An AGN is powered by the gravitational energy released from the accretion disk formed by its central super massive black hole(SBH) and typically releases its energy through relativistic jets that extend a distance far greater than the size of its core with negligible diverging angle. The typical Lorentz factor for the relativistic jet is∼ 10. Since the constitution of jets is still debated, we assume that the jets are consist of electrons and protons, with the total length from few kpc to Mpc. Based on that, we can reasonably simplify the geometry by modeling the jet as a cylinder(see Fig. 6.2) which contains a constant plasma density and magnetic field strength over a large distance. The accelerating gradient is then estimated with those characteristic parameters evaluated near the AGN central engine.
The maximum luminosity an AGN can achieve is restricted by the Eddington
BH
10-4pc
eЁ& p jet
accretion disk
Figure 6.2: The simplifiede−−p jet geometry with ignoring the divergence angle.
The plasma density and background magnetic field strength are considered as constant.
limit at which the outward radiation pressure is equal to the inward gravitational
whereσT = 8π/2(α/mc) is the Thompson scattering cross section for electron.
With a central SBH mass ∼ 108M, the maximum luminosity of AGN is ∼ 1046erg/s and the size of jet is roughly the size of the accretion disk ∼ 3Rs∼ 10−4pc, whereRsis the Schwarzschild Radius
Rs=2GM
c2 . (6.9)
For an AGN jet having the maximum luminosity implies the plasma density n 1010 Kap-lerian) speed, being of the order the inward drift speed Vinfall[81]. The corre-sponding plasma and electron cyclotron frequencies are obtained, ωp ∼ 5.6 × 108∼ 109 rad/s and ωc ∼ 1011rad/s and the ratio of ωc/ωp is about 102. The temperature is estimated as the black body temperature in the core as
T 3 × 105
L LE
1/4
M8−1/4∼ 105K. (6.12)
and the Debye length is given by λD=
T
4πne2 10−3m (6.13)
which is much smaller compared to the plasma wavelengthλp= 2πc/ωp∼ 3 m.
Thus we ensure the validity of plasma collective effect in AGN jets.
With these characteristic parameters, it is possible to estimate the accelerat-ing gradient of MPWA produced in jets. First we calculate the average strength parametera0=eE0/mecω from the following relation
a20= 4π e2
where uAGN is the total energy density in the jets and η the energy fraction imparted into the magnetowave modes. The energy densityuAGN can be easily computed from the total luminosity,
uAGN= L · t
(10−4pc)2· c · t ∼ 1046
1039 = 107erg/cm3. (6.15) Our calculation so far only consider the toy model ignoring the jet divergence.
But in fact the jet divergence happens and the magnetic field and plasma density descend as 1/r2. As a result theωc/ωpratio decreases as a function of 1/r since ωc ∝ B0 ∝ 1/r2 and ωp ∝ √n ∝ 1/r. When the magnetowaves with phase velocityvph∼ c propagate into a low ωc/ωp (vphi) region, the mode conversion process will take place[82] and the magnetowaves will be converted into the normal electromagnetic waves to keep traveling.
Based on that argument, it is possible to estimate the luminosity of magne-towave from the observed radio wave luminosity of AGN, since the frequency of magnetowave is in the range of radio wave if we take the magnetowave fre-quencyω ∼ ωc/2 for convenience. According to [83, 84], the observed differential luminosity to classify the low and high luminosity classes is
∂2
∂ν∂ΩL178MHz= 1025W Hz−1sr−1, (6.16) at frequency 178 MHz. We are safe to take the frequency as the lower bound of magnetowave frequency. Therefore the total magnetowave luminosity is given by
Lmag= 1025× 107× ωc
2· 2πerg· s ∼ 1042erg· s (6.17) and we can deduce the energy fractionη of the order of (10−3− 10−4) from the ratio of the magnetowave luminosity to the total AGN luminosity∼ 1046erg· s.
Thea0 in turn can be calculated from Eq. (6.14) a0 = 4πη the accelerating gradientG can be calculated from Eq.(4.10)
G = a20
(1−ωωc)2χeEwb∼ O(102)(eV/cm) (6.19)
with the form factorχ of order 1 and Ewb∼ 105V/cm forn0∼ 1010cm−3. We notice that the accelerating gradient G is obtained with the parameters taken in the jet rest frame.
For protons to reach energy = 1021eV in our frame, it only requires energy gain 1020 eV in the jet frame with Γ of bulk motion typically being 10. Thus under the most optimized condition, the minimum distance for the protons to accomplish = 1020 eV is 1018 cm ( 0.3 pc) in the jet frame, corresponding to = 1021 eV for 3 pc in our frame. It is quite tiny compared to the typical AGN jet length.
Chapter 7
Conclusions
We have established a novel acceleration mechanism for UHECR which is based on the wakefield excited by magnetowaves in astrophysical jets. The magne-towave itself is a medium wave and has a lower phase velocity than the speed of light. To have a good accelerating performance, we focus on the high frequency and high phase velocity whistler wave. It was shown that a high ωc/ωp ratio is the condition for MPWA, with which the dispersion relation of the whistler pulse tends to be linear with a slope close to the speed of light. We have formu-lated the nonlinear magnetowave induced plasma wakefield and confirmed it via the computer simulation. On the application to UHECR production, the mag-netowaves are generated randomly. We expect a power spectrum for UHECR resulting from the stochastic particle wakefield interactions. Regarding AGN as the working source, we have estimated the accelerating gradient by putting physical parameters of AGN and finally concluded an optimized acceleration length required for particles to ZeV.
To summarize the content of the thesis, in chapter 2, we have introduced the basic concept of plasma from its definition, dynamics and the dielectric properties for waves. Plasma is a partially ionized gas. Having the quasi-neutrality and collective behaviors, the plasma can be defined following the three criteria, λ L, g ≫ 1 and ωτ > 1. Plasma can be described by using fluid and kinetic approaches which are equivalent for solving problems. Since most problems in plasma can be solved regarding plasma as a fluid, we have used
the complete set of fluid equations to study the physics of plasma, particularly electrostatic wave and electromagnetic wave in plasma. In the last part, we introduced the plasma wakefield acceleration and the three types of driving pulses for wakefield excitation.
In chapter 3 and 4, we have studied the plasma wakefield acceleration in magnetized plasma in high and low frequency branches. In chapter 3, we dis-cuss the case with ω/ωc c in which the dispersion relation approaches the unmagnetized case and the magnetized effect can be ignored. We compared the results of plasma wakefield witha0 1 (linear) and a0≥ 1 (nonlinear) respec-tively. In the linear regime, the plasma wake goes like a sinusoidal wave with the maximum amplitude linearly proportional toa20. Whereas, in the nonlinear regime, the plasma within the pulse is totally expelled from the laser center and piled up to form a peak that leads to a sawtooth-like wakefield with amplitude proportionala0. Taking the driving pulse as a square circularly polarized pulse, we analytically derived the plasma wakefield from the second order differential equation. The maximum of plasma wakefield amplitude wasa0/
1 +a20, so the accelerating gradientG ∝ a20whilea0 1 and G ∝ a0 whilea0 1.
In chapter 4, we studied the case for pulse frequencyω < ωc(MPWA theory).
To implement the acceleration mechanism in this range, the MPWA condition ωc/ωp 1 was made. We concentrated on the whistler modes and calculated the plasma wakefield in linear and nonlinear regime. With introducing the pon-deromotive force, we have derived the linear plasma wakefield, whose amplitude contains an additional factor (1− ωc/ω) to the ordinary G obtained without an external magnetic field. In the nonlinear regime, we made the MPWA condi-tion contain γ factors. The plasma wakefield was solved from a full complete set of relativistic fluid equations and was also shown a sawtooth-like behavior.
When the strength of background magnetic fieldB0increases,G decreases be-causeωc/ω > 1, opposed to the laser case. Since there exists a singularity at 1 +φ → ωc/ωp, we would make an upper limit ona0. Beyond that the plasma becomes unstable. Then considering the upper limit ona0, we have predicted the maximum accelerating gradient that MPWA can reach.
In chapter 5, we performed the particle-in-cell simulation to verify this MPWA theory. A code named ”em1da” written by R. Sydora was used. In
our simulation, we compared two cases with different ωc/ωp ratios, 6 and 12.
With successfully self-generated whistler wavepackets, we have confirmed the excitation of plasma wakefield and the validity of our MPWA theory. We have also showed that the whistler wavepacket sustains a longer distance with a higher ωc/ωp, i.e., it is less vulnerable to the dispersion. This aspect is especially im-portant for MPWA to be a viable mechanism for terrestrial accelerator since it is essential for an accelerated particle to continuously gain energy from the plasma wakefield in order to attain a high energy[85].
Finally with the MPWA theory established, we apply the mechanism to UHECR. In chapter 6, we have shown that the power law spectrum can be deduced from the stochastic interactions between the test particle and the accelerating-decelerating phases of the wakefield. Without taking the dissipat-ing process into account, the power law index is ideally given as -2. Next we discussed the possible sources for UHECR generation and the most recent ob-servations from Pierre Auger and HiRes. So far this issue is still not settled. We invoked the AGN as a possible source and modeled the jet as a cylinder. From the parameters estimated near the AGN core and the observed luminosity of radio waves, we have obtained the accelerating gradient of MPWA in AGN jet.
It enables a particle to possibly gain energy above 1021 eV in a short distance compared to the total jet length.
In this thesis we have shown the validity of MPWA for UHECR production with a power law spectrum and a linear accelerating gradient. But as a fist step we only simulated the process with a Gaussian magnetowave profile. However it is desirable to investigate our mechanism with magneto-shocks instead, which is astrophysically more relevant. Then the investigation of the generation of magnetoshocks in the plasma outflows becomes crucial to the next step. In ad-dition, due to the involvement of the background magnetic field, MPWA should be taken as a fundamental phenomena in plasma physics. We have derived the plasma wakefield with the full relativistic fluid equations, but the other non-linear phenomena of plasma magnetowave interaction should be investigated in detail. It would be extremely exciting if proof-of-principle experiments on MPWA can be pursued. With regard to the possible physical mechanism to excite the whistler magnetowave driving pulse for experimentation, it has been
shown that a fast ion-acoustic wave can decay into a whistler wave plus an ion-acoustic wave[86]. It is therefore conceivable that such a decay process, or conversely the fusion of two ion-acoustic waves, can produce whistler wave. In-spired by this, one wonders if a similar process can occur between a light wave and a whistler wave. If so, then perhaps a laser pulses could be converted into a whistler wave pulse in a magnetized plasma under suitable conditions.
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Appendix A
Transverse Fluid
Momentum Equation
When an electromagnetic wave in magnetized plasma propagates parallel to the external magnetic fieldB0along z direction, the plasma motions associated with the EM fields can be described as,
dγβx ωc ≡ eB0/mc is the electron cyclotron frequency. It is convenient to further express the above equations in terms of the normalized vector potential a that is related to E⊥ and B⊥ from Maxwell equations,
dγβx where the total time derivative d/dt = ∂/∂t + cβz∂/∂z. Since the electro-magnetic wave becomes circularly polarized when travels along the electro-magnetic field, we assume a right-handed polarized wave that have β⊥ =βx+iβy and a = ax+iay. Then we multiply Eq. (A.3b) byi and add the result to Eq. (A.3a), we obtain the simplified form of the equation of motion forβ⊥
d
dt(γβ⊥− a) = −iωcβ⊥. (A.4)
To solve the equation, we decompose the plasma responses to the driving pulse into slow and fast parts and assume that β⊥ =β⊥sexp[ikζ] to respond to the electric field E⊥sexp[ikζ]. E⊥s is the envelop of the driving pulse and exp[ikζ] is the fast oscillation part. Assuming that βz varies according to the scale of the pulse envelop,βz=βzs, we arrive at
by insertingβ⊥into Eq. (A.4). Sequentlyβ⊥ is solved from integration by part and it reads,
where the second term on the right hand is expected to be suppressed compared to the first term (quasistatic approximation). Finally we obtain the relation betweenβ⊥ anda,
β⊥= a
|γ − ωc
ω(1 − βz) |
. (A.5)
Having this relation substituted into into the wave equation ofa in Eq. (3.3), we arrive at relativistic dispersion relation of magnetowave is obtained,
ω2=c2k2+ ωp2/γ
1− ωc/γω. (A.6)
Appendix B
Differential Equation of Nonlinear MPWA
The integration of the complete fluid equations for MPWA in (ζ, τ) coordinate system leads to
φ − γ(1 − βz) =−1, (B.1a)
n(1 − βz) =n0, (B.1b)
which is similar to the form in unmagnetized case. Therefore the relation be-tweenn, n0,γ and φ is given
n n0 = γ
1 +φ = 1
1− βz. (B.2)
We have the Poisson equation (Eq. (4.15a)) rewritten as
∂2φ
∂ζ2 = k2p(n n0 − 1)
= k2p( 1 1− βz − 1)
= k2p( βz
1− βz). (B.3)
The main difference between with and without magnetic field cases is on the γ factor. With β⊥ solved in terms ofa (Eq. (4.19))
β⊥= a
|γ − ωc
ω(1 − βz) |
, (B.4)
we have
Squaring Eq. (B.1a), we get
γ2(1− βz)2= (1 +φ)2. (B.6)
So that the combination of Eq. (B.5) and (B.6) results in
⎛
which can be rearranged as
1 +βz
The Poisson equation is finally obtained,
∂2φ