New mass limit for white dwarfs: super-Chandrasekhar type Ia supernova as a new standard candle

0.23 MB·English

by  Upasana Das

New mass limit for white dwarfs: super-Chandrasekhar type Ia supernova as a new
standard candle
Upasana Das, Banibrata Mukhopadhyay
Department of Physics, Indian Institute of Science, Bangalore 560012, India
[email protected] , [email protected]
Type Ia supernovae, sparked off by exploding white dwarfs of mass close to Chandrasekhar
limit, play the key role to understand the expansion rate of universe. However, recent obser-
vations of several peculiar type Ia supernovae argue for its progenitor mass to be significantly
super-Chandrasekhar. We show that strongly magnetized white dwarfs not only can violate the
3 Chandrasekhar mass limit significantly, but exhibit a different mass limit. Weestablish from foun-
1 dationallevelthatthegenericmasslimitofwhitedwarfsis2.58solarmass. Thisexplainstheorigin
0 ofover-luminouspeculiartypeIasupernovae. Ourfindingfurtherarguesforapossiblesecondstan-
2 dard candle, which has many far reaching implications, including a possible reconsideration of the
expansion history of the universe.
n
a
J
PACS: 97.60.Bw, 97.20.Rp, 98.80.Es,97.10.Ld, 71.70.Di
5
2

Introduction. Recently,somepeculiartypeIasupernovae: e.g. SN2006gz,SN2007if,SN2009dc,SN2003fg,have
]
−
R beenobserved[1]withexceptionallyhigherluminositiesbutlowerkineticenergies. Thekinetic energyisproportional
S to the difference between the obtained nuclear energy which arises from the synthesis of elements in the explosion
. through fusion and the binding energy of the white dwarf. Most of the light-curves of above mentioned peculiar
h
supernovae appear over-luminous, slow-rising, which do not allow them to be calibrated as standard candles. This
p
- questions the use of all type Ia supernovae in measuring distances of far away regions and hence unraveling the
o expansionhistory of the universe. However,assumingthe progenitorto be a highly super-Chandrasekharmass white
r
t dwarfreproducesthelowkineticenergiesandthusvelocitiesseenintheabovesupernovae[2]. Thisisbecausealarger
s
mass implies a larger binding energy of the star and hence a smaller velocity for the same and/or higher luminosity
a
[ (due to nuclear fusions) than that observed in a standard type Ia supernova. The progenitor masses required to
explain the above supernovae lie in the range 2.1 2.8M⊙, when M⊙ being the mass of sun, subject to the model
1 −
chosen to estimate the nickel mass [1–6]. Now naturally the following vital questions arise. Is there any fundamental
v
basis behind the formation of a highly super-Chandrasekhar white dwarf? How to address the significant violation
5
6 of Chandrasekhar mass limit? What is the ultimate mass limit of a white dwarf? Here we plan to address all the
9 above issues by exploiting the effects of magnetic field in compact objects. This will lead, as we will show, to a
5 natural explanation of the so called “peculiar” type Ia supernovae. This might eventually lead these supernovae to
. be considered as altogether new standard candles. This has many far reaching implications, including a possible
1
0 reconsiderationof the expansion history of the universe.
3 Before proceeding further, let us recall the physics of a white dwarf and its link to the type Ia supernova. When a
1 star exhausts its nuclear fuel, it converts to either of the three compact objects: white dwarf, neutron star or black
: hole,dependingontheinitialmassoftheevolvingstar. Itisgenerallybelievedthatthefateisawhitedwarfwhenthe
v
i mass ofthe initial star in its mainsequence is MMS<∼5M⊙. Such a main sequence star undergoingcollapseleading to
X
asmallvolumeconsistsofalotofelectrons. Beinginasmallvolumemanysuchelectronstendtooccupysameenergy
r states, hence making them to degenerate electrons, as the energy of a particle depends on its momentum which is
a
determinedby the totalvolumeofthe system. However,being a fermionanelectronobeysPauli’sexclusionprinciple
whichsaysthat no two fermions canoccupy the same quantum state. Hence, once upto the Fermi level,which is the
maximumallowedenergyofafermion,isfilledbytheelectrons,thereisnoavailablespacefortheremainingelectrons
in a small volume of a collapsing star, which expels the electrons to move out leading to an outward pressure. In a
white dwarf,the inwardgravitationalforce is balancedby the forcedue to outwardpressurecreatedby to degenerate
electrons. Chandrasekhar in one of his celebrated papers [7] showed that the mass of a white dwarf cannot be more
than 1.44M⊙ which sets the famous Chandrasekhar mass limit of a white dwarf.
If a white dwarf having mass close to the Chandrasekhar limit gains more mass (e.g. by accretion, when the mass
is supplied by a companion star of the white dwarf), then its mass exceeds the Chandrasekhar limit, which leads to
a gravitationalforce strongerthan the outwardforce that arises due to the degenerateelectrons. Hence, this leads to
the contraction of the white dwarf and a subsequent increase of its temperature, which is favorable for the initiation
of fusion reactions again. If the white dwarf mostly consists of carbon and oxygen, namely the carbon-oxygen white
dwarf (which is commonly the case), then nuclear fusion of carbon (and oxygen) takes place. Subsequently, within
a few seconds, a substantial fraction of the white dwarf matter undergoes a runaway reaction which releases huge
2
energy 1051ergtounbinditinanexplosion,namelytypeIasupernovaexplosion. Thiseventuallyleadstoacomplete
∼
gravitationalcollapse of the star without leaving any remnant.
As allthe commonlyobservedtype Ia supernovaeareproducedby (almost)the samemechanism, namelythe mass
of the progenitor white dwarf exceeding Chandrasekhar limit and subsequent processes, the underlying variations
of luminosity as functions of time, namely light-curves, appear alike for all the explosions. All of these supernovae
exhibitconsistentpeakluminosity,therelationbetweenthepeakluminosityandwidthofthelight-curve[8,9],dueto
the uniform mass of white dwarfs (Chandrasekharlimit) which finally explode, e.g., because of the accretionprocess.
Very importantly, the apparentstability of this value helps the underlying supernovae to be usedas standardcandles
in order to measure the distances to their host galaxies. Since these supernovae are exceptionally bright, they can be
observed across huge cosmic distances. The variation of their brightness with distance (or redshift) is an extremely
important tool for measuring various cosmological parameters, which in turn shed light on the expansion history of
the Universe. Theirenormousimportanceis selfevidentandwasbroughtinto the primefocus by the awardingofthe
Nobel Prize in Physics in 2011, for the discovery (made possible by the observation of distant type Ia supernovae)
that the universe is undergoing an accelerated expansion [10, 11].
Now we move on to our goal of establishing a new mass limit for super-Chandrasekhar white dwarfs, of whose
formationthereisnounderstandingfromfoundationallevel—acaveatbehindthehypothesisofsuper-Chandrasekhar
progenitor for the peculiar type Ia supernovae, raised by the earlier authors [2]. In fact they emphasized on the
pursuitoftheoreticalstudies inordertoassessthe hypothesis. Although,basedonanumericalcodeforstellarbinary
evolution, the rotating white dwarfs are suggested to hold mass up to 2.7M⊙ [12], a foundational level calculation is
missing and there is no estimate of mass limit of such a star either.
Inthis letter,weshowthat(highly)magnetizedwhite dwarfsnotonlycanhavemass 2.6M⊙,butalsoexhibitits
∼
ultimate limit of mass. Hence, we propose a fundamentally new mass limit for white dwarfs, which eventually helps
in explaining light-curves of peculiar type Ia supernovae. This may further lead to establishing these supernovae as
modified standard candles for distance measurement.
The motivation behind our approach lies in the discovery of several isolated magnetized white dwarfs through the
Sloan Digital Sky Survey (SDSS) with surface fields 105 109G [13], [14]. Hence their expected central fields could
−
be 2 3 orders of magnitude higher. Moreover, about 25% of accreting white dwarfs, namely cataclysmic variables
(CVs−), are found to have high magnetic fields 107 108G [15].
−
Equation of state. As the starting choice is the magnetized white dwarf, we first recall degenerate electrons
−
under the influence of magnetic field which are knownto be Landauquantized [1]. Largerthe magnetic field, smaller
is the number of Landau levels occupied [17] (see supplement [18]). Recent works [2, 3, 20] establish that Landau
quantization due to strong magnetic field modifies the equation of state (EoS) of the electron degenerate gas, which
results in a significantmodificationof the the mass-radiusrelationof the underlying white dwarf. Interestingly,these
white dwarfs are found to have super-Chandrasekhar masses. The main aim of this letter is to obtain the maximum
possible mass of such a white dwarf (which is magnetized), and therefore a (new) mass limit. Hence we look for the
regime of high density of electron degenerate gas and the corresponding EoS, which further corresponds to the high
Fermi energy (E ) of the system. This is because high density corresponds to high momentum, which implies high
F
energy (see supplement [18]). Note that the maximum Fermi energy (E ) corresponds to the maximum central
Fmax
density of the star. Consequently, conservation of magnetic flux (technically speaking flux freezing theorem which is
generallyapplicable for a compactstar)arguesfor the maximumpossible fieldof the system, whichimplies that only
the ground Landau level will be occupied by the electrons. For the expressions of density, pressure and the EoS for
such a highly magnetized system, see supplement [18]. Hence, in the limit of E >> m c2, when m is the mass of
F e e
the electronsandc the speedoflight, for a givenmagnetic field exhibiting the system ofone Landaulevel,the EoSis
2
P =K ρ , (1)
m
when P and ρ are respectively the pressure and density of the gas and the constant K is given by
m
m c2π2λ3
K = e e , (2)
m (µ m )2B
e H D
where λ = ~/m c, the Compton wavelength of electron, ~ the Planck’s constant h divided by 2π, µ the mean
e e e
molecularweightperelectron,m themassofhydrogenatomandB themagneticfieldintheunitsof4.414 1013G.
H D
×
Mass limit of white dwarfs. NowfollowingtheLane-Emdenformalism[4],weobtainthe massofthe magnetized
−
white dwarf
K 3/2
2 m
M =4π ρ , (3)
c
2πG
(cid:18) (cid:19)
3
and the corresponding radius
πK
m
R= , (4)
2G
r
when ρ is the central density of the white dwarf supplied as a boundary condition in addition to the condition that
c
dρ/dr =0 at r=0, when r is the radial distance from the center of the star such that at the surface r =R, and G is
the Newton’s gravitationconstant. See supplementary information [18] for detailed calculations.
Now the expression of ρ for a one Landau level system in the limit E = E >> m c2 is given by (see
c F Fmax e
supplement [18])
ρ = µemH B3/2. (5)
c √2π2λ3 D
e
Substituting ρ from equation(5) in equation(3), we obtainthe mass of the white dwarf, independent of ρ and B ,
c c D
given by
3/2
hc 1 10.312
M = 2G (µ m )2 ≈ µ2 M⊙, (6)
(cid:18) (cid:19) e H e
and from equations (2), (4) and (5) we obtain the radius
π2/3 hc 1/2
R= ρ−1/3 0 as ρ , (7)
27/3 (µ m )4/3 G c → c →∞
(cid:18) e H (cid:19)
which set the new limits for mass and radius. Note that the Chandrasekhar limit also corresponds to ρ and
→ ∞
R 0. For µ =2 which is the case of a carbon-oxygenwhite dwarf
e
→
M 2.58M⊙. (8)
≈
Interestingly,whilehighmagneticfieldintroduces anisotropiceffects intothe startending itto be anoblatespheroid,
this does not affect the limiting mass as the corresponding radius tends to zero. However, for lighter white dwarfs
with finite radii, super-Chandrasekharmass would havebeen achievedata lowerfield, if the star is appropriatelyset
to be a spheroid rather than a sphere, as justified earlier [3, 20].
Scaling behaviors of mass and radius of the white dwarf with its central density. Now we provide generalscaling
−
laws for the mass and radius of the white dwarf describing their variations with its central density (as is known for
non-magnetized white dwarfs proposed by Chandrasekhar)and magnetic field strength. By Lane-Emden formalism
M K3/2ρ32−nn, (9)
c
∝
where K depends on B and ρ for a magnetized white dwarf (see supplement [18]). For the extremely high density
D c
regime of the white dwarf, the EoS reduces to a polytropic form P = KρΓ (can be verified from supplement [18]),
whenΓ=1+1/n,isthepolytropicindex. Inthisregime,Γ=2,consequentlyn=1andK =K B−1 ρ−2/3,as
m ∝ D ∝ c
shown by equations (I.5), (2), (5). This finally renders M in equation (9) to be independent of ρ , as already shown
c
by equation (6). Similarly, the scaling of radius is obtained as
R K1/2ρ12−nn, (10)
c
∝
which reveals R ρ−1/3 for n=1, as already shown by equation (7).
c
∝
Justification of high magnetic field in white dwarfs. So farwe haveemployedLandauquantizationinsearchofa
−
newmasslimitgivingrisetosuper-Chandrasekharwhitedwarfs. However,theeffectofLandauquantizationbecomes
significant only at a high field B 1013G=B . How can we justify such a high field in a white dwarf?
D cr
∼ ×
Letusconsiderthecommonlyobservedphenomenonofamagnetizedwhitedwarfaccretingmassfromitscompanion.
Nowthesurfacefieldofanaccretingwhitedwarf,asobserved,couldbe 109G<<B [13]. Itscentralfieldhowever
cr
can be several orders of magnitude higher 1012 G, which is also less∼than B . Naturally, such a magnetized CV,
cr
∼
commonly known as a polar, still lies on the mass radius relation obtained by Chandrasekhar. However, in contrast
with Chandrasekhar’s work (which did not include magnetic field in the calculations), we will see that a non-zero
initialfieldinthewhitedwarf,howeverineffectiveforrenderingLandauquantizationeffects,willprovetobecrucialin
supporting the additionalmass accumulateddue to accretion. As the magnetizedwhite dwarfaccretesmass,its total
4
x 105
3
2.5
c
c
g/
270 er 2
1
of 1.5
s
nit
u
n 1
P i
0.5
0
0 5000 10000 15000
ρ in units of 109 gm/cc
FIG. 1: Equations of state — the solid line represents Chandrasekhar’s equation of state (corresponding to zero magnetic
field and hence infinitely many Landau levels), the dotted and dashed lines represent the 5-level (corresponding to very
strong magnetic field) and 1-level (corresponding to ultimate equation of state for extreme magnetic field) systems of Landau
quantization respectively. EFmax=200mec2.
mass increaseswhich in turn increasesthe gravitationalpower andhence the white dwarfcontractsin size due to the
increased gravitational pull. However, the total magnetic flux is conserved in such a process and, hence, as a result
of the above decrease in size of the star, the central (as well as surface) magnetic field also increases. Here we are
interested in the evolution of the central field, since it is this field which is primarily responsible for rendering super-
Chandrasekhar mass to the white dwarf, as justified in [3]. Since accretion is a continuous process, the deposition
of matter on the surface of the white dwarf, followed by its contraction and subsequent increase of magnetic field,
continues in a cycle. In such a process, eventually the central magnetic field could exceed B . As a result, the
cr
EoS of the electron degenerate matter gets modified as shown in Figure 1. Hence, the inward gravitational force is
balanced by the outward force due to this modified pressure and a quasi-equilibrium state is attained. In this way, a
very high magnetic field is generated, which in turn prevents the white dwarf from collapsing, thus making it more
massive. Subsequently, with the continuationof accretionthe white dwarfapproachesthe new mass limit 2.58M⊙,
∼
as obtained above, which sparks off a violent thermonuclear reaction with further accretion, thus exploding it and
giving rise to a super-Chandrasekhar type Ia supernova. The evolution of the mass-radius relation of a polar into
that of a super-Chandrasekhar white dwarf of the maximum possible mass is shown in Figure 2, along with a few
typicalmass-radiusrelationsfordifferentmagneticfieldstrengthsdescribingpossiblestarsinintermediateequilibrium
states. Theultimatewhitedwarf,correspondingtothemaximummass 2.58M⊙,liesonthemass-radiusrelationfor
∼
a one Landaulevelsystem, but the intermediate white dwarfshaving weakermagnetic fields correspondto multilevel
systems. The one Landau level system corresponds to the central magnetic field 8.8 1017 G. The intermediate
15 × 13
systems of 200-leveland 50124-levelcorrespond to central magnetic fields 4.4 10 G and 1.7 10 G respectively
(see supplement [18] for relevant formula), when E = 200m c2. This va×lue of E is f×ound to produce the
Fmax e Fmax
theoreticalmasslimitwithgoodnumericalaccuracy. NotethatonecaninprinciplegouptohigherE ,whichwill
Fmax
lead to a further decrease in the radius of the white dwarf keeping the mass practically same. In order to construct
this evolutionary track, we consider the values of ρ corresponding to the density at the ground-to-firstLandau level
c
transitionofthe respectiveEoSs,sinceonlythis choiceleadsto the maximumpossiblemass. We emphasizeherethat
the range of masses for the super-Chandrasekhar progenitors obtained from observations is not very strict. Hence,
if the new mass limit obtained by us is taken into account, one could possibly do away with the mass distribution
altogether. However, if the distribution is indeed real, then it could be attributed to the difference in accretion rates
found in different CVs.
We thank A.R. Raoof TIFR for insightfulsuggestions. B.M. acknowledgespartialsupport throughresearchGrant
No. ISRO/RES/2/367/10-11. U.D. thanks CSIR, India for financial support.
[1] R.A. Scalzo et al.,Astrophys.J. 713, 1073 (2010).
[2] D.A. Howell et al.,Nature443, 308 (2006).
[3] M. Hickenet al.,Astrophys.J. 669, L17 (2007).
[4] M. Yamanakaet al.,Astrophys.J. 707, L118 (2009).
5
102
m101
k
0
0
0
1
of 100
s
nit
u
R in 10−1
10−2
0 0.5 1 1.5 2 2.5 3
M in units of solar mass
FIG. 2: Mass-Radius relations — thepuresolid line representsChandrasekhar’s result and theone marked with filled circles
representstheevolutionarytrackofthewhitedwarfwiththeincreaseofmagneticfield. Thedot-dashed,dottedanddashedlines
represent the white dwarfs with 50124-level, 200-level and 1-level systems of Landau quantization respectively (corresponding
to increasing magnetic fields). EFmax=200mec2.
[5] J.M. Silverman et al.,Mon. Not. R.Astron. Soc. 410, 585 (2011).
[6] S. Taubenberger et al.,Mon. Not.R. Astron. Soc. 412, 2735 (2011).
[7] S. Chandrasekhar, Mon. Not.R. Astron. Soc. 95, 207 (1935).
[8] M.M. Phillips, Astrophys. J. 413, L105 (1993).
[9] G. Goldhaber et al.,Astrophys.J. 558, 359 (2001).
[10] A.G. Riess et al.,Astron. J. 116 1009 (1998).
[11] S.Perlmutter et al.,Astrophys. J. 517, 565 (1999).
[12] I.Hachisu, M. Kato, H.Saio, and H. Nomoto, Astrophys. J. 744, 69 (2012).
[13] G.D. Schmidt et al.,Astrophys.J. 595, 1101 (2003).
[14] K.M. Vanlandingham et al.,Astron. J. 130, 734 (2005).
[15] D.T. Wickramasinghe and L. Ferrario, Pub. Ast.Soc. Pac. 112, 873 (2000).
[16] D.Lai and S.L. Shapiro, Astrophys.J. 383, 745 (1991).
[17] R.K.Pathria, Statistical Mechanics (Butterworth-Heinemann, Oxford,1996).
[18] Seesupplementary material for details of thecalculations.
[19] U.Das and B. Mukhopadhyay,Phys. Rev.D 86, 042001 (2012).
[20] U.Das and B. Mukhopadhyay,Int.J. Mod. Phys. D 21, 1242001 (2012).
[21] A.Kunduand B. Mukhopadhyay,Mod. Phys.Lett. A 27, 1250084 (2012).
[22] A.R.Choudhuri,Astrophysics for Physicists (Cambridge Univ.Press, New York,2010).
6
Supplementary information for the letter
I. BASIC EQUATIONS FOR ONE LANDAU LEVEL SYSTEM OF DEGENERATE ELECTRONS
The Fermi energy level (E ) of a Landau quantized system is given by
F
2 2 2 2 4
E =p (ν) c +m c (1+2νB ), (I.1)
F F e D
when p (ν) is the Fermi momentum of νth Landau level (ν = 0,1,2.....), m the mass of electrons, c the speed of
F e
light, B the magnetic field in units of 4.414 1013G. From the condition that p (ν)2 0, the maximum number of
D F
× ≥
occupied Landau levels is given by [1]
2
EFmax 1
mec2 −
ν = . (I.2)
m (cid:16) 2B(cid:17)
D
For one Landau level system, when only ground Landau level (ν = 0) is occupied, ν = 1. Similarly, for two level
m
system, when ground (ν = 0) and first (ν = 1) levels are occupied, ν = 2, and so on. Hence, for one Landau level
m
system, density (ρ) and pressure (P) of the electron degenerate gas [2] are given by
2
µ m B E µ m
e H D F e H 2
ρ= 1 B E for E >>m c , (I.3)
2π2λ3e s(cid:18)mec2(cid:19) − −→ 2π2mec2λ3e D F F e
B m c2 E E2 m2c4 E + E2 m2c4
P = D e F F − e ln F F − e , (I.4)
4π2λ3e p(mec2)2 − ( pmec2 )!
when µ is the mean molecular weight, m the mass of proton, λ the Compton wavelength of electron. Now
e H e
eliminating E from equations (I.3) and (I.4) for an arbitrary E , we arrive at the equation of state (EoS) for a one
F F
level system given by
m c2 ρ+ Q2+ρ2
P = e ρ Q2+ρ2 Q2ln , (I.5)
2QµemH − ( pQ )!
p
when Q=µ m B /2π2λ3.
e H D e
Followingpreviouswork[3],wenowapproximatetheaboveEoSandEoSsforanyotherν byapolytropicrelation
m
P =KρΓ such that the polytropic index Γ=1+1/n is piecewise constant in different density ranges and K being a
dimensional constant. This will proveto be useful in orderto understandscaling behaviors of mass and radius of the
white dwarf with its central density.
Now in terms of Q, equation (I.3) can be rewritten as
E
F
ρ=Q . (I.6)
m c2
e
In the limit E >>m c2, we note that, ρ>>Q. Thus in this limit, equation (I.5) reduces to
F e
m c2 2ρ
e 2 2
P = ρ Q ln . (I.7)
2Qµ m − Q
e H (cid:18) (cid:26) (cid:27)(cid:19)
The logarithmic term is much smaller than the first term in the above equation and hence by neglecting it we obtain
m c2
e 2
P = ρ , (I.8)
2Qµ m
e H
which corresponds to the polytropic EoS with Γ=2.
7
II. LANE-EMDEN EQUATION AND EXPRESSIONS FOR MASS AND RADIUS
The underlying white dwarf following above EoS obeys the magnetostatic equilibrium condition
1 d B2 B~ B~
P + =F + ·∇ , (II.1)
g
ρdr 8π 4πρ
(cid:18) (cid:19)
whenristheradialdistancefromthecenterofwhitedwarf,B~ themagneticfieldinG,B2 =B~ B~,F thegravitational
g
·
force. This equation is supplemented by the estimate of mass (M) within any r given by
dM
2
=4πr ρ (II.2)
dr
approximating the star to be spherical. The white dwarf in the present context can be considered in the Newtonian
frameworkand the magnetic fieldtherein is nearlyconstantin the regimeofsmaller radii(as justified previously[3]).
Moreover,at a very largedensity, as in the limiting case to be consideredhere, the star becomes so compact as if the
magnetic field remains constant throughout. Hence, taking above facts into consideration and combining equations
(II.1) and (II.2), the equilibrium condition may be read at any r as
1 d r2dP
= 4πGρ, (II.3)
r2dr ρ dr −
(cid:18) (cid:19)
where G is the Newton’s gravitation constant. We now briefly recall the Lane-Emden formalism (see, e.g., [4]). Let
us define
ρ=ρ θn, (II.4)
c
where ρ is the central density of the white dwarf and θ is a dimensionless variable and
c
r =aξ, (II.5)
where ξ is another dimensionless variable and constant a carries the dimension of length defined as
1−n 1/2
(n+1)Kρ n
c
a= . (II.6)
4πG
" #
Thus using equations (II.4), (II.5) and (II.6) along with the polytropic form of EoS, equation (II.3) reduces to
1 d dθ
ξ2 = θn, (II.7)
ξ2dξ dξ −
(cid:18) (cid:19)
which is the famous Lane-Emden equation. Equation (II.7) can be solved for a given n, along with the boundary
conditions
θ(ξ =0)=1 (II.8)
and
dθ
=0. (II.9)
dξ
(cid:18) (cid:19)ξ=0
Note that for n < 5, θ becomes zero for a finite value of ξ, say ξ1, which basically corresponds to the surface of the
white dwarf such that its radius
R=aξ1. (II.10)
Also by combining equations (II.2), (II.4), (II.5) and (II.7) we obtain the mass of the white dwarf
ξ1
M =4πa3ρ ξ2θndξ. (II.11)
c
Z0
[1] D. Lai and S.L. Shapiro, Astrophys.J. 383, 745 (1991).
[2] A. Kunduand B. Mukhopadhyay,Mod. Phys.Lett. A 27, 1250084 (2012).
[3] U. Das and B. Mukhopadhyay,Phys. Rev.D 86, 042001 (2012).
[4] A.R. Choudhuri,Astrophysics for Physicists (Cambridge Univ.Press, New York,2010).