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1
PHYS833 Astrophysics of Compact Objects
Notes Part 5
Before considering white dwarf cooling, consider how white dwarfs are formed. The
figure below shows the evolutionary path taken by a 1 M☉ star, starting from a pre-main
sequence model (point 1). Points 2 and 3 mark the start and end of the core hydrogen
burning main sequence phase. Point 4 marks the start of the core helium flash that ends
the red giant phase. Point 5 marks the start of quiescent core helium burning. After the
end of core helium burning, helium burns in a shell, stably at first, but once the region
between the helium and hydrogen burning shells become thin, helium shell flashes (aka
thermal pulses) occur. Three of the He shell flash loops can be seen in the diagram,
labeled A, B, C. Point 6 marks where the nuclear energy production ends, and the white
dwarf cooling track begins. At the end of the evolution, mass loss has reduced the stellar
mass to 0.541 M☉.
4
C
B
A
4
3
6
Log L/L
2
5
1
1
3
0
2
-1
-2
-3
5.2
5.0
4.8
4.6
4.4
4.2
4.0
3.8
3.6
3.4
3.2
Log Teff (K)
The figure below shows how the central temperature and density change through the
evolution of the star. The numbers mark the same evolutionary points as in the first
figure. We see that the star enters the white dwarf cooling track with a central
temperature of 9 107 K. The central density is about 65% of the final central density of
2.65 106 g cm-3, and hence there is some contraction of the core during cooling. The
composition at the center is, by mass, 0.665 16O, 0.313 12C, and 0.022 heavier elements.
2
8.5
5
8.0
6
4
Log Tc (K)
7.5
3
2
7.0
6.5
1
6.0
-1
0
1
2
3
4
5
6
7
-3
Log ρc (g cm )
Are the electrons degenerate in the core at the start of cooling? If they are, then they are
non-relativistic because the density is too low for relativistic electrons.
The quick test for degeneracy is to compare the expression for degenerate electron
pressure with that for non-degenerate electrons. These give the same pressure when
ρ 
10  
 µe 
which gives a transition temperature of
53
13
= 8.3107
ρ
T,
µe
(5.1)
23
ρ 
Tnd _ d = 1.2 10   .
(5.2)
 µe 
The values for density and temperature given above give that the electrons are degenerate
( T < 0.1Tnd _ d ) and become more so as the core temperature decreases.
The first quantitative study of the rate of cooling of white dwarfs was made by Mestel1.
As is standard in stellar structure and evolution calculations, he used the diffusion
approximation for radiative transfer. Here we give the physical basis of the diffusion
equation. A detailed derivation can be found in any text book on radiative transfer.
Consider two parallel plane surfaces in a star, which are separated by one photon mean
free path, l. We assume LTE so that each surface can be treated as a blackbody. Let one
surface have temperature T + ∆T and the other surface have temperature T. The total heat
flux in the direction from the hotter surface to the colder is
5
4
H = σ ( T + ∆T ) − σ T 4 4σ T 3 ∆T .
(5.3)
Here σ is Stefan’s constant. Usually the hotter surface is deeper in the star. Hence the
outward heat flux is
1
L. Mestel, 1952, MNRAS, 112, 583
3
H = −4σ T 3
dT
l.
dr
(5.4)
The medium will contain a number of different kinds of particles that can absorb or
scatter photons but first consider the case in which there is only one kind of absorbing
particle that has cross-section σ a , and number density n. Consider a collimated light
beam of area A. The mean free path is the distance this light beam will travel before being
completely absorbed. The number of absorbers required to absorb the beam is N = A σ a .
But N = nAl. Hence l = 1 ( naσ ) .
For a mixture of many different kinds of absorbers or scatterers,
l −1 = ∑ nkσ k .
(5.5)
k
The opacity is defined to be
κ=
1
ρ
∑n σ
k
k
,
(5.6)
k
so that
κρ l = 1.
(5.7)
Using this in equation (5.4) gives
4σ T 3 dT
,
(5.8)
κρ dr
which differs from the correct result by a factor 4/3.
The correct result is
16π acr 2T 3 dT
L=−
,
(5.9)
3κρ
dr
where L is the radiative luminosity, which is the energy crossing a spherical surface of
radius r per unit time, and σ has been replaced by ac/4.
H =−
White dwarf cooling – Mestel theory
A white dwarf has an electron degenerate core with a thin non-degenerate envelope (m ≈
10-4 M☉). In the core the electrons have a large mean free path because almost all
available energy levels in the Fermi ‘sea’ are filled. This results in a high thermal
conductivity and the core is isothermal to a high degree. We can assign a unique
temperature, Tc, to the core. Because the white dwarf is supported by degenerate electron
pressure very little energy can be released by gravitational contraction. Also very little
energy can come from the thermal energy of the electrons because most of them are
already in the lowest energy states. In addition, essentially all nuclear processes are
finished (but due to hydrogen diffusing inwards H-burning can linger on2). Hence the
major source for the energy emitted from the surface in photons or neutrinos comes from
cooling of ions in the core. In the early stages of white dwarf cooling, thermal neutrino
2
I. Iben, J. MacDonald 1985, ApJ, 296, 540
4
losses are important but because they scale like T15/2 they rapidly become negligible. The
stellar luminosity is then related to the central temperature by
dT
L∗ = − M ∗CV c ,
(5.10)
dt
where CV = 3k 2 Amu is the specific heat per unit mass for a monatomic gas of atomic
weight, A.
To get a second relation between L∗ and Tc, the non-degenerate envelope is assumed to
be radiative, with Kramers’ opacity law
κ = κ 0 ρT − 7 2 .
(5.11)
The Kramers’ opacity law was originally derived for the opacity from free – free
transitions. The monochromatic opacity is of form
κν ≈ 2 10 25
Z 2 ne 1
(1 − e−hν
A T1 2 ν 3
kT
)m
2
kg −1 ,
(5.12)
where ne is the electron number density in units of electrons per m3. For the Planck
distribution, a characteristic frequency is given by hν ≈ kT . Hence the frequencyaveraged opacity will be of Kramers’ form.
The envelope is thin with very little energy generation, so that in the envelope, L = L∗, m
= M∗ to a good approximation. From the equation of radiative transfer
dT
3κρ L
=−
,
dr
16π acr 2T 3
(5.13)
dP
Gm ρ
=− 2 ,
dr
r
(5.14)
and hydrostatic equilibrium
we obtain
dT
dT dr
3κ L∗
=
=
.
(5.15)
dP
dP
16π acT 3GM ∗
dr
Using (5.11) and a perfect gas equation of state, this gives
3κ 0 µ L∗
dT
=
(5.16)
PT −15 2 .
dP 16π acℜGM ∗
Since our goal is to derive a relationship between T and P at the transition from nondegenerate to degenerate electrons, we do not need to be very precise in specifying the
surface boundary conditions. For simplicity, we take T = 0 at P = 0. Integration of
equation (5.16) gives
5
17 3κ 0 µ L∗
P2
(5.17)
4 16π acℜGM ∗
(from which we can deduce that the envelope is radiative if there are no ionization
zones).
The core-envelope interface is where electrons become degenerate i.e. where
T 17 2 =
53
 ρ 
ρ
K1   = ℜT .
µe
 µe 
In terms of temperature and electron pressure this is (in cgs units)
Pe = 2.0T 5 2 .
The total pressure (ion + electron) at the interface is then

µ 
µ
P = Pe + Pion = 1 + e  Pe = e Pe .
µ
 µion 
Hence
P = 2.0
(5.18)
(5.19)
(5.20)
µe 5 2
T
µ
(5.21)
at the core - envelope interface. Inserting this into equation (5.17) gives the central
temperature. Putting in the numerical values for the constants, we get
M
L∗ = 4 × 107 Tc 7 2
.
(5.22)
M
Using this with equation (5.10), gives
dTc
= −1.6 ×10 −34 Tc 7 2
(5.23)
dt
which has solution
 t 
Tc = 2.3 ×10 

 1 yr 
−2 5
10
.
(5.24)
From equation (5.22), we find
L∗
M  t 
= 1.86 × 1010 ∗ 

L
M  1 yr 
−7 5
.
(5.25)
In the table below, cooling times for a 0.6 M☉ white dwarf from Mestel theory are
compared with the detailed calculations.
L∗/L
10-2
10-3
10-4
Mestel Age (yr)
4 108
2 109
1010
IM85 Age (yr)
2.5 108
109
5 109
The luminosity function of white dwarfs
6
The luminosity function refers to dividing the space density of any kind of astrophysical
object into bins of bolometric magnitude or log luminosity. Represent the space density
of white dwarfs per unit interval of MBol by
φ ( M Bol ) dM Bol pc −3 M Bol −1.
(5.26)
3
The white dwarf luminosity function from the Sloan Digital Sky Survey shown below.
-2.5
log N
-3.0
-3.5
-4.0
-4.5
-5.0
8
10
12
14
16
MBol
White dwarf luminosity function from the Sloan Digital Sky Survey.
The integral of the luminosity function gives the space density of white dwarfs in the
solar neighborhood, 0.005 pc-3. About 1 in 10 stars is a white dwarf.
The average slope of the luminosity function before the sharp drop at MBol = 15.4 is
d log N dM Bol = 0.291. The bolometric magnitude is related to luminosity by
 L 
(5.27)
M Bol = 4.755 − 2.5log   .
L
 
Hence d log N d log L = −0.727. If we assume that white dwarfs form at a uniform rate
and that they all have the same mass and composition so that they all follow the same
cooling curve, then the number of white dwarfs in any bin will be proportional to the
lifetime in that bin. With these assumptions, the Mestel cooling law predicts
d log N d log L = − 5 7 = −0.714, which is remarkably close to the observed value,
considering the simplicity of the assumptions. Large deviations from the mean slope are
often attributed to variations in the rate of formation of white dwarf forming stars.
The sharp drop off in the luminosity function at MBol = 15.4 indicates a deficiency of
white dwarfs at luminosities below 5.5 10-5 L . If we use the simple Mestel cooling law,
this corresponds to a cooling time of 16.5 Gyr, which is longer than the WMAP 13.7 Gyr
age of the Universe4. If we divide by the factor 2 indicated by the IM85 calculations, the
cooling time is reduced to 8.3 Gyr. If we accurately knew the masses of the coolest white
3
4
Harris et al, 2006, AJ, 131, 571
D. Spergel et al. 2003, ApJS, 148, 175
7
dwarfs and how this mass is related to the initial mass of the progenitor stars, so that we
can add the lifetime of the progenitor, we can determine a lower limit on the age of the
Galaxy. Complications arise from the chemical evolution of the Galaxy and the lack of a
clear correlation between Galactic age and chemical abundance, not to mention mixing of
stars from different parts of the Galaxy, and the effects of the essentially unknown stellar
He mass fraction.
Calculations5 give the following results:
MWD (M☉)
τPreWD (Gyr)
τ3 (Gyr)
0.237
0.9
1.0
1.2
1.5
2.0
2.5
3.0
0.576
0.604
0.643
0.692
0.766
0.845
0.933
10.702
7.395
3.965
1.938
0.8640
0.5188
0.3161
1.041a
0.783a, 0.760b
0.820a
0.245
1.0
1.2
1.5
2.0
2.5
3.0
0.567
0.588
0.615
0.669
0.726
0.792
8.827
4.657
2.247
0.9654
0.6263
0.3719
1.0
1.2
1.5
2.0
2.5
3.0
0.545
0.569
0.584
0.632
11.426
5.917
2.758
1.147
0.721
0.4585
1.0
1.2
1.5
2.0
2.5
3.0
0.543
0.557
0.567
0.595
0.632
0.678
13.980
7.127
3.264
1.303
0.8647
0.5136
Y
0.001
0.004
0.01
0.02
a
M (M☉)
Z
0.255
0.275
Element diffusion and gravitational settling neglected.
0.916a
0.896a
0.744a
τ 3 is the time for the luminosity of the white dwarf to drop to 10
−3
L⊙ .
For a given initial mass, stars with lower Z evolve more quickly and leave higher mass
white dwarf remnants. Because stars with lower Z have lower opacity, they tend to be
smaller and hotter. The energy generation rates are higher which explains the shorter
5
Lawlor, T. & MacDonald, J. 2006, MNRAS, 371, 263
8
lifetimes. The remnants have higher masses in part because there is less time for mass
loss and in part because mass loss rates are lower.
If we plot pre- white dwarf lifetime against mass in a log-log plot, we find
1.2
Z
Z
Z
Z
1.0
=
=
=
=
0.001
0.004
0.01
0.02
0.8
Log τPreWD (Gyr)
0.6
0.4
0.2
-0.0
-0.2
-0.4
-0.6
-0.1
-0.0
0.1
0.2
Log M/M
0.3
0.4
0.5