Propagation in the Galaxy 2: electrons,
positrons, antiprotons
As we mentioned in the previous lecture the results of the
propagation in the Galaxy depend on the particle interaction
cross section. If the average amount of matter an 1 GeV
cosmic ray sees is 10.8 g/cm2 and the proton mean free path
is 100 g/cm2 only 10% of the protons would interact. For Fe
nuclei, whose cross section is about 15 times higher 80% of
the nuclei would interact, so the flux of Fe is affected much more
than the proton flux. Of course, Fe nuclei are contained in the
Galaxy much longer, which increases the effects.
Electrons and positrons have much higher electromagnetic
interaction cross sections. Proton- proton cross section at
relatively low energy is 30 mb (1 mb = 10-28 cm2) while the
basic Thomson cross section is 665 mb.
The interaction cross section is measured in barns
(1 barn = 10-24 cm2). The most used unit is a milibarn,
which is barn/1,000.
Electrons and positrons have much higher electromagnetic
interaction cross sections. Proton- proton cross section at
relatively low energy is 30 mb (1 mb = 10-27 cm2) while the
basic Thomson cross section is 665 mb.
Definition of the average interaction length – the amount of
matter in which a particle interacts once. It starts again with
the Avogadro number. The number of atoms per volume is
n = NAA. In a compound material ni = NAwi/Ai.
The average interaction length is
 E) = ( ni i))-1
In case of air the average interaction length is
 (mb)
If a proton air cross is 308 mb the average interaction
length is 79 g/cm2. The cross section increases with
energy and the interaction length decreases with energy.
The energy dependence can not be faster than ln2E.
This is the (Marcel) Froissart's bound on the interaction
cross section.
Energy loss processes for electrons and positrons
The force between two point charges at a distance R from
each other (the Coulomb force) is
This is the basic force between two
charged particles.
Ionization loss: The energy loss on excitation and ionization
of atoms of a charged particle moving through matter is
(in units of MeV/(g.cm2)) is
which comes to about 2 MeV for electrons in air or water.
Note the inverse proportion on the particle mass M.
Z and A are the average charge and mass of the medium, so
energy loss in Hydrogen is smaller than in air or water.
Compton scattering is a process in which a photon hits
one of the atoms electrons and transfers energy to it. For
electron energy loss the inverse process,
Inverse Compton scattering is very important in astrophysics.
is the differential cross section for Compton scattering where
k and k' are the photon energies before and after the scattering
and q is the primary photon energy in terms of the electron
mass. In the case of Inverse Compton scattering the cross
section is the same where k is replaced by the photon energy
in the electron frame with an account for the angle between
the electron and the photon.
For photon energy much larger than the electron mass
Bremsstrahlung: Charged particles interact with the
electromagnetic field of the nuclei of matter and produce
gamma rays. The energy loss is
The radiation length X0 gives the average amount of
matter for bremsstrahlung energy loss
For air this amount is 37 g/cm2 and it is close to that for
water. In Hydrogen this amount is 61 g/cm2.
re is the classical radius of the electron = e2/mec2
= 2.82x10-13 cm
Synchrotron radiation: Charged particles lose energy in
propagation in magnetic fields. For electrons moving randomly
in a magnetic field the energy loss is
The characteristic frequency of the radiated photons is
Synchrotron loss is
inversely proportional
to the square of the
particle mass.
Production of electron positron pairs is the opposite
process to bremsstrahlung and their cross sections are closely
related. The total cross section is
The energy distribution
of the pair particles
at high gamma ray
energy (top) and at
threshold (bottom).
In order to produce two
electrons the energy in
the CM should be
higher than 2 me.
Production of electrons and positrons in the Galaxy
depends on the models of cosmic ray containment in the
Galaxy. The classical model is called the leaky box model
where the cosmic rays move with certain mean free path
and have probability p to escape from the Galaxy at every
step. The models include interactions of the propagated
cosmic rays with galactic matter and production of neutral
mesons (mostly 0). The neutral pions decay to 2 gamma
rays that propagate in the Galaxy and generate electron
positron pairs. The electrons and the positrons also propagate
and lose energy. One has to take into account the energy
losses in propagation and the probability to escape from the
Galaxy to calculate the fluxes that are detected at Earth.
Have in mind that positrons are only produced in propagation
while a fraction of the electrons maybe accelerated at the
sources of cosmic rays. Positrons also annihilate in propagation.
For this reason one expects higher electron flux.
Positron production in proton interactions in one gram of
interstellar matter (by R.J. Protheroe). Comparison with
other calculations.
Comparison of Protheroe's calculation to older set of data.
We will discuss the new measurements in the next lecture.
Comparison of measurements to the calculation
for three different models
of electron and positron
propagation. This is also
an old data set.
The ratio of e+/e- is
supposed to decrease
with increasing energy
unless there are positrons
from the decay of dark
matter. In such a case
there would be a peak
at ½ MDM particle.
Production of antiprotons
Antiprotons are generated in collisions of accelerated cosmic
rays with galactic matter. Lets consider the production in pp
collisions
pp -> p + p + p + pbar
The center of mass energy in the case the particle 2 is at
rest is s1/2 = (m12 + m22 + 2m2E1Lab)1/2
It is obvious that in the center of mass the energy should
exceed the sum of the masses of the produced particles
for the interaction to be possible. In this particular case
the sum of all masses is 4mp = 3.75 GeV. We can now use
the definition of the CM energy to calculate what is the energy
threshold for cosmic ray protons to produce proton-antiproton
pair. The minimum energy is 6.54 GeV. At this energy the
production of proton-antiproton pair with 0 momentum is
possible, i.e. all secondary particles are at rest. In practice
one need higher energy to generate the collision.
Energy dependence of the
antiproton production cross
section in pp interactions.
Why are we interested in positrons and antiprotons?
This is philosophical question related to cosmology. Why
do we have such huge asymmetry between baryons and
antibaryons in the Universe. We should have equal amounts
of matter and antimatter after the Big Bang, but cannot see
antimatter except for positrons and antiprotons.
If we are able to see many more antiprotons than the number
produced in the Galaxy this would be a sign that we have
antimatter astrophysical objects, probably in antimatter
galaxies.
The big experiment AMS-2 that is at the space station now
(Anti Matter Search) is looking for such signals (antiprotons
and anti-Helium) as well as for ordinary cosmic rays. The first
results from AMS-2 will come out this year.