The relativistic Feynman-Metropolis-Teller theory for white dwarfs in general relativity
The recent formulation of the relativistic Thomas-Fermi model within the Feynman-Metropolis-Teller theory for compressed atoms is applied to the study of general relativistic white dwarf equilibrium configurations. The equation of state, which takes into account the -equilibrium, the nuclear and the Coulomb interactions between the nuclei and the surrounding electrons, is obtained as a function of the compression by considering each atom constrained in a Wigner-Seitz cell. The contribution of quantum statistics, weak, nuclear, and electromagnetic interactions is obtained by the determination of the chemical potential of the Wigner-Seitz cell. The further contribution of the general relativistic equilibrium of white dwarf matter is expressed by the simple formula constant, which links the chemical potential of the Wigner-Seitz cell with the general relativistic gravitational potential at each point of the configuration. The configuration outside each Wigner-Seitz cell is strictly neutral and therefore no global electric field is necessary to warranty the equilibrium of the white dwarf. These equations modify the ones used by Chandrasekhar by taking into due account the Coulomb interaction between the nuclei and the electrons as well as inverse -decay. They also generalize the work of Salpeter by considering a unified self-consistent approach to the Coulomb interaction in each Wigner-Seitz cell. The consequences on the numerical value of the Chandrasekhar-Landau mass limit as well as on the mass-radius relation of He, C, O and Fe white dwarfs are presented. All these effects should be taken into account in processes requiring a precision knowledge of the white dwarf parameters.
The necessity of introducing the Fermi-Dirac statistics in order to overcome some conceptual difficulties in explaining the existence of white dwarfs leading to the concept of degenerate stars was first advanced by R. H. Fowler in a classic paper fowler26. Following that work, E. C. Stoner stoner29 introduced the effect of special relativity into the Fowler considerations and he discovered the critical mass of white dwarfs 111In doing this, Stoner used what later became known as the exclusion principle, generally attributed in literature to Wolfgang Pauli. For a lucid and scientifically correct historical reconstruction of the contributions to the critical mass concept see nauenberg08. For historical details about the exclusion principle see also heilbron83.
where g is the Planck mass, is the neutron mass, and is the average molecular weight of matter which shows explicitly the dependence of the critical mass on the chemical composition of the star.
Following the Stoner’s work, S. Chandrasekhar chandrasekhar31 222At the time a 20 years old graduate student coming to Cambridge from India. pointed out the relevance of describing white dwarfs by using an approach, initiated by E. A. Milne milne30, of using the mathematical method of the solutions of the Lane-Emden polytropic equations emdenbook. The same idea of using the Lane-Emden equations taking into account the special relativistic effects to the equilibrium of stellar matter for a degenerate system of fermions, came independently to L. D. Landau landau32. Both the Chandrasekhar and Landau treatments were explicit in pointing out the existence of the critical mass
where the first numerical factor on the right hand side of Eq. (2) comes from the boundary condition (see last entry of Table 7 on Pag. 80 in emdenbook) of the Lane-Emden polytropic equation. Namely for , no equilibrium configuration should exist.
Landau rejected the idea of the existence of such a critical mass as a “ridiculous tendency” landau32. Chandrasekhar was confronted by a lively dispute with A. Eddington on the basic theoretical assumptions he adopted 333The dispute reached such a heated level that Chandrasekhar was confronted with the option either to change field of research or to leave Cambridge. As is well known he chose the second option transferring to Yerkes Observatory near Chicago where he published his results in his classic book chandrasekharbook. (see wali82 for details).
Some of the basic assumptions adopted by Chandrasekhar and Landau in their idealized approach were not justified e.g. the treatment of the electron as a free-gas without taking into due account the electromagnetic interactions, as well as the stability of the distribution of the nuclei against the gravitational interaction. It is not surprising that such an approach led to the criticisms of Eddington who had no confidence of the physical foundation of the Chandrasekhar work 444It goes to Eddington credit, at the time Plumian Professor at Cambridge, to have allowed the publication of the Chandrasekhar work although preceded by his own critical considerations eddington35.. It was unfortunate that the absence of interest of E. Fermi on the final evolution of stars did not allow Fermi himself to intervene in this contention and solve definitely these well-posed theoretical problems ruffinibook. Indeed, we are showing in this article how the solution of the conceptual problems of the white dwarf models, left open for years, can be duly addressed by considering the relativistic Thomas-Fermi model of the compressed atom (see Subsec. II.5 and Sec. IV).
The original work on white dwarfs was motivated by astrophysics and found in astrophysics strong observational support. The issue of the equilibrium of the electron gas and the associated component of nuclei, taking into account the electromagnetic, the gravitational and the weak interactions is a theoretical physics problem, not yet formulated in a correct special and general relativistic context.
One of the earliest alternative approaches to the Chandrasekhar-Landau work was proposed by E. E. Salpeter in 1961 salpeter61. He followed an idea originally proposed by Y. I. Frenkel frenkel28: to adopt in the study of white dwarfs the concept of a Wigner-Seitz cell. Salpeter introduced to the lattice model of a point-like nucleus surrounded by a uniform cloud of electrons, corrections due to the non-uniformity of the electron distribution (see Subsec. II.3 for details). In this way Salpeter salpeter61 obtained an analytic formula for the total energy in a Wigner-Seitz cell and derived the corresponding equation of state of matter composed by such cells, pointing out explicitly the relevance of the Coulomb interaction.
The consequences of the Coulomb interactions in the determination of the mass and radius of white dwarfs, was studied in a subsequent paper by T. Hamada and E. E. Salpeter hamada61 by using the equation of state constructed in salpeter61. They found that the critical mass of white dwarfs depends in a nontrivial way on the specific nuclear composition: the critical mass of Chandrasekhar-Landau which depends only on the mass to charge ratio of nuclei , now depends also on the proton number .
This fact can be seen from the approximate expression for the critical mass of white dwarfs obtained by Hamada and Salpeter hamada61 in the ultrarelativistic limit for the electrons
being the pressure of the Wigner-Seitz cell obtained by Salpeter in salpeter61 (see Subsec. II.3) and is the pressure of a free-electron fluid used by Chandrasekhar (see Subsec. II.1). The ratio is a function of the number of protons (see Eq. (20) in salpeter61) and it satisfies . Consequently, the effective molecular weight satisfies and the critical mass of white dwarfs turns to be smaller than the original one obtained by Chandrasekhar-Landau (see Eq. (2)).
In the mean time, the problem of the equilibrium gas in a white dwarf taking into account possible global electromagnetic interactions between the nucleus and the electrons was addressed by E. Olson and M. Bailyn in olson75; olson76. They well summarized the status of the problem: “Traditional models for the white dwarf are non-relativistic and electrically neutral … although an electric field is needed to support the pressureless nuclei against gravitational collapse, the star is treated essentially in terms of only one charge component, where charge neutrality is assumed ”. Their solution to the problem invokes the breakdown of the local charge neutrality and the presence of an overall electric field as a consequence of treating also the nuclei inside the white dwarf as a fluid. They treated the white dwarf matter through a two-fluid model not enforcing local charge neutrality. The closure equation for the Einstein-Maxwell system of equations was there obtained from a minimization procedure of the mass-energy of the configuration. This work was the first pointing out the relevance of the Einstein-Maxwell equations in the description of an astrophysical system by requiring global and non local charge neutrality. As we will show here, this interesting approach does not apply to the case of white dwarfs. It represents, however, a new development in the study of neutron stars (see e.g. PLB2011)
An alternative approach to the Salpeter treatment of a compressed atom was reconsidered in gursky2000 by applying for the first time to white dwarfs a relativistic Thomas-Fermi treatment of the compressed atom introducing a finite size nucleus within a phenomenological description (see also bertone2000).
Recently, the study of a compressed atom has been revisited in 2011PhRvC..83d5805R by extending the global approach of Feynman, Metropolis and Teller feynman49 taking into account weak interactions. This treatment takes also into account all the Coulomb contributions duly expressed relativistically without the need of any piecewise description. The relativistic Thomas-Fermi model has been solved by imposing in addition to the electromagnetic interaction also the weak equilibrium between neutrons, protons and electrons self-consistently. This presents some conceptual differences with respect to previous approaches and can be used in order both to validate and to establish their limitations.
In this article we apply the considerations presented in 2011PhRvC..83d5805R of a compressed atom in a Wigner-Seitz cell to the description of non-rotating white dwarfs in general relativity. This approach improves all previous treatments in the following aspects:
In order to warranty self-consistency with a relativistic treatment of the electrons, the point-like assumption of the nucleus is abandoned introducing a finite sized nucleus 2011PhRvC..83d5805R. We assume for the mass as well as for charge to mass ratio of the nucleus their experimental values instead of using phenomenological descriptions based on the semi-empirical mass-formula of Weizsacker (see e.g. gursky2000; bertone2000).
The electron-electron and electron-nucleus Coulomb interaction energy is calculated without any approximation by solving numerically the relativistic Thomas-Fermi equation for selected energy-densities of the system and for each given nuclear composition.
The energy-density of the system is calculated taking into account the contributions of the nuclei, of the Coulomb interactions as well as of the relativistic electrons; the latter being neglected in all previous treatments. This particular contribution turns to be very important at high-densities and in particular for light nuclear compositions e.g. He and C.
The -equilibrium between neutrons, protons, and electrons is also taken into account leading to a self-consistent calculation of the threshold density for triggering the inverse -decay of a given nucleus.
The structure of the white dwarf configurations is obtained by integrating the general relativity equations of equilibrium.
Due to 4) and 5) we are able to determine if the instability point leading to a maximum stable mass of the non-rotating white dwarf is induced by the inverse -decay instability of the composing nuclei or by general relativistic effects.
Paradoxically, after all this procedure which takes into account many additional theoretical features generalizing the Chandrasekhar-Landau and the Hamada and Salpeter works, a most simple equation is found to be fulfilled by the equilibrium configuration in a spherically symmetric metric. Assuming the metric
we demonstrate how the entire system of equations describing the equilibrium of white dwarfs, taking into account the weak, the electromagnetic and the gravitational interactions as well as quantum statistics all expressed consistently in a general relativistic approach, is simply given by
which links the chemical potential of the Wigner-Seitz cell , duly solved by considering the relativistic Feynman-Metropolis-Teller model following 2011PhRvC..83d5805R, to the general relativistic gravitational potential at each point of the configuration. The overall system outside each Wigner-Seitz cell is strictly neutral and no global electric field exists, contrary to the results reported in olson76. The same procedure will apply as well to the case of neutron star crusts.
The article is organized as follows. In Sec. II we summarize the most common approaches used for the description of white dwarfs and neutron star crusts: the uniform approximation for the electron fluid (see e.g. chandrasekhar31); the often called lattice model assuming a point-like nucleus surrounded by a uniform electron cloud (see e.g. baym71a); the generalization of the lattice model due to Salpeter salpeter61; the Feynman, Metropolis and Teller approach feynman49 based on the the non-relativistic Thomas-Fermi model of compressed atoms and, the relativistic generalization of the Feynman-Metropolis-Teller treatment recently formulated in 2011PhRvC..83d5805R.
In Sec. III we formulate the general relativistic equations of equilibrium of the system and show how, from the self-consistent definition of chemical potential of the Wigner-Seitz cell and the Einstein equations, comes the equilibrium condition given by Eq. (6). In addition, we obtain the Newtonian and the first-order post-Newtonian equations of equilibrium.
Finally, we show in Sec. IV the new results of the numerical integration of the general relativistic equations of equilibrium and discuss the corrections to the Stoner critical mass , to the Chandrasekhar-Landau mass limit , as well as to the one of Hamada and Salpeter , obtained when all interactions are fully taken into account through the relativistic Feynman-Metropolis-Teller equation of state 2011PhRvC..83d5805R.
Ii The Equation of State
There exists a large variety of approaches to model the equation of state of white dwarf matter, each one characterized by a different way of treating or neglecting the Coulomb interaction inside each Wigner-Seitz cell, which we will briefly review here. Particular attention is given to the calculation of the self-consistent chemical potential of the Wigner-Seitz cell , which plays a very important role in the conservation law (6) that we will derive in Sec. III.
ii.1 The uniform approximation
In the uniform approximation used by Chandrasekhar chandrasekhar31, the electron distribution as well as the nucleons are assumed to be locally constant and therefore the condition of local charge neutrality
where is the average atomic weight of the nucleus, is applied. Here denotes the nucleon number density and is the number of protons of the nucleus. The electrons are considered as a fully degenerate free-gas and then described by Fermi-Dirac statistics. Thus, their number density is related to the electron Fermi-momentum by
and the total electron energy-density and electron pressure are given by
where we have introduced the dimensionless Fermi momentum with the electron rest-mass.
The kinetic energy of nucleons is neglected and therefore the pressure is assumed to be only due to electrons. Thus the equation of state can be written as
Within this approximation, the total self-consistent chemical potential is given by
is the electron free-chemical potential.
As a consequence of this effective approach which does not take into any account the Coulomb interaction, it is obtained an effective one-component electron-nucleon fluid approach where the kinetic pressure is given by electrons of mass and their gravitational contribution is given by an effective mass attached to each electron (see e.g. landaubook). This is even more evident when the electron contribution to the energy-density in Eq. (11) is neglected and therefore the energy-density is attributed only to the nuclei. Within this approach followed by Chandrasekhar chandrasekhar31, the equation of state reduces to
ii.2 The lattice model
The first correction to the above uniform model, corresponds to abandon the assumption of the electron-nucleon fluid through the so-called “lattice” model which introduces the concept of Wigner-Seitz cell: each cell contains a point-like nucleus of charge with nucleons surrounded by a uniformly distributed cloud of fully-degenerate electrons. The global neutrality of the cell is guaranteed by the condition
where is the Wigner-Seitz cell density and is the cell volume.
The total energy of the Wigner-Seitz cell is modified by the inclusion of the Coulomb energy, i.e
where is given by Eq. (11) and and are the electron-nucleus and the electron-electron Coulomb energies
The self-consistent pressure of the Wigner-Seitz cell is then given by
where is given by Eq. (12). It is worth to recall that the point-like assumption of the nucleus is incompatible with a relativistic treatment of the degenerate electron fluid (see ferreirinho80; ruffini81 for details). Such an inconsistency has been traditionally ignored by applying, within a point-like nucleus model, the relativistic formulas (9) and (10) and their corresponding ultrarelativistic limits (see e.g. salpeter61).
The Wigner-Seitz cell chemical potential is in this case
ii.3 Salpeter approach
A further development to the lattice model came from Salpeter salpeter61 whom studied the corrections due to the non-uniformity of the electron distribution inside a Wigner-Seitz cell.
Following the Chandrasekhar chandrasekhar31 approximation, Salpeter also neglects the electron contribution to the energy-density. Thus, the first term in the Salpeter formula for the energy of the cell comes from the nuclei energy (15). The second contribution is given by the Coulomb energy of the lattice model (19). The third contribution is obtained as follows: the electron density is assumed as , where is the average electron density as given by Eq. (17), and is considered infinitesimal. The Coulomb potential energy is assumed to be the one of the point-like nucleus surrounded by a uniform distribution of electrons, so the correction given by on the Coulomb potential is neglected. The electron distribution is then calculated at first-order by expanding the relativistic electron kinetic energy
about its value in the uniform approximation
considering as infinitesimal the ratio between the Coulomb potential energy and the electron Fermi energy
The influence of the Dirac electron-exchange correction dirac30 on the equation of state was also considered by Salpeter salpeter61. However, adopting the general approach of Migdal et al. migdal77, it has been shown that these effects are negligible in the relativistic regime 2011PhRvC..83d5805R. We will then consider here only the major correction of the Salpeter treatment.
The total energy of the Wigner-Seitz cell is then given by (see salpeter61 for details)
Correspondingly, the self-consistent pressure of the Wigner-Seitz cell is
The Wigner-Seitz cell chemical potential can be then written as
From Eqs. (29) and (31), we see that the inclusion of each additional Coulomb correction results in a further decreasing of the pressure and of the chemical potential of the cell. The Salpeter approach is very interesting in identifying piecewise Coulomb contribution to the total energy, to the total pressure and, to the Wigner-Seitz chemical potential. However, it does not have the full consistency of the global solutions obtained with the Feynman-Metropolis-Teller approach feynman49 and its generalization to relativistic regimes 2011PhRvC..83d5805R which we will discuss in detail below.
ii.4 The Feynman-Metropolis-Teller treatment
Feynman, Metropolis and Teller feynman49 showed how to derive the equation of state of matter at high pressures by considering a Thomas-Fermi model confined in a Wigner-Seitz cell of radius .
The Thomas-Fermi equilibrium condition for degenerate non-relativistic electrons in the cell is expressed by
where denotes the Coulomb potential and denotes the Fermi energy of electrons, which is positive for configurations subjected to external pressure, namely, for compressed cells.
Defining the function by , and introducing the dimensionless radial coordinate by , where , being the electron Compton wavelength; the Poisson equation from which the Coulomb potential is calculated self-consistently becomes
The boundary conditions for Eq. (33) follow from the point-like structure of the nucleus and, from the global neutrality of the Wigner-Seitz cell , where defines the dimensionless radius of the Wigner-Seitz cell by .
For each value of the compression, e.g. , it corresponds a value of the electron Fermi energy and a different solution of Eq. (33), which determines the self-consistent Coulomb potential energy as well as the self-consistent electron distribution inside the cell through
In the non-relativistic Thomas-Fermi model, the total energy of the Wigner-Seitz cell is given by (see slater35; feynman49 for details)
where is the nucleus mass, is given by Eq. (9) and and are the electron-nucleus Coulomb energy and the electron-electron Coulomb energy, which are given by
where is the non-relativistic kinetic energy of a uniform electron distribution of density , i.e.
with defined by
The self-consistent pressure of the Wigner-Seitz cell given by the non-relativistic Thomas-Fermi model is (see slater35; feynman49 for details)
The pressure of the Thomas-Fermi model (44) is equal to the pressure of a free-electron distribution of density . Being the electron density inside the cell a decreasing function of the distance from the nucleus, the electron density at the cell boundary, , is smaller than the average electron distribution . Then, the pressure given by (44) is smaller than the one given by the non-relativistic version of Eq. (10) of the uniform model of Subsec. II.1. Such a smaller pressure, although faintfully given by the expression of a free-electron gas, contains in a self-consistent fashion all the Coulomb effects inside the Wigner-Seitz cell.
The chemical potential of the Wigner-Seitz cell of the non-relativistic Thomas-Fermi model can be then written as
Integrating by parts the total number of electrons
we can rewrite finally the following semi-analytical expression of the chemical potential (45) of the cell
where is the electron free-chemical potential (14) calculated with the average electron density, namely, the electron chemical potential of the uniform approximation. The function depends explicitly on the gradient of the electron density, i.e. on the non-uniformity of the electron distribution.
ii.5 The relativistic Feynman-Metropolis-Teller treatment
We recall now how the above classic Feynman, Metropolis, and Teller treatment of compressed atoms has been recently generalized to relativistic regimes (see 2011PhRvC..83d5805R for details). One of the main differences in the relativistic generalization of the Thomas-Fermi equation is that, the point-like approximation of the nucleus, must be abandoned since the relativistic equilibrium condition of compressed atoms
would lead to a non-integrable expression for the electron density near the origin (see e.g.ferreirinho80; ruffini81).
It is then assumed a constant distribution of protons confined in a radius defined by
where is the pion Compton wavelength. If the system is at nuclear density with fm. Thus, in the case of ordinary nuclei (i.e., for ) we have . Consequently, the proton density can be written as
where denotes the Heaviside function centered at . The electron density can be written as
where and we have used Eq. (50).
The overall Coulomb potential satisfies the Poisson equation
with the boundary conditions and due to the global charge neutrality of the cell.
By introducing the dimensionless quantities , , and replacing the particle densities (52) and (53) into the Poisson equation (54), it is obtained the relativistic Thomas-Fermi equation 2008pint.conf..207R
which must be integrated subjected to the boundary conditions , and , where .
The neutron density , related to the neutron Fermi momentum , is determined by imposing the condition of beta equilibrium
subjected to the baryon number conservation equation
In Fig. 1 we see how the relativistic generalization of the Feynman-Metropolis-Teller treatment leads to electron density distributions markedly different from the constant electron density approximation. The electron distribution is far from being uniform as a result of the solution of Eq. (55), which takes into account the electromagnetic interaction between electrons and between the electrons and the finite sized nucleus. Additional details are given in 2011PhRvC..83d5805R.
V. S. Popov et al. popov10 have shown how the solution of the relativistic Thomas-Fermi equation (55) together with the self-consistent implementation of the -equilibrium condition (56) leads, in the case of zero electron Fermi energy (), to a theoretical prediction of the -equilibrium line, namely a theoretical - relation. Within this model the mass to charge ratio of nuclei is overestimated, e.g. in the case of He the overestimate is , for C , for O , and for Fe . These discrepancies are corrected when the model of the nucleus considered above is improved by explicitly including the effects of strong interactions. This model, however, illustrates how a self-consistent calculation of compressed nuclear matter can be done including electromagnetic, weak, strong as well as special relativistic effects without any approximation. This approach promises to be useful when theoretical predictions are essential, for example in the description of nuclear matter at very high densities, e.g. nuclei close and beyond the neutron drip line.
The densities in white dwarf interiors are not highly enough to require such theoretical predictions. Therefore, in order to ensure the accuracy of our results we use for , needed to solve the relativistic Thomas-Fermi equation (55), as well as for the nucleus mass , their known experimental values. In this way we take into account all the effects of the nuclear interaction.
Thus, the total energy of the Wigner-Seitz cell in the present case can be written as
where is the experimental nucleus mass, e.g. for He, C, O and Fe we have 4.003, 12.01, 16.00 and 55.84 respectively. In Eq. (61) the integral is evaluated only outside the nucleus (i.e. for ) in order to avoid a double counting with the Coulomb energy of the nucleus already taken into account in the nucleus mass (59). In order to avoid another double counting we subtract to the electron energy-density in Eq. (60) the rest-energy density which is also taken into account in the nucleus mass (59).
The total pressure of the Wigner-Seitz cell is given by
where is the relativistic pressure (10) computed with the value of the electron density at the boundary of the cell.
The electron density at the boundary in the relativistic Feynman-Metropolis-Teller treatment is smaller with respect to the one given by the uniform density approximation (see Fig. 1). Thus, the relativistic pressure (62) gives systematically smaller values with respect to the uniform approximation pressure (10) as well as with respect to the Salpeter pressure (29).
In Fig. 2 we show the ratio between the relativistic Feynman-Metropolis-Teller pressure (62) and the Chandrasekhar pressure (10) and the Salpeter pressure (29) in the case of C. It can be seen how is smaller than for all densities as a consequence of the Coulomb interaction. With respect to the Salpeter case, we have that the ratio approaches unity from below at large densities as one should expect.
However, at low densities – g/cm, the ratio becomes larger than unity due to the defect of the Salpeter treatment which, in the low density non-relativistic regime, leads to a drastic decrease of the pressure and even to negative pressures at densities g/cm or higher for heavier nuclear compositions e.g. Fe (see salpeter61; 2011PhRvC..83d5805R and Table 1). This is in contrast with the relativistic Feynman-Metropolis-Teller treatment which matches smoothly the classic Feynman-Metropolis-Teller equation of state in that regime (see 2011PhRvC..83d5805R for details).
No analytic expression of the Wigner-Seitz cell chemical potential can be given in this case, so we only write its general expression
where and are given by Eqs. (58) and (62) respectively. The above equation, contrary to the non-relativistic formula (45), in no way can be simplified in terms of its uniform counterparts. However, it is easy to check that, in the limit of no Coulomb interaction , , and and, neglecting the nuclear binding and the proton-neutron mass difference, we finally obtain
as it should be expected.
Now we summarize how the equation of state of compressed nuclear matter can be computed in the Salpeter case and in the relativistic Feynman-Metropolis-Teller case, parameterized by the total density of the system:
(i) For a given radius of the Wigner-Seitz cell the relativistic Thomas-Fermi equation (55) is integrated numerically and the density of the configuration is computed as where is the energy of the cell given by Eq. (58).
(ii) For that value of the density, the radius of the Wigner-Seitz cell in the Salpeter treatment is
where Eq. (15) has been used. On the contrary, in the relativistic Feynman-Metropolis-Teller treatment no analytic expression relating Wigner-Seitz cell radius and density can be written.
(iii) From this Wigner-Seitz cell radius, or equivalently using the value of the density, the electron density in the Salpeter model is computed from the assumption of uniform electron distribution and the charge neutrality condition, i.e. Eq. (15). In the relativistic Feynman-Metropolis-Teller treatment, the electron number density at the boundary of the Wigner-Seitz cell is, following Eq. (53), given by
where the function is the solution of the relativistic Thomas-Fermi equation (55).
Iii General relativistic equations of equilibrium
Outside each Wigner-Seitz cell the system is electrically neutral, thus no overall electric field exists. Therefore, the above equation of state can be used to calculate the structure of the star through the Einstein equations. Introducing the spherically symmetric metric (5), the Einstein equations can be written in the Tolman-Oppenheimer-Volkoff form tolman39; oppenheimer39
where we have introduced the mass enclosed at the distance through , is the energy-density and is the total pressure.
We turn now to demonstrate how, from Eq. (69), it follows the general relativistic equation of equilibrium (6), for the self-consistent Wigner-Seitz chemical potential . The first law of thermodynamics for a zero temperature fluid of particles, total energy , total volume , total pressure , and chemical potential reads
where the differentials denote arbitrary but simultaneous changes in the variables. Since for a system whose surface energy can be neglected with respect to volume energy, the total energy per particle depends only on the particle density , we can assume as an homogeneous function of first-order in the variables and and hence, it follows the well-known thermodynamic relation
In the case of the Wigner-Seitz cells, Eq. (71) reads
where we have introduced the fact that the Wigner-Seitz cells are the building blocks of the configuration and therefore we must put in Eq. (71) . Through the entire article we have used Eq. (72) to obtain from the knowns energy and pressure, the Wigner-Seitz cell chemical potential (see e.g. Eqs. (13) and (23)). From Eqs. (70) and (71) we obtain the so-called Gibbs-Duhem relation
In a white dwarf the pressure and the chemical potential are decreasing functions of the distance from the origin. Thus, the differentials in the above equations can be assumed as the gradients of the variables which, in the present spherically symmetric case, become just derivatives with respect to the radial coordinate . From Eq. (73) it follows the relation
which can be straightforwardly integrated to obtain the first integral
The above equilibrium condition is general and it also applies for non-zero temperature configurations ( see e.g. klein49). In such a case, it can be shown that in addition to the equilibrium condition (76) the temperature of the system satisfies the Tolman isothermality condition constant 1930PhRv...35..904T; 1930PhRv...36.1791T.
iii.1 The weak-field non-relativistic limit
In the weak-field limit we have , where the Newtonian gravitational potential has been defined by . In the non-relativistic mechanics limit , the chemical potential , where denotes the non-relativistic free-chemical potential of the Wigner-Seitz cell and is the rest-mass of the Wigner-Seitz cell, namely, the rest-mass of the nucleus plus the rest-mass of the electrons. Applying these considerations to Eq. (76) we obtain
Absorbing the Wigner-Seitz rest-mass energy in the constant on the right-hand-side we obtain