MSc Physics & Astronomy
Gravitation, Astro-, and Particle Physics
Master Thesis
Contribution of nearby cosmic ray
accelerators to the positron excess
The search for positrons originating from pulsars and
supernovae
by
Dylan van Arneman
10919414
July 2020
60 ECTS
Conducted between August 2019 and July 2020
Supervisor:
Daily supervisor:
Examiner:
Abstract
Much is unknown about the origin of cosmic positrons. In 2009, the PAMELA detector measured an unexpectedly high number of extra terrestrial positrons bombarding the Earth’s atmosphere. This measurement was further supported by new measurements by the Fermi-LAT and AMS-02. In this work we try to find the origin of these positrons by developing a framework that allows us to model positron production and propagation. Specifically, we model the pro-duction of high-energy positrons created by pulsars and supernova remnants (SNR) that lie within 2 kpc of Earth. To model the positron propagation, we solve the diffusion equation for charged Galactic particles and take into account various positron energy loss mechanisms such as inverse Compton radiation, synchrotron radiation and losses by electromagnetic collisions. We ultimately use our framework to predict the positron spectrum on Earth using the most up-to-date pulsar, SNR and diffusion properties. Our results show that nearby pulsars and supernovae can contribute a significant amount of positrons to the local positron spectrum. Using our models, we estimate the maximum allowed pulsar and supernova positron production efficiencies to be 10−1 and 10−3.7, respectively. The results obtained by this work, however, should only be inter-preted as an estimate of the pulsar and SNR contribution. In order to fully determine the contribution by these pulsars and SNRs, all local sources would need to be identified and their properties would need to be measured more accurately.
Acknowledgements
First and foremost, I would like to thank my daily supervisor Oscar Macias. Thank you, not only for helping me with this project, but also for being my academic mentor throughout this year. Thank you for all the advice, tips and help that you have given me. I would also like to thank my supervisor Shin’ichiro Ando for guiding me during this project. Thank you for all the help and feedback that you have given me. I would especially like to express my gratitude to both of you for giving me the opportunity to visit the conference in Tokyo. Furthermore, I would like to thank Christoph Weniger for being my examiner. Next, I would like to thank everybody from the “Ando group” for hosting and attending the daily zoom meetings and helping us all get through these tough and uncertain times. A special thanks to all of my friends. Your support has greatly helped me to get through these troubling times. I would like to thank my fellow students, especially those from the GRAPPA room, for the stimulating, funny and enlightening conversations. Finally, I would like to thank my parents for always motivating me and supporting me.
Contents
1 General introduction 5
1.1 Introduction to cosmic rays . . . 5
1.2 Cosmic ray spectrum . . . 6
1.2.1 Primary and secondary particles . . . 7
1.3 AMS-02 detector . . . 8
1.4 Positron excess and context . . . 8
2 Cosmic ray propagation 11 2.1 Diffusion of charged particles . . . 12
2.1.1 Diffusion coefficient . . . 13
2.1.2 Solving the diffusion equation . . . 16
3 Positron and electron production 19 3.1 Electron and positron sources . . . 19
3.2 Pulsar properties and positrons . . . 20
3.2.1 Pulsar spin-down properties . . . 21
3.2.2 EM emission from pulsars . . . 23
3.2.3 Positron acceleration mechanisms by pulsars . . . 23
3.2.4 Pulsar injection spectrum . . . 27
3.3 SNR positrons . . . 28
3.3.1 Supernova mechanisms . . . 29
3.3.2 Supernova types and energies . . . 29
3.3.3 SNR injection spectrum . . . 30
3.4 Production of secondary positrons . . . 30
4 Positron and electron energy loss mechanisms 33 4.1 Positron energy loss mechanisms . . . 33
4.2 Synchrotron radiation . . . 34
4.3 Inverse Compton scattering . . . 35
4.4 Bremsstrahlung . . . 37
4.5 Electromagnetic collisions: ionization and Coulomb . . . 39
5 Positron flux results and discussion 42
5.1 Comparison with the literature . . . 42
5.1.1 Propagation setup . . . 42
5.1.2 Reproduction results and discussion . . . 44
5.2 Updated pulsar results . . . 48
5.2.1 Updated propagation setup . . . 48
5.2.2 Updated spectra results . . . 49
5.3 Injection parameter tests . . . 50
5.4 SNR positrons . . . 55
5.5 Secondary positrons . . . 57
5.6 Final discussion positron flux . . . 57
6 Conclusion 59
Chapter 1
General introduction
1.1
Introduction to cosmic rays
Every second, approximately 1000 highly energetic particles bombard each square meter of the Earth’s atmosphere [1]. This flux of charged particles of extrater-restrial origin is also known as cosmic radiation. Though cosmic rays (CRs) are most commonly understood to be composed of baryonic matter, namely pro-tons and heavier nuclei, they are also associated with other particles. Examples of such associated particles are electrons, positrons, neutrinos and high-energy electromagnetic radiation. All of the previously mentioned particles can be created or accelerated by astrophysical sources such as stars, supernovae and galactic nuclei. Interestingly, some of these particles can be detected by detec-tors on Earth. Throughout this thesis we will be considering all the particles previously mentioned as CR particles.
Much is still unknown about these cosmic particles. Their exact origins and how they were accelerated to such high energies are largely still a mystery to many physicists. Answering these fundamental questions could offer a better understanding of many areas physics, including particle physics, astrophysics and cosmology. Perhaps it could even provide indications of new interactions and exotic particles, like dark matter.
In 1912, physicist Victor Hess set out to investigate the amount of charged particles in the Earth’s atmosphere [2]. To do so, Hess flew up to a height of 5 km in a hot air balloon and measured the intensity of the surrounding radiation using electroscopes. Hess observed that as the balloon rose in altitude, so too would the measured intensity of the radiation [2]. Hess concluded that this radiation had to have come from outside of the atmosphere [2]. This resulted in more research being done into the nature of these particles and ultimately birthed the field of astroparticle physics.
In this thesis we will investigate the world of CRs. Specifically, we will research the origin of high-energy positrons and electrons that have been mea-sured on Earth. This thesis will look into the possibility of positrons being
cre-ated and/or accelercre-ated by astrophysical sources within 2 kpc of Earth. These sources will mainly be pulsars and supernova remnants (SNRs). In this thesis, Chapter 2 will examine the physics of CR propagation by studying the diffusion of particles. Next, Chapter 3 will discuss the positron production mechanisms by local sources. Chapter 4 will cover the energy loss mechanisms of electrons and positrons and Chapter 5 will present predictions of the positron flux spec-trum. Finally, Chapter 6 will discuss possible caveats related to these results and present the final conclusion.
Though the field of astroparticle physics has essentially existed since the time of Victor Hess (which is roughly 100 years ago), much about CRs and the CR spectrum is still not very well understood. The nature of CR propagation is not very straightforward. As opposed to what one might expect, CRs exhibit diffusive behaviour, which makes it more difficult to model. One difficulty is that the theoretical models that had been used to explain the phenomenon of CRs had many issues [3]. The primary issue was that there was insufficient knowledge about the structure of the magnetic fields of the Galaxy and the Solar system [3]. Another issue was regarding the lack of knowledge concerning the acceleration mechanisms of astrophysical sources [3]. Additionally, interaction cross sections between certain CR species were poorly understood [3]. Moreover, CR research is highly dependent on observational data. This data was often lacking and had large uncertainties [3].
1.2
Cosmic ray spectrum
Fortunately, there has been a lot of progress within the last 30 years. Within the past 30 years, there have now been many projects and collaborations that dedicated the necessary resources to observe CR particles [1]. Examples of such projects include DAMPE [4], the Pierre Auger Observatory [5], PAMELA [6] and others. More CR measurement projects are listed in figure 1.1. This figure contains the full CR spectrum as of 2016, and is obtained from [7]. The most relevant project for this thesis is the AMS-02 detector, which we will discuss in more detail in section 1.3. Several interesting features can be seen in the CR spectrum presented in figure 1.1. First of all, a large portion of the ’all-particle’ spectrum appears to be linear in log space. This means that the spectrum can be roughly described using a power law [1]. There are however breaks in this spectrum, namely at E ' 106 GeV and at E ' 109 GeV. These break regions
are known as the knee and ankle, respectively [1]. The break regions can be used to investigate key aspects of CRs, mainly their propagation mechanism and their origin. For example, it has been proposed that all particles whose energies lie beyond the ankle originate from extragalatic sources [1], whereas the particles before the knee originate from within the Galaxy [1]. This idea is, however, still largely uncertain.
Another interesting thing to note about the CR spectrum is that it goes up to extremely high energies. As can be seen in figure 1.1, the energy of the CR particles can exceed the energy obtained by modern day man-made particle
Figure 1.1: The full cosmic ray particle spectrum as of 2016, obtained from [7]. This spectrum features most cosmic ray types, including protons, nuclei, electrons, positrons and antiprotons.
accelerators [7]. For example, the LHC can reach energies up to roughly 13 TeV [8], whilst the CR spectrum exceeds energies of 107TeV. This means that CRs
can be used to probe physics on at otherwise so far unreachable energy range.
1.2.1
Primary and secondary particles
CR particles can be grouped into two categories: primary particles and sec-ondary particles. Primary particles are particles that originate directly from the (astrophysical) source. These particles are either created by the source and then propagate out, or have already existed in the vicinity of the source and were accelerated by the source. Secondary particles, on the other hand, are not directly produced by an astrophysical source. Instead, they are products of primary particles colliding and interacting with other matter along the way, producing new particles. This process is called spallation. Another form of
secondary production is by the decay of a cosmic particle. A nice example to discuss is the case of 10Be. 10Be is mainly produced by the spallation of cos-mic protons with the interstellar medium (ISM), and can radioactively decay to produce10B [9]. This makes both 10Be and10B secondary particles, though each are produced through different means.
1.3
AMS-02 detector
There are several detectors that can detect CRs. The most relevant detector for this thesis is the Alpha Magnetic Spectrometer (AMS-02) detector. The AMS-02 is a detector on board the International Space Station (ISS) and was launched in May 2011 [10]. It has since then been in operation and will continue to do so for the duration of the lifetime of the ISS [10]. Its main objective is to accurately measure the CR flux spectrum, and in doing so will investigate the nature of antimatter, dark matter and possible new physics [11]. The AMS-02 detector has an accuracy of approximately 1% in the TeV energy regime [11].
One of the most interesting measurements done by AMS-02 is the positron flux spectrum. In figure 1.2, we see the electron and positron flux spectrum as measured by AMS-02 [12, 13]. An interesting feature of this detection is the the discrepancy between the electron flux spectrum and the positron flux spectrum. The electron flux seems to behave in a completely different manner when compared to the positron flux at energies above roughly 3 GeV. If one were to assume that high-energy cosmic electrons and positrons were to originate from the same source, one would expect the spectra to look similar. This, however, is not the case, and could possibly indicate that high-energy cosmic electrons and positrons originate from different sources.
Aside from measuring cosmic electrons and positrons, AMS-02 also measures a wide variety of other CR particles such as nuclei and antiprotons [14, 15].
1.4
Positron excess and context
As previously mentioned, the discrepancy between the spectral shapes of the electron and positron fluxes triggered many questions. In 2009, Adriani et al. [16] reported that there seemed to be an unexpectedly high number of positrons detected by the PAMELA detector [16]. This result is unexpected, as the ratio between positrons and electrons was expected to decrease in energy, and not increase [17], as was found in the PAMELA measurements. Adriani et al. 2009 [16] suggests that this overabundance of positrons is due to extra positron cre-ation by exotic processes, such as dark matter decay and annihilcre-ation. They postulate that secondary antiparticles, such as positrons and antiprotons, could be created by the decay or annihilation of dark matter particles; this would con-tribute to the total flux spectrum [16]. However, there are also other positron sources, such as pulsars and supernova remnants. These could also produce positrons and contribute to the overall positron abundance [18]. Though the
Figure 1.2: The AMS-02 electron and positron flux as of 2019 [13]. It is inter-esting to note that the spectral shapes of the two fluxes are dissimilar. This dissimilarity could possibly indicate that CR electrons and positrons have dif-ferent origins.
contribution by pulsars was very uncertain, as none of the published pulsar models up to 2009 were able to agree with the PAMELA data [16].
Several years later, other detectors, such as Fermi-LAT and AMS, also sup-ported the claim of a positron excess [19]. In order to further investigate the source of these positrons, Abeysekara et al. 2017 [19] probed the contribution of two nearby pulsars, Geminga and Monogem, to the local positron spectrum. In their model, Abeysekara et al. 2017 [19], assumed that high-energy positrons created by pulsars would undergo homogeneous and isotropic diffusion and, by these means, propagate to Earth. Abeysekara et al. 2017 [19] found that the two nearby pulsars were not able to produce the necessary amount of positrons to explain the excess, and suggested that the extra positrons had to have come from other sources [19]. Some proposed sources were other pulsars, SNRs, mi-croquasars and once again dark matter [19].
After this, Profumo et al. 2018 [17] decided to reinvestigate the positron production by pulsars, now using a different model. Rather than assuming isotropic and homogeneous diffusion, the positrons would now diffuse through multiple media, each with a different diffusion coefficient [17]. This new model would suggest that diffusion is radially dependent, and that positrons diffuse more efficiently when far enough away from the pulsar [17]. Their results claim that when using this model, their predicted positron spectrum could explain the measured positron data [17].
revisit the positron production by pulsars and SNRs, and to investigate how much they realistically contribute to the measured positron flux spectrum. In addition, finding the origin of high-energy positrons could perhaps also help give an indication of the source of other types of cosmic antimatter, such as antipro-tons. In this thesis we will focus primarily on positrons produced by pulsars and will only be considering only homogeneous and isotropic diffusion. We expect that pulsars will be able to contribute a significant amount of positrons to the local positron spectrum.
The primary goal of this thesis is to expand upon the work of Delahaye et al. 2010 [18]. Delahaye et al. 2010 [18] modeled the positrons originating from pulsars and SNRs that lie within 2 kpc of the Earth. However, the re-sults obtained by Delahaye et al. 2010 [18] are rather outdated, as they use outdated observational and theoretical parameter values. Therefore, this thesis will seek to redo these calculations, as well as extend their analysis by including more up-to-date pulsar, SNR and ISM properties. These updated properties include pulsar and SNR distances, ages and diffusion parameters. We will also investigate how numerous other parameters related to the propagation and pro-duction of positrons, such as pulsar efficiency and spectral index, can affect the final positron spectrum. Finally, we shall compare the contribution by pulsars and those by SNRs to see how significant either source is to the overall local positron spectrum.
Chapter 2
Cosmic ray propagation
As mentioned briefly in the introduction, the propagation of cosmic ray particles is not trivial. To get an intuitive picture of the difficulties that come with CR propagation, I will introduce the following scenario. Suppose an astrophysical source A emits both photons and charged particles seemingly in the direction of another astrophysical object B. Though it would seem as if the final destination of the particles would be object B, this is not actually the case. It would be mis-leading to expect these particles to travel in a straight line, as there are several trajectory changing interactions that the particles can undergo whilst propa-gating. The particles are therefore not actually considered to be emitted in the direction of B. Figure 2.1 shows a schematic representation of the propagation of photons and charged particles throughout the Galaxy.
Figure 2.1: A schematic representation of photons and charged particles travers-ing the Galaxy. Charged particles originattravers-ing from an astrophysical source A will exhibit a random walk behaviour, as their paths are deflected by magnetic fields and interactions with interstellar material. Photons may also be deflected or absorbed, resulting in attenuation.
The photons traversing the ISM may be absorbed or deflected by molec-ular gas clouds or dust along their trajectory, resulting in the attenuation of the total number of photons measured at location B. Similarly, charged par-ticles traveling from a source through the ISM will have their paths deflected by interactions with ambient photons, magnetic fields and by collisions with interstellar gas. Similarly to the photons, the total number of charged particles measured at location B will be similarly reduced. However, unlike the pho-tons, a significant fraction of these charged particles will end up at a relatively unpredictable location, as their trajectories are guided by inhomogeneities in the Galactic magnetic field. As long as the details of the Galactic magnetic field inhomogeneities are unknown, it will be impossible to exactly predict the trajectories of charged particles.
Charged particles traveling in a magnetic field will follow a helical path along the magnetic field line, spiraling with an angular frequency [20]:
ωg=
ZeB
γmc2, (2.1)
where Ze is the charge of the particle, B is the magnetic field strength, m is the particle mass and γ is the Lorentz factor. This helical path is also shown in figure 2.2, obtained from [21]. The radius of this circle is given by the gyroradius, also called the Larmor radius [1]:
rL=
pc
ZeB, (2.2)
where p is the momentum of the particle with charge Ze. The Larmor radius is a useful quantity in the context of CRs, as it gives an indication as to how far a CR can deflect whilst propagating. The bending of the CR trajectory as a result of magnetic inhomogeneities will cause the particle to move in an unpredictable random walk behaviour, as can be seen in figure 2.1. As long as the fine details of the Galactic magnetic field inhomogeneities are unknown, we can only make statistical assumptions of their trajectories. Hence, we adopt a diffuse mechanism to describe CR propagation.
2.1
Diffusion of charged particles
In order to describe the previously mentioned random walk behaviour of par-ticles propagating through a medium, we can use the diffusion equation. It is interesting to realize that this equation is not unique to the propagation of CRs, but can in fact be used to describe many random motion effects [22]. This equation was initially solved by Syrovatskii in 1959 [23]. However, we will follow the steps described by Atoyan et al. 1995 [24] to solve this diffusion equation.
If we neglect the effects of convection and reacceleration, which is a valid simplification at energies greater than 1 GeV [18], the cosmic ray transport equation takes the following form [18]:
∂ψ
∂t = ∇ · (D(x, E)∇ψ) + ∂
Figure 2.2: Charged particles in a magnetic field will follow a helical path along the magnetic field line. The radius of this helix is called the gyroradius or Larmor radius. Figure obtained from [21].
where ψ is the differential number of particles in volume and energy, given by ψ ≡ ψ(t, x, E) =dEdVd4N [17]. D(x, E) is the diffusion coefficient and is expressed in units of area per second, b(E) denotes the energy loss rate of the CR and is unique to the type of particle. Q is the injection term, which is dependent on the production/acceleration mechanism of the particle. By solving this equation, one can ultimately obtain an expression for the CR flux as a function of energy, time and spatial position.
A useful quantity to define is the diffusion radius, given by [24]:
λ(E, t) ≡ 4 × Z E0(E,t) E D(E0) b(E0)dE 0 !1/2 . (2.4)
This expression describes how far a CR of energy E can propagate within a time t, given that it started with an initial energy E0 [24].
2.1.1
Diffusion coefficient
When investigating equation (2.3) one can quickly see that the diffusion coeffi-cient plays an integral role in the propagation of CRs. The goal of this section is to explore this diffusion coefficient in depth and discuss commonly used models used to describe this coefficient.
One of the most commonly used models for CR diffusion is to confine prop-agation to so-called ’propprop-agation zones’. The diffusion coefficient within such a zone is isotropic, homogeneous and solely dependent on the energy of the CR.
The diffusion coefficient takes the following form [18, 17]: D(E) = D0β R 1 GeV δ ≈ D0 E 1 GeV δ , (2.5)
where R is the rigidity of the particle and D0 and δ are constants. From
this point on, we will refer to D0 and δ as the diffusion parameters. Spatial
independence within the propagation zone is a valid assumption, as high-energy particles are only sensitive to larger spacial scales [3]; thus if the propagation zone is small enough, the diffusion parameters would not change. Due to the fact that the random walk behaviour of CRs is a result of inhomogeneities in the Galactic magnetic field, we can relate the diffusion coefficient to the power spectrum of these inhomogeneities [3, 25]. Two other parameters that could have effect on the diffusion are the Alfven velocity VA and the convective wind
velocity Vc, which are responsible reacceleration and convection, respectively
[18]. However, as mentioned before, we omit the effects of reacceleration and convection for simplicity.
Unfortunately, there is still a lack of theoretical understanding of these mag-netic inhomogeneities [3], so the most common way to derive the values of D0
and δ is to fit them to measured CR data.
In order to fit the diffusion parameters to the data, we must first model the relevant propagation zone. To begin, we confine the CR propagation to a cylindrical slab. This slab, schematically represented in figure 2.3, contains the Milky Way (MW) disk in the middle, surrounded by a diffusive halo [18, 9]. Here we assume that CRs can diffuse relatively large distances from the Galactic disk. The MW disk has a half-thickness of h and a radius RMW. The
surrounding halo has the same radius as the MW and extends to a vertical height L [18, 9]. For the MW, the best fit value for the half-thickness of the Galaxy is h = 4 kpc [26], whilst recent studies estimate the MW radius to be RMW' 20 − 30 kpc [18, 27]. The vertical height of the diffusion slab L ranges
in values up to 15 kpc [18]. Note that most, if not all, of the Galactic material resides within the MW disk [9], which takes up only roughly 27% of the total volume of the full propagation zone. This means that, with the assumption that CRs can diffuse throughout this slab independently of their position [9], CRs can propagate relatively far from any astrophysical objects [18].
When trying to determine the values necessary for computing the diffusion coefficient, it is necessary to distinguish primary CRs from secondary CRs. As mentioned in the introduction chapter, primary CRs originate directly from the astrophysical source of acceleration, and can therefore be more easily identi-fied as having come from a point source at a given point in time [28, 29, 30]. Secondary CRs, on the other hand, are particles that are created as a result of interactions of a primary CR. As a result of the production mechanisms of secondary particles, it is not easily possible to pinpoint an exact location in space from which these secondaries originate, unlike primary CRs.
The reason why it is important to distinguish primary CRs from secondary CRs, is namely because the ratio of measured secondary-to-primary CR flux can
Figure 2.3: The propagation zone within which Galactic CRs diffuse. This cylindrical slab contains the Milky Way disk in the middle, surrounded by a diffusive halo. The MW has radius RMW and half-thickness h. The diffusive
halo has the same radius as the MW, and a half-thickness L. CRs can diffuse throughout this slab and the diffusion coefficient within this slab is spatially independent. This image is not to scale.
give insight into the mechanisms that drive CR propagation [28]. This is mainly due to the fact that the shape of the secondary-to-primary ratio is very sensitive to the diffusion coefficient [3]. If parameter values of the diffusion coefficient were to change, the spectrum of the secondary-to-primary ratio would also change significantly. Therefore, we can use known secondary-to-primary ratios to test certain values of the diffusion coefficient.
The most used secondary-to-primary flux ratio used in this type of analysis is boron-to-carbon (B/C), in part due to the fact that this ratio is very well measured by numerous experiments [30]. Carbon is believed to be produced primarily by astrophysical sources [29], whilst boron is thought to be produced mainly through means of spallation [29, 30]. Heavy high-energy nuclei, such as carbon and oxygen, can collide with the ISM and produce boron. Other heavier nuclei can do this as well [29, 30].
The boron-to-carbon ratio is a particularly convenient measurement, as they are neighbouring elements and therefore have similar masses as well as similar interactions with matter [28]. Hence, by measuring the boron to carbon ratio, one can obtain an estimate of the average amount of ISM propagated by CRs [29]. Since boron and carbon have mainly the same energy per nucleon, one can analyze this B/C ratio and find the momentum/energy dependence of the diffusion coefficient [30]. As mentioned before, the shape of this B/C ratio is very sensitive to changes in diffusion parameters D0, δ, VAand Vc[3]. This is precisely
what one does when trying to fit B/C measurements to obtain constraints on the values of D0/L, L and δ [18, 3]. Besides boron-to-carbon, there are also
other primary-to-secondary and primary-to-primary ratios that can be used for similar analysis [3].
In this thesis we will be using the diffusion parameters obtained by previous works, as opposed to deriving them here. Specifically, we will use the diffusion parameters obtained by Delahaye et al. 2010 [18] and Song et al. 2019 [31].
This will be further discussed in Chapter 5.
2.1.2
Solving the diffusion equation
In this section we will solve the diffusion equation (equation 2.3) following the methods described by Atoyan et al. [24]. Using their approach, we can solve the diffusion equation for any (spatially independent) arbitrary diffusion coefficient, as well as energy loss and injection term. This means that this method can essentially be used for any particle type in any part of the Galaxy, regardless of astrophysical source. The solution to this equation will ultimately lead us to an expression for the CR flux.
On the condition that we contain the CR propagation within the previously defined propagation zone, spherical symmetry can be assumed. Equation (2.3) can now be rewritten as [24]:
∂ψ ∂t = D(E) r2 ∂ ∂rr 2∂ψ ∂r + ∂ ∂E(b(E)ψ) + Q. (2.6) Note that here D(E) is given by equation (2.5).
The first step is to assume that initially, all particles are uniformly dis-tributed in the source with a radius rs. At t0 = 0 the initial distribution
function is given by [24] ψ0(r, E) = ∆N (E) (4π/3)r3 s Θ(rs− r). (2.7)
Here ∆N (E) describes the source injection spectrum, which is proportional to the injection term Q, and Θ(rs− r) is the Heaviside step function. We now
define [24]: T ≡ Z E0 E dx b(x) = g(E), (2.8) which is an expression that gives an estimate of the time that it takes for a particle of some initial fixed higher energy E0 to decay and cool down to an
(arbitrarily) lower energy E. Basically, this expression can be used to determine the time duration that it took to decay to an energy E.
Next, we define the inverse function to this newly introduced g(E) term, namely E = g−1(T ) ≡ (T ). This expression will have the inverse function of equation (2.8), namely, one can input a timescale T and determine what energy the particle will decay to. This function can eventually be used to determine the initial energy of the particle, given that E and T are known.
We also rewrite the particle density in terms of [24]:
Ψ ≡ b(E)ψ(r, t, E)r. (2.9) Now, equation (2.6) can consequently be rewritten in terms of these new ex-pressions to obtain, ∂Ψ ∂t = D1(T ) ∂2Ψ ∂r2 − ∂Ψ ∂T, (2.10)
where we have set Q = 0, as we are now solely dealing with the propagation of the initial particles and thus there is no further injection. Notice also that the diffusion coefficient D(E) is now transformed to D1(T ) = D() [24].
In order to reduce this equation further into a partial differential equation of two variables, we must shift our variables in the following way [24]:
(t, T ) → (τ = T − t, z = T ). The differential equation now becomes [24]:
∂ ˜Ψ
∂z = D1(z) ∂2Ψ˜
∂r2. (2.11)
Notice that now we are using ˜Ψ(r, τ, z) ≡ Ψ(r, t, T ). The initial distribution function, equation (2.7), can now be rewritten as
˜ Ψ0= r Θ(rs− r) (4π/3)r3 s ∆N (E)b(E)|E=(τ ). (2.12)
We can also define the following variables [24]: u = Z z 0 D1(z0)dz0 and u0= Z τ 0 D1(z0)dz0. (2.13)
Finally, we define ∆u = u − u0. The solution to equation (2.11) on the
semi-infinite line r > 0 is given by [24]: ˜ Ψ = 1 2√π∆u Z ∞ 0 exp −(r − x) 2 4D∆u − exp −(r + x) 2 4D∆u Ψ0(x, τ )dx. (2.14)
To return back to the solution of the original differential equation (2.6), we must transform the initial distribution function, equation (2.7), to a delta function. We can do this by taking rs→ 0. Finally, we have now obtained an expression
for the distribution of particles as a function time, particle position and particle energy [24]: ψ(r, t, E) = ∆N (E0)b(E0) π3/2b(E)λ3 exp −r 2 λ2 . (2.15) Notice that this equation now also includes the earlier defined diffusion radius λ(E, t) (see also equation 2.4). Also notice that this expression contains both the initial particle energy E0, as well as the final particle energy E.
Finally, the CR flux of a specific particle type is obtained by multiplying ψ with the prefactor c/4π [24]. Doing this, we finally obtain:
Φ(r, t, E) = c 4π ∆N (E0)b(E0) b(E)(πλ2)3/2 exp −r 2 λ2 . (2.16)
We can now apply this equation to determine the positron flux on Earth. In the following chapters we will obtain an analytical expression for each of the terms in equation (2.16). Specifically, we will investigate the injection term ∆N (E) in Chapter 3 and the energy loss term for positrons in Chapter 4.
Chapter 3
Positron and electron
production
Now that we have derived the expression for the CR flux in the previous chapter, we can examine how positrons are produced. The goal of this chapter is to ultimately obtain an expression for the positron injection spectrum ∆N (E). Using this expression, we will be able to obtain an an estimate of the amount of high-energy positrons produced by local sources.
This chapter is structured as follows: we will be discussing positron produc-tion by pulsars in secproduc-tion 3.2 and by supernovae (SNe) in secproduc-tion 5.4. Finally, we will briefly discuss secondary positron production in section 3.4. Note that though we will not be including secondary positrons in our model, we still in-clude a section on this for completeness.
3.1
Electron and positron sources
Using equation (2.4), we can see how far electrons and positrons can propagate. An example of the diffusion radius for positrons originating from a pulsar dis-tanced 160 pc from Earth is shown in figure 3.1. In this scenario, the positrons started propagating 343 kyrs ago. From this figure, it is evident that high-energy positrons can traverse up to 1 - 4 kpc, easily reaching Earth. This result further motivates the idea that local sources, within 2 kpc, can produce a sig-nificant amount of positrons that can reach Earth. Furthermore, we see that there appears to even be a maximum distance that positrons can travel within a given amount of time. Interestingly, this distance is smaller than the max-imum allowed distance you would obtain if the particle were to be traveling in a straight line at the speed of light (dlight,max = ct > dmax,diff ). The two
main local astrophysical electron and positron sources are pulsar wind nebulae (PWNe) and supernova remnants (SNRs) [18], so we will only be considering these two sources in this thesis.
Figure 3.1: The diffusion radius, λ(E, t), for positrons originating from Geminga, placed a distance of 160 pc from Earth. In this scenario, the positrons all started propagating t = 343 kyrs ago. The purple line is the diffusion radius obtained using diffusion parameters from Song et al. 2019 [31], whereas the green line represents the diffusion radius obtained using diffusion parameters from Delahaye et al. 2010 [18]. The energy loss used for both radii is given by equation (4.26).
3.2
Pulsar properties and positrons
First, let us examine the general concept of pulsars. When a star undergoes a supernova (SN) explosion they leave behind a residue. This residue is also known as a supernova remnant (SNR). Core-collapse supernovae will leave behind either a black hole or a neutron star, whilst other types of supernovae can leave behind a white dwarf [32].
One of the earliest recorded supernova explosions was in 1054 CE, when the Crab nebula formed [33]. Centuries later, after careful observation, astronomers discovered a remnant of a dead star within this nebula. It was found that this remnant was a highly magnetized rapidly rotating neutron star, emitting pulsating electromagnetic signals [33]. This so-called pulsar is thought to have been the product of the SN explosion that formed the Crab Nebula, and is thought to inject energy in the form of relativistic particles into the Galaxy [33].
There are several types of pulsars, mainly categorized by properties related to their rotation period and their magnetic field strength [32]. Examples of pulsar types of magnetars, millisecond pulsars and radio pulsars [32]. The rotation period of a pulsar also changes over time [32].
meaning the rotation of a pulsar will slow down when given enough time. Physi-cists have theorized that the rotational energy of a pulsar is (partly) transferred to CRs by accelerating nearby particles to higher energies and forming magne-tized winds [33]. When this wind of ultrarelativistic particles comes in contact with the ISM, it produces a sphere of shocked high-energy particles, also known as a pulsar wind nebula [33]. Measured synchrotron radiation from PWNe also indicated that this is indeed the case [33]. In order to fully understand how much energy is released by these pulsars, we must first examine their spin down mechanisms and properties.
3.2.1
Pulsar spin-down properties
Pulsars are thought to lose their rotational energy due to magnetic dipole radi-ation [32]. This magnetic dipole radiradi-ation is given by [32]:
˙ Edipole= µ0 4π 2m2 3c3 2π P 4 sin2θ, (3.1) where µ0 is the vacuum permeability, m is the magnetic dipole moment of the
star, P is the spin period of the star, and θ is the angle between the spin axis and the magnetic axis of the pulsar.
The rate at which the pulsar slows down, i.e. the rate at which the rotational energy is dissipated, is given by the following expression [33]:
˙ Erot≡ − dErot dt = 4π 2 I ˙ P P3, (3.2)
where I is the moment of inertia of the pulsar, P is the spin period of the pulsar and ˙P is the rate at which the rotation of the star slows down. From observations, we know that the value of ˙E ranges between 3 × 1028 erg s−1 to
5 × 1038 erg s−1 [33].
Assuming all rotational energy lost is indeed due to magnetic dipole radia-tion, we can determine the magnetic dipole moment of the pulsar [32]:
m = − ˙Erot sin2θ 3c2 2 4π µ0 P 2π 4!1/2 . (3.3)
If we now take the magnetic field strength of a magnetic dipole to be given by [32]:
B = µ0
4πR3m, (3.4)
where R is the radius of the pulsar. Now, using the newly obtained expression (3.3), we find B = µ0 4πR3× − ˙Erot sin2θ 3c2 2 4π µ0 P 2π 4!1/2 → B = µ0 4πR3× 4π 2 I ˙ P P 1 sin2θ 3c2 2 4π µ0 P 2π 4!1/2 , (3.5)
where we have used ˙Erot defined in equation (3.2).
Finally, if we take M ' 1.35M and R ' 10 km, we find I ' 1038 kg m2
[33, 32]. Using these values, we find [33, 32]: B = 3.3 × 1015 P ˙P 1 s !1/2 1 sin θT = 3.3 × 10 19 P ˙P 1 s !1/2 1 sin θG. (3.6) This magnetic field strength can vary greatly depending on the type of pulsar. For example, magnetars can have a field strength greater than 1015 G, whilst
millisecond pulsars have a field strength of the order of 108G [33]. The average
pulsar with a PWN however has a field strength between 1012 and 1013G [33].
To obtain the age of the pulsar, we must also examine its rotation. The angular velocity of the rotation of the pulsar is given by Ω = 2πP and the time evolution of this velocity is assumed to be given by ˙Ω = −kΩn, where n is the
braking index and k is a constant. From observation, we can constrain the value of n to be between 2 and 3 [33]. If the spin-down of the pulsar is indeed a result of magnetic dipole radiation, we can take n = 3 [33]. We will assume this to be the case and n = 3 throughout the rest of this thesis.
To estimate the age of the pulsar we use the following equation [33]: τ = P (n − 1) ˙P " 1 − P0 P n−1# , (3.7)
where P0 is the initial spin period of the pulsar.
The spin period of the pulsar has the following time dependence [33]: P = P0 1 + t τ0 n−11 , (3.8)
where τ0 is the initial spin-down timescale of the pulsar and is given by [33]:
τ0≡
P0
˙
P0(n − 1)
.
τ0is sometimes also called the typical pulsar decay time τdecor the characteristic
pulsar spin-down timescale [18, 34].
This quantity is not to be confused with the characteristic age of a pulsar, which can be found by taking the limit P0 P to obtain:
τc≡
P
2 ˙P. (3.9)
Finally, another important quantity is the spin-down luminosity. For a con-stant n, the luminosity can be described by [33]:
˙ E = ˙E0 1 + t τ0 −(n+1)(n−1) , (3.10)
with ˙E0being the initial spin-down luminosity. Note that equation (3.2) is valid
for any arbitrary rotating body, whilst equation (3.10) is only in the case that n is constant.
3.2.2
EM emission from pulsars
Though it is hard to directly trace back the origin of the high-energy CR elec-trons that arrive at Earth, we can find evidence of their presence in PWNe. As mentioned in the previous section, the rotation of a pulsar will slow down over time and, in turn, release/inject energy in the form of high-energy particles. This evidence comes in the form of two EM signals: synchrotron and gamma-ray emission [33, 35]. The exact emission mechanisms of these signals will be dis-cussed in the following section, for now we will only discuss the phenomenology of these signals.
Synchrotron radiation is produced by charged particles accelerating and bending in a magnetic field. Chapter 4.2 will describe this process in more detail. As shown in section 3.2.1, pulsars can produce strong magnetic fields and thus provide the ideal environment for synchrotron emission. Given that there is a sufficient number of charged particles present near a pulsar, the PWN will emit synchrotron radiation mainly in the form of radio photons [33]. The flux of the radio synchrotron emission of the PWN can be described by a power-law [33]:
Φradio∝ να, (3.11)
where ν is the frequency of the emission and α is the spectral index of the source. Likewise, the distribution of relativistic electrons, which can also emit X-ray photons, is described by a similar power law [36]. Consequently, the distribution of these of emitted X-ray photons (not the flux)1 is also characterized by a
power law [33, 36]. The number of photons between energy E and E + dE is proportional to the photon energy by the following relation [33]:
NX−ray∝ E−γ, (3.12)
with γ being the photon index.
If we consider the total emitted power, given by [36]:
Ptot(ν) ∝ να= ν−(γ−1)/2, (3.13)
we see that we can relate the spectral index to the photon index by the following relation: [36]:
γ = 1 − 2α. (3.14)
3.2.3
Positron acceleration mechanisms by pulsars
Now that we have covered the general observational phenomena, let us dis-cuss several theoretical models that describe positron and electron acceleration. Though many of the details as to how exactly pulsars accelerate particles are unknown, there is strong observational evidence that shows that high-energy particles indeed originate from pulsars [37]. Estimates for the energies of these
1It is more convenient to describe X-ray emission in terms of number of photons, as there
particles go up to a Lorentz factor of 107[37]. Pulsars are surrounded by an atmosphere of charged particles, called the magnetosphere. Particles found in this magnetosphere undergo several interactions that will either accelerate them or cause them to emit observable EM radiation, or both [33]. The actual parti-cle acceleration is thought to occur inside of specific ’acceleration zones’ in this magnetosphere[37]. The following section will go into more detail to explain how these acceleration zones form.
Magnetosphere and acceleration zones
As mentioned before, pulsars are rapidly rotating magnetized stars. Any ro-tating magnetized body will induce electric fields. In the case of pulsars, these electric fields are so powerful that they are able to rip charged particles off the surface of the neutron star and deposit these into the atmosphere of the star [37]. This charged-filled atmosphere is called the magnetosphere [37]. This resulting overabundance of charge density produces a new electric field Emagnetosphere [37]. As a consequence of this new field, the total electric field
(Epulsar+ Emagnetosphere) is now perpendicular to the magnetic field of the
pul-sar, i.e. E · B = 0 [37]. With these conditions, the field lines will be closed and particles will not be able to escape the magnetosphere. This total electric field, however, is not perfectly homogeneous and will still contain small areas/gaps in which the total electric field and magnetic field are not perpendicular to each other [37] and the field lines will be open. These areas/gaps are the previously mentioned acceleration zones; particles could possibly be accelerated in these gaps [37], and then escape through the open field lines.
There are two main groups of models that theorize the location of these acceleration zones [37, 35]. The first group of models is the polar cap (PC) model, which predicts that the acceleration zone exists in or nearby the polar cap of the pulsar [37]. The second group of models theorizes that the acceleration zone forms only in locations where Ω · B = 0, where Ω is the direction of the rotation of the pulsar [37]. This group of models is called the outer gap model [37, 35]. Both of these models are schematically shown in figure 3.2.
In both models, the concept of pair cascades plays an important role. The idea of pair cascades is that high-energy photons can produce a high-energy electron-positron pair, which in turn will produce more high-energy photons, starting a runaway reaction that produces many high-energy photons and electron-positron pairs [35].
Both acceleration models predict the generation of pair cascades, yet their exact mechanisms differ slightly [35]. In polar cap models, pair cascades are gen-erated through photons originating from curvature radiation or inverse Compton scattering by high-energy electrons and positrons [35].
In the outer gap models, the initial charged particles and magnetic fields are not strong enough to generate curvature radiation or ICS strong enough to initiate cascade production [35]. Instead, cascades are induced through photon-photon pair production as opposed to one-photon-photon pair production [35]. In the case of old pulsars, ambient infrared photons will interact with upscattered ICS
Figure 3.2: A sketch showing theorized possible particle ’acceleration zones’ in the pulsar magnetosphere, coloured purple and blue. Note that this image shows acceleration zones predicted by both PC and OG models. The light blue shaded areas correspond to acceleration zones in older pulsars, whereas the narrow dark purple areas correspond to acceleration zones in young pulsars. The Original figure is obtained from [37].
photons and create electron positron pairs [37]. This is due to the fact that older pulsars have relatively little non-thermal X-ray photons in their environ-ment [37]. Younger pulsars however have much more X-rays, originating from synchrotron emission. These X-rays will interact with curvature photons and create electron positron pairs [37]. In either cases, there will be an electron positron cascade.
Polar cap models
The group of polar cap models can be divided into two sub-categories: free emission of particles from the pulsar surface and no free emission from the pulsar surface [37].
In the case of free emission, there are two options. In stars where the di-rection of rotation of the pulsar is in the same didi-rection as the magnetic field (i.e. Ω · B > 0), the corotation charge density at the magnetic poles will be negative [35], so it is expected to contain an over abundance of electrons. If the surface temperature of the pulsar is greater than the binding temperature of the electrons (Te≈ 105K), electrons will leave the surface and follow electric fields
and join to form a part of the magnetosphere [35]. Whilst bending due to the electric fields, the electrons will emit curvature radiation and start to induce the previously discussed pair cascades. It is interesting to note that in these models, the electric field is zero at the surface and will increase the deeper you
go into the magnetosphere [35].
If, however, the rotation of the pulsar is in the opposite direction (i.e. Ω·B < 0), the charge density above the poles will be positive [35]. This means that the magnetic poles of the pulsar will be predominantly filled with positively charged ions. If the surface temperature does not exceed the ion binding temperature (Tion ≈ 105K), ions will stay on the surface of the star. This causes a vacuum
gap to form, since the corotation charge cannot be supplied [35]. This also produces an electric field at the surface of the star, but will most likely be cancelled out due to the creation of electron positron pairs [35].
Due to these concentrations of either positively or negatively charged polar caps, there will be an electric field near the star which will accelerate particles. These particles initiate photon production (as described by the mechanisms above) and will induce the electron positron pair cascades. Particles can reach a Lorentz factor of around 107as a result of this [37]. It has been found that many
of the acceleration zones are roughly 0.5 to 1 stellar radius from the star surface [37]. These high-energy particles will ultimately escape the magnetosphere and propagate through the Galaxy.
As a pulsar gets older and converts less of its rotational energy (see equation 3.10), it will have less energy available for the acceleration of particles. The acceleration zone will move further and further from the surface of the star and will eventually not be able to accelerate particles any longer [37]. The acceleration ”cone” also gets wider [37].
Outer gap models
As mentioned earlier, in outer gap models there are vacuum gaps within the magnetosphere of the pulsar, in which particles will be accelerated. As opposed to the PC model, these gaps do not necessarily have to be near the poles of the pulsar. There will be parts of the pulsar that have a locally neutral charge density [37], above which the gaps will form. In the OG models, these vacuum gaps form as a consequence of charged particles escaping through the light cylinder (LC)2 of the pulsar. As the stellar surface area under where these
particles escaped is neutrally charged, it will not be able to supply additional particles to make up for the escaped particles; thus resulting in the formation of a vacuum gap [37]. Though some ’refilling’ of the gap is possible, it gets harder as the star ages, as there will be less energy left to accelerate particles needed to fill the gap. As a consequence of this, older pulsars will have bigger gaps [37]. Unfortunately, there is still insufficient observational data to conclusively confirm either of these two models. Hence, for the sake of the analysis in this thesis, we will be using a simplified positron injection model, which we will describe in the following section. This simplified model will use the phenomenon of pulsar spin-down as the basis for particle acceleration.
2The LC denotes the boundary at which the particles rotate slower than the speed of light.
3.2.4
Pulsar injection spectrum
As stated previously, despite the missing details on how exactly positrons are produced and accelerated,it can still be confidently assumed that pulsars do in fact produce high-energy electrons and positrons. This is mainly due to the fact that there is overwhelming observational evidence of PWNe that indeed indicate the presence of high-energy particles. This section will describe how we can use what we know about pulsar energy loss to predict the electron and positron injection spectrum.
There are two ways to describe electron/positron injection by a pulsar: burst-like and continuous (”pulsar-burst-like”). In the burst-burst-like scenario we assume a sim-plified model, in which all electron/positron injection happens over a relatively short timescale, i.e. a burst. This is very similar to electron production by SNe. In the pulsar-like scenario we assume that the emission of electron/positrons is gradual and happens over a large span of time. Though for our analysis we only use the burst-like scenario, we include pulsar-like here for completeness. Burst-like
As mentioned previously, in the burst-like scenario we assume the injection of electron/positrons happens over a very short timescale. Essentially, we assume that this timescale is negligible and all particles are injected at the same time. This is a simplification, as it removes the nuance associated with gradual injec-tion of particles. In this case, we can describe the positron/electron injecinjec-tion spectrum as the following [18]:
∆N (E) = N0 E 1 GeV −γ exp −E Ec , (3.15) where N0 is a normalization factor dependent on the energy loss of the pulsar,
γ is the electron spectral index, and Ecis the cut-off energy, an energy at which
we assume the production of electrons/positrons is heavily suppressed.
N0can be connected to the total energy loss of the pulsar with the following
equation [18]:
Z ∞
0.1GeV
dEE∆N (E) = f W0, (3.16)
where f is the fraction of spin-down energy converted to electron-positron pairs; we will also refer to this quantity as the pulsar efficiency. The positron effiency is given by fe+ = f /2. W0 is the total energy lost by the pulsar as a result of
spinning down [18]: W0= ˙Eτdec 1 + t? τdec n+1n−1 , (3.17)
where ˙E is the spin-down luminosity of the pulsar (see equation 3.10), t? is the
age of the pulsar. n is the braking index of the pulsar and τdec is again the
typical pulsar decay time. For the analysis in this thesis, we will always set the braking index n = 3.
When we fill combine expression (3.15) with equation (3.16), we are left with: Z ∞ 0.1GeV dEN0 E 1GeV 1−γ exp −E Ec = f W0. (3.18)
Putting everything together, we find the following expression for the nor-malization: N0= f ˙Eτdec 1 + t? τdec n+1n−1 × Z ∞ 0.1GeV dE E 1GeV 1−γ exp −E Ec !−1 . (3.19) There are several uncertainties that come with this expression. Firstly, the efficiency of positron production by pulsars is not very well known. Secondly, the expression for the normalization is also dependent on the pulsar age, which is also prone to uncertainties.
Pulsar-like
Pulsar-like injection case is very similar to the burst-like case, except now the normalisation carries time dependence. The injection term takes the following form [34]: ∆N (E, t) = L(t) E 1 GeV −γ exp −E Ec , (3.20) where L(t) is the time dependent luminosity of the pulsar, given by [34] (see also equation 3.10): L(t) = f ˙E(t) = f ˙E0 1 + t τdec −(n−1)(n+1) . (3.21) Here again f is the pulsar efficiency.
In the case of the pulsar-like injection, we now have a time dependent in-jection spectrum. In order to account for this, we must slightly adjust the expression for the total flux (equation 2.16). To do this, we must now integrate over the full expression as such [34]:
Φ(r, t, E) = c 4π Z t 0 ∆N (E0, t0)b(E0) b(E)(πλ(E, t0)2)3/2exp − r 2 λ(E, t0)2 dt0. (3.22) As mentioned before, however, in this thesis we will only be using the burst-like scenario to estimate the positron flux. We are interested in exploring only this scenario, and pulsar-like injection is beyond the scope of this project.
3.3
SNR positrons
As stated previously, another possible local source of high-energy positrons are supernova explosions. By observing a supernova remnant, we know that a su-pernova had to have taken place at that astrophysical location. When we refer
to SNR sources in this thesis, we are actually referring to the SNe that created the remnant. To get a good understanding of how high-energy positrons are produced by SNe, we will briefly cover the physics of supernova explosions.
3.3.1
Supernova mechanisms
A supernova can be described as follows. Once a star reaches the end of its life, it becomes more difficult for the stellar core to produce enough pressure to prevent it from gravitationally collapsing. Normally, this outwards pressure is sustained by nuclear fusion within the core. However, if the core runs out of fuel, it will no longer be able to provide the necessary outwards pressure and the star will start to collapse in on itself [32]. Eventually, the core of the star will contain a degenerate electron gas that can provide enough pressure to prevent the core from completely collapsing [32]. This type of collapse does not yet cause a supernova.
Provided that the total mass of the star is below the Chandrasekhar mass, MCh ' 1.4M , the degenerate electron pressure will be enough to sustain the
core and the star will continue to live on as a white dwarf [32].
If the final mass of the star were to exceed the Chandrasekhar mass however, the degenerate gas pressure would not be strong enough to avoid further collapse. This is the case for core-collapse SNe (CCSNe). In this scenario, the core would be compressed even further [32]. This compression continues until the density of the core is similar to (of the same order of magnitude as) the density of an atomic nucleus [32]. In which case, nuclear forces provided by the matter in the core will halt the collapse and produce an outwards shockwave [32].
This shockwave will transfer an extremely high amount of kinetic energy to the stellar matter and cause some of the matter to eject into the ISM. This whole process is known as a supernova explosion. Some of this transferred energy could go into the acceleration of electrons and positrons, thus producing CR positrons. For some SNe, roughly 99% of the released energy will in be the form of neutrinos, leaving only 1% left for the acceleration of other particles [18].
In the case of type 1a SNe (SNe1a), the supernova process is slightly different. In this scenario, the supernova is caused by the accretion of mass by a white dwarf [38, 32]. This accretion will cause the mass of the white dwarf to increase, eventually allowing it to approach the Chandrasekhar mass [32]. As the mass of the dwarf continues to increase, the high temperatures within the star activate explosive nuclear burning reactions [38]. These nuclear burning reactions cause the star to explode and release a large amount of energy [38]. This specific process of exploding and releasing of energy is called a type 1a supernova [38].
3.3.2
Supernova types and energies
As seen in the previous subsection, there are different types of SNe. Moreover, the amount of energy produced by a SN depends on the SN type. In this thesis we will only be considering two types of SNe: type 1a SNe and CCSNe. Roughly
one third of SNe are of type 1a, and the remaining two thirds are thought to be CCSNe [18].
The remnants left behind by a SN are also dependent on the SN type. Type 1a supernovae are thought to only expel matter and leave no compact object behind, unlike other SN types [18, 32]. In this case the remnant would be a white dwarf. This process is thought to transfer roughly 1051 erg into the ISM
[18].
On the contrary, CCSNe do leave behind a compact object. This object will be either a neutron star or a black hole [32]. CCSNe are produced by the collapse of a star with masses far beyond the Chandrasekhar mass, typically around 8 - 11 M [18, 32]. In the case of CCSNe with stars of mass M ≤ 20M , an
energy of roughly 1053−54 erg will be released in the explosion [18]. 99% of this
energy however will be in the form of neutrinos, leaving approximately 1051−52
erg available for other types matter [18].
3.3.3
SNR injection spectrum
SNRs have an injection term very similar to the burst-like pulsar case [18]: ∆N (E) = N0 E 1GeV −γ exp −E Ec , (3.23) where again γ is the electron spectral index and Ec is the cut-off energy. The
main difference in the SNR case is that now N0 is defined by the kinetic energy
released by the SN explosion.
For a single supernova we find [18] Z ∞
0.1GeV
dEE∆N (E) = f E?, (3.24)
where again f is the fraction of energy converted to electron and positron energy and E? is the total energy released by the explosion.
As discussed in section 3.3.2, both SNe1a and CCSNe release roughly 1051erg into the ISM (not in the form of neutrinos). Therefore, we will take E? = 1051
erg. The spectral index for SNe is thought to be roughly γ = 2, whilst the cut-off energy is of the order of 1 TeV [18]. However, only a small fraction of this total energy will be used for the positrons, roughly f = 10−4 of the total released SN energy will go into positrons [18].
In this analysis, we only want to estimate the SNR contribution, and will therefore assume that every SNR have the same normalization factor, N0 =
4.42 × 1049GeV. This comes from the assumption that E
?' 6.9 × 1047 erg and
f = 10−4 [18]. Note again that f is the efficiency for electron-positron pairs, and we assume in this thesis that the positron efficiency is given by fe+= f /2.
3.4
Production of secondary positrons
Though our model will not include secondary positrons, it is still interesting to look into some of their production mechanisms, which we will do in this section.
Secondary electrons and positrons can be produced through various mecha-nisms. The main production mechanism for secondary positrons is spallation. Specifically, the spallation of high-energy primary hadronic particles. Through nuclear interactions, high-energy hadrons can collide with interstellar material and ultimately produce positrons and electrons [39]. These primary CR hadrons are most commonly protons or helium nuclei, and the target ISM is most com-monly helium or ionized hydrogen [39]. These processes generally do not produce positrons directly. Instead, they typically produce some other baryon will decay further and has positrons as one of their final decay products.
Some examples of these types of interactions are [39]:
p + H → p + n + π+, (3.25) where the charged pion will decay into a muon, which will in turn eventually decay into an positron by the following interaction:
π+→ µ++ νµ,
µ+→ e++ νe.
(3.26) Other secondary positron production mechanisms include [39]:
p + H → X + K±, (3.27) where X is some arbitrary particle. This kaon will further decay into pions or muons, eventually producing positrons as described by equation (3.26) [39].
At lower energies, E ≤ 3 GeV, the following interaction dominates [39]: p + H → p + ∆+
∆+→ p + π0
or
∆+→ n + π+.
(3.28)
Likewise, these pions will eventually decay into positrons or electrons as well. Though charge conservation favours the production of positrons more than electrons, similar processes are still possible for the production of electrons [18]. An additional production mechanism for cosmic electrons and positrons is through triplet pair decay [2]. However, this method requires very high initial electron energies and does therefore not play a significant role in the overall positron spectrum.
As already mentioned numerous times in the previous two chapters, other secondary particles, such as heavier nuclei, can be created through similar in-teractions. The main ones being spallation and radioactive decay of primary particles [30].
The injection term for secondary particles is given by [18]: ∆N (E, x) = 4π ×X
i,j
Z
dE0Φ(E0, x)dσi,j
where the subscript i denotes the primary CR species, j denotes the target ISM particle and E is the energy of the secondary particle. Φi is the flux of the
primary particle and the cross section for the respective interaction is given by σi,j. Finally nj(x) is the density of the target ISM particle.
Provided that we know the primary particle flux, interaction cross section and target particle density, we can use equation (3.29) to determine the injection term for any arbitrary secondary particle.
Chapter 4
Positron and electron
energy loss mechanisms
As seen in Chapter 2, energy loss plays an essential role in the propagation of cosmic particles. Energy loss can directly put restrictions on how far in space a particle of a certain energy can travel and for how long they can travel. This can be seen directly in the expression for the diffusion radius, given by equation 2.4. In order to properly model what happens to a positron after escaping the pulsar magnetosphere, one must take energy losses into account.
In this chapter we will review the relevant positron and electron energy loss mechanisms. The ultimate goal of this chapter is to obtain an expression for the total energy loss of a cosmic positron traversing through the Galaxy.
4.1
Positron energy loss mechanisms
There are five main energy loss mechanisms for electrons and positrons, some of which are only relevant on high energy scales and vice versa. There are two main groups of energy loss processes: loss through collision with other particles, and energy loss through radiation. In the case of collsional losses, one can imagine a high-energy electron colliding with ISM and transferring its momentum to the target particle. Radiative losses on the other hand, will emit a photon during the energy loss process. This photon could be observable on Earth, and such an observation could be used as evidence to indicate the presence of high-energy particles in a certain environment.
The main radiative energy loss processes are synchrotron radiation, inverse Compton scattering (ICS), and bremsstrahlung. Some collisional processes in-clude ionization of ISM and Coulomb interactions with ISM. Finally, an electron can also collide with a photon to produce a triplet pair, which is another energy loss process and also produces additional secondary positrons. This last process however is only relevant on high-energy scales [2] that we will not tackle in this thesis.
4.2
Synchrotron radiation
As already briefly seen in Chapter 3, there are a lot of astrophysical sources in the Galaxy that generate magnetic fields. This makes synchrotron radiation an important energy loss mechanism when it comes to CRs. One of the most obvi-ous energy loss mechanisms to consider when dealing with CRs is synchrotron radiation.
As a positron bends in its trajectory as a result of a magnetic field, it will emit a photon. There are several ways to derive the equations that describe synchrotron radiation. Some methods, such as those given by [40], describe synchrotron radiation as a phenomenon caused by electric fields detaching from fast-moving positrons (or electrons) [40]. Others describe synchrotron radiation as being a result of interactions of positrons with virtual photons of the magnetic field [20]. In this thesis, we will be following the method and derivation provided by Blumenthal & Gould 1970 [20] as described below.
The energy loss associated with the phenomenon of synchrotron radiation can be determined analytically. To start, we should consider the non-relativistic Larmor formula (in cgs units) [41, 20]:
P = 2 3 q2a2 c3 = − dE dt, (4.1)
where q is the particle charge and a is the acceleration of the particle. We can rewrite this into its covariant form [20]:
dpσ = − 1 m2c2 2q2 3c3 dpµ dτ 2 dxσ, (4.2)
where the proper time is given by τ , pµ is the four-momentum of the particle
and m is the mass of the particle. Using the equation of motion [20], dpµ
dτ = q cFµνv
ν, (4.3)
with Fµν being the electromagnetic field tensor and vν being the four-velocity
of the particle. Finally, using q = −e and m = mefor electrons, we can combine
equation (4.3) with equation (4.2), to obtain [20]: dpσ dxσ = −2r 2 e 3c γ 2B2v2sin2θ → −dE dt = 2r2 e 3cγ 2B2v2sin2 θ, (4.4)
where we have introduced the classical electron radius re = e
2
mec2. Here θ is
the angle between the magnetic field B and the velocity v. It is convenient to rewrite equation (4.4) using the Thomson cross section σT = 8π3r2e [2]:
−dE dt =
cσT
4π γ
where we have also assumed vc ' 1.
As evident from the expression above, the synchrotron energy loss scales quadratically with the energy of the electron. Hence, this effect only becomes very relevant at high energies.
4.3
Inverse Compton scattering
Another radiative energy loss process is inverse Compton scattering (ICS). (Di-rect) Compton scattering is an interaction during which a particles scatters off a particle, resulting in the photon losing energy [1]. In this scenario, the photon will transfer some of its momentum to the particle, giving the particle additional kinetic energy [1]. This interaction is depicted in figure 4.1.
Figure 4.1: Inverse Compton scattering. Here a low-energy photon interacts with a high-energy electron, resulting in the electron transferring momentum to the photon and increasing the energy of the photon. Original figure obtained from [42].
The inverse of this process is called inverse Compton scattering. In this scenario, a high-energy electron or positron will interact with a (lower energy) photon, transferring some of its energy to the photon [1]. This process will result in the electron or positron losing energy [1].
The high-energy regime in which inelastic scattering dominates the interac-tions is called the Klein-Nishina regime [1]. In this scenario, the final energy of the scattered photon is significantly different from the energy of the photon [1]. In order to derive the energy loss equation for ICS, we will follow the methods given by [20] and Schlickeiser 2002 [2]. The positron (or electron) can interact with a photon gas of differential density dn = n(, r)d, where is the individual photon energy. For our applications, we will only consider positrons within
2 kpc of the Earth. The photon spectrum of the interstellar radiation field (ISRF) within 2 kpc is shown in figure 4.2. The ISRF contains photons from stellar sources, as well as photons from the CMB. This ISRF data was originally obtained from GALPROP [43, 44].
Figure 4.2: The interstellar radiation field (ISRF) near Geminga. This roughly represents the ISRF within 2 kpc of Earth. The ISRF in this figure contains both CMB photons as well as photons from stellar souces. CR positrons can interact with these photons and scatter them, losing energy in the process. ISRF data obtained from GALPROP [43, 44]
It is convenient to use the following notation for the final photon energy [20]: Eγ = γmec2Ee, (4.6)
where Eeis the energy of the incoming positron. The spectrum of the scattered
photons is given by [20]: dN,γ dtdEe = 2πmec 3r2 e γ n(, r)d κ(q, Γ), (4.7) with κ(q, Γ) being a dimensionless factor given by
κ(q, Γ) ≡ 2q ln q + (1 − q)(1 + 2q) +1 2 (Γq)2 1 + Γq (1 − q) , (4.8) where q ≡ Ee Γ(1 − Ee) and Γ≡ 4γ mc2. (4.9)
Here, Γ is a useful variable that gives an indication of the scattering regime.
