# Interacting Supernova Remnants: a population model for the Cherenkov
Telescope Array

Silvia Crestan

###### Abstract

The work presented in this thesis is focused on the interacting supernova
remnants (iSNRs), a class of gamma-ray emitting SNR where the radiation arise
from the interaction of the SNR with a massive molecular cloud. At the moment
only 16 iSNR are known. Before this work, there was not any population study
of these sources. Here is proposed a model for the Galactic population of
iSNRs which can be used in order to predict the number of these systems in the
Galaxy and how many of these will be detectable by the next generation of
$\gamma$-ray instruments. iSNRs are of particular interest for the particle
acceleration study because these objects have been proved to be sites of
acceleration of protons up to high energies. Furthermore, high-energy
$\gamma$-ray emission can pinpoint the presence of energetic leptons or ions
and help to constraint the acceleration efficiency and maximum energy of
accelerated particles

The model presented her was only achievable through the creation of a complete
catalog of $\gamma$-ray (both GeV and TeV) supernova remnants that for each
supernova remnants gives collect the physical and spectral information
available in litterature. Simulating and analyzing the synthetic population,
it was found that the Cherenkov Telescope Array (CTA) will be able to
duplicate the number of detected interacting systems in its survey of the
Galactic plane.

See frontespizio.pdf

## Summary

Understanding the nature of galactic sources capable of accelerating cosmic
rays (RC) is one of the main objectives of high energy astrophysics. In this
context the supernova remnants (SNR) have a key role. Since supernova remnants
are capable to convert $\sim 10\%$ of the kinetic energy of the explosion in
protons acceleration up to energies higher than $10^{15}$ $eV$ $(PeV)$, they
are considered the main sources of cosmic rays. The current generation of
Cherenkov telescopes (HESS, MAGIC, VERITAS) and $\gamma$-rays satellite
(FERMI, AGILE) revealed $\gamma$-ray emission from 37 galactic supernova
remnants compared to 295 SNR currently observed in the radio band111Green D.
A., 2019, ‘A Catalogue of Galactic Supernova Remnants (2019 June version)’
(available at ”http://www.mrao.cam.ac.uk/surveys/snrs/”). Observations of
supernova remnants at gamma-ray energies222Stefan Funk. Ground- and Space-
Based Gamma-Ray Astronomy showed that they can be divided into "Shell-like"
sources and those that are interacting with the dense molecular clouds in the
surrounding environment . "Interaction" refers to particles accelerated by
supernova shocks that penetrate the molecular cloud. For some of the
interacting supernova remnants (iSNR) the hadronic origin of the $\gamma$-ray
emission has been proven, so this sources are certainly cosmic rays
accelerators potentially up to $10^{15}eV$. This makes interacting supernova
remnants a very interesting class of sources to explain the cosmic rays
spectrum at Pev energies.

##### Purpose

The thesis purpose is to give a population model for the galactic interacting
supernova remnants to predict how many of these system will be observable by
the next generation of $\gamma$-ray instruments. The model I created will be
used in the second data challenge (DC2) and in the simulation of the galactic
plane survey (GPS) of the next Cherenkov Telescope Array (CTA). Since
interacting supernova remnants are systems composed of a supernova remnant and
a molecular cloud, I modeled separately the population of molecular clouds and
the spectrum of protons accelerated by the supernova explosions and then I
associated a probability that these systems are close enough to interact. Both
synthetic population of molecular clouds and simulated spectra reflects the
respective real populations since they are created starting from observations.
In particular, for recover the simulated spectra, I created the first
$\gamma$-ray emitting supernova remnants catalog. This catalog contains all
information relating to physical and spectral parameters and those relating to
the protons population responsible for the $\gamma$ ray emission. This kind of
informations are essential not only to classify SNR or to search for
correlation between physical and spectral features, but also because are the
starting point to perform any sort of simulation with the future high-energy
instruments. At the moment only 16 SNRs are classify as interacting in the
main catalogs of $\gamma$-ray sources (TeVCaT, 1SNRCat3331st Fermi LAT
Supernova Remnant Catalog) compared to 37 $\gamma$-ray emitting SNRs. Another
purpose is to estimate how many interacting systems CTA will detect in the
Galaxy. In conclusion, the goals are: the creation of the first $\gamma$-ray
emitting supernova remnants catalog, the creation of a population model for
the interacting supernova remnants for the DC2 and GPS and estimate how many
iSNR will be seen by CTA in the Galaxy.

##### Catalog of $\gamma$-ray Supernova Remnants

For each SNR considered I carried out an extensive literature research of
$\gamma$-ray spectra and physical characteristics. To evaluate a possible
classification based on spectral properties, I used a model-independent
approach, fitting the available data with the hadronic emission model, that
is, the relativistic proton-proton interaction followed by the $\pi^{0}$-decay
in $E_{\gamma}>100MeV$ photons. I chose this radiative model because is the
one expected for the interacting supernova remnants, object of interest of
this work. I performed the spectral fit using the NAIMA python package: this
software, through the maximum likelihood method, calculates the parameters of
the particle distribution which reproduces the observations, based on the
radiative model used. For each SNR I used a broken power law as proton
distribution and $\pi^{0}$-decay as radiative model. The main results obtained
thanks to the catalog are: confirmation also in the $\gamma$-ray energy band
that there are mainly two classes of SNRs, and the creation of the LogN-LogS
curve of the interacting supernova remnants. In principle, the slope logN-logS
curve for a sources population distributed in the Galactic plane is equal to
-1, more realistically, this curve has the expected slope only at high
integrated fluxes, at lower fluxes the logN-logS curve deviates from -1
indicating that the sample is not complete. The stretch of logN-logS with
slope -1 will provide the calibration of the population model created,
therefore the logN-logS of the synthetic iSNR must have the same slope in the
completeness region.

##### Interactiong SNR population model for the DC2 of CTA

To create a synthetic population of interacting supernova remnant, as close as
possible to what we observed in nature, I had to reproduce mainly three
aspects: the population of molecular clouds , the spectrum of cosmic rays that
interact with the nuclei of the cloud and the probability that the molecular
clouds and SNRs are close enough to interact. I realized 100 repetitions
because the model is based on random extractions of spectral parameters and
random associations between clouds and cosmic rays spectra, in order to not
introduce any sort of effects due to the extraction of a particular
realization.

##### Synthetic population of molecular clouds

I created the synthetic population of molecular clouds with realistic features
in terms of mass and distribution in the Galaxy. For the spatial distributions
I used the galaxy spiral model taken from Foucher et al 2006 and the
galactocentric distribution function from Lorimer 2006 that is compatible with
the young stellar distribution. This method gives a catalog of positions along
the spiral arm of the Galaxy. Clouds besides positions need an associate mass,
for this reason I used the mass distribution function derived from the catalog
of molecular clouds by Rice et al.444A Uniform Catalog of Molecular Clouds in
the Milky Way, Rice et al. 2016, that is a power law with a spectral index
equal to -1.72 for molecular clouds with $M_{cloud}>10^{5}$ solar masses. I
randomly associated the mass to the positions of the molecular clouds,
assuming that I could extrapolate the mass distribution function of the real
MC up to the current instrument sensitivity limit ($M_{cloud}\sim 10^{3}$).
The integral of the new mass function returns the number of molecular clouds
that should be simulated, that is 13500.

##### Synthetic spectra of cosmic rays injections

Starting from the 16 spectra of the real interacting supernova remnant, I
created the synthetic protons distributions that interact within the molecular
clouds. Thanks to the catalog realized, for each interacting supernova
remnants, all parameters of the broken power law protons injection spectra
(normalization, index before and after the break and the energy break) are
available. So for each parameters I created a Gaussian distribution with mean
and standard deviation values calculated from the real sources and then I
extracted randomly the spectral parameters values for each synthetic
interacting supernova remnants. I’m assuming that if there were more
interacting supernova remnants their spectral parameters would be distributed
following a Gaussian curve with mean and sigma value close to those I had
found. Generate the synthetic interacting supernova remnants spectra requires
besides the particle distribution, the density of the target protons. As the
other this parameter must be randomly extracted from the distribution
generated starting from the observed values of the molecular clouds taken from
Rice et al.. Using the extracted values of the protons distributions
parameters and of the clouds density, I created 13500 spectra and I randomly
associated them with molecular clouds.

##### The molecular clouds - SNR association

Certainly not all the molecular clouds have a SNR nearby, therefore I
introduced a probability function that in principle depend on the clouds mass.
The only function that reproduces the real iSNR logN-logS curve is a flat
probability at 1.5%

##### Results

Using the interactive supernova remnant population model that I created, I
estimated how many of these systems are potentially visible using the future
Cherenkov Telescope Array. I simulated and analyzed with the CTA simulation
software555http://cta.irap.omp.eu/ctools/index.html some of the realization
that I’ve made using 50h of observation for each source from 0.1 to 100 TeV .
For example in a realization I simulated 199 sources, 39 of these have TS $>$
5$\sigma$. As expected, the logN-logS of the detected iSNR follow the real
iSNR curves, but at the CTA sensitivity limit at 100 GeV-10 TeV it bends
toward a greater number of sources. Depending on realization CTA detect
between 30 and 50 synthetic iSNR. Taking into account that simulations and
analysis were done under ideal conditions (only one source for each simulation
and only with instrumental background) this means doubling the number of SNR /
MC systems already known.

###### Contents

  1. 1 Supernova remnants
    1. 1.1 Observation of supernova remnants
    2. 1.2 Evolution of supernova remnants
    3. 1.3 Particle acceleration
      1. 1.3.1 Second order Fermi acceleration
      2. 1.3.2 First order Fermi acceleration
    4. 1.4 Gamma-ray emitting processes
      1. 1.4.1 Hadronic scenario
      2. 1.4.2 Leptonic scenario
  2. 2 Interacting supernova remnants
    1. 2.1 General properties of molecular clouds
    2. 2.2 Passive molecular clouds
    3. 2.3 Molecular clouds interacting with shocks of supernova remnants
    4. 2.4 Illuminated molecular clouds
    5. 2.5 Signatures of SNR/MC interaction
  3. 3 Catalog of $\gamma$-ray Supernova Remnants
    1. 3.1 Distance and Age of supernova remnants
      1. 3.1.1 Distance
      2. 3.1.2 Age
    2. 3.2 Spectral fitting
      1. 3.2.1 Example of W44
    3. 3.3 Catalog
    4. 3.4 Some catalog results
    5. 3.5 The LogN-LogS of real interacting supernova remnants
  4. 4 CTA - Cherenkov Telescope Array
    1. 4.1 Detecting Cherenkov light
      1. 4.1.1 Status of current IACTs
    2. 4.2 CTA Telescopes
    3. 4.3 CTA performance
      1. 4.3.1 Effective area
      2. 4.3.2 Angular resolution
      3. 4.3.3 Energy resolution and energy dispersion
      4. 4.3.4 Background rate
      5. 4.3.5 The sensitivity
      6. 4.3.6 Off-Axis sensitivity
    4. 4.4 CTA Data Challenges
    5. 4.5 Galactic Plane Survey
  5. 5 Population model for interacting supernova remnants
    1. 5.1 Synthetic population of molecular clouds
      1. 5.1.1 Catalog of molecular clouds
      2. 5.1.2 Synthetic population of molecular clouds
    2. 5.2 Spectra of Cosmic rays in Molecular clouds
    3. 5.3 The molecular clouds - SNR association
    4. 5.4 Repetition
  6. 6 Simulation and analysis
    1. 6.1 Simulation
      1. 6.1.1 Spectral and spatial models
    2. 6.2 Analysis
    3. 6.3 Results
  7. 7 Conclusion

##  Chapter 1 Supernova remnants

###  1.1 Observation of supernova remnants

Supernova remnants are visible in different energy bands across the
electromagnetic spectrum, from the radio band up to TeV $\gamma$-rays. About
300 supernova remnants have been discovered in the radio band and about one
tenth of these are visible in $\gamma$-rays. The non-thermal radio spectra of
supernova remnants are interpreted as synchrotron emission from relativistic
electrons accelerated at the shock of the supernova remnant. The polarization
of the radio synchrotron emission gives information about the magnetic field
structure in the remnant. The observed X-ray emission consists of non-thermal
and thermal emission. The thermal component stems from the heated gas within
the shock, whereas the non-thermal emission is produced by electrons in the
energy range (10 - 100) TeV. Both the non-thermal x-ray and radio emission are
of leptonic origin. The observed GeV and TeV emission can be produced either
by leptons due to Bremsstrahlung and inverse Compton scattering or by hadrons
via pion decay. According to the theory of adiabatic shocks particles are
accelerated during the free expansion phase and the Sedov phase as described
in 1.2.

###  1.2 Evolution of supernova remnants

The fate of a star depends on its mass. Stars with $M\geq 8M_{dot}$ typically
end in a supernova explosion caused by the collapse of the iron core after the
star ends all the nuclear energy and the gravitational pressure surpasses the
radiation pressure. The implosion following the collapse of the core leads to
a neutron star or a stellar black hole as leftover. These supernovae are the
so called core-collapse supernovae. Also less heavy stars can explode in a
supernova if they have already reached the state of a white dwarf and they
accrete material from a massive companion star until reaching the
Chandrasekhar mass limit of about 1.3 $M_{sun}$. The typical energy released
$E_{SN}$ during the supernova explosion, although it come from two different
mechanisms, is in the order of $10^{51}\,erg$. After the explosion the stellar
ejecta expand into the ambient medium and form the so-called supernova
remnant. Typical velocities of the ejected material are $v_{eject}\approx
10^{4}kms^{-1}$ and due to the supersonic speed of the ejecta a shock front is
formed. During the expansion the shock front sweeps up the ambient medium and
the shock velocity of the forward shock decreases. As a consequence a reverse
shock, is formed. Observations of supernova remnants in different energy bands
from radio to $\gamma$-rays show that particles get accelerated at the shock
fronts during different stages in the evolution of an supernova remnant.

The evolution of a core-collapse supernova remnant can be simplified in 4
stages, as discussed in [147] and [119].
Phase I: free expansion phase: in this phase, the mass of the supernova
ejecta, $M_{ej}$, is greater than the swept-up mass, $M_{sw}$. This phase last
until $M_{ej}\sim M_{sw}$ The initial velocity $v_{0}$ of the ejected mass
$M_{ej}$ after the explosion is given by:

$v_{0}=\sqrt{\frac{2E_{SN}}{M_{ej}}}=10^{9}\,cm\,s^{-1}\,\bigg{(}\frac{E_{SN}}{10^{51}\,erg}\bigg{)}^{1/2}\bigg{(}\frac{M_{ej}}{1M_{sun}}\bigg{)}^{-1/2}$
(1.1)

is larger than the speed of sound in the medium. Therefore, a shock front is
formed. The timescale and the radius of the remnant at the end of this stage
until the beginning of the following Sedov phase depend on the density of the
ambient medium $\rho_{ISM}$. The free expansion phase ends at:

$t\simeq
200\,yr\,\bigg{(}\frac{E_{SN}}{10^{51}\,erg}\bigg{)}^{-1/2}\bigg{(}\frac{\rho_{ISM}}{10^{-24}g\,cm{{}^{-}3}}\bigg{)}^{-1/3}$
(1.2)

when the radius of the supernova remnant is:

$r=\bigg{(}\frac{3M_{ej}}{4\pi\rho_{ISM}}\bigg{)}^{1/3}=2.1\,pc\,\bigg{(}\frac{M_{ej}}{1M_{sun}}\bigg{)}\bigg{(}\frac{n_{0}}{1cm{{}^{-}3}}\bigg{)}^{-1/3}$
(1.3)

where $n_{0}$ is the number density of the ambient medium.
Phase II: Sedov Phase: Radiative losses $dE/dt$ are still negligible since
$\int(dE/dt)dt\ll E_{SN}$. In this phase kinetic and thermal energies are
conserved. The radius of the supernova remnant evolves as

$r\simeq
0.3pc\,\bigg{(}\frac{E_{SN}}{10^{51}\,erg}\bigg{)}^{1/5}\bigg{(}\frac{n_{0}}{1cm{{}^{-}3}}\bigg{)}^{-1/5}\bigg{(}\frac{t}{yr}\bigg{)}^{2/5}$
(1.4)

The shock velocity $v_{shock}$ decreases with increasing radius as:

$v_{shock}\simeq
5000\,km\,s^{-1}\,\bigg{(}\frac{r}{2\,pc}\bigg{)}^{-3}\bigg{(}\frac{E_{SN}}{10^{51}\,erg}\bigg{)}\bigg{(}\frac{n_{0}}{1cm{{}^{-}3}}\bigg{)}^{-1}$

and the heated gas reaches temperatures of about $\sim 10^{6}K$, then emits
thermal radiation visible in x-rays. At the end of this phase the shock
velocity is slowed down to $\sim 200kms^{-1}$ and the radiative losses
dominate. The outer shell of the shock decelerates first, allowing the inner
material to catch up. During the deceleration process a shock wave at the
inner edge is formed that leads to a heating of the matter in the outer shell.
The transition between this phase and the following radiation phase occurs
after at:

$t_{rad}=3\cdot
10^{4}\,yr\bigg{(}\frac{T}{10^{6}\,K}\bigg{)}^{-5/6}\bigg{(}\frac{E_{SN}}{10^{51}\,erg}\bigg{)}^{1/3}\bigg{(}\frac{n_{0}}{1cm{{}^{-}3}}\bigg{)}^{-1/3}$

Phase III: radiation phase ($t>t_{rad}$): During this stage the temperature of
the matter behind the shock decreases because of the radiation cooling is
dominant . The radial momentum of the shell is nearly constant and the
expanding shock piles up interstellar gas. The radius increases with
$R(t)\varpropto t^{1/4}$.
Phase IV: In this last phase the shock velocity drops to ($10-100kms^{-1}$ )
in about ($1-5)\cdot 10^{5}\,yr$. The remnant loses its structure and the
accumulated matter disperses into the interstellar medium.
Typical values for the phase duration, radius of the remnant, shock velocity
and swept-up material in this simplified view of the evolution of a core-
collapse supernova remnant in a homogenous environment are summarized in Table
1.1.

Phases | t (yr) | r (pc) | v (km $s^{-1}$) |  $M_{swp}$ ($M_{sun}$)
---|---|---|---|---
I-II | 90 | 0.9 | $10^{4}$ | 0.2
II-III | 22 $\cdot 10^{3}$ | 11 | 200 | 180
III-IV | 750 $\cdot 10^{3}$ | 30 | 10 | 3600
Table 1.1: Characteristic values for the phase duration, radius r, shock
velocity v and swept-up mass $M_{swp}$ for the transition between the
different phases of the evolution of a supernova remnant taken from [147]

###  1.3 Particle acceleration

Detection of non-thermal emission from supernova remnants from radio to TeV
energy range suggests them as sites of particle acceleration. It is widely
accepted that the acceleration mechanism is diffusive shock acceleration at
the shock front of the supernova remnant. The general concept was developed
contemporaneously in the late 1970s by Krymskii [112], Axford [35], Bell [38]
[39] and Blandford & Ostriker [41] as first-order Fermi acceleration. The idea
is that particles cross the shock of the supernova remnant many times,
bouncing on magnetic turbulences in the up- and down-stream regions of the
shock gaining energy. Fermi (1949) originally proposed that the acceleration
of cosmic rays is due to collisions with randomly moving magnetic fields in
the galaxy. For historical reasons, in Section 1.3.1 the so-called “second-
order” Fermi acceleration is discussed. The diffusive shock acceleration and
“first-order” Fermi acceleration occurring at strong shocks are introduced in
Section 1.3.2.

####  1.3.1 Second order Fermi acceleration

The discussion of Fermi’s approach presented is based on Longair (2011) [116].
In the second order Fermi mechanism, particles are assumed to be accelerated
thanks to stochastic collisions with the clouds (and in particular with the
magnetic field irregularities inside them, also called magnetic mirrors)
moving isotropically in the interstellar medium with a typical velocity
${v_{M}}$. Fermi showed that the particles gain energy stochastically in these
reflections and if the particles remain within the acceleration region for
some characteristic time $\tau_{esc}$, a power law distribution of particles
energies is obtained. The mirror mass is much larger than the particle’s mass
and simply assumed to be infinite for the calculation of the collision between
a particle and a moving mirror so ${v_{M}}$ is unchanged in the collision. The
energy $E^{\prime}$ of the particle in the frame of the moving cloud is

$E^{\prime}=\gamma(E+pv_{M}\cos\theta),$ (1.5)

where $\gamma=\sqrt{1/1-v^{2}_{M}/c^{2}}$ is the Lorentz factor of the mirror,
c is the speed of light and $\theta$ is the angle between the particle
trajectory and the velocity direction of the mirror. The x-component of the
particle’s momentum $p$ in the mirror frame is

$p^{\prime}_{x}=p^{\prime}\cos\theta^{\prime}=\gamma(p\cos\theta+\frac{v_{M}\cdot
E}{c^{2}})$ (1.6)

In this frame, the particle’s energy is conserved,
$E^{\prime}_{before}=E^{\prime}_{after}$, and its momentum along the
x-direction is reversed, so $p^{\prime}_{x}=-p^{\prime}_{x}$. The energy of
the particle after the collision in the observer frame is

$E^{\prime\prime}=\gamma(E^{\prime}+v_{M}\cdot p^{\prime}_{x})=\gamma\cdot
E\bigg{[}1+\frac{2\cdot v_{M}\cdot v}{c^{2}}+\frac{v_{M}}{c}\bigg{]}$ (1.7)

with the particle’s velocity $v$. The average energy gain per collision for
the particle after expanding Eq. 1.7 to second order in $v_{M}/c$ and
averaging over $\theta$ for all angles between 0 and $\pi$ reads

$\bigg{\langle}\frac{\Delta
E}{E}\bigg{\rangle}=\frac{8}{3}\bigg{(}\frac{v_{M}}{c}\bigg{)}^{2}$ (1.8)

This leads to an energy gain of second order in $v_{M}/c$ and to a power-law
energy spectrum of accelerated particles which is observed from non-thermal
sources. But Fermi’s proposed acceleration mechanism is not efficient for
random velocities of interstellar clouds with $v_{M}/c\leq 10^{-4}$.
Furthermore the mean free path of cosmic rays in the interstellar medium is
$\approx 0.1$ pc. This results in only a few collisions per year and,
therefore, the energy gain is too slow to explain the cosmic-ray energies
observed. Through this process, the spectral index depends on the
characteristics of the acceleration region. On the contrary, as it will be
shown in next section, first-order Fermi mechanism predicts a power law
spectrum with spectral index equal to -2, very close to the value required by
the cosmic ray observations.

####  1.3.2 First order Fermi acceleration

The diffusive acceleration in strong shock waves, results in an energy gain
first order of the shock velocity $v_{s}$ and gives an energy spectrum with
power-law form with a spectral index $\approx-2$. The discussion in the
following paragraphs is based on Longair (2011) [116] and illustrated in
figure 1.1. The acceleration of particles takes place at the shock front that
is formed in a supernova explosion. The shock is represented by the dashed
line in figure 1.1. The fluid velocities before and after a hydrodynamical
shock, moving with velocity $v_{s}$, are different. The particles are assumed
to propagate with a speed much larger than the shock velocity $v_{P}>>v_{s}$
and take no notice of the shock itself because their gyro radius is much
larger than the shock thickness. The upstream fluid velocity is $v_{up}\approx
v_{s}$ in the frame of the shock and the downstream velocity is
$v_{down}=v_{s}/4$ for strong shocks as illustrated in figure 1.1(b). This
dependency follows from the continuity equation

$\rho_{up}\cdot v_{up}=\rho_{up}\cdot v_{s}=\rho_{down}\cdot v_{down}$ (1.9)

Figure 1.1: The dynamics of high-energy particles in the vicinity of a strong
shock. The figure is adapted from Longair (2011).

$\rho_{up}$ and $\rho_{down}$ are the up- and downstream densities and, in the
case of strong shock, $\rho_{up}/\rho_{down}=(\tau+1)/(\tau-1)$ with $\tau$
denoting the ratio of the specific heat capacities. For a fully ionized or
monoatomic gas is $\tau=5/3$ and, hence, $v_{down}=vs/4$. In the reference
system of the upstream gas (figure 1.1(c) ), the particle velocity
distribution is isotropic and the downstream gas is moving with $3v_{s}/4$.
Particles cross the shock and gain energy by scattering head-on off the
downstream fluid. After the scattering the velocity distribution is now
isotropic in the frame of the downstream fluid and the upstream fluid has a
relative velocity of $3v_{s}/4$ in the system of the downstream fluid as
illustrated in figure 1.1(d). After crossing the shock the particles bounce
off the upstream fluid and again gain energy. Therefore, the particles gain an
amount of energy $\varpropto v_{s}/c$ with each crossing of the shock.
Performing a Lorentz transformation between the different systems and
averaging over all angles gives a fractional energy increase per shock
crossing of

$\bigg{\langle}\frac{\Delta
E}{E}\bigg{\rangle}=\frac{4}{3}\bigg{(}\frac{v_{s}}{c}\bigg{)}$ (1.10)

The approach discussed above shows that particles gain energy while crossing
the shock and scattering off magnetic turbulences in the up- and downstream
region. Bell in [38] proposed the back-reaction of the accelerated charged
cosmic rays as source for the magnetic turbulences by resonant Alfven waves.
Alfven waves are generated by particles with velocities larger than the
critical speed $v_{A}\approx B/\sqrt{4\pi\rho}$ in a medium with density
$\rho$ and a magnetic field B. For the definition of the Alfven speed and a
more in-depth discussion of plasma physics see e.g. Kulsrud (2005). Both Fermi
acceleration processes, first and second order, naturally result in a spectral
energy distribution following a power law. If the energy of a particle after
one collision is $E=\beta E_{0}$ and there is a probability $P$ that the
particle remains in the shock after the collision then there are $N=N_{0}\cdot
P^{k}$ particles left in the shock after k collisions with energies
$E=\beta^{k}E_{0}$. Eliminating k gives

$\frac{ln(N/N_{0})}{ln(E/E_{0})}=\frac{ln(P)}{ln(\beta)}$ (1.11)

and therefore

$\frac{N}{N_{0}}=\bigg{(}\frac{E}{E_{0}}\bigg{)}^{LnP/ln\beta}$ (1.12)

and the energy distribution follows a power law

$N(E)dE=constant\cdot E^{-1+(lnP/ln\beta)}$ (1.13)

The spectral index of the power law depends on the probability P that
particles remain in the shock and on the energy gain $\beta$. For diffusive
shock acceleration the argumentation in [38] gives $P=1-v_{s}/c$ and
$\beta=E/E0=1+(4v_{s})/(3c)$. Inserting these values into equation 1.13
results in an energy spectrum with power-law form and spectral index of -2

$N(E)dE\varpropto E^{-2}dE.$ (1.14)

In conclusion, first-order Fermi acceleration mechanism is much more efficient
than the second-order mechanism, since the energy gain is linear in the
velocity of the shock. Furthermore, the predicted spectrum is a simple power-
law with unique value for the spectral index, which is also close to the value
required to account for the observed cosmic ray spectrum. The only
requirements are the presence of strong shock waves and that the velocity
vectors of the high energy particles are randomized on either side of the
shock. For this reason, supernova remnants are very good candidates to be
acceleration sites for cosmic rays

###  1.4 Gamma-ray emitting processes

The study of SNRs can give detailed information about the shock propagating in
the interstellar medium, which in principle is able to accelerate particles up
to very high energies through diffusive shock acceleration, as already
discussed in section 1.3.2. Since accelerated charged particles are deflected
by the Galactic magnetic field during their propagation to Earth, direct
observation at the source cannot be performed. However, charged particles
interact with the SNR environment and produce photons from radio to TeV energy
range, which can be directly detected. Different emission processes can be
responsible for the emission of high energy photons as illustrated in figure.
1.2

Figure 1.2: Photon spectra from leptonic and hadronic emission processes as a
function of energy. Top panel: photon emission by electrons with a spectral
form of a power law with a spectral index equal to $-2$ and an exponential cut
off at 100 TeV. Synchrotron radiation (green), Bremsstrahlung (blue) and
inverse Compton scattering on the cosmic microwave background and infrared
photon fields (red dotted) and on starlight (red solid) for electrons. Bottom
Panel: $\gamma$-rays from proton-proton collisions via $\pi^{0}$-decay
(magenta). Protons are distributed as a power-law with index equal to $-2$ and
an exponential cut-off at 100 TeV. This figure is adapted from [95].

In this example the $\gamma$-ray emission is modeled for electrons and hadrons
with the spectral form of a power law with spectral index $\Gamma=-2$ and an
exponential cutoff at an energy of 100 TeV. The emission in the radio and
X-ray regime is produced by leptons via synchrotron radiation and is displayed
in green on figure 1.2. The blue curve shows the photon spectrum produced by
the Bremsstrahlung mechanism that depends on the density of the medium.
Inverse Compton scattering from high-energy electrons on photon fields, i.e.
the cosmic microwave background and starlight (red dotted and red solid curves
in figure 1.2 ), leads to emission peaking in the TeV energy band and is
produced by the same lepton population as the synchrotron emission. TeV
$\gamma$-ray emission can also be produced by high-energy protons colliding
with atoms of the interstellar medium or molecular clouds producing pions
(both neutral and charged) and in the subsequent neutral pion decay in
$\gamma$-rays as illustrated by the magenta line on the lower panel of figure
1.2. The hadronic and leptonic radiation processes are discussed in the
following Sections

####  1.4.1 Hadronic scenario

Protons (p) produce $\gamma$-ray emission colliding with interstellar
material, mainly hydrogen, and subsequent decay of secondary particles. The
main contribution originates from the decay of the neutral $\pi^{0}$ meson
into two $\gamma$ rays while charged pions decay in muons and neutrinos (with
a mean life of $2.6\cdot 10^{-8}s$):

$p+p\rightarrow\pi+X,$ (1.15)
$\pi^{0}\rightarrow\gamma+\gamma\,\,\,\,(98.8\%)$ (1.16)
$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}\,\,(99.9\%),\,\,\,\,\mu^{+}\rightarrow
e^{+}+\nu_{e}+\overline{\nu}_{\mu}$ (1.17)
$\pi^{-}\rightarrow\mu^{-}+\overline{\nu}_{\mu}\,\,(99.9\%),\,\,\,\,\mu^{-}\rightarrow
e^{-}+\overline{\nu}_{e}+\nu_{\mu}$ (1.18)

The $\pi^{0}$ has a mass of $m_{\pi^{0}}\approx 135MeV/c^{2}$ and the mean
lifetime is about $8.4\cdot 10^{-17}s$. In proton-proton collision also $\eta$
mesons are produced, those likewise have decay channels producing $\gamma$
rays, but as demonstrated by Kelner et al. [108], their contribution to the
total number of $\gamma$ rays is at most 25% depending on the photon energy.
The resulting $\gamma$-ray flux from pion-decay in an energy interval
[$E_{\gamma}$, $E_{\gamma}$ +d$E_{\gamma}$] can be calculated as:

$\frac{dN_{\gamma}}{dE_{\gamma}}=cn_{H}\int_{E_{\gamma}}^{\infty}\,\,\sigma_{inel}(E_{p})J_{p}(E_{p})F_{\gamma}(\frac{E_{\gamma}}{E_{p}},E_{p})\frac{dE_{p}}{E_{p}}$
(1.19)

where $c$ is the speed of light, $n_{H}$ the density of the medium and
$J_{p}(E_{p})$ is the proton flux at a certain proton energy $E_{p}$.
$\sigma_{inel}$ is the inelastic cross section of proton-proton interactions
and can be parametrized as:

$\sigma_{inel}(E_{p})=(34.3+1.88L+0.25L^{2})\cdot[1-(\frac{E_{th}}{E_{p}})^{4}]^{2}\,\,mb$
(1.20)

$L$ is defined as $L=ln(E_{p}/1TeV)$ and the threshold energy $E_{th}$ for
protons to produce $\pi^{0}$ mesons is $1.2\cdot 10^{-3}TeV$. The function
$F_{\gamma}(\frac{E_{\gamma}}{E_{p}},E_{p})$ is the spectrum of $\gamma$-rays
from the $\pi^{0}$\- and $\eta$-mesons decay channels for a certain proton
energy $E_{p}$:

$F_{\gamma}(x,E_{p})=B_{\gamma}\frac{ln(x)}{x}(\frac{1-x^{\beta_{\gamma}}}{1+k_{\gamma}x^{\beta_{\gamma}}(1-x^{\beta_{\gamma}})})^{4}[\frac{1}{ln(x)}-\frac{4\beta_{\gamma}x^{\beta_{\gamma}}}{1-x^{\beta_{\gamma}}}-\frac{4k_{\gamma}\beta_{\gamma}x^{\beta_{\gamma}}(1-sx^{\beta_{\gamma}})}{1+k_{\gamma}x^{\beta_{\gamma}}(1-x^{\beta_{\gamma}})}]$

with $x=E_{\gamma}/E_{p}$. The parameters $B_{\gamma}$, $\beta_{\gamma}$ and
$k_{\gamma}$, derived in [kerner], depend on L:

$B_{\gamma}=1.30+0.14L+0.011L^{2}$ (1.21)
$\beta_{\gamma}=\frac{1}{1.70+0.11L+0.008L^{2}}$ (1.22)
$k_{\gamma}=\frac{1}{0.801+0.049L+0.014L^{2}}$ (1.23)

In general, the spectral form of the proton population is conserved in the
$\gamma$-ray emission. The produced photons receive about one tenth of the
primary proton energy. For the total photon flux as described above a factor
of 1.45 is multiplied to account for the contribution by nuclei heavier than
helium present in the interstellar medium or in the molecular clouds.

####  1.4.2 Leptonic scenario

$\gamma$ rays are emitted by electrons (also positrons are included) losing
energy in magnetic fields, dense medium or scattering on photon fields. Only
photons emitted by Bremsstrahlung or those having gained energy via inverse
Compton scattering can be observed in the TeV energy range, whereas the
synchrotron radiation of electrons is typically emitted in radio and x-ray
energy bands. The same electron population is responsible for the emission
from synchrotron and inverse Compton emission, thus, synchrotron emission is
also discussed shortly.

##### Bremsstrahlung

The term Bremsstrahlung refers to electromagnetic radiation produced by the
deceleration of a charged particle when deflected by another charged particle,
typically an electron by an atomic nucleus. The moving particle loses kinetic
energy, which is converted into radiation (i.e., a photon), thus satisfying
the law of conservation of energy. In astrophysics the Bremsstrahlung from
relativistic electrons plays an important role, especially in the GeV energy
range. Following the discussion in Blumenthal & Gould [43] the energy spectrum
of the emitted photons depends on the density $n_{Z}$ and atomic number Z of
the medium. The photon spectrum $dN/dE_{\gamma}$ for a given electron energy
distribution is given by:

$\frac{dN}{dE_{\gamma}}=\int
dE_{i}N_{e}(E_{i})\cdot\frac{dN_{i}}{dE_{\gamma}}$ (1.24)

where $E_{i}$ is the electrons initial energy. The emission spectrum from a
single electron can be written, under the assumption that electrons mainly
scatter off the Coloumb field of hydrogen atoms, as

$\frac{dN}{dE_{\gamma}}=c\cdot\rho_{i}\frac{d\sigma}{dE_{i}}$ (1.25)

$\rho_{i}$ is the hydrogen density of the medium the electron propagates
through. The expression for the differential cross section
$d\sigma/dE_{\gamma}$ for the Bremsstrahlung process is:

$\frac{d\sigma}{dE_{i}}=\alpha\cdot
r_{0}^{2}\frac{f(E_{\gamma}/E_{e})}{E_{\gamma}}$ (1.26)

in which $\alpha$ is the fine structure constant, $r_{0}$ is the classical
electron radius and $f(E_{\gamma}/E_{e})$ is a energy-dependent factor. Under
the assumption that hydrogen plasma is weakly shielding $f(\epsilon)$ is given
by:

$f(\epsilon)=\Phi_{weak}\cdot\bigg{[}1+(1-\epsilon)^{2}+\frac{2}{3}(1-\epsilon)\bigg{]}$
(1.27)

with $\epsilon=E_{\gamma}/E_{e}$ and $\Phi_{weak}$ defined as

$\Phi_{weak}=4(Z^{2}+Z_{el})\left\\{ln\left[\frac{2E_{e}(E_{i}-E_{\gamma})}{E_{\gamma}}\right]-\frac{1}{2}\right\\}$

Z is the atomic number of the element the medium consists of and $Z_{el}$ is
the number of electrons. In general, the spectral form of the Bremsstrahlung
emitted by relativistic electrons in a dense medium mainly depends on the
underlying electron spectrum. Since the same electron population produced TeV
Bremsstrahlung and x-ray synchrotron radiation of a source, these two emission
mechanism are strongly related.

##### Synchrotron radiation

Synchrotron radiation is emitted when charged particles are accelerated
radially, for example electrons gyrating in a magnetic field. Only electrons
will be discussed, since the energy loss of the particle and, therefore, the
emitted radiation is suppressed for protons and ions because of much larger
particle mass. This emission process dominates the non-thermal radiation in
the radio to x-ray regime. A charged particle moving in a magnetic field with
a strength B loses the following amount of energy per time unit

$\frac{dE}{dt}=\frac{4}{3}\sigma_{T}\gamma^{2}\beta^{2}cU_{B}$ (1.28)

$U_{B}$ is the energy density of the magnetic field, $\beta=v/c$ the ratio of
the particle’s velocity and the speed of light. $\sigma_{T}$ is the Thomson
cross section $\sigma_{T}=8r_{0}^{2}/3=6.653·10^{-29}m^{2}$. Depending on the
energy of the charged particle the Thomson cross section has to be replaced by
the Klein-Nishina cross section, that can be write as
$\sigma_{K-N}=8\pi/3r_{0}^{2}(1-2x)=\sigma_{T}(1-2x)$ (where
$x=h\nu/m_{e}c^{2}$) for low energy photons. According to [126] the
synchrotron emission of an electron with energy E rises continuously up to a
maximum energy

$h\nu_{m}=1.93(E/100TeV)^{2}(B/10\mu G)keV$ (1.29)

and drops exponentially beyond this energy. The energy of the emitted photons
in a frequency range $\nu$ to $\nu+d\nu$ radiated by a distribution of
electrons in the energy range E to E + dE can be expressed as

$J(\nu)d\nu=\bigg{(}-\frac{dE}{dt}N(E)dE\bigg{)}$ (1.30)

Following [116] and substituting $E=\gamma
m_{e}c^{2}=\sqrt{\nu/\nu_{g}}m_{e}c^{2}$ with $\nu_{g}=e·B/(2\pi m_{e})$,
differentiating for dE and replacing dE/dt with 1.28 gives:

$J(\nu)\varpropto B^{(p+1)/2}\nu^{-(p-1)/2}$ (1.31)

with p is the slope of the electron energy spectrum. The electron energy
${E_{e}}$ required for x-ray radiation at a specific energy is, according to
[126],

$E_{e}=72\,\bigg{(}\frac{h\nu_{x-ray}}{1keV}\bigg{)}^{1/2}\bigg{(}\frac{B}{10\mu
G}\bigg{)}\,TeV$ (1.32)

The observation of non-thermal X-ray emission in supernova remnants thus
demonstrates the presence of high-energy electrons accelerated up to energies
of $\approx$ 100 TeV.

##### Inverse Compton effect

Inverse Compton scattering consists of the interaction of a high energy
electron with a low energy photon; the electron transfers part of its energy
to the seed photon, creating a gamma-ray. The energy loss of the electrons can
be written similar to equation 1.28 as:

$\frac{dE}{dt}=\frac{4}{3}\sigma_{T}\gamma^{2}\beta^{2}cu_{r}ad$ (1.33)

with the energy density of the photon field $u_{rad}$ replacing the magnetic
energy density. In the Thomson regime only a small fraction of the electron
energy is transferred to the photon. The spectral emissivity $I(\nu)$, under
the assumption of an isotropic photon field in the ultra-relativistic regime,
according to Blumenthal & Gould [43] calculation is:

$I(\nu)d\nu=\frac{3\sigma_{T}c}{16\gamma^{4}}\nu\bigg{[}2\nu\cdot
ln\bigg{(}\frac{\nu}{4\gamma^{2}\nu_{0}}+\nu+4\gamma^{2}\nu_{0}-\frac{\nu^{2}}{2\gamma^{2}\nu_{0}}\bigg{)}\bigg{]}d\nu.$
(1.34)

The maximal energy a photon can gain by head-on collision with an electron is
given by

$h\nu_{max}\approx 4\gamma^{2}h\nu$ (1.35)

Relevant photon fields in astrophysics are the cosmic microwave background
(CMB), infrared light emitted by dust and optical starlight. Inverse Compton
scattering of electrons on these photon fields are the standard scenario to
explain, for example, the observed TeV $\gamma$-ray radiation from some
supernova remnants.

##  Chapter 2 Interacting supernova remnants

Current Cherenkov telescopes such as H.E.S.S., MAGIC and VERITAS detected
$\gamma$-ray emission in the TeV energy range from more than twenty-five
supernova remnants, considering also the $\gamma$-ray satellite that works in
GeV energy range, $\gamma$-ray emission is detected from more than thirty SNR.
For most of the supernova remnants it is not clear whether the observed
$\gamma$-ray emission is mainly produced by accelerated leptons or hadrons.
Molecular clouds near supernova remnants provide an appropriate environment to
probe the protons acceleration by the supernova shocks, as $\gamma$-ray
emission coincident with molecular clouds proves the hadronic origin of the
emission. Leptonic emission processes, except possibly Bremsstrahlung in the
Galactic centre region, are negligible regarding the $\gamma$-ray emission of
clouds in the TeV energy band. The accelerated proton spectrum is obviously
the most interesting in order to have hints of the acceleration of cosmic rays
in SNRs and therefore, the $\gamma$-ray spectrum acquires a key role in the
search for evidences of the so called "SNR paradigm" for cosmic rays. One of
the most convincing way of proving the hadronic origin of the $\gamma$-ray
flux is the observation of the shape of the spectrum at energies below 100
MeV, where the $\pi^{0}$-decay spectrum presents a break due to the threshold
energy of the pion production, often called pion bump. At this energy there
are two key instruments (Fermi-LAT and AGILE) to proof the hadronic origin of
the gamma ray emission of some supernova remnants. Two different scenarios for
$\gamma$-ray emission from molecular clouds are reported. Section 2.2
describes the scenario of “passive molecular clouds”, clouds only embedded in
the cosmic ray sea without a individual source nearby. Molecular clouds near
cosmic ray sources such as molecular clouds interacting with the shock of a
supernova remnant are introduced in Section 2.3. First the general properties
of molecular clouds are outlined in Section 2.2.

###  2.1 General properties of molecular clouds

Molecular clouds are dense accumulations of molecular gas within the
interstellar medium, located mostly in the spiral arms. Molecular clouds are
known since the early 1980s to harbour star forming regions [42], they can be
observed directly using molecular line emission at radio wavelengths. The main
constituent is molecular hydrogen ($H_{2}$). Since $H_{2}$ does not posses a
permanent dipole moment, thus, no transitions are excited at low temperatures
as usually found in the environment of molecular clouds. For $H_{2}$ line
emission to be produced, temperatures of about 150 K are required much higher
than commonly found in molecular clouds. Hence, other molecules are used as
tracers for $H_{2}$ such as the carbon monoxide molecule (CO). The most
commonly used emission line, according to [75], is the rotational transition
${}^{12}C^{16}O(J=1-0)$ with a frequency of 115.27120 GHz
111http://www.cv.nrao.edu/php/splat/. ${}^{12}C^{16}O(J=1-0)$ has an
excitation energy of $\approx$ 5 K and a critical density of $\approx
1000cm^{-3}$. Simplified, the critical density is the density at which the
line emission gets observable. Line emissions of other transitions or
molecules have different excitation energies and critical densities, thus the
density distribution of the interstellar medium can be probed using different
molecules or different transitions as tracers. The $H_{2}$ column density and
the cloud mass can be estimated from the intensity of the CO line emission
W(${}^{12}C^{16}O(J=1-0)$) via the conversion factor $X_{CO}$ as discussed in
[46]:

$N_{H_{2}}=X_{CO}\cdot W(^{12}C^{16}O(J=1-0))$

With the exception of the Galactic centre, in the galactic plane $X_{CO}$ has
a typical value of $\approx 2\cdot 10^{20}(Kkms^{-1})^{-1}$ and can be
determined via measurements of the virial mass of the cloud, dust emission,
extinction and from GeV $\gamma$-ray emission along the Galactic plane. The
first observation of CO emission was reported by Wilson et al. (1970) [145]
observed in the galactic interstellar medium from the Orion nebula. Subsequent
radio surveys in the 1970s and ’80s established molecular clouds as major
constituents of the Galactic interstellar medium [75]. CO emission was also
detected in other galaxies [57]. The first complete CO survey of the Galactic
plane was published by Dame [62]. Since then Galactic CO emission has been
observed not only from molecular clouds and giant molecular clouds (GMCs;
molecular clouds with masses of about $\geq 10^{5}M_{sun}$ and sizes around 50
pc) but also as cold CO “bridges” between the spiral arms and extending beyond
the galactic disk.

Figure 2.1: Integrated intensity image of GRS (Galactic Ring Survey)
${}^{13}CO$ line emission, integrated over all velocities (from $V_{LSR}=-5$
to 135 $kms^{-1}$ for Galactic longitudes $l\leq 40^{\circ}$ and $V_{LSR}=$ -5
to 85 $kms^{-1}$ for Galactic longitudes $l>40^{\circ}$. The image shows that
most of the emission is confined to $b\sim 0^{\circ}$, with concentrations at
$l\sim 23^{\circ}$ and $l\sim 31^{\circ}$. A striking aspect of the image is
the abundance of filamentary and linear structures and the complex morphology
of individual clouds. The image is in units of Kkm $s^{-1}$. Image taken from
[99] )

Molecular clouds have no homogeneous density distribution as it is clearly
visible in 2.1, but a rather clotted structure with dense cores, often called
clumps. This figure shows ${}^{13}CO$ emission from the Galactic Ring Survey
[99] integrated over velocities between -5 $kms^{-1}$ and 135 $kms^{-1}$. The
magnetic field within molecular clouds can be estimated, for example by
observing the polarized emission of dust or using the Zeeman effect in HI, OH
and CN line emission. The maximum interstellar magnetic field strength is
$\approx 10\mu G$ up to densities of $300cm^{-3}$ and increases at higher
densities following a power law with index $\approx 2/3$. Discussion about
magntic field within molecular clouds and different methods for the magnetic
field estimation can be found in [60].

Figure 2.2: Three molecular clouds have different distances but the same line-
of-sight velocity $V_{R}$. The cloud 2 is located at the tangent point where
the orbital velocity $V_{MC}$ is equal to the line-of-sight velocity $V_{R}$.
The illustration is adapted from Roman-Duval [128]

The distances of molecular clouds can be determined via the so-called
kinematic distance method. The disk of the Milky Way is rotating and therefore
the Doppler shift of the observed molecular line gives the relative line-of-
sight velocity $V_{R}$ of the cloud. This is the projection of the orbital
velocity $V_{MC}$ of the molecular cloud onto the line of sight as shown for
the three molecular clouds on figure 2.3. The galactocentric radius R, (the
distance of the cloud to the Galactic centre), can be determined with rotation
curves of the galaxy [47]. The distance d of the molecular cloud to the sun
can be calculated, according to [128], as

$d=R_{0}\cdot cos(l)\pm\sqrt{R^{2}-R^{2}_{0}\cdot sin^{2}(l)}$

with the longitude position $l$ of the cloud and the galactocentric radius
$R_{0}$ of the sun. The molecular clouds 1 and 3 on figure 2.3 have different
distances but the same line-of-sight velocity. This is represent the “near-far
ambiguity” and can only be resolved with additional information, for example
those obtained by looking for absorption features in HI emission as done by
[128]. Calculate the molecular clouds mass is more delicate, since the major
contributor to the cloud mass, namely H2, is practically invisible. In order
to estimate the mass, it is necessary to rely on a tracers CO is the most used
tracer of molecular hydrogen. Assuming to know precisely the conversion factor
between the tracer and $H_{2}$ column density, then derive the mass is pretty
easy. Usually it is possible to obtain the brightness temperature vs. velocity
profile $T_{b}=T_{b}(\nu)$ of the tracer from observations. The brightness
temperature is the temperature of an object that has the same brightness (e.g.
energy emitted per unit time, area and solid angle) than a black body of that
temperature:

$I_{\nu}=\frac{2h\nu^{3}}{c^{2}}\frac{1}{\exp(\frac{h\nu}{kT})-1}$ (2.1)

where $h$ is the Planck constant, $\nu$ is the frequency, $k$ is the Boltzmann
constant and c the speed of light. On the other hand velocity is related to
physical distance as explained in the former section. Once the temperature
profile is measured, the abundance of the tracer can be derived from the
integral emissivity (W) and, under some assumptions, this quantity can be
related to the $H_{2}$ column density $N(H_{2})$, namely the number of protons
per unit area. Then having $N(H_{2})$ the mass M can be derived integrating
over the physical size of the MC:

$M(H_{2})=2m_{p}\int N(H_{2})dA$ (2.2)

where $m_{p}=1.67\cdot 10^{-27}kg$ is the proton mass, and $d$A is the surface
element. The factor two takes into account that the molecule is made of two
protons.

###  2.2 Passive molecular clouds

Cosmic rays at GeV energies remain in the Galaxy for about $\sim 10^{7}$ yr
[123]. Particles got accelerated by individual sources, but lose the
information of their origin and contribute to the “cosmic ray sea”. Molecular
clouds are embedded in this cosmic ray sea and provide “passive” targets for
cosmic ray hadrons to interact and produce $\gamma$-ray emission via neutral
pion decay. The integral $\gamma$-ray flux $F_{\gamma}$ above an energy
threshold $E$ depends on the cosmic ray flux, on the molecular cloud mass
$M_{MC}$ and on the distance $D$ according to [8]:

$F_{\gamma}(\geq E)=1.5\cdot
10^{-13}\bigg{(}\frac{E}{1TeV}\bigg{)}^{-1.75}\bigg{(}\frac{M_{5}}{D^{2}_{kpc}}\bigg{)}cm^{-2}s^{-1}$

$M_{5}$ is the mass of the molecular cloud in units of $10^{5}M_{sun}$,
$D_{kpc}$ is the distance of the cloud in kpc and this approximation is only
valid for energies much larger than 1 GeV. $\gamma$-ray emission from local
molecular clouds is observed in the GeV energy range and can be used to probe
the level of the cosmic ray sea at different locations of the Galaxy [98].
Passive molecular clouds have not been detected yet in the TeV energy range
because of the low expected fluxes and their extension, but the future
experiment CTA observatory should be capable of observing nearby clouds as
explained in [121].

###  2.3 Molecular clouds interacting with shocks of supernova remnants

Associations of supernova remnants with star-forming regions are established
$\gamma$-ray sources in GeV astronomy since the 1970s [118]. These regions are
embedded in molecular clouds and therefore the ambient medium provides an
ideal dense target material for hadrons accelerated in the supernova remnant
and $\gamma$-ray emission due to pion decay is observed. In those regions
either the supernova explodes within the dense medium, where particle
acceleration cannot take place efficiently due to the low shock speed, or the
remnant expands into a nearby cloud [9]. Supernova remnants that interact with
molecular clouds are bright sources detected with the Fermi-LAT and AGILE
satellites in the GeV energy range and are also strong in the TeV energy band
where they are observed by the major ground-based Cherenkov instruments. The
observed $\gamma$-ray spectrum reflects the spectrum of accelerated hadrons in
the shock of a supernova remnant. As will be discuss in 2.5 the interaction of
the shock of a supernova remnant with a molecular cloud is usually established
with observation of OH 1720MHz maser emission within the extent of the remnant
[72]. This type of emission is defined by compact narrow emission lines and
high effective temperatures and is produced by OH molecules colliding with
$H_{2}$ molecules. This process takes place in the compressed and heated
material behind the shock. Following Jiang et al. [100] other evidence for
supernova remnant/molecular cloud interactions includes: line broadening in
molecular emission or asymmetric profile, line emission with large high-to-low
excitation line ratio, near infrared emission due to shock excitation and
morphological agreement of the molecular emission with supernova remnant
features. A prominent example for a supernova remnant/molecular cloud
association observed both in the TeV and GeV bands [61] [79] is the supernova
remnant W28. The CO emission from this region is shown in figure LABEL:. The
TeV $\gamma$-ray emission observed with H.E.S.S. is indicated by the green
contour lines [12]. The H.E.S.S. source HESS J1801–233 is coincident with a
molecular cloud clearly visible in CO emission. The black circle indicates the
radio size and position of W28. The $\gamma$-ray emission is spatially
coincident with parts of the shell of the supernova remnant and the molecular
cloud. Also the three sources HESS J1800–240 A,B and C are coincident with
dense molecular material and the $\gamma$-ray emission observed there is
interpreted as pion decay from accelerated particles escaped from the remnant
and propagated through the interstellar medium illuminating the molecular
clouds.

Figure 2.3: The W28 region. The ${}^{12}CO$ emission is overlaid with the
4,5,6$\sigma$ green contours of the TeV $\gamma$-ray emission from H.E.S.S.
[12]

###  2.4 Illuminated molecular clouds

As said in previous sections, $\gamma$-ray production from relativistic
particles can proceed through inverseCompton (IC) scattering of ambient
photons by the energetic electrons, non-thermal Bremsstrahlung (NTB) from
collisions between relativistic electrons and ambient material, and the decay
of neutral pions formed after the collisions of energetic protons with ambient
nuclei.

Figure 2.4: Simulated broadband spectrum from SNR undergoing efficient
diffusive shock acceleration of electrons and protons. The solid magenta curve
represents synchrotron emission, the dotted blue curve is IC emission, the
dashed green curve corresponds to NTB, and the long-dashed curve represents
the $\pi^{0}$-decay emission.

Figure 2.4 presents a simulation of the broadband spectrum produced by an SNR
undergoing efficient acceleration of electrons and ions. The magenta curve
represents synchrotron emission from the relativistic electrons, and the
dotted blue curve corresponds to IC emission produced from that same electron
population up-scattering photons from the cosmic microwave background (CMB).
The dashed green curve represents NTB from the relativistic electrons
interacting with ambient material, and the long-dashed red curve corresponds
to $\gamma$-rays from the decay of neutral pions produced by collisions of the
relativistic proton component with ambient nuclei. The emission from both NTB
and $\pi^{0}$-decay scales with the ambient density $n_{0}$. As a result, in
high density environments such as those encountered in SNR/MC interactions,
significant $\gamma$-ray emission is expected if the SNR has been an active
particle accelerator. For electron-to-proton ratio for the injected particles
equal to $K_{ep}\sim 10^{-2}$, as measured locally, the $\pi^{0}$-decay
emission dominated, making such $\gamma$-ray emission an important probe of
the hadronic component of the particles accelerated by the SNR. An important
additional consideration for $\gamma$-ray production is the local photon
energy density. Contributions from starlight as well as infrared (IR) emission
from local dust can increase the IC emission and change the spectral shape of
this component due to the different effective temperatures of these photon
components. These contributions are highly dependent upon galactocentric
radius. Moreover, IR emission produced by the SNRs themselves can contribute
significantly to the IC $\gamma$-ray emission. Modeling of the broadband
spectra from such SNRs, in order to ascertain the nature of the $\gamma$-ray
emission, is complicated and has led to mixed interpretations, making the
evidence for ion acceleration controversial in some cases. However, in a
growing number of cases, $\gamma$-ray emission from some SNRs known to be
interacting with MCs seems to clearly require a significant component from
pion decay.

###  2.5 Signatures of SNR/MC interaction

Resolved studies of SNRs and their associated clouds are most easily carried
out for systems within the Milky Way, though with the increasing capacity of
new observational facilities, the studies are rapidly extending to our
neighbor galaxies, as shown by the observation of 20 shocked clouds in the
supernova remnant N132D in the Large Magellanic Cloud (LMC) [64] . To date,
studies have identified $\gamma$-ray emission from 37 galactic SNRs, about
half of which are interacting with MCs. Of these, the evidence for energetic
hadrons as the source of these $\gamma$-rays is compelling for more than 50%
of these based on energetic arguments and/or broadband spectral modeling.
Unambiguously establish whether an SNR is physically associated with a MC, is
not trivial. Several distinct criteria can be used to demonstrate a possible
SNR/MC interaction as discuss in depth by Slane et al. [134]. Morphological
traces along the periphery of the SNRs such as arcs of gas surrounding parts
of the SNR or indentations in the SNR outer border encircling dense gas
concentrations can be observed in images of SNRs. Usually such features
indicate that a dense external cloud is disturbing an otherwise spherically
symmetric shock expansion. These initial signatures need to be confirmed with
more convincing, though more rare, features like broadenings, wings, or
asymmetries in the the molecular line spectra, high ratios between lines of
different excitation state, detection of near infrared $H_{2}$ or [Fe II]
lines, peculiar infrared colors, or the presence of OH (1720) MHz masers, the
most powerful tool to diagnose SNR/MC interactions. Once the association
between an SNR and a molecular feature is established, it serves to provide an
independent estimate for the distance to the SNR through the observed Doppler
shift of the line and by applying a circular rotation model for the Galaxy.
The basic tool used to investigate cases of SNR/MC interactions is the survey
of the interstellar medium in a field around the SNR using different spectral
lines, is: atomic hydrogen emitting at $\lambda$ 21 cm to the dense shielded
regions of molecular hydrogen emitting in the millimeter and infrared ranges.
The most widely used proxy to track molecular gas is CO. This molecule has a
nonzero dipole moment, and radiates much more efficiently than the abundant
$H_{2}$ (with no dipole moment) and can be detected easily.

##### HI emission

The $\lambda$ 21 cm line of atomic hydrogen is the basic resource to observe
the environs of supernova remnants searching for candidate structures with
which the SNRs may be interacting. However, because of the high abundance and
ubiquity, confusion with unrelated gas along the line of sight represent the
major problem of this method, so the detection of an HI candidate structure
needs to be confirmed with other indicators that reinforce the hypothesis of
association. The HI can be studied either in emission or in absorption, the
latter being a very effective tool for constraining the distance to the SNR.
Moreover, the HI mapping of large fields around SNRs is an excellent tool to
explore the history of the precursor star, for example by detecting large
wind-blown bubbles around the SNRs, and to estimate the gas density of the
medium where the blast wave expands.

##### Molecular emission

As mentioned above CO studies are the most widely used tool to analyze
distribution and kinematics of cold, dense clouds with high molecular content.
${}^{12}CO$ and ${}^{13}CO$ lines in their different excitation states have
been surveyed over most of the sky “The Milky Way in Molecular Clouds”222
http://www.cfa.harvard.edu/mmw/MilkyWayinMolClouds.html by Dame et al., the
“FCRAO CO Survey of the Outer Galaxy”333http://www.astro.umass.edu/
fcrao/telescope/2quad.html by Heyer et al., the “Galactic Ring Survey”444
http://www.bu.edu/galacticring/ by Jackson et al , or the more recent “MOPRA
Southern Galactic Plane CO Survey”555www.mopra.org › data by Burton et al..
Additionally, dedicated studies using different facilities and in different
molecular transitions have been conducted towards many Galactic SNRs, such as
the case of IC 443. The SNR IC443 because of its location in a relatively
confusion free region of the outer Galaxy, is a text book case to analyze
shock chemistry, and as such it has been thoroughly studied in many molecular
transitions The morphology consisting of two semi-circular shells with
different radii is an indication of expansion in an environment with a marked
density contrast. IC443 is also an excellent example where the interaction of
the SNR with a molecular cloud almost surely gave origin to the $\gamma$-ray
emission through a hadronic mechanism. In effect, there is an excellent
concordance between VERITAS very high energy $\gamma$-ray radiation [1] and
the ${}^{12}CO$ J=1-0 cloud, see figure 2.5

Figure 2.5: The SNR IC443: left: radio-continuum at 330 MHz (in grey) with
$\gamma$-ray emission from VERITAS (contours), middle: VERITAS image of very-
high-energy $\gamma$-ray emission. Image taken from [65] Right: ${}^{12}CO$
(J=1–0; blue) and ${}^{13}CO$ (J=1–0; red) intensity maps in the -10 km
$s^{-1}$ to 10 km $s^{-1}$, overlaid with the 1.4 GHz radio continuum emission
contours [136].

##### Masers

OH (1720 MHz) masers have long been recognized as signposts for supernova
remnants molecular clouds interactions. The conditions required for the maser
formation are rather narrow, with temperatures in the range 50 - 125 K,
densities between n = $10^{3}$ \- $10^{5}$ $cm^{-3}$, and OH column densities
of $10^{16}$ \- $10^{17}$ $cm^{-2}$. As the physical conditions needed to pump
this maser are so strict, it has to be noted that their presence is sufficient
to demonstrate interaction, but its absence does not rule it out. According to
[50] , OH (1720 MHz) masers have been found in $\sim$ 10% of the known SNRs in
our Galaxy. The location of masers show, in general, very good coincidence
with density/shock tracers. In the case of W44 there is a strong correlation
between the morphology of the molecular gas and the relativistic gas traced by
synchrotron emission at centimeter wavelengths, see figure 2.6. In addition to
providing a clear indication of SNR/MC interactions, OH(1720 MHz) masers also
permit an independent estimate for the kinematic distances to the clouds, and
thus for the remnants.

Figure 2.6: Left: VLA image of W44. Circles indicate the positions of observed
OH maser emission, and cross indicates the position from which the CO profile
at the right is taken. Right: CO line profile from Cloud C in field of W44,
illustrating broad wings produced by an interaction between the cloud and the
SNR. Image taken from [134]

##  Chapter 3 Catalog of $\gamma$-ray Supernova Remnants

At the moment there is no one complete catalog of $\gamma$-ray (both GeV and
TeV) supernova remnants that for each object gives the main physical (age,
distance, positions, dimensions) and spectral information. This kind of
information are the essential to classify supernova remnants or to relate the
$\gamma$-rays emission mechanism with their features, but also because is the
starting point to generate the spectra of synthetic population of interacting
supernova remnants, (see chapter 5.2 for details). Thanks to an extensive
research 37 $\gamma$-rays emitting supernova remnants have been found. The
catalog was made recovering all physical data in catalogs already existing (A
Catalogue of Galactic Supernova Remnants by D.A Green [83], High Energy
Observations of Galactic Supernova Remnants maintained by the SNR group of the
University of Manitoba [67] , TeVcat by S. Wakely and D. Horan [143]) and all
available spectral data in the most recent literature of each instruments
collaboration considered (Fermi-LAT, AGILE, HESS, MAGIC, VERITAS) and in the
Fermi Large Area Telescope Third Source Catalog (3FGL, [2]) or in the Third
Catalog of Hard Fermi-LAT Sources (3FHL, [18]) and in every publications
containing $\gamma$-ray spectral data of supernova remnants.

Excluding data from fermi catalogs discuss in the next paragraph, the other
spectral data present in this work are taken directly from the publication
using PlotDigitizer software which allows to take a scanned image of a plot
and quickly digitize values off the plot. I chose to work in $\nu F\nu$ flux
units so all available data was conveerted in $erg\cdot cm^{-2}\cdot s^{-1}$
and in $eV$ for energies. Table 3.1 contains all data sets found divided by
instruments, symbol "$\ast$" marks those used in the spectral fitting
procedure as explained in section 3.2.

##### Fermi catalogs

The 3FGL catalog refers to high-energy $\gamma$-ray sources detected in the
first four years of the Fermi Gamma-ray Space Telescope mission by the Large
Area Telescope (LAT) in the 100 MeV–300 GeV energy range. The 3FHL catalog
instead is constructed from the first 7 years of data in the 10 GeV - 2 TeV
energy range.

In the Fermi-LAT catalogs all the objects have a classification. In the Fermi
catalogs there is a distinguish between associated and identified sources,
with associations depending primarily on close positional correspondence and
identifications requiring measurement of correlated variability at other
wavelengths or characterization of the 3FGL or 3FHL source by its angular
extent ([18]). Catalogs cited above were recovered from the data access
section of the Fermi mission
site111https://fermi.gsfc.nasa.gov/ssc/data/access/ and then from the
catalogs.fits file were extracted only the sources with the "SNR" or "snr"
classifications. The "SNR" keywords refers to identified sources, instead
"snr" refers to associated sources. In the 3FGL catalog the source photon
fluxes are reported in five energy bands (100 to 300 MeV; 300 MeV to 1 GeV; 1
to 3 GeV; 3 to 10 GeV; 10 to 100 GeV) in $erg\cdot cm^{-2}\cdot s^{-1}$
instead of the flux uncertainties are reported in $photons\cdot cm^{-2}\cdot
s^{-1}$. While in the 3FHL catalog energy limits were set to 10, 20, 50, 150,
500 GeV and 2 TeV in $photons\cdot cm^{-2}\cdot s^{-1}$ units like the flux
uncertainties.

SNR Name | FERMI-LAT | AGILE | HESS | MAGIC | VERITAS
---|---|---|---|---|---
G359.1-0.5 | (a) - [97]∗ | — |  [13]∗ | — | —
HESS J1731-347 |  [4] \- [84]∗ | — |  [86]∗ | — | —
CTB 37B | (b) - [148]∗ | — |  [10]∗ | — | —
CTB 37A | (a,b∗) - [53] \- [48]∗ | — |  [11]∗ | — | —
RX J1713.7-3946 | (a,b) - [66]∗ | — |  [15]∗ | — | —
SN 1006 (NE) |  [150] \- [58]∗ | — |  [3]∗ | — | —
SN 1006 (SW) |  [150] \- [58]∗ | — |  [3]∗ | — | —
G318.2+0.1 | — | — | — | — | —
RCW 86 | ($b^{\ast}$) [19]∗ |  |  [88]∗ | — | —
G298.6-0.0 | ($a^{\ast},b^{\ast}$) | — | — | — | —
Vela Jr. | ($a^{\ast},b^{\ast}$) - [137] | — |  [87]∗ | — | —
Puppis A | (a,$b^{\ast}$) - [149]∗ | — |  [90]∗ | — | —
IC 443 | (a,b) - [6]∗ |  [138]∗ | — |  [25]∗ |  [1]∗
Tycho | (a,b) - [33]∗ | — | — | — |  [33]∗
Cas A | (a,b) - [17]∗ | — | — |  [17]∗ |  [113]∗
Gamma Cygni | (a,b) - [71]∗ | — | — | — |  [28]∗
Cygnus Loop | ($a^{\ast},b^{\ast}$) - [125]∗ | — | — | — | —
W51 C | (a,b) - [101]∗ | — | — |  [27]∗ | —
W49 B | (a,b) - [92]∗ | — |  [92]∗ | — | —
W44 | (a,$b^{\ast}$) - [6]∗ | [51] | — | — | —
W41 | ($a^{\ast}$) - [91] | — |  [91]∗ |  [26]∗ | —
W28 north |  [61]∗ | [79] |  [61]∗ | — | —
W28 A |  [61]∗ | — |  [61]∗ | — | —
W28 B |  [61]∗ | — |  [61]∗ | — | —
W28 C |  [61]∗ | — |  [61]∗ | — | —
G349.7+0.2 | (a,$b^{\ast}$) - [89]∗ | — |  [89]∗ | — | —
HESS J1912+101 | — | — |  [82]∗ | — | —
HESS J1534-571 |  [31]∗ | — |  [31]∗ | — | —
MSH 17-39 | ($a^{\ast}$) - [56] | — | — | — | —
HB 21 | ($a^{\ast}$) - [122] | — | — | — | —
HESS J1614-518 | — | —- |  [82]∗ | — | —
W30 | ($a^{\ast}$,b) - [21] | — |  [16]∗ | — | —
3C 391 | (a,$b^{\ast}$) - [54]∗ | — | — | — | —
CTB 109 | ($a^{\ast},b^{\ast}$) - [55] | — | — | — | —
G337.0-0.1 | (a,$b^{\ast}$) - [56]∗ | — | — | — | —
S147 | (a,$b^{\ast}$) - [107]∗ | — | — | — | —
Kes17 | ($a^{\ast},b^{\ast}$) - [77] | — | — | — | —
Table 3.1: All data and catalogs data considered in this work, (a) 3FGL, (b)
3FHL. Data actually used in the fitting procedure are marked with "$\ast$"
symbol

###  3.1 Distance and Age of supernova remnants

Reliable distances for the Galactic supernova remnants are crucial in order to
obtain the other basic parameters such as for example size, age and the
explosion energy. Rarely for the supernova remnants there are unique
information about distance and age, especially because of different
measurement method. Distances and ages used in this work are taken from
literature, whose results are reported in table 3.2. For each object there is
an indication of the minimum and maximum distance and age and the exact value
used in this work. In some cases the the value chosen is not the mean value,
this is because there is strong evidence or a more recent observation which
favors a certain measure in the possible range of values.

In sections 3.1.1 and 3.1.2 are described the main ways to calculate distances
and ages of these sources.

####  3.1.1 Distance

##### The surface brightness - diameter relation

There is evidence that there should be a relationship between the surface
brightness ($\Sigma$) and the diameters of a supernova remnants ($D$). This
relation can be written as $\Sigma=aD^{\beta}$. However, the validity of this
relation has not yet been universally accepted. This method is not very
precisely in fact as reported in [52] the associated errors can reach 40% of
the measure. The reasons are related to:

  * •

The wide range of supernova explosion energies

  * •

The presence of interstellar medium around the remnant, normally higher
densities led to higher $\Sigma$

  * •

The presence of a neutron star within the remnant

  * •

Calibration uncertainties of distance and diameters

The analysis of 36 Galactic shell SNRs by Case et al. suggest that [52]
$\beta=-2.38$, but more recent analysis by Pavlovic et al. [120] set
$\beta=-4.8$, significantly steeper than those in previous works.

##### The kinematic distance

Kinematic distance measurement is based on rotation curve of the Milk Way. For
objects that follow the rotation curve well, we can derive their distances
from the observed radial velocity (e.g. HI clouds). 21 cm line observation
toward SNRs is used to build HI absorption spectra. Cold gases between
observer and SNRs will lead to absorption, gases behind an SNR will not be
seen in this SNR’s absorption spectrum (see figure 3.1 ). Therefore, the HI
absorption features are used to give a lower distance restriction and the
emissions without absorptions are used to derive an upper distance
restriction. For SNRs showing continuum emission and absorption features,
their positions are given by the velocity at which the continuous absorption
stops.

Figure 3.1: Three cases of relative positions between continuum source and HI
clouds

##### Other methods

Kassim et al. [106] found that X-ray observations could be used to calculated
the Sedov distance of shell type SNRs which are in the adiabatic expansion
phase using:

$D_{s}=8.7\cdot 10^{6}\epsilon^{0.4}_{0}P(\Delta
E,T)^{0.2}\theta^{-0.6}F^{-0.2}_{x0}T^{-0.4}$ (3.1)

Here $\epsilon_{0}$ is the initial energy of the SNR explosion in unit of
$10^{51}$ ergs,$\theta$ is the observed angular diameter of the SNR shell in
arcminutes, $F^{-0.2}_{x0}$ is the measured x-ray flux corrected for
interstellar absorption in units of $ergss^{-1}cm^{-2}$, T is the measured
thermal temperature of the x-ray emitting gas in K and function $P(\Delta
E,T)$ described the power emitting by hot electrons in a low-density plasma
via Bremsstrhalung emission. The uncertainties in determining $D_{s}$ are
dominated by the following assumption:

  * •

the SNR is in te adiabatic expansion phase

  * •

the measured x-ray temperature gives a reliable estimate of the SN shock
velocity

  * •

$\epsilon_{0}$ = 1 is approximately corrected for all the supernova

Another possibility is that some SNRs are associated with distance-known
objects, such as pulsar, HII region, CO cloud. Errors in this method are
caused by not only the distance’s errors of the associated objects, but also
the uncertainty of such association.

####  3.1.2 Age

There are mainly five ways to estimate the age of an SNR:

  * •

1\. If a supernova is associated with a neutron star, it’s age can be
estimated by using the pulsar characteristic age derived from: $t=P/2\dot{P}$,
where P is the rotation period and $\dot{P}$ is the rate of change of period.

  * •

2\. The age of a SNR can be estimated using the total explosion energy (E) and
the energy loss rate $(\dot{E})$ according to $t=e/\dot{E}$.

  * •

3\. Knowing the diameter (D) or equally the radius (R) of a supernova remnants
and the expansion velocity ($v_{s}$), the dynamical time can be calculated as
$t=CR/v_{s}$, where C is a constant that depends on the evolution phase in
which the remnant is (2/5 for the adiabatic expansion phase, 2/7 for the
radiative remnant and 1 for a freely expansion remnant) and on the expanding
velocity of the shock wave.

  * •

4\. For SNRs with a known radius (R) and thermal temperature (T) measured from
X-ray observation, the age can be obtained by $t=3.8\cdot
10^{2}R_{pc}(kT)^{-1/2}_{keV}$ yr derived from the Sedov Relation
$R=(2.026E/\rho)^{1/5}t^{2/5}$.

  * •

5\. For SNRs that already have in their radio spectra the characteristic break
at frequency $\nu_{d}$ due to synchrotron losses in a magnetic field B, the
age can be derived using the relation $t=40000B^{-1.5}\nu_{d}^{-0.5}.$

The ages estimated with these methods have uncertainties up to an order of
magnitude. There are a few supernova remnants with associated pulsars, so the
first method can be use for a very limited number of sources. For some SNR
information about R and $v_{s}$ are available so the age estimations have a
good accuracy. When this parameters are not available, the last two method can
give a plausible indication or limits on the age.

SNR Name |  $D_{min}$ \- $D_{max}$ | $D$ |  $A_{min}$ \- $A_{max}$ | $A$ | Ref.
---|---|---|---|---|---
(kpc) | (kpc) | (kyr) | (kyr) |
G359.1-0.5 | 5 - 8-5 | 7.6 | - | > 10 |  [152] [14]
HESS J1731-347 | 3.2 - 5.2 | 3.2 | 2 - 27 | 14.5 |  [73] [85]
CTB 37B | 8.3 - 11.3 | 13.2 | 0.6- 6 | 5 | [44]
CTB 37A | 6.3 - 9.5 | 9 | 1 - 100 | 16 | [44]
RX J1713.7-3946 | - | 1 | 1.6 - 2 | 1.6 |  [74] [141]
SN 1006 | - | 1.6 | - | 1 | [146]
G318.2+0.1 | - | 9.2 | - | 8 | [96]
RCW 86 | 1.2 - 2.5 | 2.5 | 1.8 - 10 | 1.8 |  [45] [20] [133]
G298.6-0.0 | 5 - 10 | 5 | - | 1 | [36]
Vela Jr. | 0.5 - 1 | 1 | 2.4- 5.1 | 3 | [29]
Puppis A | 1.3 - 2.2 | 2.2 | 3.7 - 5.2 | 4.4 |  [126] [37]
IC 443 | 0.7 - 2 | 1.5 | 3 - 30 | 10 |  [151] [115]
Tycho | 2 - 4.5 | 4.5 | - | 0.4 | [93]
Cas A | 3.3 - 3.9 | 3.4 |  | 0.3 |  [23] [69]
Gamma Cygni | 1.5 - 2.6 | 1.5 | 6- 7 | 6.6 |  [132] [142]
Cygnus Loop | 0.8 - 10 | 0.8 | 10 - 20 | 15 | [68]
W51 C | 4 - 6 | 5.6 | 16 - 30 | 30 |  [114] [124]
W49 B | 8 - 14 | 8 | 1 - 4 | 2 |  [154] [114]
W44 | 1.9 - 3.3 | 3 | - | 20 |  [51] [104]
W41 | 3.9 - 4.8 | 4.8 | 60 - 200 | 100 |  [124] [140]
W28 | 1.8 - 3.3 | 1.9 | 35- 150 | 35 |  [132] [79]
G349.7+0.2 | 18.3 - 22.4 | 11.5 |  | 1.8 |  [139] [72]
HESS J1912+101 | - | 4.1 | 70 - 200 | 100 | [135]
HESS J1534-571 | 3.5 - 6 | 3.5 | - | 10 |  [117] [32] [129]
MHS 17-39 | - | 11.8 | - | - | [49]
HB 21 | 0.8 - 1.7 | 1.7 | 5.6 - 45 | 15 |  [30] [110]
HESS J1614-518 | 1.2 - 5.5 | 1.5 | - | 30 | [130]
W30 | 3.2 - 6 | 4.6 | 15 - 28 | 20 |  [105] [22]
3C 391 | 5.4 - 7.2 | 7.1 | 4 - 9 | 9 |  [114] [109]
CTB 109 | 2.8 - 7.5 | 3.2 | 8.8 - 24 | 14 |  [111] [131]
G337.0-0.1 | - | 11 | - | 5 |  [72] [59]
S147 | 0.9 - 1.6 | 1.39 | 30- 100 | 30 |  [103] [63]
Kes17 | - | 9.7 | 2 - 40 | 14 |  [81] [78]
Table 3.2: Distances and ages of all the $\gamma$-ray emitting SNR.

In figure 3.2 are plotted distribution of distance and age.

Figure 3.2: Histogram of distance (left) ad age (right) of all the SNR.

###  3.2 Spectral fitting

Supernova remnants spectra are obtained joined together all the available
data. To eventually introduce a classification based on some spectral
properties, was used a model independent approach for the emission mechanism,
meaning use a unique model for all the supernova remnants. Some of the known
interacting supernova remnants (like W44 [80], W51 C [102], IC443 [7] ) have
in their $\gamma$-ray spectrum the so called "pion bump", so their emission is
certainly hadronic. Reasonably also the other interacting supernova remnants
emits $\gamma$-rays in the same way, therefore the $\gamma$-ray emission was
fitted using an hadronic model for all the SNRs considered. Obviously the
hadronic model isn’t a good model for all the supernova remnants but, as said
this choice excludes the possibility of introduce a artificial classification
depending on the emission model.

In order to pass from spectral points to the pion decay spectrum Naima [153]
was used, a Python package developed with the aim of computing the non-thermal
radiative output of relativistic particle distributions (protons in this
case). The systematic method discuss above, means, in this case, a broken
power law (BPL) distributions for the protons populations and the Pion decay
radiative model for all sources. Naima can derive the best-fit and uncertainty
distributions of spectral model parameters through Markov Chain Monte Carlo
(MCMC) sampling of their likelihood distributions [70]. The measurements and
uncertainties in the provided spectrum are assumed to be correct, Gaussian,
and independent Under this assumption, the likelihood of observed data given
the spectral model $S(\vec{p},E_{i})$, for a parameter vector $\vec{p}$, is:

$\mathcal{L}=\prod_{i=1}^{N}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp{(-\frac{(S(\vec{p},E_{i})-F_{i})^{2}}{2\sigma_{i}^{2}})}$
(3.2)

where ($F_{i}$,$\sigma_{i}$) are the flux measurement and uncertainty at an
energy $E_{i}$ over $N$ spectral measurements. The $\ln{\mathcal{L}}$ function
in this assumption can be related to the $\chi^{2}$ parameter as
$\chi^{2}=-2\ln{\mathcal{L}}$, so that maximization of the log-likelihood is
equivalent to a minimization of $\chi^{2}$. Since for some sources there is
more than one data set referred to the same energy range ( 100 MeV-10 GeV, 10
GeV-1 TeV, > 1 TeV bands), to be sure to consider data points with their
appropriated weight was choosen one set for energy band, in order to not
influenced the maximum likelihood fitting procedure. Below is reported an
example of the fitting procedure used for all the supernova remnants starting
from the choice of the data set.

####  3.2.1 Example of W44

W44 is a middle-aged supernova remnant ($\sim 2\cdot 10^{4}$ years old)
located at $\sim 3kpc$ from Earth in the Galactic Plane. Since this remnant is
one of the more studied there are multiwavelenght observations that revealed
interesting features. In the Radio band it is characterized by an elliptical
shell due to expansion in an inhomogeneous ISM. The OH maser emission revealed
that W44 is interacting with a molecular cloud in the south-est region.

Figure 3.3: W44 spectral data. Blue points refers to 4 years of observations
with the Fermi-LAT telescope (4 August 2008 to 16 July 2012) [7], violet is
for AGILE [51] instead red and green points are for the Fermi 3FGL and 3FHL
data.

The figure 3.3 show all the data set eventually available, 3 data set are
almost in the same energy band. I used Fermi data extract in the analysis of
Ackermann et al. [7] (blue points) because of better band sampling and
moreover the data has a smaller error bars associated than those from the
systematic analysis of the 3FGL Fermi catalog. The 10 GeV-1 TeV band is
sampled only in the 3FHL catalog, no other very-high-energy instrument has
detected W44 above 100 GeV up to now. As initialization of Naima, I set the
broken power law(BPL) as protons distribution and the $\pi^{0}$-decay as
radiative model. In addition, it is required the prior function, i.e., a
function that encodes any (eventually) previous knowledge about the
parameters, limit the amplitude and the energy break to be positive, and
spectral indices to be between -1 and 5. In appendix are are reported the
models and the prior function definitions code sections

After define the array with the initial parameters estimates, the MCMC
procedure start. The maximum likelihood optimization takes about 15 minutes
(for each SNR), it depends mainly on how much the initial BPL parameters are
close to the "real" one, and on the $nwalkers$, $nburn$, $nrun$ parameters
described in detail in [70]. Fitting results can be viewed both through plots
that illustrate also the stability of the procedure and through a table (in
ecsv format) that contains all the run information, the parameters array of
the protons population of the best fit and the maximum value of the negative
log-likelihood. Results for W44 are reported in 3.4 and 3.5.

Figure 3.4: W44 fitted spectrum Figure 3.5: Protons population of W44

###  3.3 Catalog

All the $\gamma$-ray supernova remnants considered and their position in the
Galaxy are shown in figure 3.6, red rhomb marks the interacting supernova
remnants, cyan cross is for the shell-like SNR, and the green plus is for the
undefined SNR. The procedure in section 3.2.1 was reproduced for all the
supernova remnants obtaining the parameters of the protons distribution that
is responsible (together with the proton target density fixed at 1
$proton/cm^{-3}$ ) for the neutral pion decay emission. Having all spectra
allows to get the integrated flux over the entire $\gamma$-ray band (from 100
MeV to 10 TeV ) as over any other band, for my purpose, two important band are
0.1-10 TeV and 1-10 TeV. Catalog of gamma emitting supernova remnants is
reported in table in appendix. Below is reported a brief catalog columns
description.

  1. 1.

Name: Supernova remnant common name

  2. 2.

Type: INT is reserved for supernova remnants that in the TeVCat or in the
Fermi SNR catalog [5] are considered interacting with a molecular cloud.
Instead SHELL is the tag for shell like supernova remnants according to the
TeVCat classification. No tag means that the classification is no obvious.

  3. 3.

Other Names: Other known names

  4. 4.

TeVCat: SNR name (eventually) stores in the TeVCat

  5. 5.

l: Galactic Longitude in degree unit

  6. 6.

b: Galactic Latitude in degree unit

  7. 7.

RA: Right Ascension, J200 in degree unit

  8. 8.

DEC: Declination, J200 in degree unit

  9. 9.

d: Distance of SNR in kpc

  10. 10.

Age: SNR age in years

  11. 11.

MC MASS: The molecular cloud mass if the supernova is interacting, in solar
mass unit

  12. 12.

Flux100MeV: Integrated flux from 100 MeV to 10 TeV, in
$\mathrm{erg\,s^{-1}\,cm^{-2}}$

  13. 13.

Flux100GeV: Integrated flux from 100 GeV to 10 TeV, in
$\mathrm{erg\,s^{-1}\,cm^{-2}}$

  14. 14.

Flux1TeV: Integrated flux from 1 TeV to 10 TeV, in
$\mathrm{erg\,s^{-1}\,cm^{-2}}$

Figure 3.6: Full sky view of the catalog SNRs. Red rhomb is for interacting
supernova remnant, cyan cross is for shell-like SNR and green cross are for
undefined type. Image was realized using Aladin software and Fermi-LAT data up
to 2016 [34].

###  3.4 Some catalog results

This $\gamma$-ray supernova remnants catalog contains all the relevant data to
compare the real population of SNR with the synthetic one results from the
model presented in chapter 5. Furthermore I could verify the theory according
to which supernova remnants are divided into "Shell-like" SNR that are
typically young object ($\sim 10^{2}-10^{3}yr$), expanding in a relatively low
density medium, with $\gamma$-ray emission morphology correlated with the
radio and X-ray shell, and the interacting SNRs, older ($\sim
10^{3}-10^{4}yr$) mixed morphology sources, that interact with giant molecular
clouds and with a emission morphology that correlates better with MC than with
the supernova shell. This classification must be evident in some aspects of
the SNR population, for example in the age distributions, in the spectral
shape and in the integrated flux distributions. In my catalog there are all
the useful elements to check this classification. In figure 3.7 the spectral
shape of the main exponent of the two classes is shown, any other object which
belongs to this classes follows the behavior of the example reported in figure
3.7. The W44 supernova remnant is one of the brightest SNRs on the GeV sky,
the characteristic low-energy cutoff in the energy spectrum (the pion bump)
has been detected. This observation demonstrates that the $\gamma$-ray
emission in the GeV band is dominated by pion decay. The shell-like class
comprises young SNRs that are typically less luminous at GeV energies, have
harder spectra, and are often also detected at TeV energies. The most
prominent shell-like SNR that have been detected both at GeV and TeV energies
is RXJ J1713.7-3946. The absence of thermal X-ray emission suggests that in
this region the is too low density target material for interaction of protons
potentially accelerated by the SN shock to produce neutral pions. Furthermore,
the rather hard photon index in the GeV band, strongly indicates that the
emission of this SNR has a leptonic origin. The Tycho SNR instead belong to a
class of supernova remnant that haven’t a clear classification.

Figure 3.7: Spectra of SNRs W44 (red), RXJ1713.7-3946 (blue) and Tycho
(green). All points available are plotted see table 3.1, shades indicate the
$1\sigma$ error region.

I could test if the division between interacting and shell-like supernova
remnants based mainly on other wavelengths observation is reflected also in
some other features. For example that interacting supernova remnants are older
than the shell like because of accelerated protons takes some time (it depends
on how far is the MC and on the acceleration of the particles) to reach the
molecular cloud target protons. The shell-like supernova remnants emission
instead is due to electrons inverse compton on CMB local photons. The age
distributions of the two classes are reported in figure 3.1.2. Further
indication that the two classes have a different $\gamma$-ray emission
mechanism came from the ratio between the flux value at 1 GeV and 1 TeV
(Hardness ratio HR ). As on can see in figure 3.8(b), it’s hard to explain
this hardness ratio distribution (that reflects the spectral shape) with only
one emission mechanism. As before the the two classes are not compatible at
10% significance level.

(a)

(b)

Figure 3.8: Histogram of the age of the supernova remnants with Gaussian fit
superimposed. Orange and Blue color represent shell-like and interacting SNR
respectively.

In figure 3.9 are investigated correlations between the $\gamma$-ray
luminosity $L_{\gamma}=f_{100MeV-100TeV}\cdot 4\pi D^{2}$ and the age and
between the luminosity and the hardness ratio (HR). Solid points represents
object that have a data coverage both on the GeV and on the TeV energy band.
This plot show that all the interacting supernova remnants have luminosity
above $10^{35}\,erg\,s^{-1}$ and are in general older. The only exception is
HB 21 that has $L_{\gamma}\sim 3\cdot 10^{34}\,erg\,s^{-1}$, but this
discrepancy is probably due to the lack of TeV data, that will results in
underestimate the $\gamma$ luminosity. Another interesting features is that
all the iSNRs have HR $>1$, so this objects have a spectrum with the dominant
contribution to the flux at GeV energy band, like W44. The Shell-like
supernova remnants with HR $>1$ are those have a "flat" spectrum similar to
the Tycho SNR spectrum (see figure 3.7). Note that for the W28 supernova
remnants in figure 3.9 (left) was considered the luminosity of the whole
source, instead in 3.9 (right) the source was divided in W28 north and W28
south . The HR value of W28 south (source A+B+C) is the ratio of the sum of
the flux of the three sources at 1 GeV and the sum at 1 TeV.

Figure 3.9: Luminosity (100 MeV-100 TeV) in units of $erg\,s^{-1}$ plotted as
function of the age of SNRs (left) and of the hardness ratio (right). Blue
color is for interacting supernova remnants, orange points are shell-like SNR
and gray square are the undefined ones. Solid color meaning that available
data cover almost all the energy band.

All the features presents in this section confirm the scenario that there
supernova remnants are divided in two classes. The shell like supernova
remnants that are younger and have a low intrinsic luminosity furthermore the
spectra don’t have the pion bump features (like RXJ 1713.7-3946). The $\gamma$
ray and radio map of this kind of object indicates that the emission is rather
symmetric and came from the ejected material. All this considerations
indicates that the emission mechanism is leptonic. The interacting supernova
remnants, object of interest of this work, instead have a spectrum that can be
well explain with the hadronic emission, moreover in some case the presents of
the pion bump is clearly evident. The evidence that this object are brighter
and older and the fact that the hadronic mechanism needs target protons can be
explain with the explosion of a supernova close to a molecular cloud. The
shock accelerated protons (Fermi mechanism), takes some time to reach the MC
that have density of hundreds $proton\cdot cm^{-3}$. This results in
$\gamma$-ray brighter sources with a delayed emission compares to the shell-
like supernova remnants. This scenario is confirmed looking at the
$\gamma$-ray emission map of the sources, infact the $\gamma$-ray emission is
well correlated with the molecular cloud (figure 3.10 right) instead of with
the supernova shell.

Figure 3.10: Left: Multiwavelength data for supernova remnant RX J1713.7-3946
in Galactic coordinates. The white bar is 1$\deg$ long. Data are taken from
(a) XMM-Newton observations, (b) GeV $\gamma$-rays observed by Fermi-LAT
(source class, 6.5 years of data), and (c) H.E.S.S. observations. Right: IC
443 MAGIc $\gamma$-ray map. Overlayed are 12CO emission contours, that trace
the molecular cloud (cyan), contours of 20 cm VLA radio (green), X-ray
contours from Rosat (purple) and $\gamma$-ray contours from EGRET (black) .
Image credits: [76] (left) and [24] (right).

###  3.5 The LogN-LogS of real interacting supernova remnants

The main target of this work is to create a population model for the
interacting supernova remnants and give an estimate of how many of this
sources CTA can be identify. The strategy is to realize the LogN-LogS of the
real interacting supernova remnants and compare it with those obtained from
the population model presented in 5, and then extrapolate the iSNR number at
the CTA flux sensibility. The way to check if the resulting model, is
consistent with what observed by the major experiments is to compare the LogN-
LogS curve of the real SNRs (figure 3.11) with that of the synthetic
population. The LogN-LogS curve was obtained by integrating all the
interacting SNRs spectra above 100 MeV. The associated error is the Poisson
standard deviation, $\pm\sqrt{n}$. The features of the LogN-LogS that the
population model of interacting supernova remnants have to reproduce is the
slope in the region in which the sample is complete (red region in figure
3.11). Assuming a population uniformly distributed in the galactic plane (thin
disk), thanks to a Monte Carlo simulation was found that the slope of the
LogN-LogS is equal to -1. Therefore where the slope of the real LogN-LogS
curve deviates from -1, the sample is incomplete. To calculate where the
sample is complete, I fixed one real iSNR and I fitted with a power law the
LogN-LogS between any other iSNRs, repeating for each point was obtained that
the best accordance is $-1.00998$. The red region in figure LABEL:lognlogs
indicates the resulting completeness region.

Figure 3.11: Top: spectrum of all the interacting supernova remnants. Bottom:
the LogN-LogS curve of the real interacting supernova remnants. Shade blue
represent the $\pm\sqrt{n}$ error region. Red region identify the completeness
region.

##  Chapter 4 CTA - Cherenkov Telescope Array

The Earth’s atmosphere is opaque for high-energy cosmic radiation. A direct
observations of high-energy photons is possible only in the open space, by
making use of satellites. The first gamma-ray space missions, such as SAS-2
and COS-B, were launched in 1970th. The major improvement of the sensitivity
was achieved with the EGRET instrument, and then with the AGILE and the Fermi-
LAT facilities, which are still in operation observing objects at energies up
to 50 and 300 GeV, respectively. However, the steep very-high-energy spectra
of the majority of the $\gamma$-ray emitting sources make the space TeV
observations essentially ineffective. Satellites have to deal with width
limitations (($\sim 1m^{2}$ for the Fermi-LAT) that prevent them to observe
photons more energetic than some hundreds of GeV. Ground-based observatories
instead can perform deeper, even though indirect, measurements of the high-
energy particles, by observing their secondaries particles, with Extensive Air
Shower (EAS) detectors, or their Cherenkov light, by means of Imaging
Atmospheric Cherenkov Telescopes (IACT). The first ones are designed for
particles and are basically silicon scintillators, located at high altitudes,
to encounter the shower at its maximum development. IACTs instead measure the
radiation produced in these cascades, this technique is based on the
phenomenon of the Cherenkov radiation discovered by P. Cherenkov in 1934. When
a charged particle moves in a dielectric medium with a speed higher than the
phase velocity of light in that medium, the wave front of blue Cherenkov
emission appears. In 1948, Blackett [40] found out that there is a detectable
contribution to the night sky emission (Cherenkov radiation) from air showers
of particles, produced as a consequence of the $\gamma$-ray photons and/or
cosmic rays interactions with the upper layers of the Earth’s atmosphere. It
is the Cherenkov light that can be observed from the ground using the imaging
atmospheric Cherenkov technique. A few notable experiments that implement the
IACT, the High-Energy Stereoscopic System (H.E.S.S.), the Major Atmospheric
Gamma-ray Imaging Cherenkov (MAGIC) and the Very Energetic Radiation Imaging
Telescope Array System (VERITAS) have helped to establish this field of
ground-based astronomy. The discoveries made by these ground-based
$\gamma$-ray experiments have furthered our understanding of the technique
itself as well as the violent objects in our universe that produce very high-
energy gamma-rays.

###  4.1 Detecting Cherenkov light

Passing through the atmosphere, a $\gamma$-ray (primary) interacts with its
upper layers at an altitude of $\sim$ 25 km and produces a particle shower.
The energy of the primary is so high that produced particles (mainly electrons
and positrons) are moving with velocities higher than the speed of light in
the atmosphere and, thus, emit the Cherenkov radiation. The number of
particles in the air shower reaches the maximum at an altitude of $\sim$ 10 km
and then it continuously decreases as the shower goes deeper into the
atmosphere and looses its energy. The Cherenkov radiation from the air shower
forms a cone of blue visible light (light pool) around the primary gamma ray.
Such flash of Cherenkov light lasts for about 10 ns. A typical size of a light
pool on the ground is about 250 m in diameter with a density of $\sim$ 50
Cherenkov photons per $m^{2}$ for a 1 TeV $\gamma$-ray primary. An optical
telescope located inside the Cherenkov light cone will be able to detect it,
if its dish is large enough for such low fluxes and its camera is fast for
such short signals. An effective collecting area of a single Cherenkov
telescope can be approximated by the area of the Cherenkov light pool on the
ground ($\sim 5\times 10^{4}m^{2}$). This provides a remarkable advantage to
IACTs comparing to the space $\gamma$-ray telescopes.

Figure 4.1: The schematic view of the IACTs technique

By means of the analysis of a signal detected with the IACT, it is possible to
identify properties of the primary. The size and the brightness of obtained
elliptical image corresponds to the energy of the initial $\gamma$-ray photon,
while its orientation allows us to reconstruct the direction of the primary
or, in other words, the position of emitting source. A schematic view of the
IACT technique is shown in figure 4.1 $\gamma$-rays are not the only
particles, which produce air showers: a $\gamma$-ray photon accounts for $\sim
10^{4}$ cosmic-ray hadrons. Interacting with atoms and molecules of the
Earth’s atmosphere, cosmic rays also produce energetic showers of pions,
nuclei and products of their decays. Showers originated from cosmic rays are
much more extended than the electromagnetic ones, formed by $\gamma$-ray
primaries (see figure 4.2 ).

Figure 4.2: Difference between electromagnetic and hadronic showers.

The differences between the structures of the $\gamma$-ray and hadronic
showers are imprinted on their images seen with Cherenkov telescopes. As shown
in figure 4.2, an image of an electromagnetic shower has an elliptical shape
on the camera, whereas that of an hadronic event is irregular. Analyzing
obtained images of Cherenkov pools, we can discriminate signal events
($\gamma$-ray) from the background (hadrons) with $\sim$99% efficiency (see
gamma-hadron separation method in [94]). The energy threshold of Cherenkov
telescopes is defined basically by a $\gamma$-hadron separation efficiency. An
image of an electromagnetic shower produced by a low-energy gamma ray (< 100
GeV) is faint and it can be misinterpreted as a hadronic event with an
irregular shape. Using IACTs with large mirrors, we can obtain images of a
shower of a better quality. This leads to more efficient $\gamma$-hadron
discrimination and, therefore, to lower values of the energy threshold
attainable with this telescope. Using several Cherenkov telescopes as a single
$\gamma$-ray instrument provides a number of advantages. Observing with a
number of IACT-units spaced by a distance of $\sim$ 100 m can enlarge the
collective area and increase the sensitivity of the whole instrument. In
addition, analyzing multiple images of the same shower (stereoscopic
observations) it is possible to improve the reconstruction of the primary
gamma ray, i.e. to improve the determining of its energy and initial
direction. This provides better angular and energy resolutions of a Cherenkov
instrument. Stereoscopic observations also improve the gamma-hadron separation
efficiency: e.g. by accepting only events, which simultaneously trigger all
IACTs under the light pool.

####  4.1.1 Status of current IACTs

A great improvement of ground-based Cherenkov instruments was made since a
detection of the first very-high-energy $\gamma$-ray source, the Crab Nebula,
with the Whipple 10-meter IACT in 1989 [144]. Using the legacy and experience
of the first-generation IACT projects (e.g. Whipple telescope, HEGRA, CAT,
CANGAROO), the currently operating Cherenkov telescopes such as MAGIC, VERITAS
and H.E.S.S. have been developed. An order of magnitude improvement of the
sensitivity was achieved with these instruments. The MAGIC (Major Atmospheric
Gamma Imaging Cherenkov Telescopes) instrument consists of two identical
17-meter Cherenkov telescopes spaced by 85 meters, which are situated at the
Roque de los Muchachos Observatory (La Palma, Spain). The first telescope
operates since 2004, the second unit joined in 2009. This IACT performs
observations in the energy range from 50 GeV up to 30 TeV. VERITAS is the Very
Energetic Radiation Imaging Telescope Array System of four 12-meter telescopes
separated by a 100-meter distance, working at energies from 85 GeV to $\sim$
50 TeV. This facility is in operation in Mount Hopkins (Arizona, USA) since
2007. H.E.S.S. (the High Energy Stereoscopic System) was initially developed
as an array of four 12-meter telescopes at a distance of $\sim$ 120 meters,
observing the very-high-energy $\gamma$-ray sky in the 100 GeV–100 TeV energy
band since 2002. The second stage of the H.E.S.S. project started in 2013
after a construction and commissioning of the fifth large 28-meter telescope
in the center of the array, this telescope enlarged the energy range of the
H.E.S.S. instrument down to $\sim$ 20 GeV. In contrast to MAGIC and VERITAS,
H.E.S.S. operates in the southern hemisphere (Khomas Highland, Namibia).
Taking into account a large field of view of small H.E.S.S. telescopes (5∘),
this array can be considered as the best currently operating IACT machine for
the Galactic Plane Survey. A wide range of astrophysical objects is being
investigated. A large amount of important investigations, which expanded the
frontier of our knowledge concerning the VHE sources, was already performed
with the current generation of Cherenkov instruments. However, the potential
of the IACT technique has not been fully realized yet.

###  4.2 CTA Telescopes

The Cherenkov Telescope Array (CTA) is a modern project of the next-generation
ground-based very-high-energy $\gamma$-ray facility, which aims to use all the
advantages of the IACT technique. The CTA observatory will comprise two
arrays, one in each hemisphere, with hundred IACT telescopes of 3 different
types: Large Size Telescopes (LSTs, 23-meter diameter), Medium Size Telescopes
(MSTs with diameters of 10–12 meters) and Small Size Telescopes (SSTs, 4-meter
diameter). The northern site will be in the Canary island of La Palma, the
southern one at Paranal, in Chile. The two sites will have a different
configuration due to the different geographical conformations: the northern
one will be more limited in size because of the features of the site, and will
not have SSTs. In table 4.1 is reported a scheme of the different
configuration of the two sites. However this configuration, known as baseline
configuration is not granted, the CTA community for now guarantees only for a
threshold configuration, with less telescopes as reported in table 4.1.

Figure 4.3: Layouts designed for the northern and southern array of CTA. Image from https://www.cta-observatory.org/about/how-cta-works/ |  | Baseline | Threshold
---|---|---|---
| Energy range | South North | South North
LST | 20GeV-200GeV | 4 4 | \- 4
MST | 100GeV-10TeV | 25 15 | 15 5
SST | 1TeV-300TeV | 70 - | 50 -
Table 4.1: CTA telescopes specifications

Large sized telescopes (LST) with their 23-m diameter reflector have the
lowest energy threshold among the three kinds of telescoped. The LST is
designed to be an alt-azimuth telescope a parabolic reflective surface.
Although the LST will stand 45 m tall and weigh around 100 tonnes, it will be
extremely nimble, with the goal to be able to re-position within 20 seconds.
Both the re-positioning speed and the low energy threshold provided by the
LSTs are critical for CTA studies of Galactic transient phenomena, high red-
shift active galactic nuclei and gamma ray bursts. Besides the LSTs will stand
the MSTs that will cover the central energy range from 100 GeV to 10 TeV. They
will have a modified Davies-Cotton (e.g. sub-mirrors are disposed in a
spherical configuration) mirror of 12 m diameter, with polar mount connected
to a camera of 8∘of field of view. Finally SSTs will be built and spread into
a large area. Due to the power law behavior of the spectrum in fact, the high
energy photons flux is lower. With the contribution of these SSTs CTA will
extend its energy domain up to 300 TeV. SSTs will have a diameter of 4m and a
field of view of 9∘.

###  4.3 CTA performance

For any kind of measurement one has to deal with the instrument capability. As
in equation 4.1 the detected signal $e(E^{\prime},\vec{p^{\prime}})$ is the
result of the convolution between the flux $F(\vec{p})$ of the source and the
instrument response functions $R(E^{\prime},\vec{p^{\prime}}|E,\vec{p})$.
Where $E^{\prime}$ and $\vec{p^{\prime}}$ label the detected event, and $E$
and $\vec{p}$ the physical one.

$e(E^{\prime},\vec{p^{\prime}})=\int F(\vec{p})\times
R(E^{\prime},\vec{p^{\prime}}|E,\vec{p})d\vec{p}dE$ (4.1)

The instrument response functions for CTA comprise the effective area
$A_{eff}(p,E,t)$, the point spread function $PSF(p^{\prime}|p,E,t)$ and the
energy dispersion $E_{disp}(E^{\prime},p,E,t)$:

$R(p^{\prime},E^{\prime},t^{\prime}|p,E,t)=A_{eff}(p,E,t)\times
PSF(p^{\prime}|p,E,t)\times E_{disp}(E^{\prime}|p,E,t)$

where are indicated the dependences on the physical properties of the original
event $(p,E,t)$ and of the detected one $(p^{\prime},E^{\prime},t^{\prime})$.
Each of these quantity is computed by simulating showers with Monte Carlo
methods. The CTA consortium on their website provides the results of their
simulations. [Ambrogi et al., 2016] in their paper modeled the resulting
points and derived an analytical parametrization in the energy range from 50
GeV to 100 TeV for a point like source positioned at the center of the field
of view (FoV). The plot of the fitted formula for each quantity are reported
in 4.4.

####  4.3.1 Effective area

For a single telescope the effective area is determined by the radius of the
Cherenkov light pool at ground, while for a multi-telescope system is
determined essentially by the total geometrical area. CTA bigger array, when
completed, will cover an area of $\sim 300km^{2}$. Particles of different
energies create cascades of different area (more energetic particles generate
wider showers and vice-versa) that causes an energy dependency of $A_{eff}$ .
The parametrization of [Ambrogi et al., 2016] for the CTA southern observatory
effective area yields:

$A_{eff}(x)=\frac{A}{1+Bexp(-x/C)}$

where $x=\log(E/1TeV)$, $A=4.36\times 10^{6}m^{2}$ is the saturation value of
the effective area, whereas $B=6.05$ and $C=3.99\times 10^{-1}$ define the
rate of change of $A_{eff}$ with respect to energy.

Figure 4.4: The effective collection area for gamma rays from a point-like
source as a function of energy optimised for 0.5-, 5- and 50-h observations.

####  4.3.2 Angular resolution

A point like source, when detected from any camera appears not as a point but
as a circle with a certain extension. The distribution of the detected photons
depends on the instrument, and it is known as point spread function (PSF). In
first approximation the PSF is a two dimensional Gaussian:

$PSF=exp(\frac{x_{f}^{2}+y_{f}^{2}}{2\sigma_{PSF}^{2}})$

where $x_{f}$ and $y_{f}$ are the coordinates in the camera focal plane. The
angular resolution in this context is defined as the standard deviation of the
gamma ray PSF. The angular resolution depends on the energy, the higher the
energy the smaller is the PSF. For CTA a reference value is an angular
resolution of 0.05∘ at 1 TeV, as shown in figure 4.5. The fit of the data
points produced the function in 4.2

$\sigma_{PSF}=A\cdot[1+exp(-\frac{x}{B})]$ (4.2)

here $A=2.71\cdot 10^{2}$ deg is the best angular resolution achievable with
the considered configuration and $B=7.90\times 10^{-1}$ is a scaling factor
that determines how fast $\sigma_{PSF}$ changes with energy.

Figure 4.5: Angular resolution as a function of energy of CTA south, CTA north
and a comparison with other instruments respectively.

####  4.3.3 Energy resolution and energy dispersion

The measure of the gamma photon energies is not direct, it is derived from the
energy of the photons generated in the cascades. For this reason it is called
reconstructed energy and it corresponds to the peak of a probability function,
similar to the PSF, known as Point Dispersion Function. The energy resolution,
quantifies the ability to distinguish between two energy values and it is
defined as the full width at half maximum (FWHM) of the energy distribution
centered on the reconstructed energy. CTA energy resolution should reach the
level of $\sim$6% above 1 TeV. The parametrization of [Ambrogi et al., 2016]
for the energy resolution is given by:

$\Delta E/E(x)=A\times[(x-B)^{2}+(x-B^{4})]+C$ (4.3)

with $A=6.33\times 10^{-3},B=8.34\times 10^{-1}$ and $C=6.24\times 10^{-2}$.

Figure 4.6: Energy resolution as a function of energy for CTA south and north
respectively

####  4.3.4 Background rate

The main sources of background for IACT arrays are gammas generated in the
atmosphere from the impact of protons and heavier nuclei of the CR. The
CTAconsortium simulates the arrival of these particles, assuming for them a
power law spectrum consistent with the satellites measurements. This
contribution is influenced by the zenith angle of the observation, since
higher zenith angles correspond to more layers of atmosphere that the
particles cross, enhancing the chance for interaction. Besides since the final
detection products are visible photons, all the sources (both of astrophysical
and human origin) of contamination in the optical band have to be taken into
consideration. Usually CTA background is evaluated for a dark sky, namely with
no moon, and offset from the galactic plane. The electronics noise must be
considered as well.

Figure 4.7: The (post-analysis) residual cosmic-ray background rate per square
degree vs reconstructed gamma-ray energy for CTA south and north respectively.

####  4.3.5 The sensitivity

The sensitivity of a telescope is the minimum flux detectable with a required
statistical significance, e.g. with enough number of events to distinguish the
signal from the background. For CTA the requirements to be fulfilled for each
energy bin are:

  * •

at least 5$\sigma$ significance level;

  * •

the presence of at least 10 excess events;

  * •

a signal of at least 5% of the background systematic error.

The sensitivity is then calculated simulating the flux of a source; for most
applications a point source with the spectrum of the Crab nebula is used.
Statistical limits are calculated using a maximum likelihood approach, and the
background systematics are assumed to have an uncertainty of 1%. The
sensitivity of a gamma-ray detector is determined by the effective collecting
area, the residual background and the angular resolution and hence depends
strongly on the energy. Furthermore different layouts of the array, will give
different effective areas, and consequently different sensitivities. The two
arrays of CTA will hence have different performances: the northern one will be
approximately two times less sensitive than the southern one. Moreover, these
sensitivity curves are background dependent, as one can see from the
sensitivity definition itself and hence depends on the zenith angle and other
factors that influence the background. The plots in figure 4.8 show the
computed differential sensitivity for a point-like source. Different exposure
times and zenith angles were assumed. Different configurations were also
tested: it was chosen to consider not only the differences between North and
South, but also the difference from the baseline configuration, namely the
final designed configuration and the threshold configuration (as in table
4.1). For what concerns the extended sources, the question is more delicate.
In this case the background rate increases proportionally to the factor
$\sqrt{\theta^{2}+\sigma^{2}_{PSF}}$ [Funk et al., 2013], and therefore the
sensitivity can be assumed to scale according to the relation:

$sens_{ext}=sens_{point}\cdot\frac{\sqrt{\theta^{2}+\sigma^{2}_{PSF}}}{\sigma^{2}_{PSF}}$

A reference value for CTA differential sensitivity is the sensitivity of a
point source observed at low zenith angle ($z\leq 20^{\circ}$) at 1TeV, with
an exposure time of 50 hrs, namely:

$sens(E=1TeV,z=0,t_{exp}=50h)=6.55\times 10^{-14}\frac{TeV}{cm^{2}s}$ (4.4)

Figure 4.8: Comparison of CTA sensitivity for different cases. In (a)
sensitivities for the southern array in its definitive configuration for
different observation time intervals. In (b) the sensitivity for observation
at different zenith angles. In (c) we considered the provisional threshold
configuration. Finally in (d) CTA southern and northern arrays are compared
with the main current gamma ray detectors. This last figure is from
https://www.ctaobservatory.org/science/cta-performance/

####  4.3.6 Off-Axis sensitivity

All performance parameters presented above are valid for a source located
close to the centre of the CTA field of view (FoV). The differential
sensitivity curves for a point-like source at increasing angular distances
from the centre of the FoV are shown below. The radius of the FoV region in
which the sensitivity is within a factor 2 of the one at the centre is around
2 degrees near the CTA threshold, and >3 degrees above a few 100 GeV. Angular
and energy resolution also degrade as one approaches the edge of the FoV. The
provided IRFs contain the evolution of all performance parameters with off-
axis angle.

Figure 4.9: Off-axis sensitivity of CTA south and north respectively.

###  4.4 CTA Data Challenges

CTA data challenges (DC) are a large-scale semi-realistic test of the high-
level analysis system, each with a different focus and an increasing level of
realism in particular towards instrument imperfections and processing tools.
At the moment only the first data challenge (DC-1) is complete, the second is
scheduled for next year. In DC-1 the CTA collaboration has produced a set of
simulated high-level science data (DL3) - i.e. the reconstructed properties
(direction, energy, time) of gamma-ray like showers (real gamma rays and
misclassified charged particles). In DC-1 the focus was on the validation of
the core versions of the Science tools (currently ctools and gammapy). A large
number of different science cases was tested in the science working groups
(SWGs). In addition, a systematic comparison of the two high-level analysis
tools that are currently in development within CTA was performed. In DC-1,
1980 hours were scheduled for the Southern array and 1815 hours for the
Northern array each under the assumption of a total of 1100 hours/year
observation time at each site. These hours were distributed amongst the
different KSPs, see table 4.2

KSP | Duration | Pointings
---|---|---
Galactic Plane Survey | 1020 south, 600 north | 3270
Galactic Center Survey | 825 south | 1671
Extragalactic Survey | 200 south, 300 north | 1271
AGN monitoring | 960 north | 1920
Table 4.2: Distribution of observational time among the KPS

The science working groups provided the input to DC-1 in the form of xml files
that describe the flux from a certain sky direction. These models are
consistent with existing TeV and Fermi-LAT data and in many cases include an
extrapolation to lower fluxes based on population models. The second data
challenge is motivated by the requirement of a more realist models both of
population of sources and of specific cases. In particular in the case of
supernova remnants only few new SNRs have actually been detected using a
systematic analysis of the DC-1 data. Although this is probably partially due
to the fact that the pipeline lacks a treatment of shell-type morphology, the
model used predicts many large and faint SNRs, difficult to disentangle from
large-scaled diffuse emission. Infact, in DC-1 only the population of young
SNR is modeled, instead a population model for the interacting supernova
remnants is fully missing. The model presented in section 5 will be the
population model of iSNR for the DC-2.

###  4.5 Galactic Plane Survey

CTA is an observatory available for guest observer programs. In addition, the
CTA consortium will be granted with a fixed amount of time that will be
devoted to sky surveys. One of these so called key science projects (KSP) is a
Galactic Plane Survey (GPS). This survey will achieve a sensitivity better
than 4.2 mCrab over the entire Galactic Plane and of 1.8 mCrab in the Galactic
center and is expected to lead to the discovery of many new VHE objects. The
GPS will be divided into two phases: a short term (Years 1-2) preliminary
phase, and a long term (Years 3-10) one with a total observing time of 1020
hours with CTA South and 600 hours with CTA North. The observations will
proceed in a scan of the sky, composed of subsequent pointing with a double
row strategy with a nominal pointing separation of $3^{\circ}$, like sketched
in figure 4.10. The observations will be performed at low zenith angles
($z\leq 45^{\circ}$) in order to preserve the sensitivity. That will allow to
cover the $|b|<2^{\circ}$ region, where the majority of the known sources,
like SNRs and PWNe, is located. The major scientific objectives for the CTA
GPS include the following:

  * •

discovery of new and unexpected phenomena in the Galaxy. These would include
completely new source classes, new types of transient and variability
behavior, or new forms of diffuse emission.

  * •

discovery of PeVatron candidates that are of key importance in our search for
the origin of cosmic rays. These candidates will likely require deeper,
follow-up observations to confirm and characterize their PeVatron nature.

  * •

detection of many new VHE Galactic sources (of order 300 – 500), particularly
PWNe and SNRs, to increase the Galactic source count by a factor of five or
more. The substantially increased statistics and more uniform sensitivity will
allow more advanced population studies to be performed. The ultimate goal is
to significantly advance our understanding of the origin of cosmic rays.

  * •

measurement of the large-scale diffuse VHE gamma-ray emission, to better
understand its origin in terms of inverse-Compton, $\pi^{0}$ decay, and
unresolved source components.

  * •

discovery of new VHE gamma-ray binary systems, a unique class of objects with
periodic emission on varying timescales, where physical processes are observed
from different vantage points depending on each system’s orbital inclination.
Only five such systems are currently known in the Galaxy.

  * •

production of a multi-purpose legacy data set comprising sky images and source
catalogues of the complete Galactic plane at very high energies. This dataset
will have long-lasting value to the entire astronomical and astroparticle
physics communities, far beyond the lifetime of CTA.

  * •

the GPS will produce and periodically release sky maps and catalogues

Figure 4.10 shows a simulated image of what could result from a CTA survey of
a portion of the Galactic plane using a model that incorporates SNR and PWN
source populations as well as diffuse emission. The conclusion that can be
drawn from these population estimates is that CTA can expect the detection of
many hundreds of Galactic sources that would follow from a survey of the plane
with a sensitivity at the level of a few mCrab.

Figure 4.10: Simulated CTA image of the Galactic plane for the inner region
-90∘ < $l$ < 90∘ , adopting the actual proposed GPS observation strategy, a
source model incorporating both supernova remnant and pulsar wind nebula
populations and diffuse emission.

##  Chapter 5 Population model for interacting supernova remnants

The aim of this section is create a synthetic population model of Galactic
SNR-MC systems emitting $\gamma$-rays. I had then to reproduce three aspects
of the real supernova remnants - molecular clouds systems, starting from the
$\gamma$-SNR catalog and from the Rice et al. [127] catalog of galactic
molecular clouds. First of all I had to create a synthetic population of
molecular cloud that reflects the features of the molecular clouds in the
Milky Way in terms of positions, mass and radial velocity. Then I had to
reproduce the cosmic rays injection spectra in the molecular clouds. The last
aspects which I had to consider is the real fraction of supernova remnants
with a molecular cloud nearby, it means that I had to introduce a probability
function that returns the number of interacting systems comparable to the
observed one.

###  5.1 Synthetic population of molecular clouds

####  5.1.1 Catalog of molecular clouds

Rice et al.[127] have created a catalog of 1064 massive molecular clouds
throughout the Galactic plane, using the ${}^{12}CO$ CfA-Chile survey, which
is by far the most uniform survey of molecular gas in the Galaxy. This
catalog, with 1064 massive clouds totaling $2.5\cdot 10^{8}M_{sun}$

, contains $25^{+10.7}_{-5.8}$% of the molecular gas mass of the Galaxy, and
trace several spiral arms, most notably the Sagittarius,Perseus, Outer,
Carina, and Scutum-Centaurus arms. Rice et al. derived a cloud mass spectrum,
which describes the distribution by number of clouds of different masses
expressed as a truncated power law:

$N(M^{{}^{\prime}}>M)=N_{0}[(\frac{M}{M_{0}})^{\gamma+1}-1]\\\ $ (5.1)

where $\gamma$ is an index describing how mass is distributed amongst clouds.
$\gamma$ > -2 indicates that the majority of mass is contained in massive
clouds; $\gamma$ $\approx$ -2 means that mass is roughly equally distributed
in all mass bins; and $\gamma$ < -2 indicates that low-mass clouds contain the
majority of mass. To measure mass spectra in their study, Rice et al. used the
maximum-likelihood method and they carry out this measurement region-by-
region, only considered clouds with $v_{rad}|>20$ km $s^{-1}$ located at least
20∘ from $l$ = 0∘ or $l$ = 180∘ in order to minimize kinematic distance
errors, which have a substantial effect on estimated masses. Results are
summarized in figure 5.3 and each region mass distribution function are in
figure 5.2 and 5.1

Figure 5.1: Cumulative mass functions for clouds beyond the solar circle. In
each panel, both a truncated (solid blue) and non-truncated (dashed red) power
law mass spectrum is fit. For the outer Galaxy, the non-truncated, dashed red
power law functional form is preferred Figure 5.2: Cumulative mass functions
for clouds within the solar circle. In each panel, both a truncated (solid
blue) and non-truncated (dashed red) power law mass spectrum is fit. Figure
5.3:

The authors also suggest that their catalog is complete in the solar circle
for mass above $10^{5}$ $M_{sun}$ With this restrictions the number of
molecular clouds drops from 1064 to 153. A first test for my model is to
simulate an appropriate number of molecular clouds so that by applying the
same restriction roughly the same number of clouds is obtained.

Figure 5.4: Top-down Galactic map of all clouds with circle sizes indicating
cloud masses. The solar circle is drawn in solid purple, and the tangent
circle in the inner Galaxy is drawn in dashed purple. The Sun’s location is
marked with a yellow star and the Galactic Center is marked with a black "x."

####  5.1.2 Synthetic population of molecular clouds

Taking into account the Rice et al.’s studies, I had to reproduce the
positions of the molecular clouds along the spiral arm, the molecular clouds
mass distribution and their radial velocity distribution in order to create a
realistic and normalized model.

##### Spatial distribution

The models for the spatial distribution of $\gamma$-ray sources in our galaxy
are mainly based on recent pulsar data. In general it is necessary to
distinguish between evolved distributions, that are obtained from
observational data to describe the current population of objects and birth
distributions, that are used in studies to reproduce the current population of
objects. The spatial distribution is described via the surface density of the
objects, commonly it is radial symmetric and just varies with the distance to
the Galactic center. The vertical distribution (height above the Galactic
plane) is treated separately.

###### Radial distribution

Evolved distributions: Based on the data of the current ATNF catalog Yusifov
et al. analyzed the radial distribution of pulsars in our galaxy. They found
that the surface density could be well described by a $\Gamma$-function:

$p(r)\propto(\frac{r}{R_{sun}})^{a}exp[-b\frac{r}{R_{sun}}]$ (5.2)

$R_{sun}$ denotes the position of the sun in the Galaxy. It was assumed that
$R_{sun}$ = 8.5 kpc. According to Case and Bhattacharya the distribution of
SNRs in the galaxy can also be described by formula 5.2. In table 5.1 all
values of the parameters a and b are listed.

Birth distributions: In their optimal model Faucher et al. proposed a gaussian
birth distribution that led to:

$p(r)\propto exp[-\frac{(r-R_{\mu})^{2}}{2\sigma^{2}}]$ (5.3)

Paczynski (1990) made the assumption that the birth rate just varies
exponentially with distance from the Galactic center.

$p(r)\propto e^{-r/r_{0}}$ (5.4)

Yusifov et al. instead proposed a birth distribution that follows the
distribution of OB stars (which are considered as progenitor stars of neutron
stars) in the Galaxy. A comparison of all distribution models is shown in
figure 5.5. The solid curves represent the current distributions, the dash-
dotted lines the birth distributions. In general the peak of the birth
distributions is positioned farther from the Galactic center. This holds also
for [CB98], as the distribution of SNRs can be considered as a birth
distribution of pulsars. The distribution of Paczynski is an exception,
because it is not based on current data.

Model | Abbrev. | Parameters | Birth | Evol.
---|---|---|---|---
Case and Bhattacharya (1998) | [CB98] | a=2.0 b=3.53 | $\surd$ | $\surd$
Faucher-Giguere and Kaspi (2006) | [F06] |  $R_{\mu}$=7.0 $\sigma$=1.8kpc | $\surd$ |
Paczynski (1990) | [P90] |  $r_{0}$=4.5 kpc | $\surd$ |
Yusinov and Kucuk (2004) | [YK04] | a=1.64 b=4.01 |  | $\surd$
Yusinov and Kucuk (2004) | [YK04B] | a=4.0 b=6.80 | $\surd$ |
Lorimer et al. (2006) | [L06] | a=1.9 b=5.0 |  | $\surd$
Table 5.1: Parameters of the radial distribution models. Figure 5.5:
Comparison of radial distribution models. The dot dashed lines indicate birth
distributions, the solid lines evolved distributions. The area under the
curves is normalised to 1

###### Z-distribution

It is commonly assumed that the current vertical distribution of pulsars
follows an exponential law:

$p(z)=e^{-z/z_{0}}$ (5.5)

Faucher-Giguere and Kaspi demonstrate that an initial scaleheight of
$z_{0}\approx 50$ pc reproduce the current distribution of pulsars.

###### Spiralarm modelling

Based on observations of other spirals it is known that massive stars are
mostly born in spiralarms. As it is not expected that they move far from where
they were formed during their lifetime, the spiralarm structure should be
considered in the distribution of birth positions of pulsars. A simple model
can be taken from Faucher-Giguere and Kaspi. The raw shape of each arm follows
a logarithmic spiral that is defined by:

$\Theta(r)=kln(r/r_{0})+\Theta_{0}$ (5.6)

The parameters of the four known spiralarms of the milky way are shown in
table 5.2

Spiralarm | k[rad] |  $r_{0}$[kpc] |  $\Theta_{0}$ [rad]
---|---|---|---
Norma | 4.25 | 3.48 | 1.57
Carina-Sagittarius | 3.48 | 4.25 | 4.71
Perseus | 4.89 | 4.90 | 4.09
Crux-Scutum | 4.89 | 4.90 | 0.95
Table 5.2:  Parameters of the four known spiralarms of the milky way