Hot White Dwarf Stars

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1 Bakytzhan A. Zhami K.A. Boshkayev, J.A. Rueda, R. Ruffini Al-Farabi Kazakh National University Faculty of Physics and Technology, Almaty, Kazakhstan Supernovae, Hypernovae and Binary Driven Hypernovae An Adriatic Workshop June 20-30, 2016, ICRANet, Pescara

2 Outline 1 Motivations 2 Equations of Stellar Structure: General Relativistic and Newtonian 3 Equation of State at T 0 and T = 0 4 Some Numerical Calculations 5 Analytic Expression for Mass-Radius Relation 6 Summary and Future Prospects 7 References

3 Motivations 3 / 25 Figure: Mass-radius relation for T = 0 white dwarf stars vs observational data (S.M. Carvalho et al, 2014),(P.-E. Tremblay et al., 2011)

4 Equations of Stellar Stucture 4 / 25 From spherically symmetric metric ds 2 = e ν(r) c 2 dt 2 e λ(r) dr 2 r 2 dθ 2 r 2 sin 2 θdϕ 2, (1) the equations of equilibrium can be written in the TOV form, dν(r) dr dm(r) dr dp(r) dr = 2G 4πr 3 P(r)/c 2 + M(r) [ ] c 2, (2) r 2 1 2GM(r) c 2 r = 4πr 2 E(r) c 2, (3) = 1 dν(r) [E(r) + P(r)]. 2 dr (4)

5 Equations of Stellar Structure 5 / 25 From the Eqs. (2) and (4) the total pressure can be rewritten in the following form dp(r) dr = GM(r)ρ(r) r 2 [ 1 + P(r) ] [ ρ(r)c πr 2 ] [ P(r) M(r)c 2 1 2GM(r) rc 2 (5) The Tolman-Oppenheimer-Volkoff equation completely determines the structure of a spherically symmetric body of isotropic material in equilibrium. ] 1

6 Equations of Stellar Structure 6 / 25 If terms of order 1/c 2 are neglected, the TOV equation becomes the Newtonian hydrostatic equation, dp(r) dr = GM(r)ρ(r) r 2, (6) dm(r) = 4πr 2 ρ(r) (7) dr and dφ(r) = GM(r) dr r 2 (8) used to find the equilibrium structure of a spherically symmetric body of isotropic material when general-relativistic corrections are not important.

7 Equation of State at T 0 7 / 25 The Chandrasekhar EoS is given by E Ch = E N + E e E N = A Z M uc 2 n e, (9) P Ch = P N + P e P e, (10) where A is the average atomic weight, Z is the number of protons, M u = g is the unified atomic mass, c is the speed of light and n e is the electron number density. In general, the electron number density follows from the Fermi-Dirac statistics and is determined by n e = 2 (2π ) 3 0 exp 4πp 2 dp [ Ẽ(p) µe(p) k B T ], (11) + 1 where k B is the Boltzmann constant, µ e is the electron chemical potential without the rest-mass, and Ẽ(p) = c 2 p 2 + m 2 ec 4 m e c 2, with p and m e the electron momentum and rest-mass, respectively.

8 Equation of State at T 0 8 / 25 It is possible to show that can be written in an alternative form as where n e = 8π 2 (2π ) 3 m3 c 3 β 3/2 [ F 1/2 (η, β) + βf 3/2 (η, β) ], (12) F k (η, β) = 0 t k 1 + (β/2)t 1 + e t η dt (13) is the relativistic Fermi-Dirac integral, η = µ e /(k B T ), t = Ẽ(p)/(k BT ) and β = k B T /(m e c 2 ) are degeneracy parameters. Consequently, the total electron pressure for T 0 K is given by P e = 23/2 3π 2 3 m4 ec 5 β 5/2 [ F 3/2 (η, β) + β ] 2 F 5/2(η, β). (14)

9 Equation of State at T = 0 9 / 25 When T = 0 one can write for the number density of the degenerate electron gas the following expression from the Eq. (11) n e = P F e 0 2 (2π ) 3 d 3 p = 8π P F e (2π ) 3 0 p 2 dp = (PF e ) 3 3π 2 3 = (m ec) 3 3π 2 3 x 3 e The total electron energy-density and electron pressure P e = (2π ) 3 = m4 ec 5 8π 2 3 [x e P F e 0 c 2 p 2 c 2 p 2 + m 2 ec 4 4πp2 dp (15) 1 + x 2 e (2x 2 e /3 1) + arcsinh(x e )], (16) where x e = P F e /(m e c) is the dimensionless Fermi momentum.

10 Some Numerical Calculations 10 / P erg cm T 10 4 K T 10 5 K T 10 6 K T 10 7 K T 10 8 K Figure: Total pressure as a function of the mass density in the case of µ = A Z = 2 white dwarf for selected temperatures in the range T = K.

11 Some Numerical Calculations 11 / 25 M M T 10 4 K T 10 5 K T 10 6 K T 10 7 K T 10 8 K Figure: Mass versus radius for µ = 2 white dwarfs at temperatures T = [ 10 4, 10 5, 10 6, 10 7, 10 8 ] K.

12 Some Numerical Calculations 12 / T 10 4 K T 10 5 K T 10 6 K T 10 7 K T 10 8 K Figure: Mass versus radius for µ = 2 white dwarfs at temperatures T = [ 10 4, 10 5, 10 6, 10 7, 10 8 ] K in the range R = km.

13 Some Numerical Calculations 13 / 25 M M T 10 4 K T 10 5 K T 10 6 K T 10 7 K T 10 8 K R 10 3 km Figure: Mass-radius relations of white dwarfs obtained with the Chandrasekhar EoS (dashed lines) for selected finite temperatures from T = 10 4 K to T = 10 8 K and their comparison with the masses and radii of white dwarfs taken from the Sloan Digital Sky Survey Data Release 4 (brown dots).

14 Next, we will consider static and T = 0 temperature white dwarfs.

15 Pressure and Density Relation P [dyne/cm 2 ] ρ [g/cm 3 ] Figure: Pressure and Density Relation for Degenerate Electron Gas.

16 Mass and Parameter of Compactness M M Rg/R [10-2 ] M 1.40 M R g /R [10-2 ] Figure: The solid red is NP, the dashed blue from GR.

17 Surface Gravitational Potential and Radius φ[10 18 cm 2 /s 2 ] φ[10 18 cm 2 /s 2 ] R [km] R [km] Figure: The solid red is NP, the dashed blue from GR.

18 Mass and Density Relation M M M 1.40 M ρ [g/cm 3 ] ρ [g/cm 3 ] Figure: The solid red is NP, the dashed blue from GR.

19 Mass and Radius Relation M M M 1.35 M R [10 3 km] R [10 3 km] Figure: The solid red is NP, the dashed blue from GR.

20 Analytic Expression (AE) for Mass-Radius Relation M M = R a + br + cr 2 + dr 3 + kr 4 (17) Newtonian Physics: a = 6.11 km, b = 0.664, c = km 1, d = km 1, k = General Relativity: a = km, b = 0.665, c = km 1, d = km 2, k = km 3

21 Analytic Expression vs Structure Equations M 0.8 M R [km] M 0.8 M R [km] Figure: The left panel AE vs NSSE and the right panel AE vs GRSSE (the solid red is Analytic Expression, the dashed blue lines from Stellar Structure Equations (see Ref. Carvalho, Marinho, Malheiro, 2015 ))

22 Analytic Expression vs Structure Equations M M Analytic expression (GR) General relativity Newtonian physics Analytic expression (NP) R [km] Figure: Mass and Radius Relation

23 Summary and Future Prospects The main parameters of static WDs have been found both in Newtonian Physics and General Relativity and compared. The importance of GR was shown for massive white dwarfs. The importance of finite temperatures has been shown. The analytic expression for mass-radius relation can be used. One who needs to plot mass-radius relation can use it. Work in progress... In the future we will consider effects of rotation, finite temperatures, general relativity together.

24 References 1 S.M.de Carvalho, M. Rotondo, J.A. Rueda, R. Ruffini, Phys. Rev. C (2014). 2 P.-E. Tremblay, P. Bergeron, A. Gianninas, Astrophys. J (2011). 3 G. Carvalho, R. Marinho, M. Malheiro, AIP Conference Proceedings, , (2015). 4 Landau L. D., & Lifshitz, E. M., The Classical Theory of Fields (Butterworth-Heinemann Press, Oxford, 1975). 5 Landau L. D., & Lifshitz, E. M., Statistical Physics (Pergamon Press, Oxford, 1980). 6 K. Boshkayev, J. Rueda, B. Zhami, G. Balgimbekov, Zh. Kalymova, IJMPh: CS, 41, (2016). 7 K. Boshkayev, B. Zhami et al., NAS RK, 307, 49 (2016).

25 THANK YOU FOR ATTENTION! Back to the title

Polytropic Stars. c 2

Polytropic Stars. c 2 PH217: Aug-Dec 23 1 Polytropic Stars Stars are self gravitating globes of gas in kept in hyostatic equilibrium by internal pressure support. The hyostatic equilibrium condition, as mentioned earlier, is: