Theoretical Models of Highly Magnetic White Dwarf Stars with Non-Polytropic Equation of State

Super-massive white dwarf (WD) stars in the mass range 2.4 - 2.8 solar masses are believed to be the progenitors of “super-luminous” Type Ia supernovae according to a hypothesis proposed by some researchers. They theorize such a higher mass of the WD due to the presence of a very strong magnetic field inside it. We revisit their first work on magnetic WDs (MWDs) and present our theoretical results that are very different from theirs. The main reason for this difference is in the use of the equation of state (EoS) to make stellar models of MWDs. An electron gas in a magnetic field is Landau quantized and hence, the resulting EoS becomes non-polytropic. By constructing models of MWDs using such an EoS, we highlight that a strong magnetic field inside a WD would make the star super-massive. We have found that our stellar models do indeed fall in the mass range given above. Moreover, we are also able to address an observational finding that the mean mass of MWDs are almost double that of non-magnetic WDs. Magnetic field changes the momentum-space of the electrons which in turn changes their density of states (DOS), and that in turn changes the EoS of matter inside the star. By correlating the magnetic DOS with the non-polytropic EoS, we were also able to find a physical reason behind our theoretical result of super-massive WDs with strong magnetic fields. In order to construct these models, we have considered different equations of state with at most three Landau levels occupied and have plotted our results as mass-radius relations for a particular chosen value of maximum Fermi energy. Our results also show that a multiple Landau-level system of electrons leads to such an EoS that gives multiple branches in the mass-radius relations, and that the super-massive MWDs are obtained when the Landau-level occupancy is limited to just one level. Finally, our theoretical results can be explained solely on the basis of quantum and statistical mechanics that warrant no assumptions regarding stars.


Introduction
Recently, several observations of "peculiar" Type Ia supernovae: SN 2003fg, SN 2006gz, SN 2007if, SN 2009dc seem to indicate that their progenitor stars might be super-massive WDs with masses in the range 2.4 -2.8 solar masses (M ). This mass range clearly exceeds the Chandrasekhar mass limit of 1.44 M [1] [2] [3] [4]. These supernovae are different from the more common Type Ia supernovae in that they are more luminous and the ejected matter has lower kinetic energy than that observed in a regular supernova which suggests that the ancestor is most likely a WD star that is more massive than a WD of 1.44 M -a super-Chandrasekhar mass progenitor [5]. Some researchers have called for careful screening of Type Ia supernova events in future cosmology studies due to a possible SN sample contamination from such over-luminous supernovae events; while others suggest the need for a possible reconsideration of the expansion history of the Universe [5] [6].
In their very first work on the MWDs, Das & Mukhopadhyay have hypothesized the existence of very strong magnetic fields inside the WD in order to explain such a higher mass of the star [7]. Also, recently several WDs have been discovered that have surface magnetic fields in the range 10 5 to 10 9 G [8] [9] [10] [11] [12]. Nearly 10% of all WDs are found to be magnetic with fields in excess of ~10 6 G and that the mean mass of their mass distribution is ~0.93 M , while that of non-magnetic WDs is ~0.56 M [12]. The presence of such high surface fields can lead us to hypothesize the existence of stronger interior fields, that although cannot be probed directly, to be quite a few orders of magnitude higher. This limit is set by the scalar virial theorem [13] where T is the rotational energy, W the gravitational energy, Π the internal energy and M the magnetic energy. T and Π are both positive but W is negative, therefore the maximum magnetic energy M can be comparable to W but cannot be more than it in an equilibrium condition as is seen argued in Ref. [13].
For a star of mass M and radius R, this gives us for the average maximum interior B-field, and for the WDs this limit turns out to be ~10 12 G [13].
Successful magnetic WD models with B~10 12 G at the center but with a vanishing field at the surface have been constructed by Ostriker & Hartwick [14]. Such a strong field can also be a consequence of a "frozen-in" field concept, also known as the flux conservation phenomenon, according to which the magnetic flux of a star is more or less conserved throughout its evolution and its eventual collapse to becoming a degenerate star [15] [16]. Also, Ferrario & Wickramasinghe argue, on the assumption of flux conservation, that certain main-sequence stars such as Ap and Bp stars with fields in the range ~200 -25,000 G can evolve into MWDs with dipolar fields in the range ~10 6 -10 9 G, and this field range belongs to the vast majority of known MWDs [17].
Here, we consider a strongly magnetized, relativistic, completely degenerate electron gas at 0 K T = . The star is assumed to be spherically symmetric and the magnetic field is assumed to be static and uniform throughout the star.
The purpose here is to study the effect of strong magnetic field on the momentum-space ( p -space) and DOS of electrons as well as on the EoS of matter within the star, and then obtain the mass-radius relation of such a magnetized white dwarf star. Because the electrons within the star are considered to be relativistic (except possibly those in the density regions of a thin outer crust of a WD), we choose such values of B-field that are higher than a critical value which is given by equating the cyclotron energy of the electron to its rest mass energy [13], 2 c mc ω = (2) where m is the mass and c ω is the cyclotron frequency of the electron corresponding to the critical B-field and is given by Therefore, the critical magnetic field is given by Central field values of about 10 12.3 G could be possible due to the aforementioned flux conservation if the progenitor of WD originally had a high magnetic field of the order 10 8 G to begin with [18]. The collapse of such a star into a degenerate remnant would dramatically increase the interior magnetic field since the total flux, 2 BR φ ∝ , is more or less conserved. Central field " cen B " in excess of 10 3 G has been theoretically proved to be possible within a MWD in the case of an electron gas occupying only one Landau level [19]. So, it does seem reasonable to hypothesize fields ≥ 10 13 G. Although such high interior fields seem too extreme, there can be interesting consequences by assuming much stronger magnetic fields of the order of 10 13-15 G, as is presented in this work.
In the presence of a magnetic field, an electron gas will be Landau quantized.
Here, we study the effect of magnetic field on a system of relativistic, degenerate electrons that occupy at most three Landau levels. We also address the possibility of having super-massive WDs, having very strong, static and uniform magnetic field throughout, that violate the Chandrasekhar limit. We do this by making stellar models of MWDs from a "non-polytropic" EoS and find that our theoretical results do concur with the observations.  [7]. We believe this to be an improper approach in dealing with a "non-polytropic" EoS. Not only that but a justification of their results of super-massive MWDs, from a fundamental perspective, is missing in their work. We have not done such piece-wise fits in our work and have also succeeded in providing an elementary explanation of our findings by correlating the electron momentum-space and the DOS with the EoS, and eventually, with the mass-radius relations of MWDs.
We have organized this paper as follows. In the next section, §2, we discuss the relevant equations and procedures necessary to construct models of MWDs and to correlate everything. In §3, we emphasize on the correlation between the plots of the EoS and those of the DOS. In §4, our numerical results are presented in the form of plots of mass-radius relations and a table summarizing all the mass-radius values. In the ensuing section, §5, we discuss the significant changes that take place in the momentum-space of electrons in presence of a magnetic field and correlate that to the DOS. More explanation for the super-Chandrasekhar WD star is given in §6. A comparison with the non-magnetic results is provided in §7, wherein we have retrieved the Chandrasekhar mass limit when 0 B → , i.e., in the weak-field limit. In §8, we compare our results with those of Das & Mukhopadhyay and summarize the shortcomings in DM's methods.
Lastly, we conclude our results in §9.

Relevant Equations of a Cold, Degenerate Relativistic Free Electron Gas in a Magnetic Field
On solving the relativistic Dirac equation, one obtains the energy eigenvalues of a free electron in an external static and uniform magnetic field oriented in the z-direction, which are given by [13] is the Landau quantum number and can take on values 0,1, 2, ν = which are known as Landau levels. Here, 0,1, 2, n = is the principal quantum number and 1 z σ = ± is the spin quantum number or the spin of the electron. The electrons can become relativistic in either of the following two ways [13].
1) the cyclotron energy of the electron exceeds its rest mass energy 2) the density is so high that the mean Fermi energy of an electron exceeds its rest mass energy.
The first possibility gives us the definition of the critical magnetic field cr B given in Equation (4). In the absence of a magnetic field, the density of states ( ) g E per unit volume, denoted by ( ) * g E is given by, which can be integrated from where F E is the Fermi energy of the electrons, to get the number density of electrons e n , ( ) Due to the presence of the magnetic field B, the DOS per unit volume be- is the dimensionless magnetic field, mc λ = is the Compton wavelength of an electron, Θ is the Heaviside function and D ν is the degeneracy of the Landau level ν such that It just so happens that the above equation resembles the equation of the DOS of a quantum wire [20] with the exception that the Fermi gas in a quantum wire is, of course, non-relativistic. In a quantum wire, an electron gas is confined in two dimensions (say x and y-directions) but can move freely in the z-direction.
Classically speaking, in a MWD, we essentially have the same situation of an electron confined in two dimensions (circular orbit) and free moving in the third dimension.
The upper limit of the summation can be found by setting 2 0 F p ≥ in the equation for the Fermi energy given by [13] ( ) which gives us and therefore  When the electrons fill up all the lower energy states up to the Fermi level, we get the electron number density as [13] ( ) ( ) The matter density ρ is related to the electron number density via where e µ is the mean molecular weight per electron and H m is the mass of a hydrogen atom. The electron energy density at 0 K T = is given by [13] ( ) Then the pressure of electron gas in a magnetic field is given by [13] ( ) ( The purpose of this paper is to highlight the effects of a strong magnetic field on the DOS of electrons and hence on the EoS of matter inside the star, and eventually to investigate the possibility of strong magnetic fields giving rise to super-massive WD stars that would probably explain the recent observations of the peculiar Type Ia supernovae. Considering this, we restrict our system to at most three Landau levels corresponding to 0 ν = , 1 ν = and 2 ν = which will be respectively called a one-level, two-level and three-level system similar to that  (14), (16) and (19). Each point on the plot implicitly represents a value of the Fermi energy F E , and the entire plot is gener- to ensure that the electron gas is relativistic throughout the star, except possibly for a very thin outer crust, which does not affect the main results of our work.
The next step would be to solve the equation of hydrostatic equilibrium of the star under the assumption of spherical symmetry in order to find ρ as a function of radial distance "r". The hydrostatic equilibrium equation is given by [ The pressure due to radiation is neglected here and so is the pressure due to the ionized nuclei which are non-relativistic at the densities found in a typical WD because they are much heavier than electrons. So, e P P = has been assumed throughout this paper. The above equation can be re-written as The radius R of the star is given by the first zero of the solution, while the mass M of the star is given by  Figure 1 shows the plots of the non-polytropic EoS of a cold, degenerate, relativistic electron gas in a magnetic field. Here, an explanation will be given only for and is indicated by the solid line. The last point on the curve corresponds to that number density of electrons e n which completely fills up all the available single-particle energy states of the first Landau level. We can see that the upper part of the curve is much stiffer in comparison to the lower part which is softer. To understand this behavior, we look at Equation (9) for the DOS per unit volume (hereafter referred to as just DOS) in a magnetic field and Figure 2 which are the plots of the DOS given here in units of 5 × 10 31 cm −3 erg −1 . They resemble the DOS of a non-relativistic electron gas in a magnetic field [24]. For the first Landau level corresponding to 0 ν = , we essentially have an infinite DOS at 2 1 E mc = , as can be seen from Equation (9), which then drops off as the energy of the electrons increases when they start to occupy the higher energy states. The area under this curve will give us the number density of electrons.

Correlation between the Plots of the Non-Polytropic Equations of State and Those of the Density of States
From the graph of DOS in Figure 2(a), one can see that it becomes more or less steady at a value of about 3.5 (in units of 5 × 10 31 cm −3 erg −1 ) at about For a given B-field, e n is directly proportional to the Fermi energy     the DOS is so low (about 3.5) for a significant portion of the energy range, electrons from the energy range , and hence the resulting graph of the EoS of a one-level system looks "stiffer" for almost the entire density range, except at very low densities from 0 to about 2.9 × 10 9 gcm −3 (corresponding to ) in the plot where the curve (EoS) is "softer" due to lesser electrons occupying energy states up to smaller and smaller Fermi energies (up to about ). Basically, when the DOS is low, the electrons occupy energy levels that are not close to each other (similar to the case of a particle confined in a one-dimensional box of infinite potential), and that translates into a steeper rise in pressure with density in contrast to electrons occupying very close-by energy levels when the DOS is high, which translates into a softer rise in pressure with density. So, it is a combination of low DOS and high e n that results in a steeper rise in pressure for Fermi energies which in turn results in a "stiff" EoS, or that of high DOS but low e n for that results in a "soft" EoS. Essentially, the same explanation holds for two-level and three-level systems as well except that both these systems have, respectively, two and three energy values when the denominator of Equation (9)  ) end up occupying the states in this energy range which translates into relatively lower rise in pressure with density. The reason why the rise in pressure with density is not much for Fermi energies that fall in this energy range, even when the DOS is very high, is simply because it is a very small energy range, therefore, the electrons end up occupying very close-by energy values in such a small energy range. This is exactly the reason why one sees a softer EoS for a two-level system in Figure 1(b), in the region to the right of the kink- 9 3 4. 15  , respectively. It is this region of the EoS where the pressure does not rise too much with density which is what is termed as having a "soft" equation of state. In a similar manner, we can see that there is also a softening of the EoS at very low densities corresponding to Fermi energies between Thus, there is a direct correlation between the way in which the DOS is distributed as a function of electron energies and the appearance of the EoS-graph. A similar explanation also holds true for a three-level system of a degenerate electron gas where we can see three regions of softening of the EoS (Figure 1(b)).
Once the explanation given above is understood, a seemingly surprising fact also becomes clear now when we look at Figure 1(b) of the EoS. From this figure, we see that for the one-level system, the last point on the graph corresponds to a density of about 1.17 × 10 10 gcm −3 or an electron number density of about 3.5 × 10 33 cm −3 while that for a two-level system corresponds to . So, even though, the density is the smallest for a one Landau level system for the same Fmax E , we have the largest value of pressure corresponding to that density value on the graph- can conclude that a one Landau-level system of electrons, even though less dense (at the same value of Fmax E ), will end up having much more pressure because of a steeper rise in pressure with respect to density.

The Mass-Radius Relations
In this section, we will give an explanation for the mass-radius (M-R) relations of a MWD star corresponding to 2 20 Fmax E mc = and link it to the discussion given in the previous section-Section 3. A similar explanation will also hold true for the other two cases- . On all the graphs, each dot represents a star with a particular value of central density c ρ which is chosen from the different density values from the EoS-plot. Furthermore, as one goes from right to left (on a given branch) in the plot, we find stars with decreasing values of c ρ . Here, the reader is reminded of the assumptions of spherical symmetry and a uniform and static magnetic field.

One Landau Level System
First, we take a look at Figure 3  tem. Thus, we see that a high magnetic field can result in a star, the mass of which is significantly higher than the Chandrasekhar limit.

Two Landau Level System
Next, we look at Figure 3 i.e., a star with such a small mass and radius is because of the H. Shah, K. Sebastian softening of the EoS at two places-one just to the right of the kink and the other at densities closer to zero, but more so due to the softening to the right of the kink. Such a star has within it a significant range of densities- Moving on to the upper branch, we have a star which is the most massive and has the largest radius even when its central density Due to this drop in the DOS, there will be a much steeper rise in the outward pressure due to the electrons (the degeneracy pressure). Also, the somewhat lower pressure gradient region for most stars on the upper branch exists in the outermost layers of the star which are its lowest density regions, and hence would not have a considerable effect on its hydrostatic equilibrium. . Following the same trend of explanation as given above for a two-level system, we see that a three-level system results in three branches. Again, the branches are presented in the same manner as for the two-level system with the rightmost star of the bottom-most branch having the highest central density of all the stars in all three branches. This c ρ would correspond to the last density value ( ). This star has three regions of density where the EoS is softer compared to other regions-one to the right of the highest kink, the second to the right of the lower kink and the third at very low densities. This star is barely stable due to these three regions inside it, but mainly due to the two softer-EoS regions located to the right of the two kinks in Figure 1(b), and this is why it ends up being less massive and smaller in size as well. This branch ends at some non-zero value of mass M and radius R with a particular central density value located slightly to the right of the highest kink and then we have a jump in the mass-radius values for the middle branch.

Three Landau Level System
For the middle branch, the rightmost star has ~0.37 M M and 7 3.37 10 cm R × corresponding to 9 3 7 10 g cm ρ − = × ⋅ which is located to the left of the highest kink. It has a lower mass mainly because of a range of densities within it (located to the right of the lower kink) where the pressure does not change too much as can be seen from Figure 1(b) again. The subsequent stars have lower masses as well as radii; exactly the same trend which was seen for the two-level case with the last star having a c ρ which is equal to a particular value of ρ located to the right of the lower kink (basically from the softer part of the EoS) and having a non-zero but really small mass, and then we have a jump to the topmost branch.
In the topmost branch, the rightmost star has

Magnetic Momentum-Space and the Shape of the DOS Curve
Because of the significant changes that take place in the p -space in the presence of a magnetic field, we devote one entire section to explain the magnetic p -space and the shape of the DOS curve. Those changes in p -space are responsible for the changes in the DOS, which in turn are responsible for modifying the EoS that leads to a super-Chandrasekhar WD. Figure 4 shows the p -space in the absence of a magnetic field. Here, the individual momentum components x p , y p and z p form a continuum. In the absence of B-field, the number of states in a small interval dk in k -space, where k is a wave vector, are given by [22] ( ) ( ) ( ) Here, the spacing of lattice points in k -space is taken to be 2 L π by assuming periodic boundary conditions for obtaining solutions to the wave equation in a cubic enclosure of side L and volume In the presence of B-field, we have quantization in the p x -p y plane with the radii of the circles given by where ⊥ p is the projection of p onto the p x -p y plane. The successive circles correspond to increasing values of the principle quantum number n. For a given n and z p , the number of free orbitals that coalesce into a single magnetic level i.e., the degeneracy D in the p x -p y plane is given by  where A is the area of the orbit (of course, semi-classically speaking) in the x-y plane [23].
Because of the presence of the magnetic field, the quantized three-dimensional p -space gets defined by cylindrical surfaces corresponding to successive values of quantum number n on including the third momentum direction: z p [24].
The Fermi surface is a sphere that would cut the cylinders, as illustrated in Figure 5, to give states that are occupied by electrons with a Fermi momentum belonging to a particular level i.e., 0 n = , 1 or 2. For a given B-field, as the Fermi surface grows more and more cylinders are covered by it. One can also see that the Fermi momentum in the z-direction gets smaller as the radius of the cylinder increases i.e., the highest quantum number has the smallest F p in the z-direction. Also, when the magnetic field becomes very strong, there exists only the ground Landau level corresponding to eigenvalue 0 n = i.e., only one cylindrical surface [24].  For example, when some value of 2 2 2 x y z p p p p = + + corresponding to 0 n = becomes equal to that of p ⊥ of the Landau level 1 ν = with 1 n = , then the second cylinder in p -space also starts to get occupied at that same energy. The same explanation holds for other levels also.   After every discontinuity in the DOS, we also see that the DOS drops. This can be understood by combining the facts that firstly, the momentum z p is quantized and secondly, the electron energy eigenvalues are given by Equation (5). Therefore, the higher the value of z p , the smaller the DOS in a unit energy interval, similar to that in the case of a particle in a 1-d box.

Further Explanation for the Super-Chandrasekhar Mass
From Section 4, we saw that a super-massive WD was indeed possible if it had a very strong B-field inside it. What is more interesting is that a super-massive WD was possible when the electrons had occupied only the first Landau level irrespective of the total levels in the EoS-one, two or three. This happens due to the fact that after the infinity in the DOS (in one-level system) and between the first two infinities in the DOS (in two and three-level systems), there is a drop in the DOS to a very small value which allows the electrons that occupy all the energy states up to a certain Fermi energy F E , which falls within those energy ranges, to exert a higher outward pressure even at significantly lower densities. This was particularly evident in the cases of two and three-level systems.
Also, for the same Fmax E , as the magnetic field goes down the radii of the cylinders corresponding to the quantum number n also goes down (Equation (27)). This means that the degeneracy in the p x -p y plane also decreases as can be seen from Equation (28). So, the electrons corresponding to 0 n = or 0 ν = level for two and three-level systems start occupying states with non-zero z p much earlier than those in a one-level system. Hence, such electrons start exerting higher degeneracy pressure (at the same density) sooner compared those in one-level system, and that makes the equations of state of two and three-level systems much stiffer than the one with just one Landau level occupied i.e., only 0 ν = . This can be seen from Figure 1 and Figure 5.
These two reasons in the previous two paragraphs, make the EoS related to just the first Landau level very stiff for almost the entire corresponding density range except at very low densities that, anyways, would form the outer layer of the star and therefore would not matter that much. For two and three-level systems, the EoS at the highest density also starts out as very stiff, but because of a significant range/s of densities for two/three-level systems within the star where the pressure gradient is not adequate, the star remains less massive even at those higher central densities.
Therefore, it just so happens that a MWD becomes more massive than the  . They look exactly the same proving that, for the same Fmax E , as the magnetic field goes down, the number of Landau levels increases and so the magnetic EoS approaches the non-magnetic one.  . We can also see that for 500-levels there is only one branch. Also, these M-R relations are the same as those of a non-magnetic WD, in the following two aspects [21]:

Comparison with the Non-Magnetic Results
1) Stars are much lighter and do not violate the Chandrasekhar limit.
2) The radius of a star decreases as the mass increases.
These can be explained from Figure 10. It shows that a magnetic EoS is much stiffer than a non-magnetic one. A stiffer EoS is more successful in counteracting gravity than a softer one (non-magnetic one). That is why MWDs can become more massive than their non-magnetic counterparts. It can also be seen from Figure 10 that for the same outward degeneracy pressure a non-magnetic electron gas is more dense for almost the entire pressure range. This means that gravitational pressure has a good chance of overcoming the degeneracy pressure, which is exactly what happens in a typical non-magnetic WD. In addition, the non-magnetic EoS being soft, also facilitates gravitational compression. The lower mass stars in Figure 9 will be bigger because the electron gas, at those densities, would oppose gravity effectively. Figure 11 shows a comparison of enclosed mass of a star as a function of radial distance r for both a one-level system and a 500-level system. The same central density is chosen for both stars ( .29 10 cm R × (500 levels). We can see that the star's radius in the case of a 500-level system is bigger even when its mass is much smaller or that of a one-level system is smaller even when its mass is much bigger. This can be explained, again, from Figure 10. A stiffer EoS (one-level system) is more successful in counteracting gravity, but that also means that the same degenerate gas has to support more mass. More mass means more gravitational compression. Hence, we have a compact star corresponding to a onelevel system. One can clearly see in Figure 11 that much more mass is enclosed for the same radial distance in the case of a one-level MWD; therefore it has a smaller radius.
From Figure 12 we can see that the magnetic EoS starts to approach the nonmagnetic one when the B-field decreases, even for much fewer Landau levels.
The kinks in the EoS get smaller as the number of Landau levels increases for the same Fmax E . We can conclude from all this, that as the number of Landau levels increases (or as the B-field decreases), one no longer finds the same number of branches in the M-R relations as the number of Landau levels in the EoS, but rather the M-R relations evolve in such a way that the number of branches reduces, and ultimately they coalesce to become one branch as can be seen for 500 Landau levels. This is something we can expect since the M-R relations should have just one branch in the limit 0 B → .
We can also expect this because the plot of a magnetic DOS starts to look like  Figure 13. This means that the kinks in the EoS become so small that its overall appearance is of a non-magnetic one. Basically, in Figure 5, as the number of cylinders increases for the same Fmax E (or the same sized Fermi surface), the magnetic p -space begins to look like the continuum of a non-magnetic p -space.

Comparison with an Earlier Work by Das & Mukhopadhyay (DM)
The main point to discuss in this section is the M-R relations. Because of the correlation between the DOS and the EoS, we have firmly established that a multiple (at least two and three) Landau-level system would lead to multiple branches in the mass-radius relations. This is very different from what DM are proposing [7]. This discrepancy arises due to their incorrect approach in dealing with the EoS. They have done "piece-wise" power-law fits (or polytropic fits as they call them) of the form P Kρ Γ = , where 1 n n + Γ = , to the EoS and have wrongly assumed that each individual fit, with particular values of K and Γ , holds throughout the star. Then they have proceeded to numerically solve the equation of hydrostatic equilibrium for ρ as a function of radial distance r, and thus have obtained the mass and radius of the star with one polytropic relation throughout using Equations (21)- (25). This power-law/polytropic relation of the EoS works very well only for a non-magnetic WD where just one such relation would hold [21].
This approach of DM has a serious flaw, in that, they did not make use of the would be wrong to assume otherwise.
One can also derive Equations (14) and (19) solely on the basis of quantum mechanics as well as statistical mechanics, and thus does not involve an assumptions concerning stars. They together form an "equation of state" for a relativistic degenerate gas no matter where it is found [25]. Once an EoS is generated, that itself describes how the pressure varies with density within a star that has a particular value of a static and uniform B-field within it. So, when a stellar model is generated with a particular value of central density c ρ , which is chosen from the different density values on the EoS-plot, one has to start with the pressure value corresponding to that chosen value of density at the center of the star.
Then the remaining EoS-plot dictates how the pressure varies with density until we reach the surface of the star. We have made proper use of the EoS and have not done "piece-wise" or "polytropic" fits.

Conclusions
In this work, we have established a crucial correlation between the DOS and the non-polytropic EoS of a cold, relativistic, degenerate electron gas in a magnetic field which is required to understand the possibility of having super-massive WD stars with strong magnetic fields inside them. We have found how the distribution of single-particle states as a function of electron energies in the presence of a strong magnetic field dramatically changes the EoS of matter inside a MWD star, and renders the star super-massive. In particular, it is only when the electrons occupied the first Landau level that the star became heavier than the Chandrasekhar limiting mass. The drop in the DOS to a very small value in the presence of a B-field for an extended range of energy values is vital in making the pressure gradient very steep, and thus helping to increase the outward degeneracy pressure of the electrons. This was particularly evident in the case of a one Landau-level system.
Although, our results are based on the same assumptions (spherical symmetry, and uniform and static B-field) as made by Das & Mukhopadhyay, they satisfactorily address the observational finding that a magnetic field does lead to an overall higher mean mass of WDs [12]. Such a higher mass of the magnetic WD can only be understood if one looks at the fundamental physics involved, as is done in this work. We have also found that for two and three-level systems, there are the same number of branches in the M-R relations as there are levels. This, again, was found to be directly related to the DOS. For these levels, the star had density regions inside it where the pressure gradient was not enough to make that star massive even when the central densities were high. The region/s of densities seen to the right of the kink/s in the EoS played a crucial role in this. One may be puzzled by the discontinuity in the mass-radius values of stars in the branches of the two and three-level systems even as the central densities of stars, starting from the bottom-most branch to the topmost branch, smoothly varied from right to left across the kink/s in their equations of state, however, we have to understand the fact that these stars (stellar models) are built from scratch, guided by the EoS and its dependence on the DOS in the presence of a magnetic field. This is strictly a theoretical model with the assumptions of spherical symmetry and uniform and static B-field. So, it does seem that for multiple Landau levels (at least for two and three levels), there are these jumps in the stellar massradius values as the electron occupancy keeps on decreasing to lower levels in a given EoS.
Because we humans cannot reach the stars to perform experiments on them, a critical tool for understanding what might be happening within the stars is via an EoS developed through theoretical models such as that of a cold, relativistic, degenerate electron gas in a magnetic field, as is done in this work.
We believe that we have convincingly tackled this problem of presence of magnetic field inside a star and its consequences from a fundamental perspective.
Most importantly, we have succeeded in answering the question-"What is so special about a magnetic field that it makes a white dwarf super-massive?"