Strongly magnetized white dwarf

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Gavin Potter
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1 IUCAA, PUNE Supervisor : Prof. Dipankar Bhattacharya December 16, 2014 Science with LAXPC/ASTROSAT

2 White Dwarfs Origin: Low / intermediate mass (< 10M ) stellar remnants Components: Carbon-Oxygen (or Helium) Stability: Electron degeneracy pressure provides support against Gravity Density: High (typically 1M squeezed within the volume of earth) M ch [de Carvalho et al. 2014]

3 Magnetic WDs Probable origin : Magnetic ux freezing in the stellar evolution processes [Ruderman, 1972] Possible dynamo processes in common envelope phase in binary systems [Potter and Tout, 2010] [Wickramasinghe and Ferrario, 2005]

4 WD : SNIa progenitor By accreting from binary companion, WD crosses M Ch. Rapid contraction triggers thermonuclear explosion. SNIa characteristic light curve (standard candle) is used to calibrate the distance of galaxies

5 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]

6 Eects of magnetization Lorentz force in the hydrostatic equilibrium Modication of EoS due to Landau quantization [Pathria : Statistical Mechanics]

7 Quantized EoS: Phase space integral B=0 B 0 2 h d 3 p = 1 ( ) 2 ( ) p 3 π 2 λ 3 e me c d p me c ν 2eB g h 2 ν dpz = 2β ( (2π) 2 λ ν 3 gν d e x 3 2β F ρ = µem νm H (2π) 2 λ 3 e ν=0 g νxe(ν) ] 1 + x 2 3 ( F sinh 1 2βme c2 x F P = νm (2π) 2 λ 3 e ν=0 g ν(1 + 2νβ)η P = ρ EF µe m P H = ρ ( ) EF µe m H + P β β E F Mass density ρ = µem H 1 [ 3π 2 λ 3 e Pressure P = πm4 c 5 x 3h 3 F (2x 2 3) F Pressure Gradient pz me c xe(ν) 1+2νβ ) ) ρ(β) ρ(0) quantized energy Level (ν) E F =2m e c E F =10m e c β = B B c ; B c = T E F =2m e c 2 E F =10m e c 2 ( P) β ɛ F ( P ) ɛ F β ν ( β)

8 Equilibrium structure stellar structure eq.s The hydrostatic force( balance eq. : j ) B 1 ρ P = Φg + 1 ρ Poisson equation: 2 Φ g = 4πGρ Maxwell equation (with σ ):. B = 0 and B = µ o j EoS: P = P(ρ) or P = P(ρ, B ) virial condition: 3Π + W + M = 0 where, Π = PdV ; W = ρφ g dv and M = B 2 2µ 0 dv

9 Assumptions and method T T F stationary ( t 0) axisymmetric, i.e. φ 0. non-rotating. The source of the magnetic eld (i.e. the current distribution) is conned within the white dwarf. σ HSCF : integral formalism (Hachisu, 1986; Tomimura & Eriguchi, 2005) 1 µ e m H E F + Φ g = M(u) + C here, u = A φ.r sin θ, Φ g ( r ) = G ρ( r ) r r d 3 r, A φ ( r ) sin φ = µ 0 4π jφ ( r ) sin φ r r d 3 r.

10 Equilibrium conguration B core =1 ( )T; E Fmax =6.1m e c z/r eq ξ/r eq P=P(E F ) P=P(E F,B) β ρ/ρ max θ =0 θ =π/4 a) θ =π/2 b) 7 B core =1 ( )T; E Fmax =6.1m e c 2 VC ɛ 4 56 F 3 2 c) 1 r/r eq Number of Grid points

11 Mass-radius relation 16 B dependent Mass-Radius relation 2.0 B core =1 ( )T; E Fmax =6.1m e c Radius ( 10 6 m) M/M ρ c (kg.m 3 ) B core =0 B core =1 B c 1.5 R/R 10 2 z/r eq ξ/r eq 4 B core =6 B c B core =1 B c B core =10 B c B core =100 B c B core =1000 B c Mass (M/M ) [Bera & Bhattacharya, 2014]

12 Mass-radius relation Newtonian GR : Das & Mukhopadhyay, B dependent Mass-Radius relation M/W= M/W=0.1 Radius ( 10 6 m) B core =0 B core =1 B c B core =6 B c B core =1 B c B core =10 B c B core =100 B c 0 B core =1000 B c Mass (M/M )

13 Additional mass 50 B core =1 B c % of additional mass B core =6 B c B core =1 B c B core =10 B c B core =100 B c B core =1000 B c Radius ( 10 6 m) B dependent Mass-Radius relation 2.0 B core=0 B core=1 B c B core=6 B c B core=1 B c B core=10 B c B core=100 B c M/M ρ c (kg.m 3 ) B core=1000 B c Mass (M/M ) R/R M/ W f m.f g f g 2 a) θ =π/2 b) θ =π/2 E Fmax =6.1m e c 2 E Fmax =7.0m e c 2 Fmax e E =8.0m c 2 r/r eq r/r eq ρ/ρ max ρ/ρ max c) θ =0 r/r p

14 E ects of Landau quantized EoS EFmax =6.1me c EoS P = P (ρ) Bcore = T P = P (ρ, B ) EFmax =59me c P = P (ρ) Bcore = T P = P (ρ, B ) ρc δ(ρ/ρmax) M/M b) 0 θ =0 θ = π/2 log10(ρρ10 ) ξ/req M /W e-03 z/req Bcore =1 ( )T ; EFmax =6.1me c2 a) Req /10 m Rp /Req δ(ρ/ρmax) central conditions -4.76e-03 θ =0 θ = π/2 r/req c) r/rθ VC

15 Stability? Congurations may be prone to several dynamical instabilities Typical time scales: τ Alfven 0.1 s τ viscous s r τ ohmic ( RWD ) s

16 Observables in Xray Gravitational Redshifted line emission from stellar surface. 6.4 kev Fe line shift [in ev] M/W= M/W= Mass (M ) Magnetic characteristics can be inferred from post-shock accretion column/ cyclotron line for objects like polar.

17 Summary WD can support a larger mass in the presence of a strong magnetic eld. (additional mass upto 0.5M, when M /W 13%) At the maximum strength of the magnetic eld, the impact of Landau quantization on the stellar structure is not signicant. Existence of such object can be veried from X-ray observations.

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

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