Some characteristics of plasma in a white dwarf
Vanea COVLEA*, Alexandru JIPA, Marius CĂLIN*, Oana RISTEA, Cătălin RISTEA, Tiberiu EȘANU, Călin BEȘLIU, Ionel LAZANU
Atomic and Nuclear Chair, Department of Stucture of Matter, Atmospheric and Earth Physics, Astrophysics, Faculty of Physics, University of Bucharest,
Romania
* [email protected], *[email protected]
Introduction
The end point in the evolution of main-sequence stars with an initial mass of (1 - 8)M consists of a
degenerate core after the ejection of significant amount of matter in the form of a planetary nebula.
The core will predominantly be made of helium or carbon. In such a remnant, the pressure support
in the core is provided by an ideal (non-interacting) gas of degenerate electrons, whereas most of
the mass density is contributed by a non-degenerate gas of carbon or helium ions. The Fermi
energy of the electrons in such a system is higher than the kinetic temperature of the system and
hence the electrons can be taken to be a zero-temperature Fermi gas.
We have in such a white dwarf a degenerate electrons plasma. We try to calculate some specific
parameters of plasma such as: Debye length, Langmuir frequency, Debye number, etc. Our
calculation is based on white dwarfs with a mass between (0,8 - 1,4) M  a temperature from
10,000K to 30,000K and a core made of carbon (µe = 2).

M M
6
9
3
36
-3
18 -1
R(x10 m) ρ (x10 kg/m ) ne (x10 m ) ωp (x10 s )
0.8
0.9
1
1.1
1.2
1.3
1.4
6.94
6.181
5.421
4.641
3.802
2.824
1.413
1.42
2.01
2.979
4.747
8.639
21.081
168.05
0.4244
0.6008
0.8905
1.419
2.5824
6.3012
50.2212
Te (K)
2
R  0.0126 R  
 e 
53
M 
 
 M 
with the Chandrasekhar’s mass
Debye length
Debye number
Results
Langmuir frequency
Electron mean free path
Coulomb effective cross section
1 3
  M 

1  
  M Ch 
M Ch



 2
 1.435M  

 e
1
lD 
e
ND
43 12
 0 k BTe
n




2
λD (x10-14 m) ND (x10-6)
M/MO
0.8
0.9
1
1.1
1.2
1.3
1.4
lmfp
Te = 20,000K
λD (x10-14 m) ND (x10-6)
M/MO
1.0593
0.8903
0.7313
0.5793
0.4294
0.2749
0.0973
4l3D
n
3
p 
Te = 15,000K
2.1132
1.7759
1.4588
1.1556
0.8566
0.5483
0.1942
0.8
0.9
1
1.1
1.2
1.3
1.4
Te = 25,000K
λD (x10-14 m) ND (x10-6)
3.878
3.262
2.642
2.1184
1.573
1.0074
0.3562
0.8
0.9
1
1.1
1.2
1.3
1.4
M/MO
1.4981
1.2591
1.0342
0.8192
0.6073
0.3887
0.1377
σ (x10 m )
3,898
17,325
0,9745
0,6237
0,4331
Te = 30,000K
λD (x10-14 m) ND (x10-6)
M/MO
1.297
1.0904
0.895
0.709
0.5259
0.3367
0.1192
5.977
5.0234
4.126
3.2676
2.4228
1.5543
0.5492
0.8
0.9
1
1.1
1.2
1.3
1.4
λD (x10-14 m) ND (x10-6)
M/MO
1.6749
1.4077
1.1562
0.9159
0.679
0.4346
0.1539
8.3527
7.0201
5.7652
4.5668
3.3862
2.1666
0.7668
0.8
0.9
1
1.1
1.2
1.3
1.4
1.8348
1.542
1.2666
1.0034
0.7438
0.4761
0.1686
10.9807
9.2272
7.5795
6.0047
4.4512
2.8484
1.0082
Debye length and Debye number at different temperatures of a white dwarf for different mass ratio
Te=10,000K
ne 2
me 0
1

n
  r
2
C
M/Mo
lmfp(x10-19 m)
0.8
0.9
1
1.1
1.2
1.3
1.4
60,448
42,700
28,824
18,079
0,9935
0,4071
0,0510
Te=15,000K
lmfp(x10-19 m)
13,600
96,072
64,853
40,676
22,354
0,9160
0,1149
Te=20,000K
Te=25,000K
Te=30,000K
lmfp(x10-19 m)
lmfp(x10-19 m)
lmfp(x10-19 m)
241,792
170,800
115,299
72,316
39,743
16,285
0,2043
377,754
266,866
180,149
112,990
62,096
25,445
0,3192
544,047
384,310
259,430
162,715
89,424
36,643
0,4597
The electron mean free path for white dwarf plasma at different temperatures
Debye length vs. mass ratio
Some conclusion
For a white dwarf can be calculated the plasma
parameters, taken into account the specific
properties of such dense matter.
From these preliminary calculations, the
degenerate electron plasma of a white dwarf is a
cold, extreme dense and strong coupled plasma.
Debye number vs. mass ratio
2
The values of the Coulomb effective cross section for electron in
white dwarf plasma
Data analysis
Te = 10,000K
-18
rC (x10 m)
11,139
74,263
55,697
44,558
37,131
10,000
15,000
20,000
25,000
30,000
3.6855
4.3852
5.3374
6.739
9.0901
14.2005
40.0915
Radius, density of white dwarf, number density of electrons and Langmuir frequency
The mass - radius relation for white dwarfs may be estimated using the
usual algebraic approximation to the differential equations of stellar
structure and an analytical approximation to the equation of state for
degenerate electron gas.
-10
Langmuir frequency vs. mass ratio
The electron mean free path at different
temperatures
Bibliography
1. T. Padmanabhan – Theoretical Astrophysics, Volume II: Stars and Stellar Systems,
Cambridge University Press, 2000
2. Arnab Rai Choudhuri – Astrophysics for Physicist, Cambridge University Press,
2010
3. Malcolm S. Longair – High Energy Astrophysics, 3rd Edition, Cambridge University
Press, 2011
4. S. Chandrasekhar – An Introduction to the Study of Stellar Structure, The University
of Chicago Press, 1938
Acknowledgment: This work was supported by a grant: PN-II-ID-PCE 34/05.10.2011
of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI