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

Prasanta Bera Strongly magnetized white dwarf

Prasanta Bera Strongly magnetized white dwarf

Wickramasinghe, D. T. and Ferrario, L. (2005).The origin of the magnetic �elds in white dwarfs.MNRAS, 356:1576�1582.

Prasanta Bera Strongly magnetized white dwarf