1
Compact Stars – White Dwarfs, Planets, Neutron Stars
White dwarf structure dominated by P (ρ) for degenerate
electron gas.
(n + 1) K 1/2 (1−n)/(2n)
ξ1
R=
ρc
4πG
n/(n−1)
(n + 1) K
(3−n)/(n−1)
(3−n)/(1−n)
2
0
M = 4πR
−ξ1 θ1
ξ1
4πG
(n + 1) K 3/2 (3−n)/(2n) 2 0 ρc
−ξ1 θ1 .
= 4π
4πG
Non-relativistic γ = 5/3:
K=
2/3
2
3π
5
h̄2
5/3
(NoYe)5/3 = 1013Ye
cgs
me
−1/6
R = 1.121 × 104ρc,6
(2Ye)5/6 km
−3
R
1/2
5
(2Y
)
M.
M = 0.496ρc,6 (2Ye)5/2 M = 0.701
e
104 km
(1)
Relativistic γ = 4/3:
K=
1/3
2
3π
4/3
h̄c (NoYe)4/3 = 1.24 × 1015Ye
4
−1/3
R = 3.35 × 104ρc,6 (2Ye)2/3 km
M = 1.457 (2Ye)2 M.
cgs
(2)
2
Very low density (Thomas-Fermi regime γ = 10/3):
10/3
31/3π 3e2 4πNo 10/3 3h̄c h̄c 6
13 12
K=
cgs
=
1.05
×
10
10
A
A
m e c 2 e2
5/3
12
2/3
ρc km
R = 1.18 × 105
A
5
12
M = 0.001915
ρ3c M
A
5/2 9/2
A
R
M .
= 2.88 × 10−8
12
104 km
(3)
Physical reasoning behind the Chandrasekhar mass:
Consider N degenerate fermions in a star of radius R, so
that number density n ∝ N R−3. Momentum of a fermion is
∼ h̄n1/3 and Fermi energy is EF ∼ h̄cN 1/3R−1. The gravitational energy per fermion is ∼ −GM mB R−1 if M = N mB .
The total energy is
E = EF + EG = h̄cN 1/3/R − GN m2b /R.
Equilibrium is reached when this is minimized. Both terms
scale as 1/R.
When E is positive, E can be decreased by increasing R.
This decreases EF so that eventually the fermions become nonrelativistic: then EF ∼ p2F ∼ R−2. This then decreases faster
than EG, so E becomes negative. However, as R → ∞, E → 0.
This implies there is a minimum of E at a finite value of R.
When E is negative, E can be decreased without bound by
decreasing R so that no equilibrium state is possible and a
black hole forms.
3
The maximum baryon number for equilibrium is determined
by setting E = 0:
Nmax ∼
h̄c
Gm2b
!3/2
∼ 2 × 1057
Mmax ∼ Nmax mB ∼ 1.5 M.
Note that the mass is independent of the fermion’s mass.
The radius at equilibrium is set by the condition EF ≥ mc2,
h̄c
R≤
mc2
h̄c
Gm2B
!1/2
∼
5 × 103 km,
3 km,
m = me
m = mn .
At sufficiently high density, neutronization and pyconuclear
reactions can occur. Thus, both A and N −Z will increase with
density. The neutronization threshold for 56Fe is about 109 g
cm−3. At this density, the Fermi energy of an electron is about
mcc2 +3.695 MeV, the threshold for the inverse beta-decay 56Fe
+ e− →56 Mn + νe. The Mn immediately electron captures:
56 Mn + e− →56 Cr + ν . The Cr is stable until densities above
e
10
−3
10 g cm are reached.
Lighter nuclei have other thresholds: 4He is at 20.6 MeV,
12 C is at 13.4 MeV, 16O is at 10.4 MeV and 20Ne is at 7.0
MeV. The loss of electrons softens the EOS: the Chandrasekhar
mass decreases. A white dwarf at these densities will begin to
10
gravitationally collapse. Thus the maximum density <
∼ 10 g
cm−3, with a minimum radius >
∼ 1500 km.
4
Electrostatic corrections
In a degenerate system in which the nuclei are ordered in
either a solid or liquid, there is Coulomb energy associated with
the ordering. If the nuclei are equally spaced and surrounded
by a uniform density electron gas, the interaction energy per
electron is
9 4π 1/3 2/3 2 1/3
9 Ze2
=−
Z e ne
Ec/Z = −
10 Rc
10 3
where ne = 3Z/(4πRc3). The corresponding pressure is
1/3
4π
∂E
/Z
3
4/3
c
Z 2/3 e2ne .
Pc = n2e
=−
∂ne
10 3
In the extreme relativistic limit this is just a constant fraction
(a few percent) of the degeneracy pressure:
Pc
25/3 3 1/3 e2 2/3
Z .
=−
Pd
5
π
h̄c
In the non-relativistic limit, Pc becomes more important at
lower and lower densities:
Pc
mee2Z 2/3
.
=−
1/3
2
Pd
(2ne) πh̄
At low enough density, Pc = −Pd:
3
2
2
Z me e
2
−3
ne,c =
ρc ' 0.4Z g cm
,
2π 3h̄6
and the total pressure vanishes. For iron this is about 250 g
cm−3, which is not the laboratory value of 7.86 g cm−3 because
it is incorrect to treat the e− gas as uniform.
5
The electron Fermi energy, modified by Coulomb potential
p2F
.
EF = −eV (r) +
2me
is constant in space, otherwise electrons would move to a lower
EF . The electron density is
8π 3
8π
p
=
[2me (EF + eV (r))]3/2 .
F
3
3
3h
3h
The potential is determined by Poisson’s equation
ne =
∇2V = 4πene + nuclear contribution
where the last term is effectively a delta function at the origin. Omitting it for r > 0, we have the boundary condition:
rV (r) → Ze as r → 0. The electric field should vanish at
the outer boundary, since this volume must be overall neutral:
dV /dr|Rc = 0. Poisson’s equation in spherical geometry is
d2φ φ3/2
=
,
dx2
x1/2
Ze2φ (r)
EF + eV (r) =
, x=r
r
Eq. (4) has boundary conditions
(4)
128Z 1/3 mee2
.
2
9π 2
h̄
dφ φ (xo)
,
φ (0) = 1,
φ (xo) ≡
=
dx xo
xo
where xo corresponds to r = RcR. The latter condition can
be seen by evaluation of Z = 4π ner2dr over the entire volume. The equation Eq. (4) has a unique solution, when φ0(0) =
−1.588071, in that as xo → ∞, φ(xo) → 144x−3
o → 0. Oth0
erwise, for larger values of φ (xo), φ doesn’t vanish anywhere
0
6
and diverges as x → ∞. At some point, the second boundary
condition will be satisfied.
The pressure at the outer boundary is that of free particles
4/3 2 4 φ (x ) 5/2
128Z
8π
m
e
1
o
e
2 e2
5 (R ) =
Z
P =
.
p
c
2
10π
x
15h3me F
9π 2
h̄
o
The density is the total mass divided by the volume:
3
3
2
2
4AmB Z 2mee
3AmB 128Z mee
=
.
ρ=
2
2
4π 9π 2
3
h̄ xo
πh̄ xo
For low densities, the solution approaches the unique solution
10/3, with K given by Eq. (3).
φ(xo) → 144x−3
o . Thus, P ∝ ρ
Mass-Radius Relation for Degenerate Objects
The mass-radius diagram for cold compact objects is shown
in the figure: the solid lines are the limiting expressions Eqs
(1–3), the dashed line is the full result, for 12C. The maximum
radius configuration has the properties, approximately, of the
planet Jupiter.
In the relativistic limit, for radii much smaller than 5000
km, the equation of state will deviate from that of a γ = 4/3
gas. Electron capture will reduce Ye and the value of the Chandrasekhar mass. Therefore, a regime where dM/dρc < 0 will
exist. Such a regime is dynamically unstable. At sufficiently
high density, where nuclear forces become important, the effective value of γ will increase, the mass will reach a minimum
value (Mmin ' 0.01 M, where R ' 300 km), and stability is
restored. As the central density increases further, dM/dρc > 0.
This is the neutron star regime. As the mass increases, and the
radius shrinks, general relativity, which we have heretofore ignored, becomes important. The most important feature that
general relativity introduces is that at densities well in excess of
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the nuclear saturation density, ρs = 2.7 · 1014 g cm−3, the mass
reaches a maximum value, in the range 1.5-3 M. Larger density configurations are once again dynamically unstable. The
maximum mass is discussed in a subsequent lecture.
Cooling of white dwarfs
The interior of a white dwarf has energy transport dominated by conduction. The electrons are extremely degenerate,
nowever, so they must have very large mean free paths. The
thermal conductivity is very high. The temperature gradient
must be rather small. The interior is roughly isothermal. Near
the surface, isothermality breaks down as the opacity increases.
The surface regions are diffusive, with a temperature gradient
dT
3 κρ L
=−
.
dr
4ac T 3 4πr2
At the high densities, Kramer’s opacity is dominant: κ =
κoρT −3.5 , with κo ' 4.3 × 1024Z(1 + X) cm2 g−1. With hydrostatic equilibrium,
dP
4ac 4πGm (r) T 6/5
=
.
dT
3
κo L
ρ
8
The surface layer is thin, so m(r) = M . Using the nondegenerate pressure P = NoρkT /µ, and eliminating ρ, we have
4ac 4πGM kNo 7.5
T dT.
P dP =
3 κo L µ
Integrating from P = 0 at T = 0 to the interior,
r
2 4ac 4πGM µ 3.25
ρ=
T
.
8.5 3 κoL kNo
The surface approximation breaks down in the interior when
matter becomes degenerate. This occurs when the non-degenerate
pressure equals the degenerate pressure at radius r∗ where one
has ρ∗ and T∗. This results in
3/2
ρ∗ = 2/4 × 10−8T∗ Ye−1 g cm−3,
M 3.5
µYe2
5
T∗ ergs−1.
L = 5.7 × 10
Z (1 + X) M
4 , but inThis is similar to the blackbody law L = 4πR2σTef
f
volves the interior temperature, not the visible temperature of
7/8
the surface. It suggests a relation like Tef f ∝ T∗ . From L and
M , and the composition, one can deduce T∗. L of 10−2 − 10−5
L imply T∗ = 106 − 107 K, ρ∗ < 103 g cm−3 and
RNokT∗ < −2
R − r∗
10 .
' 4.25
R
GM µ ∼
The energy that is radiated as thermal energy by the white
dwarf is the residual ion thermal energy, since the electrons are
degenerate and the star can’t release any gravitational energy.
For a monatomic non-degenerate ion gas, with cv = 3k/2 the
total thermal energy is (with T∗ = T )
3 No k
T M.
U=
2 A
9
The cooling rate is L = −dU/dt. Using L = CM T 7/2,
3 kNo −5/2
−5/2
.
T
− To
t = to +
5 AC
Taking T << To, the cooling time is
3 NokT M
3 Nok CM 5/7
τ=
=
.
5 A L
5 CA
L
For L ∼ 10−3 L, we obtain τ ∼ 109 yr.
It is interesting to compare the cooling theory with observations. Like cars on a highway, the slower they go, the more congested the freeway (or vice versa). The number of white dwarfs
of a given luminosity should relate to their relative abundance,
especially if the birth rate of white dwarfs has been roughly
constant in time. The luminosity function is φ(L) which is the
space density of white dwarfs per unit interval of log L. Thus,
with a uniform production rate,
d log (L) −1
.
φ (L) ∝
dt
If τ ∝ L−α, where our theory suggests α ' 5/7, one finds
log φ = −α log L + constant.
It turns out this is approximately matched by observations,
−4 L . Theoretical corrections to the specific
until L <
∼ 10
heat of very cold white dwarfs imply that α → 0 below this
luminosity, but observations actually reveal that α << 0 when
−4.5 . This deficit of white dwarfs of low luminosity is
L<
∼ 10
due to the finite age of the galactic disc. The cooling time of the
white dwarfs where the sudden drop in α occurs then yields an
estimate of the age of the galactic disc, about 10 billion years.