Hot White Dwarf Stars - ICRANet · PDF fileHot White Dwarf Stars Bakytzhan A. Zhami K.A. Boshkayev, J.A. Rueda, R. Ru ni Al-Farabi Kazakh National University Faculty of Physics and

Hot White Dwarf Stars

Bakytzhan A. ZhamiK.A. Boshkayev, J.A. Rueda, R. Ruffini

Al-Farabi Kazakh National UniversityFaculty of Physics and Technology, Almaty, Kazakhstan

Supernovae, Hypernovae and Binary Driven HypernovaeAn Adriatic Workshop

June 20-30, 2016, ICRANet, Pescara

Bakytzhan Zhami et al., An Adriatic Workshop, ICRANet Hot White Dwarf Stars

Outline

1 Motivations

2 Equations of Stellar Structure: General Relativistic andNewtonian

3 Equation of State at T 6= 0 and T = 0

4 Some Numerical Calculations

5 Analytic Expression for Mass-Radius Relation

6 Summary and Future Prospects

7 References


Motivations 3 / 25

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


Equations of Stellar Stucture 4 / 25

From spherically symmetric metric

ds2 = eν(r)c2dt2 − eλ(r)dr2 − r2dθ2 − r2 sin2 θdϕ2 , (1)

the equations of equilibrium can be written in the TOV form,

dν(r)

dr=

2G

c24πr3P(r)/c2 + M(r)

r2[1− 2GM(r)

c2r

] , (2)

dM(r)

dr= 4πr2

E(r)

c2, (3)

dP(r)

dr= −1

2

dν(r)

dr[E(r) + P(r)] . (4)


Equations of Stellar Structure 5 / 25

From the Eqs. (2) and (4) the total pressure can be rewritten inthe following form

dP(r)

dr= −GM(r)ρ(r)

r2

[1 +

P(r)

ρ(r)c2

] [1 +

4πr2P(r)

M(r)c2

] [1− 2GM(r)

rc2

]−1(5)

The Tolman-Oppenheimer-Volkoff equation completely determinesthe structure of a spherically symmetric body of isotropic materialin equilibrium.


Equations of Stellar Structure 6 / 25

If terms of order 1/c2 are neglected, the TOV equation becomesthe Newtonian hydrostatic equation,

dP(r)

dr= −GM(r)ρ(r)

r2, (6)

dM(r)

dr= 4πr2ρ(r) (7)

anddΦ(r)

dr=

GM(r)

r2(8)

used to find the equilibrium structure of a spherically symmetricbody of isotropic material when general-relativistic corrections arenot important.


Equation of State at T 6= 0 7 / 25

The Chandrasekhar EoS is given by

ECh = EN + Ee ≈ EN =A

ZMuc

2ne , (9)

PCh = PN + Pe ≈ Pe , (10)

where A is the average atomic weight, Z is the number of protons,Mu = 1.6604× 10−24 g is the unified atomic mass, c is the speedof light and ne is the electron number density. In general, theelectron number density follows from the Fermi-Dirac statistics andis determined by

ne =2

(2π~)3

∫ ∞0

4πp2dp

exp[E(p)−µe(p)

kBT

]+ 1

, (11)

where kB is the Boltzmann constant, µe is the electron chemicalpotential without the rest-mass, andE (p) =

√c2p2 + m2

ec4 −mec

2, with p and me the electronmomentum and rest-mass, respectively.


Equation of State at T 6= 0 8 / 25

It is possible to show that can be written in an alternative form as

ne =8π√

2

(2π~)3m3c3β3/2

[F1/2(η, β) + βF3/2(η, β)

], (12)

where

Fk(η, β) =

∫ ∞0

tk√

1 + (β/2)t

1 + et−ηdt (13)

is the relativistic Fermi-Dirac integral, η = µe/(kBT ),t = E (p)/(kBT ) and β = kBT/(mec

2) are degeneracy parameters.Consequently, the total electron pressure for T 6= 0 K is given by

Pe =23/2

3π2~3m4

ec5β5/2

[F3/2(η, β) +

β

2F5/2(η, β)

]. (14)


Equation of State at T = 0 9 / 25

When T = 0 one can write for the number density of thedegenerate electron gas the following expression from the Eq. (11)

ne =

∫ PFe

0

2

(2π~)3d3p =

8π

(2π~)3

∫ PFe

0p2dp =

(PFe )3

3π2~3=

(mec)3

3π2~3x3e

(15)The total electron energy-density and electron pressure

Pe =1

3

2

(2π~)3

∫ PFe

0

c2p2√c2p2 + m2

ec4

4πp2dp

=m4

ec5

8π2~3[xe

√1 + x2e (2x2e /3− 1) + arcsinh(xe)] , (16)

where xe = PFe /(mec) is the dimensionless Fermi momentum.


Some Numerical Calculations 10 / 25

1 10 100 1000 104 105 1061012

1014

1016

1018

1020

1022

Ρ @ g�cm3D

P@e

rg�c

m3 D

T=108

K

T=107

K

T=106

K

T=105

K

T=104

K

Figure: Total pressure as a function of the mass density in the caseof µ = A

Z = 2 white dwarf for selected temperatures in the rangeT = 104 − 108 K.



0 5 10 15 20

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R @103 kmD

M

M�

T=108

K

T=107

K

T=106

K

T=105

K

T=104

K

Figure: Mass versus radius for µ = 2 white dwarfs at temperaturesT = [ 104, 105, 106, 107, 108] K.



10 1005020 3015 70

0.05

0.10

0.20

0.50

1.00

R @103 kmD

M

M�

T=108

K

T=107

K

T=106

K

T=105

K

T=104

K

Figure: Mass versus radius for µ = 2 white dwarfs at temperaturesT = [ 104, 105, 106, 107, 108] K in the range R = 10− 100× 103 km.



T=108

K

T=107

K

T=106

K

T=105

K

T=104

K

0 5 10 15 20

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R @103 kmD

M

M�

Figure: Mass-radius relations of white dwarfs obtained with theChandrasekhar EoS (dashed lines) for selected finite temperatures fromT = 104 K to T = 108 K and their comparison with the masses and radiiof white dwarfs taken from the Sloan Digital Sky Survey Data Release 4

(brown dots).Bakytzhan Zhami et al., An Adriatic Workshop, ICRANet Hot White Dwarf Stars

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


Pressure and Density Relation

0.01 10 104 107 1010 1013

1012

1017

1022

1027

1032

ρ [g/cm3]

P[dyne/cm

2]

Figure: Pressure and Density Relation for Degenerate Electron Gas.


Mass and Parameter of Compactness

0.001 0.010 0.100 10.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Rg/R [10-2]

M

M⊙

0.6 0.8 1.0 1.2 1.41.30

1.35

1.40

1.45

1.50

Rg /R [10-2]

M

M⊙

Figure: The solid red is NP, the dashed blue from GR.


Surface Gravitational Potential and Radius

500 1000 5000 1040

1

2

3

4

5

6

R [km]

φ[1018cm

2/s2]

300 400 500 600 7003.0

3.5

4.0

4.5

5.0

5.5

6.0

R [km]φ[1018 cm2 /s2 ]



Mass and Density Relation

104 106 108 1010 10120.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

ρ [g/cm3]

M

M⊙

109 1010 1011 10121.30

1.35

1.40

1.45

1.50

ρ [g/cm3]

M

M⊙



Mass and Radius Relation

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R [103 km]

M

M⊙

0.5 1.0 1.5 2.0 2.5 3.01.20

1.25

1.30

1.35

1.40

1.45

1.50

R [103 km]

M

M⊙



Analytic Expression (AE) for Mass-Radius Relation

M

M�=

R

a + bR + cR2 + dR3 + kR4(17)

Newtonian Physics:a = 6.11 km, b = 0.664,c = 2.2610−5km−1, d = −1.5010−9km−1, k = 1.3510−12

General Relativity:a = 14.65 km, b = 0.665,c = 2.1710−5km−1, d = −1.3810−9km−2, k = 1.3410−12km−3


Analytic Expression vs Structure Equations

0 5000 10000 15000 20 000 250000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R [km]

M

M⊙

0 5000 10000 15000 20000 250000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R [km]

M

M⊙

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 ))


Analytic Expression vs Structure Equations

Analytic expression (GR) General relativityNewtonian physics Analytic expression (NP)

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R [km]

M

M⊙

Figure: Mass and Radius Relation


Summary and Future Prospects

The main parameters of static WDs have been found both inNewtonian Physics and General Relativity and compared. Theimportance 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, finitetemperatures, general relativity together.


References

1 S.M.de Carvalho, M. Rotondo, J.A. Rueda, R. Ruffini, Phys.Rev. C. 89. 015801 (2014).

2 P.-E. Tremblay, P. Bergeron, A. Gianninas, Astrophys. J. 730.128 (2011).

3 G. Carvalho, R. Marinho, M. Malheiro, AIP ConferenceProceedings, 1693. 030004-1, (2015).

4 Landau L. D., & Lifshitz, E. M., The Classical Theory ofFields (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, 1660129 (2016).

7 K. Boshkayev, B. Zhami et al., NAS RK, 307, 49 (2016).


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Hot White Dwarf Stars - ICRANet · PDF fileHot White Dwarf Stars Bakytzhan A. Zhami K.A. Boshkayev, J.A. Rueda, R. Ru ni Al-Farabi Kazakh National University Faculty of Physics and