = /VD = /VE = max( /V1, /V2, /V3). A null factor appears in Eq. (91a) to saveaesthetics. In the special case, /T → 0, Eq. (86a) reduces to a second-degreeequation whose solutions are /V01 = /V02 = 1/3, while the related function isotherwise divergent as a → +∞. In general, the extremum points of VDWisothermals ( /T ≤ 1) occur at /V = /VB (minimum) and /V = /VD (maximum),/VB ≤ /VD. As /T → 0, /VB → 1/3, /VD → +∞, where, in all cases, 1/3 < /VB ≤1 ≤ /VD.
B Intersections between real and VDW
isothermals of equal temperature
With regard to selected real and VDW isothermals of equal temperature,three distinct intersections occur in presence of saturated vapour i.e. below thecritical temperature, which are coincident at the critical temperature. Vapourpressure maintains constant in presence of liquid phase i.e. /VA ≤ /V ≤ /VE,where an infinitesimal vapour and liquid mass fraction characterizes the ex-treme reduced volume values, /VA and /VE, respectively. In presence of a solephase, liquid ( /V < /VA) or gas ( /V > /VE), related real and VDW isothermalbranches coincide.
940 R. Caimmi
The intersections between real and VDW isothermals of equal tempera-ture (within the range where they are different) can be determined as in-tersections between horizontal lines and VDW isothermals on the Clapeyronreduced plane. Two intersections necessarily occur at ( /VA, /pA) and ( /VE, /pE).The third one, ( /VC, /pC), necessarily lies between the other two where, in ad-dition, /pA = /pC = /pE. The coordinates of the extremum points, ( /VB, /pB),minimum, and ( /VD, /pD), maximum, must necessarily satisfy the following in-equalities: /VA ≤ /VB ≤ /VC; /VC ≤ /VD ≤ /VE; /pB ≤ /pC ≤ /pD. It can be seen thatregions, ABC, CDE, bounded by real and VDW isothermals with equal temper-ature, have equal areas e.g., [15] Chap. VIII, §85), which yields the followingrelation e.g., [5]:
/pC =8
3
/T
/VE − /VA
ln3 /VE − 1
3 /VA − 1− 3
/VA /VE
; (92)
where, for a selected isothermal, the unknowns are /pC = /pA = /pE, /VA, and/VE.
The reduced volumes, /VA, /VC, /VE, see Fig. 3, may be considered as inter-sections between a VDW isothermal ( /T < 1) and a horizontal straight line,/p = /pC, on the Clapeyron reduced plane. In other words, /VA, /VC, /VE, are thereal solutions of the third-degree equation:
/V 3 −(
1
3+
8
3
/T
/pC
)/V 2 +
3
/pC
/V − 1
/pC
= 0 ; (93)
which has been deduced from Eq. (82), particularized to /p = /pC. Related solu-tions may be calculated using Eqs. (91). The last unknown, /pC, is determinedfrom Eq. (92).
An inspection of Fig. 3 shows that the points, A and E, are located on theleft of the minimum, B, and on the right of the maximum, D, respectively.Keeping in mind the above results, the following inequality holds: /VA ≤ /VB ≤1 ≤ /VD ≤ /VE, which implies further investigation on the special case, /VC = 1.The particularization of VDW equation of state, Eq. (82), to the point, C = C1,assuming /VC1 = 1, yields:
/T =/pC1 + 3
4; (94)
and Eq. (93) reduces to:
/V 3 − (1 + 2b) /V 2 + 3b /V − b = 0 ; (95a)
b =1
/pC1
; (95b)
with regard to the generic third-degree equation, Eq. (87), the three solutions,x1, x2, x3, satisfy the relations e.g., [23] Chap. 9:
x1 + x2 + x3 = −a1 ; (96a)
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 941
x1x2 + x2x3 + x3x1 = a2 ; (96b)
x1x2x3 = −a3 ; (96c)
where, in the case under discussion:
a1 = −1− 2b ; a2 = 3b ; a3 = −b ; (97a)
x1 = /VA ; x2 = /VC1 = 1 ; x3 = /VE ; (97b)
and the substitution of Eqs. (97) into two among (96) yields:
/VA = b−√b2 − b ; (98a)
/VE = b+√b2 − b ; (98b)
finally, the combination of Eqs. (94), (95b), and (98) produces:
/VA =1− 2
√1− /T
4 /T − 3; /T ≤ 1 ; (99a)
/VE =1 + 2
√1− /T
4 /T − 3; /T ≤ 1 ; ; (99b)
which, together with /VC1 = 1, are the abscissae of the intersection pointsbetween a selected VDW isothermal on the Clapeyron reduced plane and thestraight line, /p = /pC1 , in the special case under discussion.
The substitution of Eqs. (99) into (92), the last related to /p = /pC1 viaEq. (94), yields:
/T√1− /T
ln3− 2 /T + 3
√1− /T
3− 2 /T − 3√
1− /T= 6 ; (100)
which, keeping in mind the limit:
limx→0
[1
xln
1 + x
1− x
]= 2 ; (101)
holds only for the critical isothermal, /T = 1. Accordingly, the abscissa ofthe intersection point, C, between a selected VDW isothermal and related realisothermal, see Fig. 3, cannot occur at /VC = 1 unless the critical isothermalis considered. Then the third-degree equation, Eq. (93), must be solved in thegeneral case ( /VC 6= 1) by use of Eqs. (91).
C The limit of zero absolute temperature
Ideal gas equation of state, expressed by Eq. (1), in the limit of zero absolutetemperature, T = 0 K, implies (i) pressure attains any value, p > 0, provided
942 R. Caimmi
V → 0, and (ii) volume attains any value, V > 0, provided p→ 0. Accordingly,related ideal isothermal on the Clapeyron plane, (OV p), tends to positivecoordinate semiaxes. The same holds for reduced variables on the Clapeyronreduced plane, (O /V /p).
VDW equation of state, expressed by Eq. (2), in the limit of zero absolutetemperature, T = 0 K, implies (i) pressure attains any value, p > −A/B2,provided V → NB, and (ii) volume attains any value, V > NB, providedp → −A/B2; where NB = Vc/3; A/B2 = 27pc; via Eqs. (3); (5); respectively.Accordingly, related VDW isothermal on the Clapeyron plane, (OV p), tendsto subdomains, V > Vc/3; p > −27pc; or, in reduced variables, /V > 1/3;/p > −27; respectively.
With regard to VDW isothermal in the limit of absolute zero reducedtemperature, /T → 0, extremum points occur at /VB → 1/3 (minimum) and/VD → +∞ (maximum), as inferred in Appendix A and, in addition, /pB → −27(minimum) and /pD → 0 (maximum), according to above considerations. Insummary, B→ B0 ≡ (1/3,−27) and D→ D0 ≡ (+∞, 0).
The real isothermal ( /VA ≤ /V ≤ /VE) in the limit of absolute zero reducedtemperature, /T → 0, can be determined via Eq. (92) keeping in mind /VB ≥ /VA
and /VD ≤ /VE. The result is /pC → 0.
In general, intersections between VDW isothermals and horizontal lines,/p = /pC, on the Clapeyron reduced plane, (O /V /p), are solution of a third-degreeequation expressed by Eq. (93) which, in the special case of the horizontal axis,/pC = 0, reduces to a second-degree equation, as:
8
3/T /V 2 + 3 /V − 1 = 0 ; (102)
and related solutions read:
/V =9∓
√81− 96 /T
16 /T; (103)
where real solutions imply /T ≤ 81/96 = 27/32 and the special case, /T =27/32, yields a VDW isothermal which is tangent to the horizontal axis on( /VB, 0). If, in particular, /pC = 0 and /T → 0, Eq. (93) reduces to a first-degreeequation, as:
3 /V − 1 = 0 ; (104)
and related solution reads /V = 1/3, as expected.
To get further insight, let a variable, x, be defined as:
81− 96 /T = x2 ; /T =81− x2
96; 0 ≤ x < 9 ; (105)
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 943
where the limit of zero absolute reduced temperature, /T → 0+, relates tox→ 9−. The substitution of Eq. (105) into (103) yields:
/V =9∓ x
16(81− x2)/96=
96
16
9∓ x(9 + x)(9− x)
=6
9± x; (106)
with regard to VDW isothermals, /T = (81 − x2)/96. In the limit of zeroabsolute reduced temperature, x→ 9−, the above solutions read /V → 6/18 =1/3 and /V → +∞, as expected.
VDW isothermals are plotted in Fig. 12 for integer x, 0 ≤ x ≤ 8, wherevalues related to negative pressure or sufficiently small volume are not shownfor sake of clarity. An inspection of Fig. 12 shows regions, ABC, CDE, boundedby real and VDW reduced isothermals (e.g., Fig. 3), exhibit equal areas e.g.,[15] Chap. VII, §85 which increase as reduced temperature decreases and tendto infinite as /T → 0.
More specifically, the trend is described by the following relations:
according to above considerations where, in general (e.g., Fig. 3):
/VA ≤ /VB ≤ /VC ≤ /VD ≤ /VE ; /T ≤ 1 ;
/pB ≤ /pA = /pC = /pE ≤ /pD ; /T ≤ 1 ;
according to the results of Appendix B.The limit of zero absolute reduced temperature, /T → 0, implies (i) region
ABC tends to a rectangular triangle of catheti, AB = /pA − /pB = 27 andAC = /VC− /VA → +∞; (ii) region CDE tends to a triangle of height, /pD− /pC =/pD − /pE → 0, and basis, /VE − /VC → +∞; and (iii) regions ABC and CDEexhibit infinite area of same order, where their ratio equals unity.
D Fractional macroisothermals
In considering whether fractional macroisothermals, YT = (TP/TG) = (mP/mG)(φ/m) = const via Eqs. (37) and (40), hence φ/m = const, are a viable alter-native with respect to fractional isoenergetics, XT = φ = const via Eqs. (39)and (42), for determining isoenergetics or macroisothermals, attention shall berestricted in finding a counterexample.
With regard to UU macrogases, the following relation holds [5]:
φ
m=
(2m+5)y2−3
2(y3+m); 1 ≤ y < +∞ ;
2(y3+m)(2+5m)y−3my3
; 0 < y ≤ 1 ;(107)
944 R. Caimmi
where y = 1 implies φ/m = 1, independent of m, according to Eq. (43). Re-lated fractional macroisothermal reads:
(2m+ 5)y2 − 3 = 2(y3 +m) ; 1 ≤ y < +∞ ; (108a)
2(y3 +m) = (2 + 5m)y − 3my3 ; 0 < y ≤ 1 ; (108b)
which, after some algebra, may be cast under the form:
m =
2y3−5y2+3
2(y2−1)= 2y2−3y−3
2(y+1); 1 ≤ y < +∞ ;
2y(1−y2)3y3−5y+2
= −2y(y+1)3y2+3y−2
; 0 < y ≤ 1 ;(109)
and the function, m = m(y), can be studied.Special values are the following:
m(0) = 0 ; m(1) = −1 ; limy→+∞
m(y) = limy→+∞
y = +∞ ; (110)
where, in addition to the origin, zeroes of m(y) are solutions of the second-degree equation, 2y2 − 3y − 3 = 0, hence y = (3 ∓
√9 + 24)/4, and the zero
within the domain, y ≥ 1, reads:
m
(3 +√
33
4
)= 0 . (111)
On the other hand, vertical asymptotes relate to solutions of the second-degree equation, 3y2 + 3y − 2 = 0, hence y = (−3 ∓
√9 + 24)/6, and the
vertical asymptote within the domain, 0 ≤ y ≤ 1, reads:
limy→y∓0
m(y) = ±∞ ; y0 =−3 +
√33
6. (112)
The above results define the sign of m(y) all over the domain, as:
0 ≤ y ≤√
33− 3
6; m(y) ≥ 0 ; (113a)
√33− 3
6≤ y ≤
√33 + 3
4; m(y) ≤ 0 ; (113b)
√33 + 3
4≤ y < +∞ ; m(y) ≥ 0 ; (113c)
accordingly, the mass ratio, m, can assume both signs.But m ≥ 0 by definition, which implies homodirection axis ratios, y, within
the range defined by Eq. (113b), cannot occur along the fractional macro-isothermal, φ/m = 1. For this reason, it would be better dealing with frac-tional isoenergetics, where the above mentioned inconvenient does not takeplace.
Received: September 1, 2019; Published: September 22, 2019
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 945
Figure 2: Same as in Fig. 1 (right panel), where the occurrence (within the areabounded by the bell-shaped dashed curve) of saturated vapour is considered.Above the critical isothermal (T/Tc = 1) the trend is similar with respect toideal gases. Below the critical isothermal and on the right of the bell-shapeddashed curve, gas still behaves as an ideal gas. Below the critical isothermaland on the left of the bell-shaped dashed curve, liquid shows little changein volume as pressure rises. Within the area bounded by the bell-shapeddashed curve, liquid phase is in equilibrium with saturated vapour phase. Adiminished volume implies smaller saturated vapour fraction and larger liquidfraction at constant pressure, and vice versa. VDW equation of state is nolonger valid in this region. The dashed curve (including the central branch) isthe locus of intersections between VDW and real isothermals, the latter beingrelated to constant pressure where liquid and vapour phases coexist. Endingpoints are (1/3, 0), left; (+∞, 0), middle; (+∞, 0), right. The dotted curve isthe locus of VDW isothermal extremum points. Ending points are (1/3,−27),left; (+∞, 0), right.
946 R. Caimmi
Figure 3: VDW and real isothermals with equal temperature, T/Tc = 20/23.The two curves coincide within the range, V ≤ VA and V ≥ VE. VDWisothermal exhibits two extremum points: minimum, B, and maximum, D,while real isothermal is flat within the range, VA ≤ V ≤ VE. Configurationsrelated to VDW isothermal within the range, VA ≤ V ≤ VB (due to tensionforces acting on particles, yielding superheated liquid), and VD ≤ V ≤ VE
(due to the occurrence of undercooled vapour), may be obtained under specialconditions, while configurations within the range, VB ≤ V ≤ VD, are alwaysunstable. Volumes, VA and VE, correspond to maximum value in presenceof sole liquid phase and minimum value in presence of sole vapour phase,respectively. Regions, ABC and CDE, exhibit equal area.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 947
Figure 4: Reduced macropressure, /pG, and reduced macrotemperature, /TG,vs reduced volume, /VG, for κ = 0.90, 0.95, 1.00, 1.05, 1.10, 1.15, from bottomto top, where κ = M2
G for reduced macropressure (full curves) and κ = MGχc
for reduced macrotemperature (dotted curves).
948 R. Caimmi
Figure 5: Fractional isoenergetics related to UU macrogases for /X−1T =
20/23, 20/22, 20/21, 20/20, 20/19, 20/18, from bottom to top. The contribu-tion from the term, /X−1
T / /X−1V , is shown by dotted curves. Normalization
values are (XVn, Xpn, XTn) = (0.8425, 53.15545, 10).
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 949
Figure 6: Reduced macrotemperature, /TG, vs reduced variable, /XV, re-
lated to UU macrogases for /M−1G /V
1/3G = 1 and /X−1
T = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top. A similar trend is exhibited for crit-
ical fractional isoenergetic, /X−1T = 1, and /M−1
G /V1/3
G = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top.
950 R. Caimmi
Figure 7: Fractional isoenergetics related to HH macrogases for /X−1T =
20/23, 20/22, 20/21, 20/20, 20/19, 20/18, from bottom to top, where cases,ΞG/ΞP = 0.25, 0.50, 1.00, 2.00, 4.00, are superimposed. The dashed curve(including central branch) is the locus of intersections between HH fractionalisoenergetics and horizontal lines yielding regions of equal area. The dottedcurve is the locus of HH fractional isoenergetic extremum points. Plotting/X−1
p vs /X−1V yields fractional isoenergetics similar to VDW isothermals shown
in Fig. 2, where extremum points are lying below the critical point.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 951
Figure 8: Reduced macrotemperature, /TG, vs reduced variable, /XV, re-
lated to HH macrogases for /M−1G /V
1/3G = 1 and /X−1
T = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top, where cases, ΞG/ΞP = 0.25, 0.50,1.00, 2.00, 4.00, are superimposed. A similar trend is exhibited for criti-cal fractional isoenergetic, /X−1
T = 1, and /M−1G /V
1/3G = 20/23, 20/22, 20/21,
20/20, 20/19, 20/18, from bottom to top.
952 R. Caimmi
Figure 9: Fractional isoenergetics related to HN/NH macrogases for /X−1T =
20/23, 20/22, 20/21, 20/20, 20/19, 20/18, from bottom to top, where cases,ΞG/ΞP = 0.25, 0.50, 1.00, 2.00, 4.00, are superimposed. The dashed curve(including central branch) is the locus of intersections between HN/NH frac-tional isoenergetics and horizontal lines yielding regions of equal area. Thedotted curve is the locus of HN/NH fractional isoenergetic extremum points.Plotting /X−1
p vs /X−1V yields fractional isoenergetics similar to VDW isother-
mals shown in Fig. 2, where extremum points are lying below the critical point.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 953
Figure 10: Reduced macrotemperature, /TG, vs reduced variable, /XV, re-
lated to HN/NH macrogases for /M−1G /V
1/3G = 1 and /X−1
T = 20/23, 20/22,20/21, 20/20, 20/19, 20/18, from bottom to top, where cases, ΞG/ΞP = 0.25,0.50, 1.00, 2.00, 4.00, are superimposed. A similar trend is exhibited for crit-ical fractional isoenergetic, /XT = 1, and /M−1
G /V1/3
G = 20/23, 20/22, 20/21,20/20, 20/19, 20/18, from bottom to top.
954 R. Caimmi
Figure 11: Comparison between VDW critical isothermal (full), HH criticalfractional isoenergetic (dotted), and HN/NH critical fractional isoenergetic(dot-dashed). With regard to ordinary fluids, vapour and liquid phase coexistwithin the bell-shaped region bounded by the dashed curve and, in addition,X−1
V = V,X−1P = p. More extended (along the horizontal direction) bell-
shaped regions are expected for both HH and HN/NH fractional isoenergetics.The critical point, by definition, reads ( /XVc , /Xpc , /XTc) ≡ (1, 1, 1). Differentletters denote expected location of different astrophysical systems. Caption:EG - elliptical galaxies; S0 - lenticular galaxies; SG - spiral galaxies includingbarred; IG - irregular galaxies; DS - dwarf spheroidal galaxies; GC - globularclusters; CG - galaxy clusters; WC - wholly gaseous clouds where stars neverformed; WG - (hypothetical) wholly gaseous galaxies where stars never formed.
From micro to macro cosmos. A century after van der Waals’ Nobel Prize 955
Figure 12: VDW isothermals, /T = (81−x2)/96, plotted for integer x, 0 ≤ x ≤8, from top to bottom where the first is tangent to the horizontal axis and thelast lies outside the box. The region of negative pressure or sufficiently smallvolume is not shown for sake of clarity. In the limit of zero absolute reducedtemperature, /T → 0+, x → 9−, VDW isothermal reads /p > −27; /V → 1/3;and /p→ −27; /V > 1/3. Other curves and captions as in Fig. 2.
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