CONTENTS 2
Contents
1 Supernova basics and observational classification 3
2 Introduction to astrophysical fluids 19
3 Hydrodynamic Equations 31
4 Surface waves 43
5 Perturbations at a two-fluid interface 54
6 Density fluctuations and Jeans instability 69
7 Shock waves and supernova envelope expansion 79
8 Shocks jump conditions and envelope expansion 87
9 Particle acceleration in SNRs 98
10 Massive star evolution 110
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 3
1 Supernova basics and observational classification
1.1 Main Course Topics:
1. Supernovae: classification and basic properties
2. Astrophysical Fluids
3. Hydrodynamic instabilities
4. Shocks and particles acceleration
5. Supernovae models and observations
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 4
1.2 Literature
• Prialnik, D., An Introduction to The Theory of Stelalr Structure and
Evolution, 2000 simple introduction into the subject
• Landau & Lifshitz, Fluid Mechanics, 1987 cover most fluid topics
• some other books and web sources indicated in the slides
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 5
1.3 What is a supernova ?
Stars undergoing a tremendous explosions, during which their lumi-
nosity becomes comparable to that of an entire galaxy called supernovae.
Figure 1: Left: Multiwavelength X-ray image of the remnant of Kepler’s Supernova, SN
1604. (Chandra X-ray Observatory); Right: Nova Persei (exploded 1901) from US Naval
Observatory in Flagstaff
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 6
Historically, nova was the name for an apparently new star, e.g. brighten
suddenly by many orders of magnitude. Nova characteristically increases
in brightness some ten magnitude. The rise is very rapid.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 7
1.4 Supernova vs nova
Figure 2: From Zeilik and Gregory, Introductory Astronomy and Astrophysics, 1998
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 8
1.5 SN 1987A
Figure 3: NASA Image of 1987A supernova: The above two photographs are of the
same part of the sky. The photo on the left was taken in 1987 during the supernova
explosion of SN 1987A, while the right hand photo was taken beforehand. Supernovae
are one of the most energetic explosions in nature, making them like a 1028 Megaton
bomb (i.e., a few octillion nuclear warheads).
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 9
1.6 Historical observations of supernovae
Supernovae have been witnessed in our galaxy but only in the histori-
cal past. Now we observe supernova remnants.
Notable supernovae in the Milky Way (our galaxy):
• 1006 - the brightest ever recorded (1/4 Moon!)
• 1054 -the best known (Crab nebula nowadays), first recorded Japanese,
but also by Koreans and then Chinese
• 1572 -Tycho’s supernovae (Tycho Brahe)
• 1604 -a student of Tycho, Kepler, also observed a supernova
• 1680 - Cassiopeia A (allegedly seen by Flamsteed, the first Royal
Astronomer for England) is a gaseous remnant that could have been
a under-luminous supernovae (the last discovered supernovae in
our galaxy).
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 10
1.7 First image of a supernova?
SN 1054, the best known a Crab
nebula, (M1);
It was probably also recorded by
Anasazi Indian artists (in present-
day Arizona and New Mexico), as
findings in Navaho Canyon and
White Mesa
Milton (1978) points that on the
morning of July 5, 1054 the cres-
cent moon came remarkably close to
the supernova, as seen (only) from
Western North America.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 11
1.8 Observations of supernovae: How frequent ?
Figure 4: Image of SN 1987A
There were no supernova
observed in our Galaxy since
1680, although supernovae oc-
cur once per hundred years in
a single spiral galaxy like the
Milky Way.
1987A from the Large Mag-
ellanic Cloud is probably the
best studied supernova of all
times (see Figure 4).
Before 1987 around 30 su-
pernovae were observed yearly,
now it is around a few hundred
per year
For the list of the latest discovered supernovae see: this webpage.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 12
1.9 The latest supernova in the Galaxy...
Cassiopeia A Image: X-ray: NASA/CX-
C/SAO; Optical: NASA/STScI; Infrared:
NASA/JPL-Caltech
The last nearby supernova
explosion occurred in 1680,
It was thought to be just
a normal star at the time,
but it caused a discrepancy
in the observer’s star cata-
logue which historians finally
resolved 300 years later, af-
ter the supernova remnant
(Cassiopeia A) was discov-
ered and its age estimated.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 13
1.10 Observational classification of supernovae
Supernovae explosions are divided into two types in accordance with
spectral properties (observation of some spectral lines in absorption):
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 15
1.12 Lightcurves of supernovae
Schematic light curves for super-
novae of Types Ia, Ib, II-L, II-P (from
http://www.talkorigins.org/faqs/supernova/)
V-band Magnitudes of Type Ia
(from the Perlmutter group (Su-
pernova Cosmology Project)) Note
that absolute magnitude are almost
identical
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 16
1.13 Supernova location within Galaxies
• Type Ia supernovae appear in all sort of galaxies: spiral, elliptical,
and irregular. They tend to avoid the arms of spiral galaxies (regions
of star formation), and hence explode from old (long-lived) stars.
The only type to explode in elliptical galaxies (no star formation)
• Type II supernovae have never appeared in elliptical galaxies, occa-
sionally in irregular galaxies, but mostly in the arms of spiral galax-
ies, e.g. stars leading to Type II supernovae are short-lived =⇒
massive stars
Note: Tycho Brahe and Kepler both witnessed Type Ia supernovae.
1987A was Type II-P.
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 17
1.14 Supernova progenitors
Type Ib and Ic only seem to explode in the arms of spiral galaxies (like
type II but unlike Ia). Also are likely to be associated with massive stars
probably Wolf-Rayet stars.
Type Ib and Ic are also strong radio sources unlike Type Ia (presumably
collision of a shock with the media =⇒ strong mass loss)
• Type II and Type Ib/c are likely to represent the explosions of mas-
sive stars
• Type Ia are likely to be a part of double star systems (to increase
white dwarf above Chandrasekhar limit)
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1 SUPERNOVA BASICS AND OBSERVATIONAL CLASSIFICATION 18
1.15 Example: supernova progenitor
Figure 6: The Hubble Space Telescope’s Advanced Camera for Surveys acquired
these images of a small field inside M51. The left image, taken in January, shows the
field prior to the eruption of Supernova 2005cs. The right image, taken on July 11th,
shows the magnitude-14 supernova. 2005cs supernovae progenitor is a red supergiant
of 7-10 solar masses
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 19
2 Introduction to astrophysical fluids
2.1 Description of N-particle system
Generally a system of N particles can be easily described by the sys-
tem of 6 × N Hamiltonian equations. However, as soon as N ∼ 10
24
À 1
(Avogadro number), the solution of the system becomes impossible to find
and alternative (simplified) methods of description should be applied.
There are a number of simplifications possible:
• Test particles description: Exact solution for M ¿ N particles while
the rest of the particles are treated as an external slow varying me-
dia.
• Fluid (HD) description of plasma: plasma is assumed to be a con-
tinues media at L À l, where L is a scale of processes to consider,
and l is the mean free path of a particle in a plasma.
• Classical kinetics is the study of the relationship between motion
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 20
and the forces affecting motion introducing statistical tools for de-
scription.
Other methods (or combination of the above) are possible.
2.1.1 Fluid Description
Fluid dynamics studies the motion of fluids (liquids and gases) on
macroscopic scale, a fluid is regarded as a continuous medium.
The mathematical description of the state of a moving fluid is via macro-
scopic parameters fluid density ρ, fluid velocity
−→
v , and for instance pres-
sure p.
All macroscopic parameters are functions of position
−→
r and time t .
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 21
2.2 Conservation of matter
Figure 7: Fluid volume V
and surface element d
−→
s .
Let us consider a fluid with mass density ρ
in volume V (Figure 7). The mass of fluid is
simply
M =
Z
V
ρdV
where integration is over the volume V.
The mass of the fluid flowing in unit time
through an surface element d
−→
s is
ρ
−→
v d
−→
s
(
>0 outflow
<0 inflow
Given that there are no sources or sinks of fluid inside the volume V
(i.e. no chemical or nuclear reactions to produce/absorb fluid), the total
mass of fluid flowing out of the volume V in unit time is
I
∂V
ρ
−→
v d
−→
s
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 22
where ∂V denotes integration over the surface bounding volume V .
The decrease per unit time in the mass of fluid in the volume V can be
written as a time derivative
−
∂
∂t
Z
V
ρdV
Provided that the mass in volume V is conserved, one can equate the two
expressions, so we can write
∂
∂t
Z
V
ρdV = −
I
∂V
ρ
−→
v d
−→
s (2.1)
The surface integral can be transformed to volume integral using Green’s
formula
I
∂V
ρ
−→
v d
−→
s =
Z
V
∇(ρ
−→
v ) dV
Therefore Equation (2.1) can be re-written:
Z
V
µ
∂ρ
∂t
+ ∇(ρ
−→
v )
¶
dV = 0
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 23
note that ∇
−→
v ≡ div
−→
v .
Since the equation above is true for any volume, the integration must
vanish:
∂ρ
∂t
+ ∇(ρ
−→
v ) = 0 (2.2)
This is the equation of continuity.
Expanding second term in the continuity equation, one finds
∂ρ
∂t
+ ρ∇
−→
v +
−→
v ∇ρ = 0
The vector ρ
−→
v =
−→
j is called mass flux density.
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 24
2.3 Equation of motion
Figure 8: Fluid volume and
pressure.
Let us again consider some volume (a ele-
ment of fluid) in a fluid (Figure 8). The total force
acting on this volume is equal to the integral of
the pressure
−
I
∂V
pd
−→
s
where p(
−→
r (t), t) is the pressure. The integral is
taken over the surface ∂V bounding the volume
V .
Transforming this integral into a volume integral, one finds
−
I
∂V
pd
−→
s = −
Z
V
∇pdV
where ∇p ≡ grad p is the pressure gradient.
In other words, we can say that a force −∇p acts on unit volume of
the fluid. We can now write the equation of motion of a volume element in
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 25
the fluid, recalling second Newton law, m
−→
a =
−→
F , we can write for the fluid
element:
ρ
d
−→
v
dt
= −∇p (2.3)
where d/dt denotes the rate of change of velocity of a given fluid volume
and not the rate of change of the fluid velocity at a fixed point in space.
2.3.1 Lagrangian and Eulerian descriptions
If we ‘follow the fluid’ (as above in Equation (2.3)), an arbitrary function
f changes as we track a particular fluid element, called Lagrangian fluid
element. Such derivative is called the convective derivative or Lagrangian
derivative.
Description at given (fixed) spatial positions (as we used in the conti-
nuity equation (2.2) ) is called Eulerian description.
Let us relate these two descriptions. Consider function f (t,
−→
r (t)) of a
fixed Lagrangian fluid element. The derivative of f with respect to time t
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 26
can be written
d f
dt
=
∂f
∂t
+
∂f
∂
−→
r
d
−→
r
dt
since
d
−→
r
dt
=
−→
v , we can write
d f
dt
=
∂f
∂t
+
−→
v
∂f
∂
−→
r
or in operator form as relation between Lagrangian and Eulerian
derivatives
d
dt
=
∂
∂t
+
−→
v ·
∂
∂
−→
r
(2.4)
The convective derivative (2.4) gives the Lagrangian rate of change in
time in terms of Eulerian measurements.
Using equation (2.4), the equation of motion (2.3) can be written:
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 27
∂
−→
v
∂t
+ (
−→
v · ∇)
−→
v = −
1
ρ
∇p (2.5)
This equation of fluid motion was first obtained by Euler in 1755 and
is called Euler equation.
If the fluid is in the gravitational field additional force ρ
−→
g , where
−→
g is
acceleration due to gravity, acts on any unit volume. Equation (2.5) can
be re-written:
∂
−→
v
∂t
+ (
−→
v ∇)
−→
v = −
1
ρ
∇p +
−→
g
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 28
2.4 Incompressibility
Consider a closed surface ∂V in a fluid. The net volume rate of the
fluid that is leaving volume V is given by the surface integral over ∂V
I
∂V
−→
v d
−→
s
Using Gauss’s integral theorem, one finds
I
∂V
−→
v d
−→
s =
Z
V
∇
−→
v dV
For incompressible fluid the volume rate should be zero. Since this must
be true for all possible volumes in the fluid, we can write
∇ ·
−→
v = 0 (2.6)
This is so-called incompressibility condition.
Note: Deriving the equation of motion, we have ignored the processes
of energy and momentum dissipation due to internal friction (viscosity)
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 29
and heat exchange between different parts of fluid (thermal conductivity).
Such incompressible fluids are called ideal.
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2 INTRODUCTION TO ASTROPHYSICAL FLUIDS 30
2.5 Fluid equations in Lagrangian description
Using Equation (2.4), we can write our first hydrodynamic equation
(continuity equation) with Lagrangian derivative:
∂ρ
∂t
+ ρ∇
−→
v +
−→
v ∇ρ =
Eulerian
z }| {
µ
∂
∂t
+ (
−→
v ∇)
¶
ρ +ρ∇
−→
v =
Lagrangian
z}|{
dρ
dt
+ρ∇
−→
v = 0 (2.7)
Similarly, we can write equation of motion including gravity
ρ
Lagrangian
z}|{
d
−→
v
dt
= ρ
Eulerian
z }| {
µ
∂
∂t
+ (
−→
v ∇)
¶
−→
v = −∇p + ρ
−→
g
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3 HYDRODYNAMIC EQUATIONS 31
3 Hydrodynamic Equations
3.1 Fluid vorticity
Newtonian gravity is a conservative force, so the gravitational accel-
eration may be expressed as the gradient of a scalar-valued gravitational
potential ψ(
−→
r ):
−→
g = −∇ψ
Assuming that ρ = constant, the hydrodynamic equation of motion can be
written
∂
−→
v
∂t
+ (
−→
v · ∇)
−→
v = −∇
µ
p
ρ
+ ψ
¶
To simplify the equation above, let us consider
−→
v × (∇ ×
−→
v ) term. We
can re-write using the vector triple product property
1
−→
v × (∇ ×
−→
v ) = ∇(
↓
−→
v ·
−→
v ) −
↓
−→
v (∇ ·
−→
v )
1
For any three vectors
−→
A × (
−→
B ×
−→
C) =
−→
B(
−→
A ·
−→
C) −
−→
C(
−→
A ·
−→
B)
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3 HYDRODYNAMIC EQUATIONS 32
where notation
↓
−→
v explicitly states that
−→
v should be differentiated. The first
RHS term is ‘half derivative’, where only the first
−→
v (and not the second)
should be differentiated. Indeed, since
∇
−→
v
2
= ∇(
−→
v ·
−→
v ) = ∇(
↓
−→
v ·
−→
v ) + ∇(
−→
v ·
↓
−→
v ) = 2∇(
↓
−→
v ·
−→
v ),
we can write
∇(
−→
v ·
↓
−→
v ) =
1
2
∇
−→
v
2
The term
↓
−→
v (∇ ·
−→
v ) can be re-arranged, so that the differential operator ∇
acts to the right
↓
−→
v (∇ ·
−→
v ) = (∇ ·
−→
v )
↓
−→
v = (
−→
v · ∇)
−→
v
Combining these two expression, we can write
−→
v × (∇ ×
−→
v ) =
1
2
∇(
−→
v
2
) − (
−→
v · ∇)
−→
v
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3 HYDRODYNAMIC EQUATIONS 33
and re-arranging
(
−→
v · ∇)
−→
v =
1
2
∇(
−→
v
2
) −
−→
v × (∇ ×
−→
v ) =
1
2
∇(
−→
v
2
) + (∇ ×
−→
v ) ×
−→
v
Using this mathematical expression, the equation of motion becomes:
∂
−→
v
∂t
+ (∇ ×
−→
v ) ×
−→
v = −∇
µ
p
ρ
+ ψ+
1
2
−→
v
2
¶
where
−→
ω ≡ ∇ ×
−→
v is called fluid vorticity. To derive equation for fluid vor-
ticity, let us apply ∇× to both parts of the equation of motion, so we can
derive
∂
−→
ω
∂t
+ ∇ × (
−→
ω ×
−→
v ) = 0
Consider the second term. Since
∇×(
−→
ω ×
−→
v ) = ∇×(
↓
−→
ω ×
−→
v )+∇×(
−→
ω ×
↓
−→
v ) =
↓
−→
ω (
−→
v ·∇)−
−→
v (∇·
↓
−→
ω )+
−→
ω (∇·
↓
−→
v )−
↓
−→
v (
−→
ω ·∇)
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3 HYDRODYNAMIC EQUATIONS 34
which can be simplified using ∇ ·
−→
v = 0 for incompressible fluid and ∇(∇ ×
...) = 0. Re-arranging, we have
∇ × (
−→
ω ×
−→
v ) = (
−→
v · ∇)
−→
ω − (
−→
ω · ∇)
−→
v
Finally one finds the equation for vorticity
∂
−→
ω
∂t
+ (
−→
−→
v · ∇)
−→
ω = (
−→
ω · ∇)
−→
v (3.1)
In Lagrangian description, the vorticity equations has a more simple
form
d
−→
ω
dt
= (
−→
ω · ∇)
−→
v
In 2D fluids (Figure 9), the term (
−→
ω · ∇)
−→
v vanishes and we have
d
−→
ω
dt
=
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
ω = 0
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3 HYDRODYNAMIC EQUATIONS 35
Figure 9: Left: US Satellite image of Katrina hurrcane.Right: Jupiter’s Great Red Spot
as seen by a Voyager spacecraft from NASA/JPL-Caltech.
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3 HYDRODYNAMIC EQUATIONS 36
Hence, in the absence of friction, vorticity is conserved along trajectories.
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3 HYDRODYNAMIC EQUATIONS 37
3.2 Energy equation
The first law of thermodynamics states that
dQ = dU + pdV
where
dQ is the heat (energy) that has been added to the system
dU is the change in the internal energy
pdV is the work done by the system
We want to adapt the first law of thermodynamics to a continuous fluid
in which there may be motions.
Let us first introduce q and ε – the heat added and the change of
internal energy per unit of mass:
dQ = δmdq, dU = δmdε
The volume per mass unit is 1/ρ, so that the last term in the first law
of thermodynamics can be written pδmd(1/ρ). Putting these three terms
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3 HYDRODYNAMIC EQUATIONS 38
together one finds:
dq = dε + pd
µ
1
ρ
¶
and per unit of time
dq
dt
=
dε
dt
+ p
d
dt
µ
1
ρ
¶
Using the conservation of matter (our first hydrodynamic equation in La-
grangian form, see Equation 2.7), one finds
p
d
dt
µ
1
ρ
¶
= −
p
ρ
2
dρ
dt
=
p
ρ
∇ ·
−→
v
we can derive the energy conservation equation
ρ
dε
dt
= ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = −p∇ ·
−→
v + ρ
dq
dt
(3.2)
where the term ρdq/dt is the rate of heat gain/loss per unit volume, which
could be either external to fluid (e.g. energy loss due to radiation) or
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3 HYDRODYNAMIC EQUATIONS 39
internal (e.g. due to thermal conduction within the fluid), and the term
p∇ ·
−→
v is the work done per unit volume.
3.2.1 Heat conduction
Let us consider a small volume of fluid. The heat flux
−→
F inside a
continuous fluid can be assumed to be proportional to the temperature
gradient ∇T
−→
F = −k∇T
where T(
−→
r , t ) is fluid temperature, k is the thermal conductivity coefficient
of fluid. The heat loss rate from a volume of fluid is equal to the heat flux
integrated over the bounding surface ∂V of the volume V
I
∂V
−→
F d
−→
s =
Z
V
∇ ·
−→
F dV
Using −ρdq/dt = ∇ ·
−→
F , one finds in Euler’s coordinates:
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3 HYDRODYNAMIC EQUATIONS 40
ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = ∇(k∇T) − p∇ ·
−→
v (3.3)
This is the equation for energy conservation including thermal con-
duction.
For incompressible fluid, ∇ ·
−→
v = 0 and for fluid with thermal conduc-
tivity k = 0, one can simplify the equation (3.3), so we have in Lagrandian
and Eulerian descriptions
dε
dt
=
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = 0
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3 HYDRODYNAMIC EQUATIONS 41
3.3 Hydrodynamic equations
To sum up hydrodynamic equations, we can write all equations to-
gether (Eulerian coordinates)
mass:
µ
∂
∂t
+ (
−→
v · ∇)
¶
ρ = −ρ∇
−→
v
momentum: ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p + µ4
−→
v
energy: ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = ∇(k∇T) − p∇ ·
−→
v
where µ4
−→
v is the viscosity term (we did not derive) and µ is the
dynamic viscosity.
The equations are known as the Navier-Stokes Equations. These
equations describe how the velocity, pressure, temperature, and density
of a moving fluid are related. The equations were derived independently
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3 HYDRODYNAMIC EQUATIONS 42
by G.G. Stokes, in England, and M. Navier, in France, in the early 1800’s.
The equations are nonlinear and very difficult to solve.
In case of ideal fluid, e.g. ν = 0, k = 0, and ∇ ·
−→
v = 0, the equations
are simplified to:
mass:
µ
∂
∂t
+ (
−→
v · ∇)
¶
ρ = 0
momentum: ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p
energy:
µ
∂
∂t
+ (
−→
v · ∇)
¶
ε = 0
In both cases, these equations are incomplete to describe the fluid, the
equation of state of the fluid needs to be added to complete the system.
In case of ideal gas, the pressure has simple expression p = (γ − 1)ρε.
Pulsars and Supernova I
4 SURFACE WAVES 43
4 Surface waves
4.1 Surface gravity waves
Figure 10: Fluid surface and
coordinates used.
Let us consider the free fluid surface in
a gravitational field
−→
g (e.g. ‘stellar surface’).
−→
g is antiparallel to
−→
z and is perpendicular
to the fluid surface in Figure 10.
Hydrodynamic equations can be written
(assuming ideal fluid)
µ
∂
∂t
+ (
−→
v · ∇)
¶
ρ = 0
ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p + ρ
−→
g
In addition, if there is no vorticity inside of the fluid [note that the vorticity
equation (3.1) says that velocities induced remain irrotational], we can
Pulsars and Supernova I
4 SURFACE WAVES 44
assume that velocity can be presented as a gradient of a scalar function
−→
v = −∇ϕ,
where ϕ(
−→
r , t ) is the velocity potential.
Combining second hydrodynamic equation and the relation (
−→
v ·∇)
−→
v =
1
2
∇(
−→
v
2
)+(∇×
−→
v )×
−→
v , where we note that when
−→
v = −∇ϕ, ⇒ ∇×
−→
v = 0,
hence one finds
−
∂∇ϕ
∂t
+ ∇
µ
1
2
−→
v
2
¶
= −∇
µ
p
ρ
+ ψ
¶
,
where ψ is gravitational field potential. This equation can be re-written in
the form of an integral
∇
F(t) - some constant in space
z }| {
µ
−
∂ϕ
∂t
+
1
2
−→
v
2
+
p
ρ
+ ψ
¶
= 0,
where F(t) is a constant in space, but can be a function of time. For the
geometry (Figure 10), the gravitational potential is ψ = gz.
Pulsars and Supernova I
4 SURFACE WAVES 45
Assume further:
1. Let a constant pressure p
0
act on the surface of fluid
2. We will consider small perturbations, so that
−→
v
2
≈ 0
3. The constant p
0
acting on the surface can be eliminated by re-
defining the velocity potential ϕ
Adding to ϕ a quantity independent on coordinates p
0
t/ρ, e.g. ϕ → ϕ +
p
0
t/ρ, we simplify equation for F(t)
F(t) =
µ
−
∂ϕ
∂t
+ gz
¶
= 0.
Figure 11: Fluid surface perturbation with amplitude ξ(x, z, t)
Pulsars and Supernova I
4 SURFACE WAVES 46
Without loss of generality, we can set F(t) = 0, since F(t) could always
be absorbed into velocity potential. Hence at a perturbed fluid surface
z = ξ, we can write
µ
−
∂ϕ
∂t
+ gz
¶
¯
¯
¯
¯
z=ξ
= 0 =⇒ ξ =
1
g
∂ϕ
∂t
¯
¯
¯
¯
z=ξ
where ξ is the amplitude of a small perturbation (see Figure 11).
Since the perturbation is small, we can assume that the vertical com-
ponent of the velocity v
z
of the points at the surface is simply the time
derivative of ξ, so we have
v
z
=
dξ
dt
'
∂ξ
∂t
Note that dξ/dt ' ∂ξ/∂t due to smallness of ξ. Using the definition of the
velocity potential, z-component of fluid velocity is v
z
= −
∂ϕ
∂z
, therefore one
finds at z = ξ
−
∂ϕ
∂z
¯
¯
¯
¯
z=ξ
=
∂ξ
∂t
=
1
g
∂
2
ϕ
∂t
2
¯
¯
¯
¯
z=ξ
Pulsars and Supernova I
4 SURFACE WAVES 47
Finally we have the following equation at the surface z = ξ:
∂ϕ
∂z
¯
¯
¯
¯
z=ξ
+
1
g
∂
2
ϕ
∂t
2
¯
¯
¯
¯
z=ξ
= 0 (4.1)
In addition due to incompressibility of fluid ∇ ·
−→
v = 0, we have the second
equation for velocity potential ϕ:
4ϕ = 0, (4.2)
where 4 is the Laplace operator or Laplacian, 4 ≡ ∇
2
, e. g. the diver-
gence of the gradient of velocity potential.
The system of equations (4.1, 4.2) describe small perturbations at fluid
surface.
Let us now find the solution of the equations (4.1, 4.2). Consider waves
on the surface of a fluid whose area is unlimited and the wavelength is
small in comparison with the depth of the liquid. So we seek for a solution
of the form
ϕ(z, x, t) = f (z)cos(kx − ωt),
Pulsars and Supernova I
4 SURFACE WAVES 48
where ω is the angular frequency and k is the wavenumber of a wave
prorogating in x direction.
From Equation (4.2), one finds
d
2
f
dz
2
− k
2
f = 0
The solution of this equation, which vanishes at z → −∞ is ∼ exp(kz).
Therefore, we can write
ϕ(z, x, t) = A exp(kz)cos(kx − ωt)
where A is a constant. Note that the wave amplitude decreases exponen-
tially with depth, hence the waves are surface waves.
Pulsars and Supernova I
4 SURFACE WAVES 49
4.2 Dispersion relation, group and phase velocities
Using Equation (4.1) and substituting the solution ϕ(z, x, t), we can
find
ω
2
= k g (4.3)
Equation (4.3) is the dispersion relation between the wavenumber k and
the wave frequency ω for surface gravity waves. As we see from the
dispersion relation the presence of g plays the key role.
The phase and group velocities of the surface gravity waves
phase velocity : v
phase
≡
ω
k
=
r
g
k
=
s
gλ
2π
group velocity : v
grou p
≡
dω
dk
=
1
2
r
g
k
=
1
2
s
gλ
2π
where λ = 2π/k is the wavelength.
Pulsars and Supernova I
4 SURFACE WAVES 50
The dispersion relation helps us to understand some important prop-
erties of waves. In the above, we assumed deep water while deriving the
wave dispersion relation. When the water becomes shallow the wave-
length should decrease. At the same time, the wave energy flux v
grou p
A
2
should remain constant along the wave path. So when v
grou p
decreases,
the wave amplitude A must increase (Figure 12).
Surface gravity waves are common for stellar surfaces. For example
the solar g-mode or gravity waves are density waves confined to the in-
terior of the Sun below the convection zone. f-mode or surface gravity
waves are also gravity waves occurring at or near the photosphere.
The surface gravity waves are neither longitudinal nor transverse. In-
deed, using our solution for ϕ(z, x, t), we can calculate two non-vanishing
Pulsars and Supernova I
4 SURFACE WAVES 51
Figure 12: Shallow water waves off the coast of Lima, Peru.
components of velocity using
−→
v = −∇ϕ
v
x
= −
∂ϕ
∂x
= +Ak exp(kz) sin(kx − ωt)
v
z
= −
∂ϕ
∂z
= −Ak exp(kz) cos(kx − ωt)
Hence, the wave on the surface is the superposition of transverse v
z
6= 0
Pulsars and Supernova I
4 SURFACE WAVES 53
Figure 13: Animation of surface gravity waves (from Physics-Animation.com).
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 54
5 Perturbations at a two-fluid interface
Figure 14: Perturbation ξ at the
interface of two fluids.
Similar to the previous case (fluid-
empty space interface), let us consider the
interface of two incompressible and irro-
tational fluids. Writing the fluid velocity
−→
v = −∇ϕ and introducing the velocity po-
tential ϕ, we find (see previous lecture)
∇
F(t)
z }| {
µ
−
∂ϕ
∂t
+
1
2
−→
v
2
+
p
ρ
+ ψ
¶
= 0
where F(t) is a constant in space, and ψ =
gz is the gravitational potential.
Let us also assume the two fluids (see Figure 14) to have uniform
velocities U and U
0
in x-direction, and surface z = 0 (horizontal plane)
separates the two fluids with densities ρ (fluid below) and ρ
0
(fluid above).
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 55
Our aim is to find evolution of the perturbation ξ(x, z, t).
From the equation for F(t), we can express the pressures:
above: p
0
= − ρ
0
µ
−
∂ϕ
0
∂t
+
1
2
v
0
2
+ gz
¶
+ const’(t)
below: p = − ρ
µ
−
∂ϕ
∂t
+
1
2
v
2
+ gz
¶
+ const (t)
where all values with
0
are for the fluid above the interface.
Equating two pressure expressions, we derive at z = ξ
ρ
µ
−
∂ϕ
∂t
+
1
2
v
2
+ gξ
¶
= ρ
0
µ
−
∂ϕ
0
∂t
+
1
2
v
0
2
+ gξ
¶
+ K(t) (5.1)
where K(t) is a new constant in space.
Although K(t) can, in principle, be a function of time, here it has to
be constant due to boundary condition that the perturbations vanish away
from the interface at all times.
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 56
Let us find K using the boundary without perturbation. For unper-
turbed conditions, i.e. ξ = 0, we have
ϕ = 0, ϕ
0
= 0,
v = U, v
0
= U
0
.
Using the pressure balance p
0
= p (see Equation 5.1), one finds
K = ρ
U
2
2
− ρ
0
U
02
2
Let us now consider the case of small ξ. Given that the total velocities
are
−→
v = U
−→
e
x
− ∇ϕ,
−→
v
0
= U
0
−→
e
x
− ∇ϕ
0
, then we can Taylor expand v
2
and
v
02
retaining only zero and first order terms
v
2
=
¡
U
−→
e
x
− ∇ϕ
¢
2
= U
2
− 2U
∂ϕ
∂x
+ ...
v
02
=
¡
U
0
−→
e
x
− ∇ϕ
0
¢
2
= U
02
− 2U
0
∂ϕ
0
∂x
+ ...
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 57
where
−→
e
x
is the unit vector in x direction. Substituting these expansions
into the equation for pressure balance (Equation 5.1), we can write:
ρ
µ
−
∂ϕ
∂t
−U
∂ϕ
∂x
+ gξ
¶
= ρ
0
µ
−
∂ϕ
0
∂t
−U
0
∂ϕ
0
∂x
+ gξ
¶
We are not going to look for a solution for ξ, but only seek the disper-
sion relation. Hence let us consider the small perturbations ξ, ϕ, ϕ
0
that
are plane wave moving along x
ξ =A exp(−iωt + ikx),
ϕ =C exp(−iωt + ikx + kz),
ϕ
0
=C
0
exp(−iωt + ikx − kz),
where the sign before kz was chosen in such a fashion that the perturba-
tions vanish as we go far away from the interface.
If ϕ and ϕ
0
are perturbations of the velocity potential, then the total
velocity potential (including velocities U and U
0
) can be written by re-
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 58
defining the potential:
−U x + ϕ −→
total
z}|{
ϕ
−U
0
x + ϕ
0
−→
total
z}|{
ϕ
0
The Lagrangian derivative of the displacement ξ is
2
dξ
dt
=
∂ξ
∂t
+U
∂ξ
∂x
On the other hand the vertical component of the fluid velocity
dξ
dt
= v
z
= −
∂ϕ
∂z
2
Note that because of the finite U we cannot write
dξ
dt
'
∂ξ
∂t
as we did in the previous lecture.
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 59
hence
below the surface: −
∂ϕ
∂z
=
∂ξ
∂t
+U
∂ξ
∂x
above the surface: −
∂ϕ
0
∂z
=
∂ξ
∂t
+U
0
∂ξ
∂x
Substitution of Fourier components for ξ, ϕ
0
, ϕ into the above equa-
tions and into the equation for pressure balance yields
iA(−ω + kU) = −kC
iA(−ω + kU
0
) = +kC
0
ρ(−iC(−ω + kU) + gA) = ρ
0
(−iC
0
(−ω + kU
0
) + gA)
The system of these three equations can be solved for ω(k) eliminating
arbitrary constants A, C, and C
0
and we finally find:
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 60
ρ(−ω + kU)
2
+ ρ
0
(−ω + kU
0
)
2
= k g(ρ − ρ
0
)
This is called dispersion relation for the two-fluid interface problem.
The equation is quadratic and can be solved to find explicit expres-
sion for ω(k):
ω(k)
k
=
ρU + ρ
0
U
0
ρ + ρ
0
±
·
g
k
ρ − ρ
0
ρ + ρ
0
−
ρρ
0
(U −U
0
)
2
(ρ + ρ
0
)
2
¸
1/2
(5.2)
Let us use this solution (5.2) to investigate a few special cases.
Example: Let us consider a special case of surface gravity waves but
single fluid, e.g. U = U
0
= 0 and ρ > ρ
0
= 0, then we have from Equation
(5.2):
ω(k)
k
= ±
r
g
k
which is exactly as per last lecture (c.f. Equation 4.3).
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 61
5.1 Rayleigh-Taylor instability
Let us consider two fluids to be at rest so that equilibrium holds
U = U
0
= 0
but we now assume that we have heavier fluid above a lighter fluid, i.e.
ρ < ρ
0
.
Intuitively, one feels that such equilibrium will be unstable.
The solution of the dispersion relation (5.2) with U = U
0
= 0 is
ω(k)
k
= ±
s
g
k
ρ − ρ
0
ρ + ρ
0
for ρ < ρ
0
, Im(ω/k) 6= 0. Hence ω = Re(ω) + i Im(ω) and the displacement
becomes
ξ = A exp(−iωt + ikx) = A exp
(
Im
(
ω
)
t
)
exp(−i Re
(
ω
)
t + ikx),
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 62
For Im(ω) > 0, we have an exponential growth of initially small perturba-
tion ξ with time, i.e. instability.
This instability is called Rayleigh-Taylor instability and the dispersion
relation says that the criteria for this instability is
ρ < ρ
0
(5.3)
Rayleigh (1883) did for fluids in a gravitational field. Taylor (1950)
adopted the problem for the situation of accelerating fluids.
The animation illustrates the process (see Figure 15). The instability
is crucial in supernova explosions, where dense plasma is on top of the
less dense gas after the iron core collapse in a massive star. RT ‘fingers’
are evident in supernova nebulaes (Figure 16).
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 63
Figure 15: Raleigh-Taylor instability is illustrated via simulations. From
http://www.maths.manchester.ac.uk/∼mheil/MATTHIAS/Fluid-Animations/Fluid-
Animations.html
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 64
Figure 16: Composite image of the Crab nebula from NASA webpage. Note the
structure of the expanding material.
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 65
5.2 Kelvin-Helmholtz instability
Let us now consider what happens if U and U
0
are not zero. Let us
assume that ρ > ρ
0
, so the system is Rayleight-Taylor stable.
When the expression winthin the square root in Equation 5.2 is nega-
tive
g
k
ρ − ρ
0
ρ + ρ
0
−
ρρ
0
(U −U
0
)
2
(ρ + ρ
0
)
2
< 0
the frequency ω has imaginary part, so an instability appears when
ρρ
0
(U −U
0
)
2
>
g
k
(ρ
2
− ρ
02
) (5.4)
This instability is know as the Kelvin-Helmholtz instability (Helmholtz
1868, Kelvin 1871)
and causes an interface between two fluids to ‘wrinkle’ (see animation
Fig 17). Kelvin-Helmholtz-Instability is characterized by two flows running
in opposite directions. At the intersection of both flows, turbulence is nat-
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 66
urally introduced (see Figure 18). Note that tracer articles were added to
highlight the dynamics (see animation Fig 17) involved in this process.
We notice from (5.4) that Fourier components satisfying
k > g
(ρ
2
− ρ
02
)
ρρ
0
(U −U
0
)
2
become unstable, i.e. the shorter wavelength.
If g = 0, instability criteria (5.4) is always satisfied.
Both KH and RT instabilities play important role in the evolution of
supernova, the instabilities tend to lead to developed turbulence. This
has important consequences for the flame in Type Ia supernova changing
in the character of the burning.
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 67
Figure 17: Simulations and animation by Christoph Federrath, Max-Planck-Institute for
Astronomy in Heidelberg
Pulsars and Supernova I
5 PERTURBATIONS AT A TWO-FLUID INTERFACE 68
Figure 18: Left: KH instability in Saturn atmosphere. Right: Kelvin-Helmholtz-
Instability in M87 extragalactic jet (Hubble Telescope).
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 69
6 Density fluctuations and Jeans instability
6.1 Acoustic or density waves
Let us consider a homogeneous perfect gas of density ρ
0
and pressure
p
0
. Hydrodynamic equations (2.2,2.5) for a compressible gas (∇
−→
v 6= 0 )
without gravity can be written
∂ρ
∂t
+ ∇(ρ
−→
v ) = 0 , (6.1)
ρ
µ
∂
∂t
+ (
−→
v · ∇)
¶
−→
v = −∇p . (6.2)
Suppose we have a perturbation of pressure p = p
0
+ δp and the corre-
sponding density perturbation being ρ = ρ
0
+ δρ.
We now want to find out how these perturbations will evolve. These
perturbations give rise to a velocity field
−→
v = δ
−→
v , where we assume no
flows in the gas
−→
v
0
= 0.
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 70
Let us now substitute these expressions into the continuity equation:
∂
∂t
(ρ
0
+ δρ) + ∇
¡
(ρ
0
+ δρ)δ
−→
v
¢
= 0
Since ρ
0
= constant, one finds
∂δρ
∂t
+ ρ
0
∇ · δ
−→
v = 0, (6.3)
where, again, we kept only linear terms, i.e. we linearized the equation.
From the Euler’s equation (6.2), one finds
(ρ
0
+ δρ)
µ
∂
∂t
+ (δ
−→
v · ∇)
¶
δ
−→
v = −∇(p
0
+ δp)
Linearizing this equation, we obtain the following
ρ
0
∂δ
−→
v
∂t
= −∇δp (6.4)
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 71
To solve these two linear equations (6.3,6.4), we need an expression re-
lating δp and δρ. Such relation can be found using the equation of state.
For a perfect gas, we note that the motion is adiabatic (no time to
exchange energy with the surroundings), hence the small change in pres-
sure δp is related to a small change in density δρ.
δp =
∂p
∂ρ
¯
¯
¯
¯
p=p
0
δρ
Using this relation between pressure and density perturbations, the
system of linearized equations becomes
∂δρ
∂t
+ρ
0
∇ · δ
−→
v = 0
∂δ
−→
v
∂t
= −
1
ρ
0
∂p
∂ρ
¯
¯
¯
¯
p=p
0
∇δρ
Further, let us differentiate the second equation above with respect to
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 72
time,
∂
∂t
and insert
∂δρ
∂t
from the first equation
∂
2
δ
−→
v
∂t
2
= −
1
ρ
0
∂p
∂ρ
¯
¯
¯
¯
p=p
0
∇
∂δρ
∂t
=
∂p
∂ρ
¯
¯
¯
¯
p=p
0
4δ
−→
v
Finally, we obtain a wave equation for δ
−→
v :
∂
2
δ
−→
v
∂t
2
− c
2
s
4δ
−→
v = 0, (6.5)
where the constant c
2
s
=
∂p
∂ρ
¯
¯
¯
p=p
0
is the sound speed.
3
Using adiabatic gas law p = p
0
(ρ/ρ
0
)
γ
, the sound speed can be ex-
pressed via the parameters of unperturbed gas (fluid)
c
2
s
=
∂p
∂ρ
¯
¯
¯
¯
p=p
0
= γ
p
0
ρ
0
= γ
k
B
T
m
,
3
Show that a wave equation can be derived for density fluctuations δρ too.
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 73
where we used the ideal gas law p
0
= ρ
0
k
B
T/m, k
B
is the Boltzmann
constant , m is the gas particle mass.
Let us find a dispersion relation ω(k) for sound waves. Consider Fourier
components of velocity perturbation
δ
−→
v ∝ exp(−iωt + i
−→
k
−→
r )
Substituting it into the wave equation (6.5), one finds
ω
2
= c
2
s
k
2
(6.6)
which is the dispersion relation for sound waves in a homogeneous
medium.
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 74
6.2 Jeans instability
In a more general case, the gas or a fluid is a self-gravitating media.
Following Jeans (1902), let us consider perturbations in a uniform infinite
gas and introduce the gravitational potential ψ = ψ
0
+ δψ, where ψ
0
is
unperturbed and δψ is a perturbation of gravitational potential.
The unperturbed part should satisfy the hydrostatic equation, i.e. d
−→
v /dt =
0:
0 = −∇p
0
− ρ
0
∇ψ
0
where ψ
0
can be determined by the Poisson’s equation, which relates the
gravitational potential and the mass density
4ψ
0
= 4πGρ
0
where G is the Newton’s constant of gravitation. However, these equa-
tions do not have nontrivial uniform solutions for infinite gas. Indeed, as-
suming ψ
0
= constant, one immediately finds that ρ
0
= 0.
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 75
Nevertheless, we still want to use perturbation analysis and assume
that there is quasi-equilibrium or equilibrium due to additional forces, e.g.
rotating fluid. This is sometimes referred as ‘Jeans swindle’.
4
The perturbation of the equation of continuity gives (retaining only first
order terms)
∂δρ
∂t
+ ρ
0
∇δ
−→
v = 0
Linearizing the Euler’s equation with gravity included and subtracting the
equilibrium ∇p
0
= −ρ
0
∇ψ
0
, we find
ρ
0
∂δ
−→
v
∂t
= −c
2
s
∇δρ − ρ
0
∇δψ
Similarly subtracting 4ψ
0
= 4πGρ
0
in Poisson’s equation, one obtains
4δψ = 4πGδρ
4
For more detailed mathematical considerations see .
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 76
Let the perturbations vary as exp(−iωt + i
−→
k
−→
r ), i.e. plane waves :
δρ −→δρ exp(−iωt + i
−→
k
−→
r )
δ
−→
v −→δ
−→
v exp(−iωt + i
−→
k
−→
r )
δψ −→δψexp(−iωt + i
−→
k
−→
r )
then one can write the system of linear equations
− ωδρ + ρ
0
−→
k δ
−→
v = 0
− ρ
0
ωδ
−→
v = −c
2
s
−→
k δρ − ρ
0
−→
k δψ
− k
2
δψ = 4πGδρ
Combining these three equations and eliminating δρ, δψ, δ
−→
v , we readily
find the dispersion relation for sound waves in self-gravitating fluid
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 77
ω
2
= c
2
s
k
2
− 4πGρ
o
= c
2
s
(k
2
− k
2
J
) (6.7)
where k
2
J
= 4πGρ
o
/c
2
s
. For all k < k
J
, ω has to be imaginary, so we
have instability. This is so-called Jeans instability.
In other words, unstable perturbations of the wavelength (size)
λ > λ
J
=
2π
k
J
= c
s
s
π
Gρ
0
have self-gravity that can overpower the excess pressure, so the pertur-
bations grow.
The corresponding critical mass, Jeans mass, a spherical volume hav-
ing a diameter equal to the Jeans wavelength becomes
M
J
=
4π
3
ρ
0
µ
λ
J
2
¶
3
=
π
5/2
6
c
3
s
G
3/2
ρ
1/2
0
Pulsars and Supernova I
6 DENSITY FLUCTUATIONS AND JEANS INSTABILITY 78
Using that c
2
s
= γk
B
T/m, we find
λ
J
=
s
γπk
b
T
mGρ
0
and M
J
=
π
5/2
6ρ
1/2
0
µ
γk
B
T
mG
¶
3/2
Example: Assuming we have 1 hydrogen atom per 1 cm
3
at tempera-
ture 100 K, Jeans mass is around 10
36
kg.
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 79
7 Shock waves and supernova envelope expansion
7.1 Wave steepening into shock wave
Figure 19: One dimen-
sional wave.
When the wave amplitude is not small, it is
no longer valid to break up all fluid variables into
the unperturbed and perturbed parts as we did
for e.g. density waves.
Let us go back to Euler equation and con-
sider one-dimensional for density waves, e.g.
−→
v = (v,0, 0) (Figure 19). Euler equation be-
comes
∂v
∂t
+ v
∂v
∂x
= −
1
ρ
∂p
∂x
(7.1)
To solve the full problem, we have to combine equation (7.1) with the other
hydrodynamic equations. To start, let us consider simpler equation:
∂v
∂t
+ v
∂v
∂x
= 0. (7.2)
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 80
To solve equation (7.2), we consider curves
dx
dt
= v in (x, t) plane. The
total time derivative of v along a curve in (x, t) plane is given by
d
dt
v(x(t), t) =
∂v
∂t
+
∂v
∂x
dx
dt
Comparing to Equation (7.2), one finds that
dv
dt
= 0 along any curve
dx
dt
= v.
These curves are called characteristics or characteristic curves. The
method to solve partial differential equations (PDEs) of the first order via
a set of ordinary differential equations (ODEs) is called method of charac-
teristics.
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 81
7.2 Method of characteristics
Let us consider F(x
1
, x
2
, x
3
,... , x
N
) a function of N variables and a
partial differential equation of the first order
a
1
∂F
∂x
1
+ a
2
∂F
∂x
2
+ a
3
∂F
∂x
3
+ ... + a
N
∂F
∂x
N
= 0
where a
1
, a
2
, a
3
,... , a
N
are functions of x
1
, x
2
, x
3
,... , x
N
but not of F.
The solution of this equation via method of characteristics can be
found from the following system of ordinary differential equations (ODEs):
dx
1
a
1
=
dx
2
a
2
= .. . =
dx
N
a
N
Given initial (or boundary) conditions on F, one can integrate the system
of ODEs to find F.
5
5
For additional details on the method see
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 82
As an example, let us consider the following PDE
∂v
∂t
+ v
0
∂v
∂x
= 0, (7.3)
where v(x, t) is an unknown function subject to the following initial condi-
tion v(x, t = 0) = exp(−x
2
), and v
0
is a constant. Following the method of
characteristics, we write
dt
1
=
dx
v
0
=⇒
Z
x
x
0
dx = v
0
Z
t
0
dt =⇒ x = x
0
+ v
0
t
where x
0
is the constant of integration. Moreover, x
0
defines x and which
characteristic curve you are on at t = 0. Hence we can write
x
0
= x − v
0
t
and the general solution of equation (7.3) is arbitrary function of x
0
and
hence arbitrary function of x−v
0
t. The choice of the function is, in general,
determined by either initial or/and boundary conditions.
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 83
For a given initial condition v(x, t = 0) = exp(−x
2
), we have
v(x, t) = exp
¡
−(x − v
0
t)
2
¢
which is a particular solution of equation (7.3). This solution says that
initial ‘wave’ moves to the right when v
0
is positive, e.g. the wave has a
constant amplitude along our characteristic curves.
Figure 20: Steepening of a wave. Sketch of the solution for three time moments
0 < t
1
< t
3
.
In case of non-linear equation (7.2), the solution has a similar form
6
v(x, t) = v(x − vt)
6
Verify that this is the solution of Equation (7.2)
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 84
which says that the larger velocity v, the larger amplitude of the wave. The
solution can be seen in figure 20. Hence, the nonlinear term in Euler’s
equation, (
−→
v ·∇)
−→
v , determines the steepening of a wave. The steepened
wave-front is known as a shock and such waves are called shock waves.
So far, we have ignored the pressure term ∇p in our Euler equation
(7.2). Let us know retain this term
∂v
∂t
+ v
∂v
∂x
= −
1
ρ
∂p
∂x
,
and write 1-dimensional continuity equation
∂ρ
∂t
+
∂vρ
∂x
= 0 .
These two equations can be re-written in more symmetric form. When
p = p(ρ) and introducing density dependent sound speed c
2
s
(ρ) = ∂p/∂ρ,
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 85
and χ = lnρ to shorten notations, one finds
∂v
∂t
+ v
∂v
∂x
= −c
2
s
∂χ
∂x
,
∂χ
∂t
+ v
∂χ
∂x
= −
∂v
∂x
(7.4)
Now if we multiply the continuity equation by c
s
(ρ), we can combine these
two equations (7.4) to obtain
∂v
∂t
+ (v + c
s
)
∂v
∂x
= −c
s
µ
∂χ
∂t
+ (v + c
s
)
∂χ
∂x
¶
One can see now that the disturbance moves with speed u + c
s
. If the
wave amplitude is small, the small u can be neglected, so the density
fluctuations prorogate with constant speed c
s
. However, the larger the
amplitude and the velocity u, the stronger the nonlinearity. The character-
istic velocities
dx
dt
= v ± c
s
Pulsars and Supernova I
7 SHOCK WAVES AND SUPERNOVA ENVELOPE EXPANSION 86
give the trajectories of the fluid elements along which the quantities
v ±
Z
c
s
ρ
dρ
are conserved and are also known as Reimann invariants. Note that in
case of linear waves v was small.
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 87
8 Shocks jump conditions and envelope expansion
8.1 Structure of shock waves
Figure 21: A schematic dia-
gram of a shock wave.
A shock wave is a region of small thick-
ness over which different fluid dynamical
variables change rapidly. Let us consider a
mathematical discontinuity across which dif-
ferent HD variables jump.
The figure (21) shows a shock propagat-
ing into an undisturbed medium of density ρ
1
and pressure p
1
in the frame in which the
shock is at rest.
Let us consider 1D hydrodynamic equations (2.2,2.5,3.2) and re-write
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 88
them in conservative form
∂ρ
∂t
+ ∇
x
(ρv) = 0,
∂ρv
∂t
+ ∇
x
(ρv
2
+ p) = 0,
∂E
∂t
+ ∇
x
([E + p]v) = 0,
(8.1)
where E = ρε + ρv
2
/2 is the the total energy per unit volume and x is
along the shock propagation. We will consider adiabatic process, so that
p ∝ ρ
γ
, where γ is the adiabatic index [recall the sound waves section].
The equation of state for a perfect gas p = (γ − 1)ρε closes the system
(8.1).
From Equations (8.1) under steady conditions, the mass flux, the mo-
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 89
mentum flux and the energy flux should be conserved:
ρv = const,
ρv
2
+ p = const,
(E + p)v = const,
At the discontinuity, the conservation of the fluxes gives the relation be-
tween fluid parameters upstream and downstream from the shock:
ρ
1
v
1
=ρ
2
v
2
,
ρ
1
v
2
1
+ p
1
=ρ
2
v
2
2
+ p
2
,
1
2
v
2
1
+
γp
1
(γ − 1)ρ
1
=
1
2
v
2
2
+
γp
2
(γ − 1)ρ
2
,
(8.2)
where
1
denotes upstream region.
If we eliminate pressure p
2
and velocity v
2
in the downstream region,
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 90
one can find the shock compression ratio
ρ
2
ρ
1
=
(γ + 1)M
2
2 + (γ − 1)M
2
(8.3)
where
M ≡
v
1
c
s1
where c
s1
=
p
γp
1
/ρ
1
is the sound speed in the upstream medium 1. The
dimensionless quantity M is known as the Mach number. For shocks
M > 1, otherwise sound waves produced by the shock will prorogate into
medium 1 faster than shock, and shock will disappear.
Indeed Equation (8.3) can be written
ρ
2
ρ
1
=
(γ + 1)
2/M
2
+ (γ− 1)
when M → 1, ρ
2
/ρ
1
→ 1, i.e. shock disappears;
when M > 1, ρ
2
/ρ
1
> 1 and growths with M.
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 91
Because ρv = const, a shock involving a stronger compression has to
move faster.
Eliminating ρ
2
and v
2
, the pressure ratio is
p
1
p
2
=
(γ + 1) − (γ − 1)ρ
2
/ρ
1
(γ + 1)ρ
2
/ρ
1
− (γ− 1)
Equations relating variables before and after shock are known as Rankine-
Hugoniot conditions, (Rankine, 1870; Hugoniot, 1889).
The density compression has a maximum limiting value for M → ∞:
ρ
2
ρ
1
M→∞
−−−−→
γ + 1
γ − 1
Specifically for monoatomic gas, γ = 5/3, the maximum density compres-
sion is ρ
2
/ρ
1
= 4. For diatomic gas, e.g. air, γ = 1.4, so the ratio becomes
ρ
2
/ρ
1
= 6.
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 92
8.2 Supernova remnant expansion
Three stages of supernova remnant expansion:
• Free expansion, i.e. ejecta expands without deceleration r ∝ t
• Adiabatic or Sedov-Taylor phase, energy is conserved
• Radiative phase, i.e. dissipation of energy into inter-stellar medium
The free expansion phase (see e.g. animation of 1987A expansion) is
independent of the nature of the supernova explosion, without decelera-
tion. The evolution only depends on E the initial energy. At this stage the
velocity of ejected shell is 10
4
km/s and the mass swept-up negligible in
comparison with the ejecta supernova mass.
When the amount of gas swept up becomes comparable to the ejecta
mass, the kinetic energy is transferred to the swept up gas.
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 94
8.3 Spherical blast waves
Figure 23: Infra-red image is SN 1572, often called "Tycho’s Supernova." From NASA.
A sudden explosion within a localised region of gas leads to a blast
wave spreading from the centre of the explosion. For example, Tycho’s
supernova (Figure 23) shows a blast wave, which appears almost spheri-
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 95
cal.
The front of the blast is obviously a shock wave across which the
Rankie-Hugoniot conditions must be hold. Sedov (1946,1959) and Taylor
(1950) solved the problem of a blast evolution assuming blast evolution in
a self-similar fashion.
Let us consider an energy E is suddenly released in an explosion in
ambient medium density ρ
0
, and pressure p
0
∼ 0.
Let λ be a scale parameter giving the size of the blast wave at time t
after after the explosion. λ may depend on t , E and ρ
0
and the only way
to combine them is
λ =
µ
Et
2
ρ
0
¶
1/5
(8.4)
The radius of the shell of gas inside the spherical blast has to evolve in
the same way as λ, so
r
s
(t) = ζ
0
µ
Et
2
ρ
0
¶
1/5
(8.5)
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 96
where ζ
0
is a dimensionless constant
7
.
The velocity of expansion is
v
s
(t) =
dr
s
(t)
dt
=
2
5
r
s
(t)
t
=
2
5
ζ
0
µ
E
ρ
0
t
3
¶
1/5
Hence the size of the spherical blast increases as ∝ t
2/5
and the velocity
front goes down as ∝ t
−3/5
.
This solution is known as Sedov-Taylor expansion or the ‘Blast Wave’,
which is an adiabatic expansion phase in the life cycle of supernova. A
typical supernova ejects about 1M
¯
with initial velocity ∼ 10
4
km/s, so
E ∼ 10
44
J, assuming density ρ
0
= 2 × 10
−21
kg/m
3
, we have (assuming
ζ
0
= 1)
r
s
(t) ' 0.3
µ
t
year
¶
2/5
parsec v
s
(t) ' 10
5
µ
t
year
¶
−3/5
km/s
7
ζ
0
is a function of the ratio of the specific heat of air at constant pressure to the specific heat of gas at
constant volume. it appears close to 1 e.g. for air ≈ 1.0 − 1.1.
Pulsars and Supernova I
8 SHOCKS JUMP CONDITIONS AND ENVELOPE EXPANSION 97
where t is in years. The equations above (and Sedov-Taylor approxima-
tion) are valid for time less than 10
5
years (due to cooling). Although the
equations are incorrect (due to initial conditions) for t < 100 years.
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 98
9 Particle acceleration in SNRs
9.1 Supernova Remnants (SNRs) and cosmic rays
Figure 24: Cosmic ray
spectrum.
It could have been the start of the story...
C.T.R. Wilson observed radiation with ionization
chamber experiment (1902) in a railway tunnel
near Peebles, Scotland. However, concluded that
the radiation cannot be cosmic.
Hence, the story starts in 1912. Austrian physi-
cist V. Hess measured radiation level in a balloon
experiment. He took a balloon to 5 km and ob-
served what he termed as ‘penetrating radiation’
coming from space. Hess received the 1936 No-
bel Prize for his discovery of cosmic rays (Figure
24).
Using a cloud chamber invented by C.T.R. Wilson, Dimitry Skobelzyn
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 99
(1929) observed the first ghostly tracks left by cosmic rays.
Fermi (1949) explained the acceleration of cosmic-ray particles by re-
flection on moving magnetic clouds. This model naturally explains inverse
power-law distributions.
There are direct observations of energetic particles in supernova rem-
nants, e.g. X-ray Synchrotron emission, or TeV Emission (Figure 25).
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 101
9.2 Fermi acceleration - kinematics
Consider a relativistic particle reflecting from a moving ‘mirror’. If
−→
v
s
is
the velocity of the shock structure (e.g. magnetic field acting as a mirror)
then the change in particle energy E for one collision is
∆E = −2E
−→
v
s
·
−→
v
∥
c
2
where c is the speed of light. Assuming isotropic distribution of energetic
ultra-relativistic particles v ' c, we find that each time shock gives
〈∆E〉
E
=
4
3
v
s
c
to an energetic particle.
Let us now assume that there is a large number of shocks (or magnetic
mirrors) moving randomly in all directions (e.g. Fermi acceleration model).
The probability of head-on collision is proportional to v + v
s
, while the
probability of overtaking collision is proportional to v − v
s
.
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 102
Taking into account the probabilities the average gain per two colli-
sions (head-on and tail-on) becomes
〈∆E〉 =
v + v
s
2v
∆E −
v − v
s
2v
∆E ≈ 2
v
2
s
c
2
E
So there is an average gain is 〈∆E〉 > 0.
The energy change proportional to the velocity of the shock is first
order Fermi acceleration; proportional to the square is called second order
of Fermi acceleration (original Fermi model).
The average rate of energy gain can be written
dE
dt
=
〈∆E〉
τ
= 2
v
2
s
τc
2
E
where we introduced ‘collisional’ time τ. So the energy increasing expo-
nentially in time E(t) = E
0
exp(2v
2
s
t/τc
2
).
Let E = bE
0
be the average energy of the particle after one collision
and P be the probability that the particle remains within the acceleration
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 103
region after one collision. Then after k collisions, there are N = N
0
P
k
particles with energies above E = b
k
E
0
. Eliminating k one finds
N
N
0
=
µ
E
E
0
¶
ln P/ ln b
therefore we find for the distribution
N(E) ∝ E
−1+ln P/ ln b
,
e.g. the energy distribution of particles has a power-law form.
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 104
9.3 Canonical model of shock acceleration
Figure 26: Energetic particle diffusion across a shock.
Let us consider the idealised problem of particle acceleration by a
shock wave of plain (1D) geometry propagating in a medium containing
small-scale inhomogeneities of magnetic field which scatter fast particles
(Figure 26) or animation (Figure 27)
As long as the scattering of fast particles N(E, x) is strong enough to
ensure isotropy assumption, we can write the kinetic equation for the fast
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 105
(relativistic) particles in the form of stationary diffusion-advection equa-
tion
8
:
∂
∂x
vN =
∂
∂x
D
∂N
∂x
+
1
3
∂v
∂x
∂
∂E
EN (9.1)
where N(E, x) is the energy distribution of particles, v(x) - velocity of a
stream, D is spatial diffusion of energetic particles. We also assumed for
simplicity that particle speed is ∼ c >> v
s
, c is the speed of light. Therefore
E ' pc, where p is the momentum of energetic particles.
Let us integrate equation (9.1) over x, using the following boundary
conditions
N(E) = N
1
(E) and v = v
1
at x → −∞
N(E) = N
2
(E) and v = v
2
at x → +∞
We want to find N
2
(E) - the unknown distribution of particles, while N
1
is
the initial (not accelerated) distribution upstream.
8
see for example review Treumann, R. A.; Jaroschek, C. H.
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 107
After the integration one finds:
v
2
N
2
(E) − v
1
N
1
(E) + 0 − 0 =
1
3
(v
2
− v
1
)
∂
∂E
EN
2
(E, x)
so the fast particle distribution downstream, N
2
(E) is given by an ordinary
differential equation
E
dN
2
dE
+
R + 2
R − 1
N
2
=
3R
R − 1
N
1
(E) (9.2)
where R = v
1
/v
2
= ρ
2
/ρ
1
is the shock compression ratio.
Since R > 1, the general solution of Equation (9.2) is
N
2
(E) =
3
R − 1
E
−δ
Z
E
E
0
N
1
(E
0
)E
0δ−1
dE
0
+ const E
−δ
where δ = (R + 2)/(R − 1) is the spectral index of particle distribution.
When N
1
(E) = N
0
δ(E − E
0
), e.g. injection of particles of energy E
0
,
one finds that
N
2
(E) ∝ E
−δ
, E > E
0
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 108
i.e. inverse power-law with spectral index dependent on compression ra-
tio.
For strong shocks we have R = 3−4, so the spectral index δ = 2− 2.5.
In cosmic rays observed at the Earth, the spectrum of cosmic-ray ions
is un-broken power-law for energies 10
9
− 10
16
eV with δ ' 2.6.
Supernova shocks are the one of the most popular mechanisms to
explain the data. In addition, X-ray (Figure 28) and radio-emission from
supernova remnants indicate the presence of energetic particles.
Pulsars and Supernova I
9 PARTICLE ACCELERATION IN SNRS 109
Figure 28: Multiwavelength composite image of the remnant of Tycho’s supernova,
SN 1572. Note X-ray emission.
Pulsars and Supernova I
10 MASSIVE STAR EVOLUTION 110
10 Massive star evolution
Pulsars and Supernova I
Type II supernova progenitors
==> Type II and Type Ib/c are likely to
represent the explosions of massive stars
The Hubble Space Telescope's Advanced Camera for Surveys acquired these
images of a small field inside M51. The left image, taken in January, shows
the field prior to the eruption of Supernova 2005cs. The right image, taken
on July 11th, shows the magnitude-14 supernova.
=> 2005cs supernovae progenitor is a red supergiant of 7-10 solar masses
10 MASSIVE STAR EVOLUTION 111
Pulsars and Supernova I
Evolution of massive stars: late phases
The pre-explosive life of a massive star is governed by
simple principles (recall the A2/A3 courses on stellar
evolution).
Pressure [radiation + ideal gas + degenerate electrons
(
later stages
) ] = the force of gravity
where
P is the pressure.
Assuming
what is
the central density versus temperature?
Suggestions?
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Evolution of massive stars: late phases
The pre-explosive life of a massive star is governed by
simple principles (recall the A2/A3 courses on stellar
evolution).
Pressure [radiation + ideal gas + degenerate electrons
(
later stages
) ] = the force of gravity
where
P is the pressure.
For a polytrophic models central pressure and
density have simple relation
where
φ(n) is 4.9, 10.7, and 16.15 for n=0,1.5, and 3
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Central temperature of massive stars
It is convenient to define an abundance variable,
Yi
The ideal gas pressure
For a given
polytropic index
(recall politropic models)
it follows:
So the evolution of a star has a simple
central
temperature and density
dependency through its
entire evolution.
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Simulations: central temperature of massive stars
Evolution of the central temperature and density in
stars of 15 and 25
solar masse
from birth as hydrogen-
burning stars until iron-core collapse
(From Woosley, Heger, and Weaver, 2002)
Note nonmonotonic
behaviour is observed
when nuclear
fuels are ignited.
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Stellar evolution as seen from its core (from Prialnik, 2000)
p-p and CNO cycle
Helium burning into carbon
Carbon burning
Si burning
Oxygen burning
degeneracy
photodisintegration
Electron-positron pair
production
Schematic illustration
of the evolution of
stars according to
their central
temperature-density
tracks
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Fate of 1-8 solar masses stars
As a consequence of the mass loss, stars with initial
mass in the range 1-8 solar masses shed and are left
with C-O cores of mass between 0.6-1.1 solar masses.
These cores will subsequently develop into white dwarfs.
/although degenerate these stars are in no danger of
explosion/
Infrared image of the
Helix Nebula, taken by
the Spitzer space
telescope, 2007
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Fate of stars heavier than
~8 solar masses
Common properties:
-Electrons in the
core do not become
degenerate until the
final burning stages
-The luminosity
(close to Eddington
critical limit)
remains almost
constant
- Mass loss plays an
important role
Figure: Eta Carinae
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Stars ~8-11 solar masses
Common properties:
Stars below a certain mass (around 8 solar masses) develop
degenerate core and do not ignite carbon burning
In stars heavier than ~11 solar masses, carbon and neon both ignite
nondegenerately near the centre of the star.
Exact evolution are dependent on element abundances and
treatment of convection. The carbon-oxygen core and later oxygen-
neon core is surrounded by thermally pulsating thin helium shell. If
“superwind” develops these stars might develop into oxygen-neon
white dwarf without a supernova explosion.
Note: Crab supernova had a progenitor star in 8-11 solar mass range
(Nomoto et al, 1984 )
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Stars ~11-100 solar masses
Complete all stages of
burning, including Si
burning in hydrostatic
equilibrium and formation
of iron core.
Effects of partial
degeneracy and off-
centre ignition of fuels
may exist in the stars up
to 15 solar masses
Note: the evolution is
sensitive to metallicity.
Figure: Shell structure of
pre-supernova star
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Stars ~11-100 solar masses: iron core instability
If the mass of the core exceeds Chandrasekhar mass
(1.46M
Sun
), the core contracts
Electrons are captured by heavy nuclei => less electrons
=> smaller pressure => accelerates infall
Temperature rises quickly => photodisintegration of iron
nuclei
γ(100MeV) +
56
Fe 13
4
He + 4n.
/do not confuse with photofission/
The reaction absorbs ~2MeV per nucleon => severe
energy loss => still almost a free infall => temperature
continue rise => photons disintegrate helium into protons
and neutrons => free protons capture free electrons =>
neutrons => collapse stopped by neutron degeneracy
(density reaches 10
18
kg m
-3
)
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Stars ~11-100 solar masses: neutron stars
Neutron core (~40 km in
diameter) is created, so called
neutron star.
Landau (1932), Baade & Zwicky
(1934) and Oppenheimer &
Volkoff (1939) theoretically
predicted existence of neutron
stars.
Hoyle (1946) suggested the
instability associated with
photodisintegration of iron to be
triggering mechanism for
supernova explosions.
=>Type II or Ib/Ic supernovae
Figure from
http://www.astronomy.o
hio-state.edu/~ryden/
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Example: collapse of iron cores
The energy source of a supernova is gravitational,
the core (~ 1.5M
sun
) collapses from an initial white
dwarf radius (~ 0.01R
sun
) to the final radius ~20km
of the neutron star.
a) What is the amount of gravitational energy
released?
b) What is the amount of energy absorbed in
nuclear processes ( assume ~7MeV per nucleon)?
c) What is the amount of energy radiated if
supernova luminosity is 10
10
Suns?
d) What is the kinetic energy of the ejected
envelope for 10 solar mass envelope (ejecta speed
is ~10000km/s) ?
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Example: collapse of iron cores
The amount of gravitational energy released during
the collapse is around 3x10
46
J
The energy absorbed in nuclear processes is around
ten percent.
The radiated energy is around 1%.
The amount of kinetic energy in ejecta is a few %.
The “missing energy” is neutrinos p + e
-
→ n + ν
e
10
57
neutrinos can be released amounting 10
46
J of
energy (90% of the energy release)
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Detection of 1987A supernova neutrinos
Figure:The neutrino burst detected by
Kamiokande. 11 neutrinos were detected
in 13 seconds. (Totsuka, 1991)
Theoretical predictions
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Figure:
Initial-final
mass
function of
nonrotating
stars of
solar
composition
(Woosley et
al, 2002)
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Figure:
Initial-final
mass
function of
nonrotating
stars of
primordial
star
(Woosley et
al, 2002)
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Figure:
Chandra X-ray
image of X-ray
image of the
same field of
view, again
showing the
nucleus of NGC
1260 and SN
2006gy
Pair-instability supernovae
Heger and Woosley, Pulsational pair
instability as an explanation for the most
luminous supernovae, Nature,
Nov 15, 2007
Following helium burning, the star contracts at
an accelerated rate => raising the
temperature => electron-positron pairs.
Nuclear energy generation from C and Ne
is insufficient to halt this contraction=> The
collapse has already become dynamic and the
star overshoots the temperature and density
that might have provided hydrostatic
equilibrium => Implosion becomes explosion.
pair-instability supernovae can produce a
nearly solar distribution of elements from
oxygen through nickel with a large deficit of
nuclei with odd nuclear charge (N, F, Na, Al, P,
etc.) and creates no elements heavier than
zinc
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From inflow to outflow
Neutrino energy deposition
is believed to be the energy
source for the explosion
Multidimensional calculations are
essential in order to reveal the
convective flow responsible
for boosting the neutrino
luminosity of the protoneutron
star. This flow also increases
the efficiency of neutrino
absorption.
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Problems with
current models:
eject too much
neutron-rich
nucleosynthesis.
instabilities that will
affect propagation
=> effective mixing
of elements
(red supergiants that
have not lost a lot
of mass,
experience the
greatest degree of
mixing and
clumping)
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Numerical simulations of shock formation
After Romero et al, (1996)
Figure: Velocity snapshots Figure: Density snapshots
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Supernova lightcurves
The light curve of a supernova
from a massive star has three
parts whose relative proportions
vary depending upon the mass of
the hydrogen envelope (if any), its
radius, the explosion energy, and
the mass of
56
Ni produced in the
explosion.
1) Shock breaks out through the
surface (first EV, later optical
emission)
2) The plateau appears as hydrogen-
rich zones expand and cool below
about 5500K. This ‘‘recombination
wave’’ propagates inwards in mass
3) After the hydrogen has
recombined, radioactivity is the
source of energy
In type II-L
56
Co and some other
radioactivities form a tail.
In type-Ib and type-Ic
supernovae,
56
Co dominates
entire display.
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Supernova spectra
Figure:Spectrum of the type-II-
p
SN 1992H (Filippenko,
1997) compared with a nonlocal
thermodynamic equilibrium
calculation of a supernova explosion.
The spectrum of common
type-II plateau supernovae
near peak luminosity is given
by a quasi-thermal continuum.
Temperature is 0.1-10MK
Type-Ic supernovae are
similar in many ways to type
Ib but lack a distinctive He I
absorption line possibly due
to actual absence of He
or insufficient mixing.
Note: If the star has lost its
hydrogen envelope before
exploding then there is no
plateau.
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Type Ia Supernova
Hoyle & Fowler (1960) were
the first to discover that
thermonuclear burning in an
electron-degenerate stellar
core might trigger an
explosion.
Thermonuclear disruptions
of white dwarfs, either
consisting of carbon and
oxygen with a mass close
to the Chandrasekhar
mass, or of a low-mass
C+O core mantled by a
layer of helium.
Ia rise to maximum light in a period
of approximately 20 days
Light curve due to
radiative decay (
56
Ni
56
Co
56
Fe)
Note: degeneracy pressure is
independent of temperature=> the
white dwarf is unable to regulate
the burning process in the manner of
normal stars=> explosive fusion
reaction accelerated by RT
instability
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