Violation of Chandrasekhar Mass Limit: The Exciting Potential of Strongly Magnetized White Dwarfs
Abstract
We consider a relativistic, degenerate, electron gas under the influence of a strong magnetic field, which describes magnetized white dwarfs. Landau quantization changes the density of states available to the electrons, thus modifying the underlying equation of state. In the presence of very strong magnetic fields a maximum of either one, two or three Landau level(s) is/are occupied. We obtain the massradius relations for such white dwarfs and their detailed investigation leads us to propose the existence of white dwarfs having a mass , which overwhelmingly exceeds the Chandrasekhar mass limit.
Essay written for the Gravity Research Foundation
2012 Awards for Essays on Gravitation
Introduction
Chandrasekhar first showed the maximum possible mass for a white dwarf (WD) to be [1]. Later, several magnetized WDs have been discovered with surface fields of G [2], [3], [4], [5], [6]. Anticipating that the central field, which cannot be probed directly, will be much stronger than that on the surface, Ostriker and Hartwick [7] constructed models of WDs with G at the center but with a much smaller field at the surface. Recent observations of peculiar Type Ia supernovae  SN2006gz, SN2007if, SN2009dc  seem to suggest superChandrasekhar mass WDs as their most likely progenitors [8]. These WDs, found in binary systems, stabilize by rotating and accreting matter [9]. In this essay we propose a mechanism by which the WDs exceed the Chandrasekhar mass limit in presence of strong magnetic fields at their centers. In a simple theoretical framework existence of such stars has been reported recently [10].
Here we consider strongly magnetized WDs having interior magnetic field G. These Landau quantized electronic systems have at the most one, two or three occupied Landau level(s) (LL(s)). We obtain an exciting possibility of a mass of WDs .
Equation of state and solution procedure
Basic equations
The Landau quantized energy states of a free electron in a uniform, static, magnetic field are given by [11]
(1) 
where denotes the LL, given by
(2) 
when being the principal quantum number, , the momentum of the electron along the axis, the magnetic field, a critical magnetic field defined by
(3) 
where is the charge of the electron, the speed of light, the restmass of the electron, the Planck’s constant. Our interest is to study the effect of on the relativistic, degenerate electron gas.
The Fermi energy of electrons in units of for a given is given by
(4) 
where is the Fermi momentum in units .
As , the maximum number of occupied LLs
(5) 
where , the dimensionless maximum Fermi energy of the system.
Following [11] we write the electron number density
(6) 
where the Compton wavelength of the electron ; the matter density
(7) 
where is the mean molecular weight per electron (which we choose to be ) and the mass of hydrogen atom; the electron degeneracy pressure
(8) 
where
(9) 
Procedure
From equation (5) we see that if , then is small, implying electrons are restricted to the lower LLs. Since we are interested in this regime, depending on the specific values of and , we have onelevel (), twolevel () and threelevel () systems. By eliminating numerically from equations (7) and (8), we obtain the relation which is the equation of state (EOS). Figure 1 shows EOSs for the different cases given in Table 1.
Table 1
Parameters for the equations of state in Figure 1.

in units of G  


1  1.5  0.066 
2  2  0.75  0.033 
3  0.5  0.022  
1  199.5  8.81  
20  2  99.75  4.40 
3  66.5  2.94  
1  19999.5  882.78  
200  2  9999.75  441.38 
3  6666.5  294.26  

The hydrostatic equilibrium of the WD in presence of constant magnetic field is described by [12]
(10) 
with the boundary conditions:
(11) 
and
(12) 
where is the central density of the WD.
The radius of the WD is obtained by solving equation (10) and is the value of where density goes to zero. The mass of the WD, which is approximated to be spherical, is obtained by integrating the following equation:
(13) 
Constructing white dwarfs
Equations of State
Let us consider the case . The solid curve in Figure 1(b) is free of any kink (only ground LL occupied). The dotted and dashed curves have one kink (ground and first LLs occupied) and two kinks (ground, first and second LLs occupied) respectively. Any kink represents a transition from a lower to upper LL.
Massradius relations
Table 2
Parameters for the fitting function for the equation of state shown in Figure 1(d).
in units of gm/cc 
in CGS units  

0 0.096 
2.9  4.055 
0.096 0.307  2.4  1.284 
0.307 1.128  2.1  0.914 
1.128 2.117  0.35  1.105 
2.117 2.956  4/3  0.522 
2.956 3.842  2.0  0.246 
3.842 4.651  0.35  2.225 
4.651 6.116  4/3  0.496 
6.116 7.37  2.0  0.142 

Figure 2 shows the massradius relations, where each point in the curves corresponds to a WD with a particular value of which is supplied as a boundary condition.
In Figure 2(a) we see that as increases, both and increase and then at higher values of , becomes nearly independent of . We note that the most massive WD on this curve has a mass 2.3 , which corresponds to the maximum density point of the solid curve in Figure 1(b). This denotes the density at which the ground LL is completely filled. Thus a WD with this and a magnetic field strength of G has a mass greater than the Chandrasekhar limit (for details see [13]).
The turning point in Figure 2(b), after attending the maximum mass , corresponds to the kink in the corresponding EOS given in Figure 1(b). During the transition from ground to first LL, the radius and mass both decrease with increasing . Then there is a brief range of densities where the mass increases as the radius decreases and ultimately at very high densities the radius is nearly independent of the mass.
In Figure 2(c) we see a repetition of the features in Figure 2(b) twice, which corresponds to the two kinks in the corresponding EOS in Figure 1(b). (The maximum mass occurs at that where the ground LL is completely filled and transition to the first LL is about to start.)
For comparison let us recall the LaneEmden solution for the classical nonmagnetic case, which assumes a polytropic EOS
(14) 
giving rise to the following relations [12]:
(15) 
and
(16) 
This means that if , increases with , if , is independent of , if , increases with and if , is independent of . This is exactly what is observed in Figure 2, if the EOSs given in Figure 1 are fitted by the polytropic EOS, adopting constant values of in different density ranges (see Table 2). For instance in Figure 1(b) (dashed line) and Figure 2(c), at very low densities, and up to the turning point density (the kink in the EOS) the radius keeps increasing with mass and then becomes nearly independent of mass when . Then suddenly drops to a small value 0.35 which marks the onset of the transition from ground LL to first LL. In this region the pressure becomes independent of density, revealing an unstable zone in the EOS. As the density increases further, approaches the relativistic value of . In this regime we see that the radius decreases slightly as the mass does not change significantly. Next takes up a value of 2 and the above phenomena will be repeated.
Discussions
In Figure 3 we put together our results along with that of Chandrasekhar. Figure 3(a) clearly indicates that as the magnetic field decreases, the EOS approaches Chandrasekhar’s EOS.
Figure 3(b) represents the corresponding massradius relations. We observe that as the magnetic field decreases, the massradius relation approaches that obtained by Chandrasekhar. We also note that as the magnetic field increases the WDs become more and more compact in size and the probability that they will have masses exceeding the Chandrasekhar limit also increases.
Figure 4 shows the variation of mass as a function of density within a magnetized WD for three different magnetic fields. In all the cases, we note that, by the time the density falls to about half the value of , the mass has already crossed the Chandrasekhar limit as indicated by the horizontal line. Hence the effect of magnetic field is restricted to the high density regime, where the field remains essentially constant even in reality. This justifies our choice of constant field.
Conclusions
We summarize the important findings of this work as follows:

A transition from the lower to upper LL represents a kink in the EOS. If lies at the kink appearing at the ground to first LL transition, the WD has the maximum possible mass.

The most interesting result obtained is that there are possible WDs whose mass exceeds the Chandrasekhar limit and is found to be about . For instance, for and G, we obtain such a WD with gm/cc. Interestingly, the nature of massradius relations does not depend on the value of , however determines how relativistic the system is. For instance, the Chandrasekhar limit is not exceeded for , no matter what is.

As the magnetic field increases the WDs become more compact in size.

The minimum magnetic field required to have a WD is G. The magnetic field of the original star of radius , which collapses into a WD of radius cm with the above field at its center, then turns out to be G, based on the flux freezing theorem. Existence of such stars is not ruled out [14].

In principle, the strong magnetic field causes the pressure to become anisotropic [15], [16]. As a result, the combined fluidmagnetic medium develops a magnetic tension [17], leading to a deformation in the WDs along the direction of the magnetic field. The WDs hence could adopt a flattened shape [13].

The flattening effect due to magnetic field leads to superChandrasekhar WDs even for smaller magnetic fields. Such relatively weakly magnetized WDs are more probable in nature.

We end by addressing the possible reason for not observing such a high field yet in a WD. This could be due to the magnetic screening effects on the surface of the WD if it is an accreting one. In this case the current in the accreting plasma depositing on the surface, presumably creates an induced magnetic moment of sign opposite to that of the original magnetic dipole, thus reducing the surface magnetic field of the WD, without affecting the central field. Hence by estimating the surface field alone one should not assume the rest.
Acknowledgments
This work was partly supported by the ISRO grant ISRO/RES/2/367/1011.
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