1 Degenerate Fermi gas systems: white dwarf and neutron star (pulsar) Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: October 31, 2018) Subrahmanyan Chandrasekhar, FRS (October 19, 1910 – August 21, 1995) was an Indian origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars. Chandrasekhar was the nephew of Sir Chandrasekhara Venkata Raman, who won the Nobel Prize for Physics in 1930. Chandrasekhar served on the University of Chicago faculty from 1937 until his death in 1995 at the age of 84. He became a naturalized citizen of the United States in 1953. http://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar In 1930, Subramanyan Chandrasekhar, then 19 years old, was on a sea voyage from India to Cambridge, England, where he planned to begin graduate work. Chandrasekhar was interested in exploring the consequences of quantum mechanics for astrophysics. During his trip, he analyzed how the density, pressure, and gravity in a white dwarf star vary with radius. For a star like Sirius B Chandrasekhar found that the Fermi velocity of inner electrons approaches the speed of light. Consequently he found it necessary to redo the calculation of the Fermi energy taking relativistic effects into account. Chandrasekhar deduced that a high-density, high mass star cannot support itself against gravitational collapse unless the mass of the star is less than 1.4 solar masses. This finding was quite controversial within the astronomical community and it was 54 years before Chandrasekhar was awarded the Nobel Prize for this work. 1. Overview
27
Embed
White dwarf and neutron star revised 10-31-18bingweb.binghamton.edu/~suzuki/ThermoStatFIles/11.7 FD White dw… · white dwarf. White dwarfs are the end state for most stars, and
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Degenerate Fermi gas systems: white dwarf and neutron star (pulsar)
Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton
(Date: October 31, 2018)
Subrahmanyan Chandrasekhar, FRS (October 19, 1910 – August 21, 1995) was an Indian
origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for
Physics for key discoveries that led to the currently accepted theory on the later evolutionary
stages of massive stars. Chandrasekhar was the nephew of Sir Chandrasekhara Venkata Raman,
who won the Nobel Prize for Physics in 1930. Chandrasekhar served on the University of Chicago
faculty from 1937 until his death in 1995 at the age of 84. He became a naturalized citizen of the
(c) Companion of Sirius: first white dwarf (Sirius B)
M = 2.0 x 1030 kg (≈ the mass of sun)
R = 6.0 x 106 m (a little shorter than the Earth)
(d) Crab pulsar (neutron star)
M = 1.4 Msun = 2.78 x 1030 kg
R = 1.2 x 103 m.
(e) Chandrasekhar limit
The currently accepted value of the Chandrasekhar limit is about 1.4 Msun (2.765×1030
kg).
3. Kinetic energy of the ground state of fermion
The kinetic energy of the fermions in the ground state is given by
3/22
0
2
)3(25
3
5
3
V
N
mNNU
f
fFfG ℏ
,
where Nf is the number of fermions, and m0 is the mass of the fermion. The pressure P is calculated
as
3/5
0
23/22
3/22
0
2
)()3(5
1
)3(25
3
3
2
3
2
V
N
m
V
N
mV
N
V
UP
f
ff
G
ℏ
ℏ
using the formula of P in the non-relativistic limit. Note that emm 0 for the white dwarf where
electron (spin 1/2) is a fermion, and nmm 0 for the neutron star where neutron (spin 1/2) is a
fermion.
The kinetic energy of fermions in the ground state can be rewritten as
4
2
2 2/3
30
5/3 5/32 22/3
2 2 2
0 0
3(3 )
45 2
3
3 9( ) 1.10495
10 4
f
G f
f f
NU N
mR
N N B
m R m R R
ℏ
ℏ ℏ
where
3/5
0
2
10495.1 fNm
Bℏ
.
The volume V is expressed by
3
3
4RV
.
where R is the radius of the system. We find that P becomes increases as the volume V decreases.
Here we note that the density of the system, , is given by
3
3
4R
M
V
M
.
and M is the total mass of the system. The number density nf for fermions is defined as
fff
fff
fmNm
N
M
N
V
M
V
Nn
.
The average nearest neighbor distance between fermions can be evaluated
3/13/1
1
f
f
m
nd
Note that the more detail of the mass fm will be discussed in the discussion of white dwarf. fm
is the mass per fermion.
4. Gravitational self energy
5
We calculate the potential energy of the system.
Suppose that M(r) is the mass of the system with radius r.
3
3
4)( rrM
, drrrdM 24)( .
The potential energy is given by
R
r
rdMrGMU
0
)()(.
Noting that
34
3
R
M
,
the potential energy is calculated as
R
A
R
GMR
Gdr
r
rGU
R
5
3
5
1)4(
33
)4( 252
0
52
,
where
r dr
MHrL dMHrL
6
5
32
GMA
and G is the universal gravitational constant.
5. The total energy
Fig. A balance between the gravitational force (inward) and the pressure of degenerate Fermi
gas
The total energy is the sum of the gravitational and kinetic energies,
2)(
R
B
R
ARfE nonreltot
Gravitational force
Fermi gas pressure
RR0
Etot
0
7
Fig. Non-relativistic case. The plot of totE as a function of R. totE has a local minimum at
0RR , leading to the equilibrium state.
From the derivative of fnonrel(R) with respect to R, we get the distance R in equilibrium.
02
)(32
R
B
R
ARf
dR
dnonrel ,
or
A
BRR
20 .
or
5/31/3
2 21/3 81
16 e p
ZRM C
Gm Am
ℏ (see the detail below).
Thus, for the nonrelativistic degenerate Fermi gas, there is a balance between the gravitational
force (inward) and the force due to the degenerate Fermi gas pressure, leading to a stable radius
R0.
((Summary))
Compression of a white dwarf will increase the number of electrons in a given volume.
Applying the Pauli’s exclusion principle, this will increase the kinetic energy of the electrons,
thereby increasing the pressure. This electron degeneracy pressure supports a white dwarf against
gravitational collapse. The pressure depends only on density and not on temperature.
Since the analysis shown above uses the non-relativistic formula )2/( 0
2mpF for the kinetic
energy, it is non-relativistic. If we wish to analyze the situation where the electron velocity in a
white dwarf is close to the speed of light, c, we should replace )2/( 0
2mpF by the extreme
relativistic approximation Fcp for the kinetic energy.
As V is decreased with Nf kept constant, the Fermi velocity increases,
3/1
2
0
)3
(V
N
mv
f
F
ℏ .
8
in the non-relativistic case.
6. Relativistic degenerate Fermi gas
The Fermi energy of the non-degenerate Fermi gas is given by
3/22
0
2
)3(2 V
N
m
f
F ℏ
.
where Nf is the number of fermions. As V 0, F increases. Then the relativistic effect
becomes important. The relativistic kinetic energy is given by
222
0
2mccmpc
When cmp 0 ,
kccp ℏ
where m0 is the mass of fermion and mf is the mass per fermion. Note that
ℏc
ddk
,
ℏck
.
The density of states:
d
c
V
c
d
c
Vdkk
VdD
2
3
2
2
3
2
3
14
8
24
)2(
2)(
ℏℏℏ,
or
2
332)(
ℏc
VD ,
3
332
0
2
332
03
1)( Ff
c
Vd
c
VdDN
FF
ℏℏ ,
or
9
3/13/133
ff
F
nc
V
Nc ℏℏ .
where fn is the number density of fermions,
V
Nn
f
f .
The total energy in the ground state is obtained as
4
332
0
3
332
04
1)( FG
c
Vd
c
VdDU
FF
ℏℏ .
Using the expression of Nf, UG can be rewritten as
3/13/12 )()3(4
3
4
3
V
NcNNU
f
fFfG ℏ .
The pressure P is calculated as
3/43/123/43/12)()3(
4
1)()3(
4
1
3f
fGG ncV
Nc
V
U
V
UP ℏℏ
.
using the formula of P in the relativistic limit. The total mass M is denoted as
ff mNM ,
where fm is the mass per fermion (electron in white dwarf) (such as the mass of protons and
neutrons per electron). Note that fm is not always equal to the mass of each fermion (m0) (such
as electron). Since f
f
MN
m , we get
10
R
MA
R
M
m
c
R
M
m
c
mR
M
m
McU
f
f
ff
G
3/43/4
3/4
3/1
3/4
3/4
3/1
3/12
3/1
3
3/12
)4
9(
4
3
4
3)3(
4
3
)
3
4()3(
4
3
ℏ
ℏ
ℏ
which is proportional to R/1 , where
3/43/4
3/1 43937.1)4
9(
4
3
ff m
c
m
cA
ℏℏ
.
Since the gravitational energy is
R
GM
5
32
,
the total energy (relativistic) is given by
R
GMAM
R
GM
R
AMRfE reltot
23/423/4
5
3
5
3)(
.
R
frelHRL
M<M0
M>M0
11
Fig. Relativistic case. Schematic plot of frel(R) vs R for M>M0 and M<M0. When M = M0, frel(R)
= 0. For M>M0, the total energy decreases with decreasing R, leading to the stable state
near R = 0. For M<M0, the total energy decreases with increasing R, leading to the stable
state near R = ∞.
For M>M0, R tends to zero, while for M<M0, R tends to increase. The critical mass M0 is evaluated
from the condition,
2
0
3/4
05
3GMAM .
or
2/3
2
2/3
3/4
2/3
3/40 )(71562.3
39895.23
43937.15
G
c
mGm
c
m
c
GM
fff
ℏℏℏ
.
((Example))
The interior of a white-dwarf star (electrons as fermion) is composed of atoms like 12C (6
electrons, 6 protons, and 6 neutrons) and 16O (8 electrons, 8 protons, and 8 neutrons), which contain equal numbers of protons, neutrons, and electrons. Thus,
2f pm m
( pnp
npmmm
mm2
6
66
for 12C, pnp
npmmm
mm2
8
88
for 16O,
where mp and )( pn mm are the proton mass and neutron mass. Then we have
M0 = 1.72148 Msun.
The currently accepted numerical value of the limit is about 1.4 Msun (Chandrasekhar limit).
((Mathematica))
12
((Note)) Planck mass
The Planck mass is nature’s maximum allowed mass for point-masses (quanta) – in other
words, a mass capable of holding a single elementary charge. The Planck mass, denoted by mPlanck,
is defined by
G
cm
anckPl
ℏ = 1.220910 x 1019 GeV/c2.
where c is the speed of light in a vacuum, G is the gravitational constant, and ℏ is the Dirac
constant.
7. White dwarf with electron as fermion: non-relativistic case
In the white dwarf, a fermion is an electron. So we have
emm 0 .
The mass fm per electron can be described in terms of atomic number Z, and mass number A (the
sum of the numbers of protons and neutrons) as follows. Since there are Z electrons, mass fm per
electron can be evaluated as
pf mZ
Am .
where mp is the mass of proton and we neglect the mass of electrons.
Clear "Global` " ;
rule1 G 6.6742867 1011, me 9.1093821545 10
31,
eV 1.602176487 1019, mn 1.674927211 10
27,
mp 1.672621637 1027, 1.05457162853 10
34,
Msun 1.988435 1030
, c 2.99792458 108;
M03.71562
2 mp 2
c
G
3 2. rule1;
r M0 Msun . rule1
1.72148
13
((Note))
Number of protons = Z, mass of protons, pZm
Number of neutron = ZA mass of neutron, nmZA )(
Number of electrons = Z mass of electron Zme
where Z is the atomic number and A is the atomic mass. The mass fm per fermion is
penpf mZ
AZmmZAZm
Zm ])([
1
since mmm pn
Fig. Image of Sirius A (bright star in the center) and Sirius B (white dwarf, very small spot in
the figure) taken by the Hubble Space Telescope. Sirius B, which is a white dwarf, can be
seen as a faint pinprick of light to the lower left of the much brighter Sirius A.
Then the kinetic energy UG (in the non-relativistic case) can be given by
2
3/5
2
23/2
2
3/523/2
)4
9(
10
3
)4
9(
10
3
R
B
m
M
Rm
R
N
mU
fe
f
e
G
ℏ
ℏ
,
and
16
3/5
3/523/2)
4
9(
10
3
femm
MB
ℏ .
The equilibrium distance R is given by
3/52
3/12
3/5
3/5
23/1
2
3/5
2
23/23/4
0
16
81
16
81
)2
3(
2
pe
fe
fe
Am
Z
GmM
mGmM
m
M
mGM
A
B
RR
ℏ
ℏ
ℏ
.
or
5/31/3
2 21/3 81
16 e p
ZRM
Gm Am
ℏ
where
pf mZ
Am
Thus we have the relation
1/3
RM constant
for the non-relativistic case. The more massive a white dwarf is, the smaller it is. The electrons
must be squeezed closer together to provide the greater preasure needed to a more massive white
dwarf.
((Example))
17
The interior of a white-dwarf star is composed of atoms like 12C (6 electrons, 6 protons, and 6 neutrons) and 16O (8 electrons, 8 protons, and 8 neutrons), which contain equal numbers of
protons, neutrons, and electrons.
pf mm 2 m0 = me.
In this case we have
3/5
23/1
2
0
3/1
)2(16
81
pe mGmRMC
ℏ
= 9.00397 x 1016 kg1/3 m
The radius R is proportional to M-1/3. When M is equal to the mass of sun, Msun, then we have
3/10
sunM
CR = 7.16028 x 103 km.
which is almost equal to the radius of Earth (6371 km). The number density is
3
03
4
1
R
M
mmV
Nn
ff
f
f
= 3.86549 x 1035 /m3.
The Fermi energy of the electrons is
3/222
)3(2
f
e
F nm
ℏ
= 1.93497 x 105 eV.
The Fermi velocity is
2 2/3(3 )F F f
e e
v k nm m
ℏ ℏ
=2.6089 x 108 m/s
The Fermi temperature is
92.245 10FF
B
T Kk
The density is
18
V
Nm
V
M ff = 1.2931 x 109 kg/m3.
The average distance between fermions is
3/1
fm
d = 1.37277 x 10-12 m.
((Mathematica)) Numerical calculation for white dwarf