#### White dwarfs with a surface electrical charge distribution: equilibrium and stability

Eur. Phys. J. C
White dwarfs with a surface electrical charge distribution: equilibrium and stability
G. A. Carvalho 1
José D. V. Arbañil 0
R. M. Marinho Jr. 1
M. Malheiro 1
0 Departamento de Ciencias, Universidad Privada del Norte, Avenida Alfredo Mendiola 6062 Urbanización Los Olivos , Lima , Peru
1 Departamento de Física, Instituto Tecnológico de Aeronáutica , São José dos Campos, SP 12228-900 , Brazil
The equilibrium configuration and the radial stability of white dwarfs composed of charged perfect fluid are investigated. These cases are analyzed through the results obtained from the solution of the hydrostatic equilibrium equation. We regard that the fluid pressure and the fluid energy density follow the relation of a fully degenerate electron gas. For the electric charge distribution in the object, we consider that it is centralized only close to the white dwarfs' surfaces. We obtain larger and more massive white dwarfs when the total electric charge is increased. To appreciate the effects of the electric charge in the structure of the star, we found that it must be in the order of 1020 C [ ] with which the electric field is about 1016 [V/cm]. For white dwarfs with electric fields close to the Schwinger limit, we obtain masses around 2 M . We also found that in a system constituted by charged static equilibrium configurations, the maximum mass point found on it marks the onset of the instability. This indicates that the necessary and sufficient conditions to recognize regions constituted by stable and unstable equilibrium configurations against small radial perturbations are respectively d M/dρc > 0 and d M/dρc < 0.
1 Introduction
In recently years, observations reveals about the existence
of some peculiar super-luminous type Ia supernovae (SNIa)
[
1–4
]. These types of supernovae are of particular interest,
since their explosions are the most predictable and, often, the
brightest events in the sky. In some works is suggested that
a possible progenitor of such a peculiar SNIa is the
superChandrasekhar white dwarf, which exceeds significantly the
standard Chandrasekhar mass limit 1.44M . The masses
estimated for these super-Chandrasekhar white dwarfs are
between 2.1 and 2.8M , see, e.g., [
5,6
].
In the General Relativity scopes, these particular objects
have attracted the attention of several authors, who in
different works, propose diverse models in order to explain these
super-Chandrasekhar white dwarfs. For instance, we found
works where are considering white dwarfs with a strong
magnetic field [
7,8
], in rotation and with different topologies for
magnetic field [
9–13
] and in the presence of an electrical
charge distribution [14].
In what concerns to magnetized white dwarfs, in [
7,8
] are
found that a uniform and very strong magnetic field can
overcome the white dwarf maximum mass up to ∼ 2.9 M . Albeit
these objects exceed significantly the Chandrasekhar mass
limit, they suffer from severe stability issues as discussed in
the literature [
15–19
]. In [
9–13
] are found that white dwarfs
with rotation and different topology for the magnetic field
could reach masses up to 5M .
In electrically charged white dwarf [
14
], Liu and
collaborators found that the charge contained in white dwarfs
affects their structure, they have larger masses and radii than
the uncharged ones. In [
14
], the electric charge distribution
is considered such as the charge density is proportional to
the energy density, ρe = αρ (α being a proportional
constant). The choice of this charge distribution is an appropriate
assumption in the sense that a star with a large mass could
contain a large quantity of charge. Liu et al. found that the
charged white dwarfs may attain values up to 3.0M .
The study developed in [
14
] belongs to a large group of
works where charged stars are investigated. Within them,
we found some research where the influence of the
electrical charge distribution at the stellar structure of
polytropic stars [
20–24
], incompressible spherical objects [
25–
28
], anisotropic stars [29] and strange stars [30,31], and in
the radial stability against small perturbation [32,33] and
in gravitational collapse [34] are investigated. It is
important to mention that, although charged stars are anisotropic
objects, these maintain their spherical symmetry, since the
energy momentum tensor components T00(r ) and T11(r ) are
The interior background space-time of the star, in
Schwarzschild coordinates, it is assumed of the following
form
ds2 = − eν dt 2 + eλdr 2 + r 2 dθ 2 + sin2 θ dφ2 ,
where the functions ν = ν(r ) and λ = λ(r ) dependents on
the radial coordinate r only.
The Maxwell–Einstein equations in such line element lead
to the following set of stellar structure equations:
dq
dr = 4πρer 2eλ/2,
dm q dq
dr = 4πr 2ρ + r dr ,
d p m q2
dr = − ( p + ρ) 4πr p + r 2 − r 3
dν 2 d p 2q dq
dr = − ( p + ρ) dr + 4πr 4( p + ρ) dr
,
eλ + 4πqr 4 ddqr , (5)
equals, being r the radial coordinate (see, e.g., [35–37]). In
[38], Maurya et al. have obtained feasible stellar models of
charged compact stars by assuming three different solutions
for static spherically symmetric metric, and they have
validated their model by comparing it with some observational
parameters of compact objects, which are candidates to be
strange stars.
In this work, we investigate the influence of the surface
electric charge in the stellar structure and stability of white
dwarfs. We consider that the electric charge at the star’s
surface follows a Gaussian distribution, such as is considered in
[30,31]. This consideration is realized since at white dwarfs’
atmospheres the electrons and ions could play an
important role producing strong electric fields [39]. An important
point that must be analyzed in charged stars is the Schwinger
limit. It states that when electric fields attains values around
∼ 1.3 × 1016 [V/cm], these electric fields interact with the
vacuum thus producing electron-positron pairs. Due to this
phenomena, the electric fields decay to a value below the
threshold [40]. It is important to point out that in neutron
stars the magnitude of the electric field necessary to
appreciate effects on the structure of these stars are at the order of
1019V/cm [
20,30
], i.e., much higher than Schwinger limit.
This has been always advocated as a physical restriction to
consider charge effects on the structure, at least, of neutron
stars. Thus, in this article the Schwinger limit is analyzed.
The article is divided as follows. The stellar structure
equations, the equation of state and the electric charge profile are
described in Sect. 2. In Sect. 3 the stellar equilibrium
configurations of charged white dwarfs are presented. The stability
of these stars under small radial perturbations is investigated
in Sect. 4, using the results derived from the static method. In
Sect. 5 we discuss about the charge-radius relation found for
the white dwarfs under analyze. Finally, we conclude in Sect.
6. Throughout the present work, we use geometric units.
2 Stellar structure equation, equation of state and
electric charge profile
2.1 Stellar structure equations The charged fluid contained in the spherical symmetric object is described by the energy-momentum tensor:
Tμν = (ρ + p)uμuν + pgμν
1 1
+ 4π F μγ Fϕγ − 4 gμν Fγβ F γβ ,
with ρ and p being the energy density and the pressure fluid,
respectively. Moreover, uμ, gμν and F μγ stand the fluid’s
four velocity, the metric tensor and the Faraday–Maxwell
tensor, respectively.
(1)
(2)
(3)
(4)
(6)
(7)
(8)
(9)
(10)
(11)
where the metric component eλ is given by the equality:
eλ =
2m
1 − r
q2 −1
+ r 2
.
The variables q, m and ρe showed before represent
respectively the electric charge, the mass within the sphere of
radius r , and the charge density. Equation (5) gives the
Tolman–Oppenheimer–Volkoff equation [41,42] modified to
the inclusion of the electric charge [
43
].
2.2 Equation of state It is considered that the pressure and the energy density of the fluid contained in the spherical object follow the relations: 1
p(kF ) = 3π 2 h¯3 0
kF
1
ρ(kF ) = π 2 h¯3
0
kF
k4
k2 + me2
dk,
m N μe 3
k2 + me2k2dk + 3π 2 h¯3 F
k ,
where me is the electron mass, m N represents the nucleon
mass, h¯ denotes the reduced Planck constant, μe states the
ratio between the nucleon number and atomic number for
ions and kF depicts the Fermi momentum of the electron,
review [
44,45
]. Equation (8) is the electric degeneracy
pressure and the first and second terms of the right hand side of
Eq. (9) are respectively the electron energy density and the
energy density related to the rest mass of the nucleons.
It is important to mention that for numerical and analytical
analysis the Eqs. (8) and (9) are using in the form
p(x ) = ε0 f (x ),
ρ(x ) = ε0g(x ),
where
1
f (x ) = 24
1
g(x ) = 8
(2x 3 − 3x ) x 2 + 1 + 3asinhx
(2x 3 + x ) x 2 + 1 − asinhx
+ 1215.26x 3,
being ε0 = me/π 2λe3 and x = kF /me the dimensionless
Fermi momentum, with λe representing the electron
Compton wavelength. In the last two equations, it is taking into
account μe = 2.
2.3 Electric charge profile
We assume that the interior and the atmosphere of the white
dwarfs are made of respectively by degenerate and
nondegenerate matter. This hypotheses is consistent since the energy
density decays toward the white dwarf’ surface, where it
drops to zero. At this point the white dwarf must have a
nondegenerate atmosphere (review, e.g., [
46, 47
]). For the white
dwarfs under analyzes, we consider that the majority amount
of charged is concentrated at the surface of star. As
aforementioned, the surface electric charge of the white dwarf could
develop an important task, producing superficial strong
electric fields [39]. Thus, following [30], we model the
distribution of the electric charge in terms of a Gaussian, hence the
charge distribution can be regarded as the following ansatz:
ρe = k exp
−
(r − R)2
b2
,
being R the radius of the star in the uncharged case. The
constant b is the width of the electric charge distribution, in
this case, we consider b = 10 [km]. For small width of this
layer, the stellar structure of the white dwarf does not change
significantly.
In order to determine the constant k, we require to the
equality:
σ =
0
∞
4π r 2ρedr,
with σ being the magnitude proportional to the electric
charge distribution. σ would represents the total charge of the star
if we was working out in flat space-time. Using Eqs. (14) in
(15), it yields:
4π k = σ
√π b R2
2
+
√π b3
4
−1
(14)
(15)
(16)
2.4 Numerical method and boundary conditions
The stellar structure equilibrium Eqs. (3), (4), (5) and (7),
together with the Eqs. (8) and (14), are integrated using the
Runge–Kutta fourth order method, for different values of σ
and ρc. These equations are integrated from the center r = 0
towards the surface of the star r = R. The integration of
the aforementioned equations begins with the values in the
center:
m(0) = 0,
p(0) = pc,
q(0) = 0,
and
ρe(0) = ρec.
ρ (0) = ρc,
The integration of the stellar structure equations ends when
the star’s surface is attained, p( R) = 0. At the surface of the
object the interior solution matches smoothly with the
vacuum exterior Reissner–Nordström line element. This
indicates that the inner and outer metric function are related by
the equality
eν(R) = e−λ(R) = 1 −
2M Q2
R − R2 ,
(17)
(18)
where M and Q being the total mass and total charge of the
sphere.
3 Stellar equilibrium configurations of charged white
dwarfs
Figure 1 shows the behavior of the total mass M/M with
the central energy density, for six values of σ . The considered
central energy densities are in the interval 2 × 106 [g/cm3]
to 4 × 1011 [g/cm3]. The lower limit ρc = 2 × 106 [g/cm3]
is the mean density for white dwarfs and in the upper limit
ρc = 4 × 1011 [g/cm3] the neutron drip limit is reached, i.e.,
in this point, the white dwarf turn into a neutron star. In the
cases where σ ≤ 0.8 × 1020 [C], we note that the total mass
grows with the central energy density until attain a maximum
mass point, after that point, the mass starts to decrease with
the grows of the density of energy center. In turn, in the case
σ = 1.0 × 1020 [C] the mass grows monotonically with the
central energy density, i.e., we do not found a turning point.
On the other hand, we observe that exist a dependency of
the total mass with the parameter σ . For a larger σ more
massive stars are obtained. As can be noted in Fig. 1, we obtain
more massive white dwarfs using σ = 1.0 × 1020 [C]. In this
case, the mass whose respective electric field saturates the
Schwinger limit (∼ 1.3 × 1016 [V/cm]) is 2.199M , this is
attained in the central energy density 1.665 × 1011 [g/cm3].
This mass is between the masses estimated for the
superChandrasekhar white dwarfs, 2.1–2.8M [
5, 6
]. From this,
we understand that a surface electric field produces
considerable effects in the masses of white dwarfs. In addition, it is
0.005
0.004
9
Log(ρc[g/cm3])
2.2
2.0
important to mention that the grow of the mass with σ can be
understood since σ is related with the total charge contained
in the star. The charge produces a force which helps to the
one generated by the radial pressure to support more mass
against the gravitational collapse.
In Fig. 2 the total mass as a function of the radius for
few values of σ is observed. In the uncharged case σ = 0.0
the curve is close to attain the typical Chandrasekhar limit,
1.44M [
48, 49
], however, in the charged case σ = 0 we
found masses that overcome this typical limit. For instance,
in the case σ = 1.0 × 1020 [C] the mass whose electric field
saturates the Schwinger limit is around 2.199M . Again, we
mention that high values of white dwarf masses (around the
super-Chandrasekhar white dwarf masses) can be achieved
taking into account a surface electrical charge at the white
dwarf.
With the purpose to observe that the electric charge is
only distributed near the star’s surface, the pressure
profile inside the white dwarf as a function of the radial
coordinate is plotted in Fig. 3, where few values of σ and
ρc = 1010 [g/cm3] are considered. In figure we can note
that the pressure decays monotonically toward the baryonic
surface, when this is attained the pressure grows abruptly
due to the beginning of the electrostatic layer, after this point
the pressure decrease with the radial coordinate until attain
the star’s surface ( P = 0). Thus, through this result, we can
clearly note that the electric charge is distributed as a
spherical shell close to the surface of the white dwarf.
The behavior of the electric field in the star is showed
in Fig. 4. On figure is employed five different values of σ
and ρc = 1010 [g/cm3]. As can be seen, in each case
presented, the electric field exhibit a very abrupt increase from
zero to 1015−16 [V/cm], thus indicating that the baryonic
surface ends and starts the electrostatic layer. Once the electric
surface is distributed like a thin layer close to the surface,
the electric field sharply weaken with the grow of the radial
coordinate such as is showed in figure.
It is important to emphasize that the electric field could be
reduced, once taking into account the change that the
electric potential screening may suffer with the increment of the
radial distance (see [
50
]). The analysis of such a situation is
left for future investigation.
It is worth mentioning that the electric field found in the
charged white dwarfs cases are 104 times lower than those
found in charged strange stars ones (see, e.g., [30–32]). This
can be understand since white dwarfs have very larger radii
than the strange stars.
The values for σ employed, the maximum mass values
found in the range of central energy densities considered,
with their respective central energy densities, total radius,
total charge and electric field are shown in Table 1.
4 About the radial stability of charged white dwarf
Inspired in the study of the turning-point method for
axisymmetric stability of uniformly rotating relativistic stars with the
angular momentum fixed [
51
] (see also [
52
] for a detailed
discussion about that theme), in [32] is shown that the
stability of charged objects could be investigated using the
results obtained from the hydrostatic equilibrium equation.
The authors in [32] found that along a sequence of charged
stars with increasing central energy density and with fixed
total charge, the maximum mass equilibrium configuration
states the onset of instability. Thus, likewise, in this section
we use the results derived from the equilibrium
configurations to determine the maximum mass point which marks
the start of the instability.
The total charge of white dwarf as a function of the central
energy density is plotted in Fig. 5, taking into account five
different values of σ . In figure we can see that along the sequence
of equilibrium configurations with increasing central energy
density the total charge is nearly constant. In this case, we
understand that the maximum mass point must marks the
onset of instability. In other words, the regions made of
stable and unstable charged white dwarfs shall be distinguished
through the relations d M/dρc > 0 and d M/dρc < 0,
respectively.
E [V/cm]
Additionally, in Fig. 5, it can be observed also that the
electrical charge that produce considerable effects in the structure
of white dwarfs is around 1020 [C]. This amount of charge
is similar to those found in the studies, for instance, of
polytropic stars [
20, 21
], incompressible stars [
25–28
], strange
stars [30–32] and white dwarfs [
14
], where, certainly, the
electric charge is considered.
5 Universal charge-radius relation and maximum total
charge of white dwarfs
In [
53
], Madsen demonstrates that the electrical charge of
a spherically symmetrical static object is limited by the
creation of electron-positron pairs in super critical electric fields.
Taking into account a timescale τ << ∞, the net positive
charge Q is directly proportional to the square of the star’s
radius Rkm, i.e.,:
Q = βe Rk2m,
with β and e being the proportionality constant and charge
of a proton, respectively. The proportionality constant β is
directly related with the timescale chosen τ , for lower
timescale a larger β is derived. Madsen found that for a timescale
(19)
τ = 1.0 s is obtained β = 7.0 × 1031, and for τ = 1.0 ×
10−10 s, i.e., a typical weak interaction timescale, β = 1.68×
1032.
For white dwarfs we can consider the electromagnetic
interaction, consequently, the timescale can be regarded to
be around 10−18 s, thereby the constant β becomes equal to
3.34 × 1033. In this case, we obtain that the more massive
white dwarf (see Table 1) would allow the maximum charge:
Qmax ≈ 5.0 × 1020 [C].
Thus, the quantity of charge found in the most massive white
dwarf (Q = 2.045 × 1020 [C]) is under the maximum charge
limit of Eq. (20).
(20)
6 Conclusions
The static equilibrium configurations and the stellar radial
stability of charged white dwarfs are investigated in this
work. Both studies are analyzed through the results derived
from the hydrostatic equilibrium equation, the Tolman–
Oppenheimer–Volkoff equation, modified to include the
electrical part. For the interior of white dwarfs, we consider
that the equation of state follows the employed for the fully
degenerated electron gas [
44,45
]. In addition, we assume a
Gaussian distribution of charge of ∼ 10 [km] thickness close
the star’s surface. It is important to mention that the interior
solution match smoothly to the exterior Reissner–Nordström
vacuum solution.
We observe that for larger total charge, more massive
stellar objects are found. For instance, the increment of the total
charge from 0 to 2.058 × 1020 [C] allows to increase the total
mass in approximately 55.58%, growing from 1.416 M
to 2.203 M . This increment in the mass of the star is
explained since the electric charge acts as an effective
pressure, thus helping the hydrodynamic pressure to support more
mass against the gravitational collapse. It is worth
mentioning that for the total electric charge 2.058 × 1020 [C],
we found that the Schwinger limit is saturated for a white
dwarf with ∼ 2.2 M . This total mass is within the
interval of white dwarf considered as super-Chandrasekhar white
dwarfs [
5,6
]. From the aforementioned, we can understand
that a surface distribution of charge could plays an
important role in the existence of the super-Chandrasekhar white
dwarfs.
On the other hand, the stability against small radial
perturbations of charged white dwarfs are analyzed using a
sequence of equilibrium configurations with increasing
central energy density, where these spherical objects are
constituted by an equal total electric charge. In this types of
sequences, the maximum mass point marks the onset of the
instability, see [32]. From this, we can say that the regions
where lay stable and unstable white dwarfs can be
distinguished by the inequalities d M/dρc > 0 and d M/dρc < 0,
respectively.
Acknowledgements Authors would like to thank Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior-CAPES and Fundação de
Amparo à Pesquisa do Estado de São Paulo-FAPESP, under the thematic
project 2013/26258-4, for the financial supports. GAC also thanks to
Professors Dr. P. J. Pompeia and Dr. P. H. R. S. Moraes for discussion
and support under the preparation of this work.
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