Chapter 4
SHOCK WAVES IN ASTROPHYSICS
1. Physics of the shock waves
6.1. What are the shock waves?
Shock waves are found in many high energy astrophysical phenomena and play a
key role in many different astrophysical environments like star formation in the spiral
arms of galaxies, the high velocity outflows from young stars, extra-galactic radio
sources and compact galactic nuclei, and in the interaction of the solar wind with
the magnetic field of the Earth 1 .
It is a general property of perturbations in
a gas that they are propagated away from
their source at the speed of sound in the
medium. It is therefore obvious that if a
disturbance is propagated at a velocity
greater than the speed of sound, the
disturbance cannot behave like a sound
wave at all.
There is a discontinuity between the regions behind and ahead of the
disturbance and the latter region can have no prior knowledge of its imminent
arrival-the sound waves which would transmit the information are propagated at a
speed less than that of the disturbance.
These discontinuities are called shock waves. They commonly arise in
situations such as a) explosions or b) where gases flow past obstacles at supersonic
velocity or, equivalently, c) objects move supersonically through a gas. The basic
phenomenon is the flow of a gas at a supersonic velocity relative to the
local velocity of sound.
1
6.2. The basic properties of plane shock waves
We assume that there is an abrupt
discontinuity between the two regions
of fluid flow. In the undisturbed
region ahead of the shock wave, the
gas is at rest with pressure p1 , density
1 and temperature T1 . Behind the
shock
wave,
the
gas
moves
supersonically and its pressure, density
and temperature are p 2 ,  2 and T2 ,
respectively. It is more convenient to
analyze the shock waves in a
reference frame moving at velocity
U in which the shock wave is
stationary. Then, the undisturbed gas
flows towards the discontinuity at
velocity v1  U and, when it passes
through it, its velocity becomes v 2 in the
moving reference frame.
The behavior of the gas passing through the shock wave is described by a set
of conservation relations. First, mass is conserved on passing through the discontinuity
and hence
2
1 v1   2 v 2 .
(1)
Second, the energy flux, i.e., the energy passing per unit time through unit
area parallel to v1 is continuous. It is one of the standard results of fluid dynamics that
 1 


the energy flux through a surface normal to the vector v is v v 2  w  , where w is the
2

enthalpy per unit mass, w   m  pV , where  m is the internal energy per unit mass and
V   1 is the specific volume, i.e. the volume per unit mass. We consider only plane
shock wave, which are perpendicular to v1 and v 2 . Therefore the conservation of
energy flux implies
1
2


1
2


1 v1  v12  w1    2 v 2  v 22  w2  .
(2)
Notice that it is the enthalpy per unit mass and not the energy per unit mass
 which appears in this relation. The reason for this is that, in addition to internal
energy, work is done on any element of the fluid by the pressure forces in the fluid
and this energy is available for doing useful work. Another way of looking at this
relation is in terms of Bernoulli’s equation of fluid mechanics in which the quantity
1 2
1
p
v  w  v 2   m  is conserved along streamlines, which is the case for normal
2
2

flow through the shock wave.
Finally, the momentum flux through the shock wave should be continuous.
For the perpendicular shocks considered here, the momentum flux is given by
p  v 2 and hence
p1  1 v12  p2   2 v 22 .
(3)
Notice that the pressure p , being the force per unit area, contributes to the
momentum flux of the gas. Equations (1)-(3) are the three conservation equations
which are often referred to as the shock equations.
3
For simplicity we shall study the case of shock waves in a perfect gas for which
the enthalpy is w  pV / 1 , where  is the ratio of the specific heats. One can obtain
many important results for this case. First of all, we define the mass flux per unit area as
j  1 v1   2 v 2 . Then, from Eq. (3), which describes the conservation of the momentum,
we immediately find
j2 
p2  p1
.
V1  V2
(4a)
In addition, we obtain an expression for the velocity difference:
v1  v 2  j V1  V2  
 p2  p1 V1  V2  .
(4b)
The next step is to find the ratio V2 /V1 as a function of p1 and p 2 for a perfect gas.
We begin with the equation of conservation of the energy flux Eq. (2) and substitute as
follows:
1 2
1
1
1
v1  w1  v 22  w2 , j 2V12  w1  j 2V22  w2 .
2
2
2
2
Using Eq. (4b), this expression reduces to
w1  w2   1 V1  V2  p 2  p1   0 .
2
We can now substitute the perfect gas expression w  pV / 1 to obtain
V2   1 p1    1 p2
,

V1   1 p1    1 p2
(5)
4
which gives the relation between the pressures and specific volumes on either side of the
shock. We can immediately find the relation between T2 and T1 from the perfect gas law,
p1V1 p 2V2
,

T1
T2
T2 p2V2 p2   1 p1    1 p2
.


T1
p1V1
p1   1 p1    1 p2
Also using Eq. (5) we can eliminate V2 from the expression for the flux density j ,
obtaining
j2 
  1 p1    1 p2 .
2V1
From this equation we can find the velocities of the gas in front of and behind the
shock,
V   1 p1    1 p 2 
V
. (6)
v  j V  1   1 p1    1 p2  , v 22  j 2V22  2
2   1 p1    1 p 2
2
2
2
1
2
2
1
It is most convenient to write these results in terms of the Mach number
M of the shock wave, which is defined as M 1  v1 / c1 , where c1  p1 / 1 is the
velocity of sound for the undisturbed gas. Thus
M 12 
v12
.
p1V1
From Eqs. (6) we obtain the pressure ratios as
p 2 2M 12    1

,
  1
p1
5
and the density ratio
 2 v1   1 p1    1 p 2
 1
.



1 v 2   1 p1    1 p 2   1  2
2
M1
Finally we find the temperature ratio



T2
2M 12    1 2    1M 12

.
T1
  12 M 12
It is useful to look at these ratios in the limit of very strong shocks, M 12  1 .
Then
p 2 2M 12  2   1 T2 2   1M 12
,
,
.



p1
  1 1   1 T1
  12
These results show that in the limit of very strong shocks, the temperature and
pressure can become arbitrarily large but the density ratio attains the finite value
 1
. For example, a monatomic gas has   4 / 3 and hence  2 / 1  4 in the limit of
 1
very strong shocks. These results demonstrate how efficiently strong shock waves can
heat gas to very high temperatures and this is found to be the case in supernova
explosions and supernova remnants.
6
What exactly is happening in the
shock front? It is apparent that the
undisturbed gas is both heated
and accelerated when it passes
through the shock front, and this is
mediated by the atomic or molecular
viscosity of the gas. It can be
shown that the acceleration and
heating of the gas takes place over
a physical scale of the order of the
mean free path of the atoms,
molecules or ions of the gas. Thus,
the shock front is expected to be
very narrow and the heating very
strong over this short distance.
6.3. The supersonic piston
A common situation in high energy astrophysics is one in which an object is
driven supersonically into a gas, or equivalently, a supersonic gas flows past a
stationary object.
A
useful
illustrative
example is that
of a piston
driven
supersonically
into
tube
containing
stationary gas.
A shock wave forms ahead of the piston and the gas behind the shock moves at
the velocity of the piston v p  U . In the frame of reference of the shock front, which
moves at some yet unknown velocity v s , the velocity of inflow of the stationary gas is
7
v1  v s and the gas behind the shock moves at velocity v 2 . As yet we do not know
v1 and v 2 but we know that their difference v1  v 2  U .
Using Eq. (4b) we find
v1  v 2  U 
 p2  p1 V1  V2  .
Substituting for V2 and squaring we find that the above equation can be written in
terms of the pressure ratio p2 / p1 :
 p2   p2 
U 2     1U 2 
     2    1
  1 
  0.
2 p1V1  
2 p1V1 
 p1   p1  
2
We can now write p1V1  c12 , where c1 is the velocity of the sound in the
undisturbed medium, and solve for p2 / p1 :
p2
   1U 2 U    12 U 2 
 1

1 

p1
c1 
4c12
16c12 
1/ 2
.
The velocity v1  v s follows from Eq. (6);
V1
c12 
p2 
v    1 p1    1 p2  
  1    1  .
2
2 
p1 
2
1
Substituting for p2 / p1 gives

  12 U 2 
vs 
U  c12 

4
16


 1
8
1/ 2
.
This is a very neat result since it determines the thickness of the layer of shocked
gas ahead of the piston for any supersonic velocity U . Let us look at the case of a very
strong shock wave U  c1 . Then
vs 
 1
2
U.
Thus the ratio of the position of the shock front to the position of the piston is
v s / U    1 / 2 . For a monatomic perfect gas   5 / 3 and hence v s / U  4 / 3 . Thus,
all the gas which was originally in the tube between x  0 and the position of the shock
wave is squeezed into a smaller distance v s  U t . It follows that the density increase
over the undisturbed gas is  2 / 1  v s / v s  U     1 /   1 .
This simple calculation
gives some feel for what
is
observed
when
supersonically moving
gas
encounters
an
obstacle or is ejected
into a stationary gas.
Ahead of the obstacle
there is a shocked region
which runs ahead of the
advancing piston. This is
what is expected to
occur when a supernova
ejects a sphere of hot
gas into the interstellar
medium. It also shows
that there is a stand-off
distance of a shock front
from a blunt object
placed in the flow.
9
2. Fundamentals of gamma-ray astrophysics
6.4. Basic concepts
Astronomy is the attempt to gather information from cosmic objects through
the detection of particles (photons, leptons, hadrons, neutrinos etc.), emitted or
affected by them. Astrophysics is the attempt to use our knowledge of the physical laws
such as we know them on the Earth to propose theories that can represent adequately the
physical events behind the astronomical observations 2 .
Since astronomers try to detect particles, it is
convenient to start by introducing as a basic concept
the number of particles incident per unit of surface
area per unit of solid angle per unit of time arriving
at a given, unspecified detector. We will call this basic
quantity the intensity of the particles, and we will
denote it by I .
In general we will use a subscript to indicate the type of particles, e.g. I  and
I p denote the intensities of gamma rays and protons, respectively.
Once the intensity is introduced, we can define the particle flux as
F   I cos d ,

where the angle  is determined by the direction of motion of the particles with respect
to the normal to the area, and the integration is performed over the solid angle. For
isotropic radiation the flux is F  I and the number of particles per unit volume is
N
4
I.
v
In most cases v  c is a good approximation because we deal with relativistic
particles.
10
Normally, we will have particles with different energies, so it is useful to
introduce a particle energy distribution N (E ) such that

N   N ( E )dE .
0
The number of particles with energies greater than E is obtained just by
integrating from E . In a similar way, the integrated flux density is

F  E    F ( E )dE .
E
The luminosity of a source located at a distance d that radiates isotropically is
given by
L E   4d

2
 F ( E )dE ,
E
where d is the distance to the source.
The energy density of the particles is
w
Emax
 EN ( E )dE .
Emin
The energy flux is obtained from this expression just by multiplying by c / 4 , if
we deal with relativistic particles or photons.
Let us consider now that a flux of particles of type a , with velocity v a ,
interacts with some target formed by particles of type b within a volume dV . The
number of particle interactions of a given type, dN i , occurring in a time dt in the
volume dV will be proportional to the number of particles b in the volume dV and
11
to the number on incident particles that traverse the cross section dA of that volume
in the time dt :


dN i  d i nb0 dV na v a dt  .
In this expression n b0 and n a are the densities of target and incident particles in a
coordinate system with the target at rest. The differential cross section
d i characterizes the number of reactions of type i occurring per unit of time in unit
volume for a unit flux density of incident particles and unit density of the target. It is
measured in units of area, the standard unit being the mb (i.e. 10 3 Barn, 1
Barn= 10 24 cm 2 ).
The total cross section for a given interaction  i is the sum over all possible
momenta of the resulting particles after the interaction. Both  i and d i are relativistic
invariants. The total cross section  tot is obtained by summing the cross sections of all
possible processes that occur upon the interaction of particles a and b , i.e.
 tot    i .
i
The relative probability of a given reaction channel is given simply by
p   i /  tot .
In the case of gamma-ray emission, if the generation of the gamma rays is due to

the interaction of particles of type i with a given intensity I i Ei , r  with a target of

density nr  , we can write the intensity of the radiation from the resulting gamma-ray
source as




I  E    nr  Ei , E I i Ei , r dEi dr ,
l E
12

 
where l defines the direction along the line of sight (i.e. l  r / r ). The emissivity of the
gamma-ray source is defined as




q E , r    nr  Ei , E I i Ei , r dEi ,
E
in such a way that
 
I  E    q E , r dr .
l
6.5. Synchrotron radiation
A relativistic particle moving
in a magnetic field will emit
photons
within
an
angle
 ~ mc 2 / E of its direction of
motion.
In a magnetic field B an
electron moves along a helical path
with an angular frequency  B given
by
eB me c 2
B 
me c E
13
The radiation spectrum of the electron is
given by
P( E ) 
3e 3
E
B
2
Ec
me c

 K  d ,
5/3
E / Ec
where E  h is the energy of the
radiation, B  B sin  ,  is the pitch angle
and K 5 / 3 is the modified Bessel function of
the second kind.
The characteristic energy of the photons is given by
2
3h eB  E 

 .
Ec 
4 me c  me c 2 
The maximum of P (E ) occurs at Emax  1.9  10 11 B / Gauss E / GeV  GeV.
2
We see that only for extremely energetic particles and strong magnetic fields we can get
gamma-ray photons from synchrotron radiation.
The total energy rate loss by synchrotron radiation of an electron moving in a
field B can be obtained by integrating the radiation spectrum. The result is
2  e2
 dE 
  e   c
2
 dt  syn 3  me c
2
 2 2
 B  ,


where   E e / me c 2 is the Lorentz factor of the particle.
Introducing
the
Thomson
cross
section
 T  8e 4 / 3me2 c 4  0.665  10 24 cm 2 and averaging over an isotropic pitch angle
distribution, the expression for the energy losses can be set in the following convenient
form
14
4
 dE 
  e    T cwmag  2 ,
 dt  syn 3
where wmag  B 2 / 8 is the magnetic energy density and B is measured in Gauss.
Let’s assume that we have a homogeneous and isotropic power-law electron
distribution in a random magnetic field, given by
N e E e dE e  K e E e p dE e .
Then the resulting spectrum is given by

E
3e 3 K e

p
I E  
B
d
r
E
dE

e
e
 E
Ec
me c 2
l

with
3h eB  Ee 


Ec 
4 me c  me c 2 
 K  d ,
5/3
E / Ec
2
. By introducing a new variable of integration
  E / Ec  4me cE / 3heB Ee / me c
 4me c 
1

dE e   me c 2 
2
 3heB 


2 2
,
 4me c 
Ee

 
2
me c
 3heB 
1/ 2
E1 / 2 1 / 2
,
1/ 2
E1 / 2 3 / 2 d , and by making the assumption that the end
points of the electron energy spectrum do not contribute (this means that the lower limit
on the E e integration may be replaced by zero), we obtain
e3
I E   a p 
me c 2
 3e 


 4m 3 c 5 
e


 p 1 / 2
B ( p 1) / 2 K e LE( p 1) / 2 ,

where L   dr is the characteristic size of the emitting region and a ( p )  a p (0) , where

l
15

a p ( x)   
p 1
2
x


K 5 / 3  dd .
It can be shown that a ( p ) is given by
 3 p  1   3 p  19   p  5 
2 ( p 1) / 2 3



12   12   4 

.
a( p) 
 p7
8   p  1

 4 
a ( p ) is 0.147, 0.103, 0.0852 and 0.0742 for p  1.5, 2, 2.5 and 3, respectively.
The emission is a power law with index   ( p  1) / 2 .
6.6. Inverse Compton interactions
The scattering of relativistic electrons on
soft photons can produce gamma rays.
The intensity of the radiation from this
process when the soft photon field has a density

n ph E ph , r  is
I IC E   

l E

  E , E , E I E , r n E

0
e
ph
e
e
ph

ph



, r dEe dE ph dr .
We can introduce a parameter   Ee E ph / me c 2 such that for   1 the
scattering is classical. In such a limit the cross section can be approximated by the
Thomson cross scattering  T and the average energy of the emerging photons will be
16
2
E 
4
E ph  2 ,
3
where E ph is the average energy of the target photons. The energy losses for an electron
in a photon field of energy density w ph when   1 can be approximated as
 dE 
  e    T cw ph 2  2  10 14 w ph 2 eV/s.
 dt  IC
We see that at this energies the ratio of synchrotron to IC cooling times is simply
t IC / t syn  wmag / w ph .
If the incident electron spectrum is a power law and the photon field can be
approximated by a mono-energetic distribution, then we obtain

1
I  E   n ph L T me c 2
2
IC

1 p
4

 E ph 
3

( p 1) / 2
K e E( p 1) / 2 .
Here, L is the typical source dimension, and E ph and n ph are average values
for the photon energy and photon number density in the source.
If the photon field is thermal radiation, then
n ph
E   E
ph
ph
me c

    exp E
/ me c 2
2
2
2
3
c

1
,
ph / kT   1

where c  2me c  h is the Compton wavelength of the electron. By assuming again a
1
homogeneous
and
isotropic
power-law
form N e E e   K e E e p , with K e a constant, we have
I e E e  
c
K e E e p .
4
Therefore we obtain for I IC
17
electron
distribution
of
the
r02
( p 5) / 2
I  E  
LK e kT 
F  p E( p 1) / 2 ,
2 3 2
4  c
IC
where r0  e 2 / me c 2 is the classical electron radius, T is the temperature and
 p 5  p 5
2 p 3  p 2  4 p  11
 

2   2 

,
F ( p) 
 p  32  p  1 p  5
where  is the Riemann function. For p  1.5, 2 and 2.5, F ( p )  3.91, 5.25 and 7.57,
respectively.
Notes
1
The physics of shock waves is discussed in many books. Classical references are L.D.
Landau and E.M. Lifshitz, Fluid mechanics, Oxford, England: Pergamon Press, 1987 and
Ya. B. Zel′dovich and Yu. P. Raizer, Physics of shock waves and high-temperature
hydrodynamic phenomena, New York: Academic Press, 1966-1967. The presentation in
this notes mainly follow M. S.
Longair, High energy astrophysics, Cambridge:
Cambridge University Press, 1992.
2
The presentation of the astrophysics of the gamma-rays is based on G. E. Romero and
K. S. Cheng, Fundamentals of gamma-ray astrophysics, in K. S. Cheng and G. E. Romero,
editors, Kluwer Academic Publishers, Dordrecht, Boston, London, 2004.
18