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Strongly magnetized white dwarf
Prasanta Bera
IUCAA, PUNE
Supervisor : Prof. Dipankar Bhattacharya
December 16, 2014Science with LAXPC/ASTROSAT
Prasanta Bera Strongly magnetized white dwarf
White Dwarfs
Origin: Low / intermediate mass (< 10M�) stellar remnantsComponents: Carbon-Oxygen (or Helium)Stability: Electron degeneracy pressure provides supportagainst GravityDensity: High (typically 1M� squeezed within ∼ the volumeof earth)
Mch →
[de Carvalho et al. 2014]
Prasanta Bera Strongly magnetized white dwarf
Magnetic WDs
Probable origin :
Magnetic �ux freezing in the stellar evolution processes[Ruderman, 1972]
Possible dynamo processes in common envelope phase inbinary systems [Potter and Tout, 2010]
[Wickramasinghe and Ferrario, 2005]Prasanta Bera Strongly magnetized white dwarf
WD : SNIa progenitor
By accreting from binary companion, WD crosses & MCh.Rapid contraction triggers thermonuclear explosion.
SNIa characteristic light curve (standard candle) is used tocalibrate the distance of galaxies
Prasanta Bera Strongly magnetized white dwarf
Super-Chandrasekhar mass WD
A few SNIa (e.g. 2003fg) are more luminous than usual;suggests → white dwarf with M > 2M�Possibilities:
Double degenerate [Moll et al., 2014]
Single degenerate
rapid rotation
electrically charged WD [Liu et al., 2014]
strong internal magnetic �eld [Das and Mukhopadhyay, 2012]
Prasanta Bera Strongly magnetized white dwarf
E�ects of magnetization
Lorentz force in the hydrostatic equilibrium
Modi�cation of EoS due to Landau quantization
[Pathria : Statistical Mechanics]
Prasanta Bera Strongly magnetized white dwarf
Quantized EoS:
B=0 B 6=0
Phase space integral 2h3
∫d3p = 1
π2λ3e
∫ (p
mec
)2d(
pmec
) ∑ν2eBh2gν∫dpz = 2β
(2π)2λ3e
∑ν gν
∫d(
pzmec
)Mass density ρ = µemH
13π2λ3e
x3F ρ = µemH2β
(2π)2λ3e
∑νmν=0 gνxe(ν)
Pressure P = πm4c5
3h3
[xF (2x2F − 3)
√1 + x2F − 3 sinh−1 xF
]P = 2βmec
2
(2π)2λ3e
∑νmν=0 gν(1 + 2νβ)η
(xe(ν)1+2νβ
)Pressure Gradient
−→∇P = ρ
µemH
−→∇EF
−→∇P = ρ
µemH
−→∇EF +
(∂P∂β
)EF
−→∇β
0.8
1.0
1.2
1.4
1.6
ρ(β)
ρ(0)
EF =2me c2
EF =10me c2
0.0 0.5 1.0 1.5 2.01ν (∝β)
0.0
0.2
0.4
0.6
0.8
( Pβ)εF
( PεF
)β
EF =2me c2
EF =10me c2
10010 5 3 2 1quantized energy Level (ν)
β = B
Bc
; Bc = 4.4× 109T
Prasanta Bera Strongly magnetized white dwarf
Equilibrium structure
stellar structure eq.s
The hydrostatic force balance eq. :1ρ
−→∇P = −
−→∇Φg + 1
ρ
(−→j ×−→B)
Poisson equation:∇2Φg = 4πGρ
Maxwell equation (with σ →∞):−→∇ .−→B = 0
and−→∇ ×
−→B = µo
−→j
EoS:
P = P(ρ) or P = P(ρ, |−→B |)
virial condition:
3Π + W + M = 0where, Π =
∫PdV ; W =
∫ρΦgdV and M =
∫B2
2µ0dV
Prasanta Bera Strongly magnetized white dwarf
Assumptions and method
T � TF
stationary ( ∂∂t → 0)
axisymmetric, i.e. ∂∂φ → 0.
non-rotating.
The source of the magnetic �eld (i.e. the current distribution)is con�ned within the white dwarf.
σ →∞
HSCF : integral formalism (Hachisu, 1986; Tomimura & Eriguchi, 2005)
1
µemHEF + Φg = M(u) + C
here, u = Aφ.r sin θ, Φg (−→r ) = −G∫ ρ(−→r ′)|−→r −−→r ′|d
3−→r ′,
Aφ(−→r ) sinφ = µ04π
∫ jφ(−→r ′) sinφ′
|−→r −−→r ′| d3−→r ′.
Prasanta Bera Strongly magnetized white dwarf
Equilibrium con�guration
0.0 0.5 1.0 1.5ξ/Req
0.5
1.0
1.5
z/R
eq
0.00
0.12
0.24
0.36
0.48
0.60
0.72
0.84
0.96
Bcore=1×(4.4×109 )T ; EFmax=6.1me c
2
103 104 105 106
Number of Grid points
10-6
10-5
10-4
10-3
|VC
|
P=P(EF )
P=P(EF ,B)
0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
β
a)
θ=0
θ=π/4
θ=π/2
0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
ρ/ρ
max
b)
0.0 0.2 0.4 0.6 0.8 1.0r/Req
1234567
εF
c)
Bcore =1×(4.4×109 )T ; EFmax=6.1me c
2
Prasanta Bera Strongly magnetized white dwarf
Mass-radius relation
0.0 0.5 1.0 1.5 2.0
Mass (M/M¯)
0
2
4
6
8
10
12
14
16
Radiu
s (×
106
m)
Bcore=0
Bcore=0.01×Bc
Bcore=0.06×Bc
Bcore=1×Bc
Bcore=10×Bc
Bcore=100×Bc
Bcore=1000×Bc
0.0
0.5
1.0
1.5
2.0
R/R
¯×1
0−2
B dependent Mass-Radius relation
108 109 1010 1011 1012 1013 1014 1015 1016 1017
ρc (kg.m−3 )
0.0
0.5
1.0
1.5
2.0
M/M
¯
0.0 0.5 1.0 1.5ξ/Req
0.5
1.0
1.5
z/R
eq
0.00
0.12
0.24
0.36
0.48
0.60
0.72
0.84
0.96
Bcore=1×(4.4×109 )T ; EFmax=6.1me c
2
[Bera & Bhattacharya, 2014]
Prasanta Bera Strongly magnetized white dwarf
Mass-radius relation
0.0 0.5 1.0 1.5 2.0
Mass (M/M¯)
0
2
4
6
8
10
12
14
16
Radiu
s (×
106
m)
Bcore=0
Bcore=0.01×Bc
Bcore=0.06×Bc
Bcore=1×Bc
Bcore=10×Bc
Bcore=100×Bc
Bcore=1000×Bc
B dependent Mass-Radius relationM/W=0.0
M/W=0.1
NewtonianGR : Das & Mukhopadhyay, 2014
Prasanta Bera Strongly magnetized white dwarf
Additional mass
0.0 0.2 0.4 0.6 0.8 1.0r/Req
1.00.80.60.40.20.00.20.4
fm .fg
|fg |2
a) θ=π/2
EFmax=6.1me c
2
EFmax=7.0me c
2
EFmax=8.0me c
2
0.0 0.2 0.4 0.6 0.8 1.0r/Req
0.0
0.2
0.4
0.6
0.8
1.0
ρ/ρ
max
b) θ=π/2
0.0 0.2 0.4 0.6 0.8 1.0r/Rp
0.0
0.2
0.4
0.6
0.8
1.0
ρ/ρ
max
c) θ=0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
M/|W|
10
0
10
20
30
40
50%
of
addit
ional m
ass
Bcore=0.01×Bc
Bcore=0.06×Bc
Bcore=1×Bc
Bcore=10×Bc
Bcore=100×Bc
Bcore=1000×Bc
0.0 0.5 1.0 1.5 2.0
Mass (M/M¯)
0
2
4
6
8
10
12
14
16
Radiu
s (×
106
m)
Bcore=0
Bcore=0.01×Bc
Bcore=0.06×Bc
Bcore=1×Bc
Bcore=10×Bc
Bcore=100×Bc
Bcore=1000×Bc
0.0
0.5
1.0
1.5
2.0
R/R
¯×1
0−2
B dependent Mass-Radius relation
108 109 1010 1011 1012 1013 1014 1015 1016 1017
ρc (kg.m−3 )
0.0
0.5
1.0
1.5
2.0
M/M
¯
Prasanta Bera Strongly magnetized white dwarf
E�ects of Landau quantized EoS
central conditions EoS ρc M/M� Req/106m Rp/Req |M /W | | VC |
EFmax=6.1mec2 P = P(ρ) 145.2618 1.7496 2.8020 0.665 0.1247 5.4088× 10−6
Bcore = 4.414× 109 T P = P(ρ,B) 145.3913 1.7506 2.8057 0.673 0.1245 3.1047× 10−5
EFmax=59mec2 P = P(ρ) 136860.3 1.8995 0.3338 0.680 0.1295 9.3629× 10−06
Bcore = 4.414× 1011 T P = P(ρ,B) 137128.2 1.9008 0.3411 0.705 0.1289 1.9038× 10−05
0.0 0.2 0.4 0.6 0.8 1.0ξ/Req
0.0
0.2
0.4
0.6
0.8
1.0
z/R
eq
a)δ(ρ/ρmax)
-4.76e-03
0
1.45e-03
0.0 0.2 0.4 0.6 0.8 1.0r/Req
δ(ρ/ρmax)
b)
θ=0
θ=π/2
0.0 0.2 0.4 0.6 0.8 1.0r/Rθ
0.2
0.0
0.2
0.4
0.6
0.8
1.0
log 1
0(ρ1
ρ0)
c)θ=0
θ=π/2
Bcore=1×(4.4×109 )T ; EFmax=6.1me c
2
Prasanta Bera Strongly magnetized white dwarf
Stability ?
Con�gurations may be prone to several dynamical instabilities
Typical time scales:
τAlfven ∼ 0.1 s
τviscous ∼ 1017 s
τohmic ∼ ( r
RWD)21018 s
Prasanta Bera Strongly magnetized white dwarf
Observables in Xray
Gravitational Redshifted line emission from stellar surface.
0.8 1.0 1.2 1.4 1.6 1.8
Mass (M¯)
0
5
10
15
206.4
keV
Fe lin
e s
hift
[in e
V]
M/W=0.0
M/W=0.1
Magnetic characteristics can be inferred from post-shockaccretion column/ cyclotron line for objects like polar.
Prasanta Bera Strongly magnetized white dwarf
Summary
WD can support a larger mass in the presence of astrong magnetic �eld.(additional mass upto ∼ 0.5M�, when M /W ∼ 13%)
At the maximum strength of the magnetic �eld, theimpact of Landau quantization on the stellar structure isnot signi�cant.
Existence of such object can be veri�ed from X-rayobservations.
Prasanta Bera Strongly magnetized white dwarf